Quantum Mechanics and Fundamentality: Naturalizing Quantum Theory between Scientific Realism and Ontological Indeterminacy (Synthese Library, 460) 3030996417, 9783030996413

Table of contents :
Introduction
Part I: Realism
Part II: Ontology
Part III: The Wave Function
Part IV: Indeterminacy
Contents
Part I Realism
1 The Unreasonable Effectiveness of Decoherence
1.1 Introduction: Decoherence Theory
1.2 The (Lack of) Description for Subsystems
1.2.1 Decoherence Does Not Transform “Pure States into Mixtures of Well-Localized States”
1.2.2 Undesired Consequences: The Status of Classical Objects in Decoherence Theory
1.2.3 Interpreting Decoherence
1.2.3.1 Decoherence and the Realist Interpretations of Quantum Mechanics
1.2.3.2 Relation with the Measurement Problem
1.3 No Dynamics for the Subsystem S – Ehrenfest Theorem
1.4 The Basis Problem: Does Decoherence Really Help?
1.4.1 Example: Quantum Brownian Motion
1.5 Conclusion
References
2 Quantum Fundamentalism vs. Scientific Realism
2.1 Introduction
2.2 Ontology and Dynamics
2.3 Is Primitive Ontology All There Is?
2.4 Primitive Ontology with Wave Function Realism
2.5 Primitive Ontology Without Wave Function Realism
2.6 Conclusion
References
3 On the Principles That Serve as Guides to the Ontology of Quantum Mechanics
3.1 Introduction
3.2 Setting Up the Debate: Meta-ontological Principles and Quantum Mechanics
3.3 The Dynamical Matching Principle
3.4 The Minimal Divergence Norm
3.5 Conclusion: Eliminative Reasoning and the Role of the DMP and the MDN in the Quantum Ontology Debate
References
4 The Quantum World as a Resource. A Case for the Cohabitation of Two Paradigms
4.1 Introduction
4.2 Information Theory as a Resource Theory and the Information-Theoretic Interpretation of QT
4.3 World Change, Conceptual Change and Kuhn Loss
4.4 Progress, Objectivity and Scientific Realism
4.4.1 Progress
4.4.2 Objectivity and Scientific Realism
Bibliography
5 Quantum Ontology: Out of This World?
5.1 A Different World?
5.1.1 A Free Particle Moving in One Dimension
5.1.2 Simple Harmonic Oscillator
5.1.3 Projectile
5.1.4 Two Particles in 1D
5.1.5 Controversy
5.2 Relation to Our World
5.3 Relation to Quantum Mechanics
5.4 Proposal
References
6 Why Might an Instrumentalist Endorse Bohmian Mechanics?
6.1 Introduction
6.2 Instrumentalism and Representations of the Unobservable
6.3 Functions of Representations of the Unobservable
6.3.1 Highlighting or Revealing Connections Between Phenomena
6.3.2 Promoting Understanding of How Phenomena Interrelate
6.3.3 As Heuristics for Theory (or Model) Construction
6.3.4 Illustrating or Revealing Connections Between Theories (or Laws or Models)
6.4 Conclusion
References
Part II Ontology
7 Beables, Primitive Ontology and Beyond: How Theories Meet the World
7.1 Introduction
7.2 David Bohm, Pluralism and Infinitism
7.3 Local and Non-local Beables
7.4 The Primitive Ontology Approach
7.5 Beyond the PO: Minimalism and Fundamental Ontology
7.6 Conclusion
References
8 All Flash, No Substance?
8.1 The GRW Dynamics
8.2 The Tails Problem and Fundamental Ontology
8.3 GRW0 and Primitivism
8.4 GRWm and GRWf
8.5 Massy Gluts and Flashy Gaps
References
9 Does the Primitive Ontology of GRW Rest on Shaky Ground?
9.1 Introduction
9.2 The PO Approach to GRW
9.3 Accessible and Non-accessible Mass
9.4 The Indeterminate Primitive Ontology
9.5 Conclusions
References
10 Towards a Structuralist Elimination of Properties
10.1 Introduction
10.2 A New Take on Naturalized Metaphysics
10.2.1 The Traditional View: Thick Object-Oriented Metaphysics Grounded on Properties
10.2.2 Objections
10.2.3 Fundamental Entities Without Fundamental Properties: Thin Objects-Oriented Metaphysics Grounded on Structure
10.2.4 Advantages and Replies to Objections
10.3 Structuralism
10.3.1 Arguments from Quantum Mechanics and Other Motivations
10.3.2 Objections
10.3.3 Comparison between the Thin Objects View and Structuralism
10.3.4 Advantages of the Thin Objects View Over Structuralism
10.4 Conclusion
References
11 Quantum Ontology Without Wave Function
11.1 Realism Without Wave Function
11.2 Ontology
11.3 Contextuality
11.4 Relations
References
12 The Relational Ontology of Contemporary Physics
12.1 Introduction
12.2 Relationality in Quantum Mechanics
12.3 The Relationality of Symmetries
12.4 Relationality in Quantum Field Theory
12.5 Relationality in General Relativity
12.6 Relationality in Quantum Gravity
12.7 Conclusion: The Relational Nature of Contemporary Physics
References
13 Explicit Construction of Local Hidden Variables for Any Quantum Theory Up to Any Desired Accuracy
13.1 Introduction
13.2 The Generic Realistic System
13.3 A Deterministic Theory for Every Quantum Model
13.4 Special Features. Locality
13.5 About Bell's Theorem
13.6 Note Added: Quantum Weirdness
13.7 Model Building
13.8 Concluding Remarks
References
Part III The Wave Function
14 Wave Function Realism and Three Dimensions
14.1 Introduction
14.2 The Single World Universe
14.3 The Many-Worlds Universe
14.4 Our World
14.5 Summary: Three Dimensional Aspects of Universal and World Wave Functions
References
15 Reality as a Vector in Hilbert Space
References
16 Platonic Quantum Theory
16.1 Introduction
16.1.1 The Measurement Problem
16.1.2 A New Approach
16.2 The Classical Case
16.2.1 Ontic States
16.2.2 Epistemic States
16.2.3 Random Variables
16.2.4 C*-Algebras
16.2.5 State Maps
16.2.6 The Classical Measurement Process
16.3 Quantum Theory
16.3.1 Ontic Random Variables
16.3.2 Incommensurable Probability Distributions
16.3.3 Quantum State Maps
16.3.4 Noncommutative C*-Algebras
16.3.5 The Gelfand-Naimark-Segal (GNS) Construction
16.3.6 Pure State Maps and Mixed State Maps
16.3.7 Quantum Dynamics
16.3.8 Textbook Quantum Theory
16.3.9 Hilbert-Space Ingredients as Gauge Variables
16.4 The Measurement Process
16.4.1 Pre-Measurement
16.4.2 Decoherence by the Environment
16.4.3 Wave-Function Collapse
16.5 Conclusion
16.5.1 The Platonic Interpretation
16.5.2 Questions of Uniqueness
16.5.3 Comparison with Other Approaches
References
17 cat alive and cat dead Are not Cats! Ontology and Statistics in ``Realist'' Versions of Quantum Mechanics
17.1 Introduction: The Need for an Ontology
17.2 Statistics and Ontology in the Many-Worlds Interpretation
17.2.1 The Naïve Many-Worlds Interpretation
17.2.2 A Precise Many-Worlds Interpretation
17.2.3 The Pure Wave Function Ontology
17.3 Ontologies for the Spontaneous Collapse Theories
17.4 Ontologies for the de Broglie-Bohm Theory
17.5 Comparison of Ontologies
References
18 Cosmic Hylomorphism vs Bohmian Dispositionalism
18.1 Introduction
18.2 Bohmian Dispositions
18.2.1 Bohmian Primitive Ontology
18.2.2 Instantaneous powers
18.3 The No-Successor Problem
18.3.1 State-State Powers
18.3.2 State-Process Powers
18.4 Cosmic Hylomorphism
18.4.1 Powers with Aristotelian-Timing
18.4.2 Cosmic Form
References
19 The Governing Conception of the Wavefunction
19.1 What the Wavefunction Must Do
19.1.1 Three Examples
19.1.2 Two Types of Why Questions
19.1.3 Why Reason Why Questions Must Be Answered
19.1.4 The Governing Conception of the Wavefunction
19.2 What the Wavefunction Could Represent
19.2.1 The Governing Conception and Epistemic Accounts of the Wavefunction
19.2.2 The Governing Conception of the Wavefunction and Configuration Space Realism
19.3 Conclusion
References
20 Representation and the Quantum State
20.1 Introduction
20.2 How Quantum States May Be Represented
20.3 Quantum States Are Not Physical Entities
20.4 Quantum States Are Not Physical Magnitudes
20.5 A Quantum State Does Not Represent Any (Intrinsic) Physical Properties
20.6 A Quantum State Is an Extrinsically Physical Property of a System
20.7 Is a Quantum State Representational?
20.8 What Quantum States May Represent, and Why This Makes Them Modal
20.9 Representationalism and the Quantum State
References
Part IV Indeterminacy
21 Quantum Mechanics Without Indeterminacy
21.1 Quantum Metaphysical Indeterminacy
21.1.1 Metaphysical Indeterminacy
21.1.2 Quantum Indeterminacy
21.1.3 Quantum Location
21.2 Indeterminacy in GRW?
21.2.1 The GRW Theory
21.2.2 The Problem of Tails
21.3 Remaining Issues
21.3.1 Other Interpretations
21.3.2 Emergent Indeterminacy
References
22 Derivative Metaphysical Indeterminacy and Quantum Physics
22.1 Introduction
22.2 Metaphysical Indeterminacy
22.3 Fundamentality and Derivativeness
22.4 Against Barnes
22.5 Conclusions
References
23 Explicating Quantum Indeterminacy
23.1 Quantum Indeterminacy: The Challenges
23.2 The Role of Indeterminacy in Quantum Mechanics
23.3 The Structure of Quantum Indeterminacy
23.4 The Conceptual Landscape
23.5 Remaining Issues
References
24 Defending the Situations-Based Approach to Deep Worldly Indeterminacy
24.1 Introduction
24.2 The Darby/Pickup Account
24.3 Corti's Objection
24.4 Reply to Corti
24.5 Conclusion
References
25 Metaphysical Indeterminacy in the Multiverse
25.1 Introduction
25.2 Decoherence-Based EQM (DEQM)
25.3 Metaphysical Indeterminacy in World Nature
25.4 Two Approaches to MI
25.4.1 Metaphysical Supervaluationism
25.4.2 Determinable-Based MI
25.5 Supervaluationist vs. Determinable-Based Treatments of MI in World Nature in DEQM
25.5.1 The Argument from Imprecise Histories
25.5.2 The Argument from Interference
25.5.3 The Argument from Nonfundamental MI
25.5.4 The Argument from `Unfamiliar Properties'
25.5.5 The Argument from Quantum Modal Realism
25.6 Conclusion
References
26 Fundamentality and Levels in Everettian Quantum Mechanics
26.1 Introduction
26.2 Frameworks for Fundamentality in Everettian Quantum Theory
26.3 Explanatory Levels in Everettian Quantum Theory
26.3.1 The Fundamental Level and the Multiverse Level
26.3.2 The Multiverse Level and the Everett World Level
26.3.3 The Everett World Level and the Special-Science Levels
26.4 Levels of Laws in Everettian Quantum Theory
26.5 Novel Features of Everettian Levels
26.6 Conclusion
References

Citation preview

Synthese Library 460 Studies in Epistemology, Logic, Methodology, and Philosophy of Science

Valia Allori   Editor

Quantum Mechanics and Fundamentality Naturalizing Quantum Theory between Scientific Realism and Ontological Indeterminacy

Synthese Library Studies in Epistemology, Logic, Methodology, and Philosophy of Science Volume 460

Editor-in-Chief Otávio Bueno, Department of Philosophy, University of Miami, Coral Gables, USA Editorial Board Members Berit Brogaard, University of Miami, Coral Gables, USA Anjan Chakravartty, Department of Philosophy, University of Miami, Coral Gables, USA Steven French, University of Leeds, Leeds, UK Catarina Dutilh Novaes, VU Amsterdam, Amsterdam, The Netherlands Darrell P. Rowbottom, Department of Philosophy, Lingnan University, Tuen Mun, Hong Kong Emma Ruttkamp, Department of Philosophy, University of South Africa, Pretoria, South Africa Kristie Miller, Department of Philosophy, Centre for Time, University of Sydney, Sydney, Australia

The aim of Synthese Library is to provide a forum for the best current work in the methodology and philosophy of science and in epistemology, all broadly understood. A wide variety of different approaches have traditionally been represented in the Library, and every effort is made to maintain this variety, not for its own sake, but because we believe that there are many fruitful and illuminating approaches to the philosophy of science and related disciplines. Special attention is paid to methodological studies which illustrate the interplay of empirical and philosophical viewpoints and to contributions to the formal (logical, set-theoretical, mathematical, information-theoretical, decision-theoretical, etc.) methodology of empirical sciences. Likewise, the applications of logical methods to epistemology as well as philosophically and methodologically relevant studies in logic are strongly encouraged. The emphasis on logic will be tempered by interest in the psychological, historical, and sociological aspects of science. In addition to monographs Synthese Library publishes thematically unified anthologies and edited volumes with a well-defined topical focus inside the aim and scope of the book series. The contributions in the volumes are expected to be focused and structurally organized in accordance with the central theme(s), and should be tied together by an extensive editorial introduction or set of introductions if the volume is divided into parts. An extensive bibliography and index are mandatory.

Valia Allori Editor

Quantum Mechanics and Fundamentality Naturalizing Quantum Theory between Scientific Realism and Ontological Indeterminacy

Editor Valia Allori Philosophy Department Northern Illinois University Naperville, IL, USA

ISSN 0166-6991 ISSN 2542-8292 (electronic) Synthese Library ISBN 978-3-030-99641-3 ISBN 978-3-030-99642-0 (eBook) https://doi.org/10.1007/978-3-030-99642-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Introduction

This book is a collection of essays devoted to exploring different aspects in the foundations and the philosophy of quantum theory. They range from issues about its compatibility of scientific realism to questions about the ontology of the theory; from questions about what is fundamental to questions about the nature of the wave function and of quantum objects. Accordingly, the book is divided into four parts, which however share some overlap in topics.

Part I: Realism The first part deals with issues regarding scientific realism and quantum theories. Quantum mechanics is a fundamental physical theory. In virtue of this, scientific realists would think that it can inform us about the nature of the world. Nonetheless, for a very long time, many thought that quantum mechanics forced to anti-realism. Against the classical view that macroscopic objects and their properties can be understood as composed by microscopic objects governed by clear and precise laws, the Copenhagen school, championed by many of the founding fathers of quantum theory, promoted the idea that the quantum world is intrinsically incomprehensible and unknowable. All physics can do is predict experimental results, being unable to provide a satisfactory description of reality. That is, we should abandon the naïve idea behind scientific realism that theories can help us understand the world. It is commonly accepted that the problem to solve to make quantum theory compatible with realism is the measurement problem. Quantum theory is a theory about an object, the wave function, which evolves in time according to an equation called the Schrödinger equation. It is a mathematical fact that, given that this equation is linear, sums of solutions are also solutions. That is, they also represent physically possible states of affairs. If quantum theory is complete, it describes both macroscopic and macroscopic bodies, and if microscopically one has a sum of solutions (a superposition), then this may propagate macroscopically. This is the typical case of the Schrödinger cat. Assume a cat is in a box where a radioactive v

vi

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atom is hooked up to a vial of poison so that if the atom decays the vial breaks and the cat dies. If the initial wave function is a superposition of the states “decayed” and “undecayed,” then the final state will be a superposition of “dead” and “alive” cat. However, we never observe such macroscopic superpositions: the cat is either dead or alive. In other words, if the wave function evolves according to the Schrödinger equation, and the theory is complete, there are unobserved macroscopic superpositions. If we want to think of quantum theory as compatible with realism, we have to fix this problem. This is traditionally taken care of in the standard formulation of quantum theory by the collapse rule: it is postulated that whenever there is a measurement (like us opening the box and observing the state of the cat), the wave function no longer evolves according to the Schrödinger equation but randomly collapses into one of the terms of the superpositions. However, this is unsatisfactory: what physical processes are to be considered measurements? While some have searched for a solution of this problem and connected others outside of quantum theory, others have argued that decoherence, roughly the interaction of a system with its environment, can help us solve them from within the theory: the system is effectively “measured” by its interaction with the environment, so that the superpositions effectively collapse into one of its terms. Davide Romano in Chap. 1 criticizes this perspective. One of the challenges is that quantum theory needs to provide a wave function for the original system after the interaction with the environment, starting from the wave function of the system composed of the original system and its environment. Some have argued that decoherence can provide us with such a wave function. Romano instead shows that this is not the case. Moreover, standard quantum theory has the problem of explaining why the wave function can be written as a function of different variables (position, momentum, etc.), but everyday objects seem to be located in space. It has been maintained that decoherence can explain why the position basis is privileged, but Romano maintains that the argument is circular, if applied to the standard theory: positions are introduced as a privileged in the system-environment interaction. If we leave standard quantum mechanics aside, other solutions of the measurement problem are fully compatible with a realist interpretation: the pilot-wave theory (also known as de Broglie-Bohm theory, or Bohmian mechanics), the spontaneous collapse or spontaneous localization theory (which also goes under the name of dynamical reduction theory, or GRW theory), and the many-worlds theory (also popular under the name of Everettian mechanics). Roughly put, Bohmian mechanics is a theory in which the description of the system is provided by the wave function that is supplemented by particles’ positions; in the GRW theory, the wave function no longer evolves according to the Schrödinger equation to allow for the collapse to be built in the dynamics rather than postulated; Everettian mechanics instead takes seriously the idea that all possible terms of the superpositions are actual and realized in different “words” which no longer interact with one another. Granting that our belief in scientific realism is justified, one natural question is how these theories connect the physics with the metaphysics. This is a matter of controversy. Physical theories use mathematics in their formulations; therefore, if these theories describe the world, one needs to identify a correspondence rule

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to identify which mathematical entities represent physical objects, their properties, and their laws of evolution. Within the classical domain, this correspondence was straightforward, because it was clear to Newton what his theory was supposed to be about (particles), and there was an obvious mathematical way of representing their nature (points). That is, Newton went from his metaphysical hypothesis to the mathematical formalism. Instead, in quantum mechanics the opposite is true: we have a mathematical formalism that works very well in predicting experimental results, that is, there is a correspondence between the formalism and the world at the macroscopic level, but we need to physically interpret it in the microscopic domain: what do the various variables mean, from the point of view of describing the nature of the world? Some variables that appear in the quantum formalism are the same as in the classical case, and thus they seem to have an obvious correspondence: t is time, m is mass, V or F represents the interaction. But interaction between what? There is an evolution equation for the wave function, so presumably this is the object representing what physical objects are made of. It would be nice if the wave function could be represented as an oscillation in three-dimensional space, just like electromagnetic fields. However, this is not possible, as the wave function, at best, can be interpreted as oscillating in a space with much higher dimension, the so-called configuration space. The view according to which the wave function in this high-dimensional space is the fundamental ontology has been dubbed in the literature as “wave function realism.” If we take this route, then we have the problem of making sense of this connection given that the wave function lives in some abstract space, rather than three-dimensional space. Motivated in part from this problem, some have contended that the wave function is not the right kind of mathematical object to represent matter, only some kind of “local beables” can suitably do that. Roughly, a mathematical object is a be-able, as opposed to observable, in the sense that it represents something; the locality condition refers to the fact that it is in three-dimensional space rather than in a space of higher dimensions. A particular way of implementing this view is the so-called “primitive ontology” framework. The idea is that the wave function should not be seen as representational but rather it should be thought as having another role in the theory. In particular, some have argued that the wave function is more similar to a law of nature or a property than to an object, but it is not straightforward to spell out what this is supposed to mean. On this basis, in Chap. 2, Matthias Egg critically engages with the primitive ontology approach, which he dubs “quantum fundamentalism.” He argues that it is in tension with the basic idea of scientific realism if one pushes the view to the conclusion that the wavefunction is not real. In fact, while quantum fundamentalism seems to entail that we are more justified in believing in the entities which belong to the fundamental ontology rather than those which are not in it, scientific realism instead pushes in the opposite direction. That is, given we have stronger evidence from them, it encourages us to think that we are more justified in believing in nonfundamental objects than in fundamental ones. As an alternative to the primitive ontology approach, we have seen, one finds wave function realism. A natural question is whether there are some principles we

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can use to determine which ontology should we choose for quantum theory. This is discussed by Vera Matarese in Chap. 3. She presents and critically analyzes two of the principles that have been presented in the literature: the dynamical matching principle and the minimal divergence norm. The former asserts that the fundamental structure of the world should match the structure of the dynamical laws of the theory and has been used to support wave function realism; the latter asserts that one should choose the theory that minimizes the difference between what the theory says the world is like and how the world appears, and it has been taken to support the primitive ontology approach. Matarese argues that this is not straightforward and that both principles can be used to support either view. Thus, she concludes, ultimately these principles cannot guide our choice of ontology but rather they are best regarded as useful tools to restrict the space of plausible ontologies. In addition to the three theories discussed above, there are other proposals. For instance, the information-theoretic account maintains that quantum theory should be understood as providing constraints over possible experimental outcomes. It has been argued that this approach is compatible with realism because it provides a mind-independent, objective description of reality. Laura Felline critically engages with this approach in Chap. 4, by comparing it with Bohmian mechanics and the GRW theory. While the proponents of each approach argue that their framework is to be preferred, Felline instead maintains that, in a Kuhnian way, there is no common standard to which one could compare them with one another: they ask different questions, and provide different answers; they have different aims, and they provide different solutions. In this way, they both can provide us with different but equally important insights. Within the information-theoretic framework, the wave function is taken to be epistemic, that is, it represents out knowledge of the system rather than the system itself: for instance, when we make a measurement on a system, the wave function changes because we acquire new information. It has been argued that there are strong reasons, such as the PBR theorem, which disqualify epistemic views of the wave function. Under this assumption that the wave function has to represent something real (it is ontic), in Chap. 5 Travis Norsen explores the question regarding the best way of thinking about the wave function. He first criticizes the various proposals, from wave function realism to the approach that it is a law of nature or a property, and he proposes to take more seriously the possibility of constructing a theory in which the information contained in the wave function is captured by some sort of orchestrated field, like it is (imperfectly) done in his theory of exclusively local beables. One may think that all these controversies about the compatibility between realism and quantum theory is a waste of time because realism is doomed after all. If so, why should we care about the theories discussed, which take realism for granted? Darrell P. Rowbottom argues that even if one is not convinced that scientific realism is true, theories with a clear ontology may be useful regardless. In Chap. 6 he shows that also an instrumentalist should prefer a theory with a clear ontology, like Bohmian mechanics. In fact, even if the theory were to fail to be approximately true (for instance because the notion of approximate truth is not well defined, or because

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the theory is nonrelativistic), Bohmian mechanics possesses significant theoretical virtues. Rowbottom argues that they can be valuable, among other things, in the understanding of how phenomena are connected with one another. In this way, even if one does not endorse scientific realism, one can get the next best thing, namely understanding, by endorsing Bohmian mechanics.

Part II: Ontology Setting the worries about realism aside or granting that there is value in having a clear model also from the point of view of the anti-realist, one can move to more specific questions about ontology. We already mentioned wave function realism, which regards the wave function as the fundamental ontology of the worlds, as well as the primitive ontology approach, which views quantum theories to be about a three-dimensional microscopic ontology rather than the wave function. Then we have space-time state realism, according to which the fundamental ontology of a quantum mechanical world consists of a state-valued field evolving in fourdimensional space-time. There are also many other approaches based on the notion of local beables, and the one dubbed “minimalist ontology,” which seem to come from similar perspectives: the former maintains that the ontology of a theory needs to be local, and the latter argues for an ontology of point particles. Given these similarities, all these proposals have been grouped under the same category. However, they are different. Andrea Oldofredi, in Chap. 7, compares and contrasts them. Oldofredi underlies how some local beables, such as electromagnetic fields, may not be suitable primitive ontologies. This is so because the primitive ontology framework, in addition to having the requirement of a local (i.e., three-dimensional) ontology, also gives importance to symmetry properties, which would be violated if electromagnetic fields were included in the primitive ontology. In addition, it is noted how the primitive ontology program is more flexible than the minimalist ontology framework as the latter postulates a fundamental ontology of matter points, while the former leaves open the possibility for other types of ontologies such as matter fields or spatiotemporal events (“dubbed flashes”). This is evident in the framework of the various GRW theories. In fact, while Bohmian mechanics is easy to interpret as a theory of particles, in the primitive ontology framework one cannot really talk about “the” GRW theory, because in this theory there is only the wave function, and the wave function, not being a three-dimensional object, is not a suitable primitive ontology. Rather, different possible primitive ontologies for the same GRW wave function evolution have been proposed: a continuous matter density field, flashes, or even particles. To each primitive ontology corresponds a distinctive GRW-type theory: GRWm, GRWf, and GRWp, respectively. In Chap. 8, Elizabeth Miller challenges the viability of the matter field and the flash ontology. As Egg criticized the primitive ontology program from a scientific realist perspective, Miller argues that these ontological choices for the GRW theory undermine the motivation to endorse the primitive ontology view

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to start with. The argument is that if one resists a wave-function-only-GRW-theory because there are superpositions, these superpositions still remain in GRWm and GRWf, albeit in a subtle form which is worth investigating. In the literature, the proponents of the primitive ontology program have always taken for granted that the primitive ontology had determinate values. Instead, Cristian Mariani, in Chap. 9, argues that in the context of GRWm it would be more appropriate to think of the primitive ontology as indeterminate in the sense of lacking definite properties. In fact, it is argued, this would allow us to make sense of the so-called “inaccessible mass” problem. In GRWm, the matter density function is many-to-one: two different wave functions describing physically different situations can generate the same matter density. In the literature, the locutions “accessible” and “inaccessible” have been introduced to describe the different physical situations represented by the same matter density. However, if the inaccessible matter density is to be taken ontologically seriously, Mariani argues that it then remains unclear what characterizes the two physically distinct situations. He therefore proposes that by thinking of the primitive ontology as indeterminate, one could still retain the proper reductive explanatory schema of the primitive approach. Be that as it may, traditionally, the way in which scientific theories give us a picture of the world is spelled out in terms of specifying what the fundamental objects are, what their fundamental properties are, and what laws applies to them. However, there are departures from this general trend, some of which come as ways of responding to the so-called pessimistic meta induction argument against scientific realism. One of the strongest arguments for realism claims that if the theory is successfully reproducing the data, then we have grounds of believing it to be true. However, the pessimistic meta induction objection roughly states that empirical success cannot be an indication of truth since all past theories which have been successful turned out to be false. One of the responses to this challenge is structural realism, which urges us to rethink of our understanding about ontology. In fact, in this approach structure is all we are justified to believe, since structure has been preserved through theory change, and it is responsible of the theories’ success. There are various forms of structuralism, some of which have been motivated by features of quantum theory, such as entanglement: the fact that composite systems have no individual wave function suggests that they have no individuality. This elminativism is a radical form of structuralism in which there are relations without relata: objects are not fundamental, relations are. All there is, is structure. Given the necessity of explaining how relations can exist without relata, some have proposed more moderate structuralist perspectives which relax this constraint and generally maintain that structure and objects are both fundamental components of the world. Valia Allori, in Chap. 10, proposes a structuralist understanding of the properties of fundamental objects. The proposal is that these objects have no other fundamental property than the one needed to specify their nature. The basic idea is that properties are in the laws, rather than in the objects, and laws may be understood as suitable structural relations between these objects. This account is then compared and contrasted with various types of structuralism, and it is argued that it is superior in various respects: it shares the main motivations of more traditional structuralist

Introduction

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views, but it does not have the corresponding problems. In the quantum domain, this approach is compatible with both the primitive ontology approach and the wave function realism. In fact, according to wave function realism, the wave function is fundamental while everything else, including objects and their properties suitably, emerges from it, while in the primitive ontology framework all properties are in the law, including the wave function. Another perspective on quantum mechanics with a strong structuralism component is relational mechanics. In this view, defended by Carlo Rovelli in Chap. 11, the wave function does not describe objects nor properties, but it is merely a calculational device. The fundamental ontology is relational, discrete, and relative. The world is a collection of facts interrelated with one another. Facts are identified as relations by interactions between other facts. There are no fundamental objects and no fundamental properties, and all our talk about them emerges from the relational ontology. The properties described by these facts are determinate only when systems interact, and they are relative, as they can always be thought of as relations between two entities. In Chap. 12, Francesca Vidotto builds up on this and argues that relational mechanics is supported by the most promising proposals for fundamental physical theories, from classical mechanics to gauge theories, and from general relativity to quantum field theories. Vidotto argues that all these theories, from the past to the present one, suggest an ontology of relations, constructing an argument which seems akin to the structuralist response to the pessimistic meta induction problem: A relational ontology has been preserved through theory change from past to current theories; therefore, we are justified in believing in the existence of such relations. Even if she does not explore this in detail, Vidotto remarks that this relational ontology helps in dissipating the worries connected with quantum nonlocality. It has been extensively argued, but it still remains controversial, that Bell has proven that nonlocality is a fact of nature. Briefly, his argument is the following: Bell, starting from the reasoning of Einstein, Podolsky, and Rosen (EPR), considered a particle source which emits pairs of spin-correlated particles in opposite directions. Experimental results of spin measurements on the two sides display perfect anticorrelations of outcomes. Since in quantum theory each particle acquires a definite spin-property only upon measurement, when the experimenter on one side finds spin up, say, along one direction, her act of measurement makes the particle on the other side acquire the opposite property, and this violates locality of interaction. To avoid nonlocality, Bell, following EPR, concluded that the spin properties have to exist before any measurement. He then proposed an inequality which would be satisfied by a such a theory but violated in quantum mechanics. Subsequent experiments established convincingly that the quantum mechanical predictions are correct, showing, in the eyes of many, that the world is nonlocal. Relational mechanics, it is proposed by Vidotto, avoids this conclusion in virtue of its relational ontology: quantum theory is nonlocal only because we think of single particles as existing independently, while this is not the case in this framework. In Bell’s proof there is also the so-called hypothesis of statistical independence, which is usually taken for granted. It states that the experimental settings do

xii

Introduction

not depend on the distribution of the additional variables. One way to make this assumption false (and thus invalidate the nonlocality conclusion) is allowing for superdeterministic theories. Bell regarded them as conspiratorial, but this is the route that Gerard t’Hooft considers in Chap. 13. He proposes a local, determinist theory, which he dubs the “cellular automaton interpretation” of quantum theory. However, this is not an interpretation of a theory but rather a new theory itself. The quantum formalism is shown to be derivable within this theory, and it is argued that quantum mechanics is best seen as a useful tool, rather than a theory with some ontological import. Also, it is shown that this theory is local because it violates the hypothesis of statistical independence. Nonetheless, t’Hooft argues that denying this assumption is less troublesome than many have thought.

Part III: The Wave Function The third part of this collection is devoted to the wave function: should we think of it as a possible ontology of the theory, or does it have some other role in the theoretical framework? We have already seen that some approaches, such as wave function realism, take it to represent the fundamental ontology of the world. Other perspectives instead regard, in their own ways, the wave function very differently than wave function realism. Some quantum theories seem to go better with some of these approaches than other. For instance, the primitive ontology approach is the straightforward way of interpreting Bohmian mechanics, while wave function realism seems to be better suited for the GRW theory or Everettian mechanics. The reason is rather obvious: in the original formulation of these two theories there is no object other than the wave function, and adding another entity as the ontology of the theory, as the primitive ontologist suggests, makes them less simple. Nonetheless, wave function realism has been resisted by proponents of Everettian mechanics for some of the reasons we have mentioned but also for others. Lev Vaidman, who is a defender of the many-worlds theory, in Chap. 14 proposes an approach in which the wave function is objective real, like in the case of wave function realism, but not fundamental. He argues that the reason for which we have to admit that the wave function is in configuration space is to account for the nonlocal properties of the physical objects. He, however, maintains that our experience of life in three-dimensional space supervenes on the portion of the wave function defined in three dimensions. He then concludes that the many-world picture is forced upon us if we wish to avoid nonlocality in such a deterministic picture. Another possible way of defending the many-world picture is given by Sean Carroll, in Chap. 15. His approach follows wave function realism in considering the wave function as fundamental but differs in regard of which type of mathematical object one should think the wave function is. Carroll defends the view that the fundamental ontology of the world is given by the wave function, which is best understood as a vector in Hilbert space evolving according to the Schrödinger

Introduction

xiii

equation. Carroll argues that everything else suitably emerges from this: from the laws of physics, which are determined by the Hamiltonian, to three-dimensional space itself. At the opposite side of the spectrum of possibilities is Jacob Barandes. If Carroll argues that the Hibert space picture is the essence of quantum theory, in Chap. 16, Barandes instead maintains that the Hilbert formalism, including the wave function, is just a tool which has been useful. It is merely the shadow that we, prisoners in Plato’s cave, are led to believe is real. He therefore proposes what he dubs the “Platonic interpretation” of quantum theory in which systems are represented by sets of properties (as for instance, positions and momentum) whose motion is described in terms of the Gelfand-Naimark-Segal construction, which Barandes regards as equivalent to the fire in Plato’s cave allegory. Whether one can satisfactorily recover our three-dimensional picture of the world from a fundamental ontology has been extensively discussed in the literature. In this collection, Jean Bricmont contributes to this debate with Chap. 17, emphasizing how the wave function is fundamentally unlikely a physical field, which prevents them to be considered suitable ontologies. Furthermore, he argues against Everettian mechanics by providing an argument of how this theory is unable to recover the statistics of the results: Bricmont argues that the measure required to give rise to the correct predictions is unjustified and thus needs to be postulated in an ad hoc manner. The question which however remains open is about the nature of the wave function: if we think that the wave function does not represent physical systems, then what does it represent, if anything? As anticipated, some of the approaches discussed above share the attitude of considering the wave function as not representing anything material. However, they disagree in how exactly we are supposed to think of the wave function. A part of them, like relational mechanics, the cellular automaton interpretation, and the Platonic approach, maintain that the wave function does not represent anything in the world, neither matter, nor something else. It is merely a useful tool to formulate the theory. Others, like the proponents of the information theoretic approach, consider the wave function as representing something, but not representing something about the world. Rather, they think that it represents something about our knowledge of it. In other words, in their view, the wave function is epistemic, rather than ontic. Some others, like the primitive ontologists and the proponents of the minimalist ontology program, lean towards considering the wave function as nomological. That is, they think that the wave function is ontic, rather than epistemic. However, instead of representing matter, it represents something to describe the way in which matter moves. This idea can be spelled out in different ways: some regard the wave function as exemplifying a dispositional property of matter, some others instead think more generally in terms of the wave function akin to a potential or the Hamiltonian. Naturally, approaches may overlap: for instance, two approaches may agree in considering the wave function akin to a potential, but disagree about whether potentials are real or not (see, e.g., the primitive ontology approach and the Platonic interpretation).

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There are several open questions when considering each of these perspectives. As noted already, the primitive ontology program is particularly suited to discuss Bohmian mechanics. Dispositionalism is the view according to which, within this framework, the wave function represents a dispositional property of the Bohmian particles. In Chap. 18, William Simpson and John Pemberton propose an argument against the dispositional account of the wave function. They show that the dispositions fail to determine the particles’ trajectories, and they propose instead “powers with Aristotelian timing,” that is, powers that persists through time. They then argue that this cosmic hylomorphism, or a teleological process, is preferrable to dispositionalism on a variety of grounds, first of all its simplicity. More generally, Nina Emery, in Chap. 19, explores the consequences of thinking of the wave function as part of the law, proposing what she dubs “the governing conception” of the wave function, which she links to the idea that the wave function is whatever explains the behavior of physical objects. She observes that depending on what is meant by explanation, one is led to different types of questions: either questions about the reasons why something has happened, or reasons about why we should expect to observe a given phenomenon. Emery argues that we should understand the governing conception of the wavefunction as providing an answer to the former type of questions. She argues that this perspective puts constraints on wave function realism, in addition to ruling out pragmatic approaches. Often pragmatic perspectives on the wave function are characterized as maintaining that the wave function does not represent anything. Rather, it is a useful tool to describe what an experimenter should expect to observe on their lab. In contrast with this characterization, in Chap. 20, Richard Healey discusses the various understandings of thinking of the wave function as representational. Then he defends the view that the wave function represents an objective relational property of a given physical system that describes neither its intrinsic physical properties nor anyone’s epistemic state. Rather it represents the objective probabilities to certain physical events involving the system.

Part IV: Indeterminacy Apart from question about ontology, another mysterious aspect of quantum theory is its probabilistic character. In the original theory, the wave function evolved according to two evolution equations: the deterministic Schrodinger equation when the system was unperturbed, while it would randomly collapse into one of the possible solutions during experimental circumstances. This indeterminism of the law is lost in deterministic theories such as the pilot-wave theory and the manyworlds theory, but nonetheless remains in spontaneous collapse theories. Other questions are about the status of probabilities in deterministic theories, and whether they can be accounted for. This aspect has already been mentioned in Bricmont’s paper, who was arguing against Everettian mechanics being able to reproduce the quantum probabilities.

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xv

More metaphysical questions have to do with the origin of the indeterminacy we see at the level of observation. Is it epistemic or ontological? As we have seen, Mariani’s paper put forward the possibility of having an indeterminate ontology for GRWm to account for the meaning of the indeterminate portion of the matter density field. On a more general note, in the framework of standard quantum mechanics, it is often argued that there is genuine metaphysical indeterminacy regarding the properties we associate to quantum objects. In other words, observables fail to have definite properties. In fact, in standard quantum mechanics, there is just the wave function evolving according to the Schrödinger equation, and physical objects get ascribed properties by the so-called “eigenvalueeigenstate” link: the properties of a given system are given by the eigenvalues of a suitable self-adjoint operator. When the system is not an eigenstate of the operator corresponding to the property which is being measured, then the property is indeterminate. That is, it does not have any determinate value. There are various accounts to characterize metaphysical indeterminacy. One is the so-called the “determinable-based” approach according to which a state of affairs is metaphysically indeterminate if it involves determinable properties without having a unique determinate. In a less precise language, the idea is roughly that the property in question could have a value, but it does not have a unique one: there could be many (“glutty”) or none at all (“gappy”). Consider for instance the property of having the spin “up” or “down” in a given direction. A singlet state, which is a superposition of spin “up” and “down” in the same direction, would be an example of metaphysical indeterminacy, as the system has no definite spin-inthat-direction property. However, as advocated by the so-called “sparce view,” the eigenvalue-eigenstate link is compatible with there being no such properties. People have objected to this view as being unreasonable, as it seems to imply that particles have no locations. In Chap. 21, David Glick, the proponent of this approach, defends it against this charge, arguing that the sparse view eliminates quantum metaphysical indeterminacy without the alleged radical implications. If one moves outside of standard quantum theory, it has been argued that there is genuine metaphysical indeterminacy in the GRW theory. The GRW theory, understood as a theory about the wave function, suffers from the so-called problem of the tails. The wave function, evolving according to a nonlinear stochastic equation, spontaneously localizes. This is the way the theory solves the measurement problem. However, after the spontaneous collapse, the wave function will not localize into a precise point but will have tails extending to infinity. That is, the wave function will not be in an eigenstate of the position operator, so the eigenvalue-eigenstate link will fail to provide any location to the system described by the wave function. To solve this problem, people have proposed to substitute the eigenvalue-eigenstate link with something else. Among the proposals, one find the so-called “vague” link, according to which a system can acquire properties in degrees, depending on how close they are from the eigenvalue of the relevant operator. This seems an instance of genuine metaphysical indeterminacy. Glick however argues that this is not the only way to interpret this situation. He instead shows that this indeterminacy can be instead seen as an instance of vagueness, and that it can be understood not as metaphysical but

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rather as representational, namely as having to do with the way we represent the world. Alessandro Torza reconstructs Glick’s argument that standard quantum theory does not entail metaphysical indeterminacy: since quantum theory shows no evidence of metaphysical indeterminacy, then there is not any. However, Torza argues that this implicitly assumes that there is no derivative metaphysical indeterminacy. In Chap. 22, Torza proposes an alternative way of defining the meaning of metaphysical indeterminacy: a system is indeterminate if and only if there is at least a property that the system neither has nor lacks. He first shows that his approach is suitable for quantum theory, and then he argues that there are good reasons to believe that quantum metaphysical indeterminacy is indeed derivative: the quantum formalism can be embedded in different logical spaces, namely spaces of possibilities, a (fundamental) classical logical space, and a (nonfundamental) nonclassical, or quantum, logical space; since metaphysical indeterminacy arises in the latter but not in the former, it is argued that it is derivative. Peter Lewis, in Chap. 23, suggests thinking about metaphysical indeterminacy starting from physics itself. Lewis argues that if one starts from the usual Hilbert space formalism, one can understand indeterminacy as being similar to the one defined by Torza. Also, he notices that there are two distinctive ways of ascribing properties to systems, starting from the eigenvalue-eigenstate link: a classical and a non-classical one. For instance, a classical attribution would be to say that a non-eigenstate of z-spin lacks both z-spin up and z-spin down properties, while a nonclassical one would be to say that the system has indeterminate z-spin. While the former has been employed by the determinate-based approach, the latter is found in Torza’s framework. They both lead to indeterminacy; nonetheless, Lewis remarks, they are not the only possible combinations: one in fact may ascribe properties classically in Torza’s account, or non-classically in the determinablebased account, and in both cases, we would not have any indeterminacy. It is an open question which combination one should adopt, and Lewis argues it should be settled analyzing whether it is fruitful within physics. Another approach rival to the one we have seen so far is the supervaluationist account, according to which a system is metaphysically indeterminate whenever there are multiple possibly admissible, exhaustive, and exclusive states of affairs, and it is indeterminate which of these obtains. George Darby and Martin Pickup, in Chap. 24, argue for an approach which is inspired by this. Their account suitably modifies supervaluationism in terms of situations which partially (rather than totally) describe the actual world. Their approach has been recently criticized in the literature, and in this paper, they provide a reply. So far, we have seen examples of metaphysical indeterminacy from standard quantum mechanics and the GRW theory, both of which had to do with properties having indeterminate values. In Everettian mechanics, superpositions states are interpreted differently than in the other quantum theories: they do not ascribe indeterminate properties to the same system, rather they represent many systems each with a different determinate property. So, one may be tempted to conclude that there is no room form indeterminacy in Everettian mechanics. Nonetheless,

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for the theory to be successful, one needs to ensure that the various systems do not longer interact (they decohere), and therefore can be considered independent “Everettian worlds.” Everettians have advocated that decoherence can ensure that the interference between these systems is negligible. However, this allows for indeterminacy to come back in: interference is not completely absent, just so small to allow, for all practical purposes, ascribing definite properties to each world. Claudio Calosi and Jessica Wilson, in Chap. 25, provide new arguments for the existence of genuinely metaphysical indeterminacy in Everettian mechanics adding to the literature on the subject. They also argue that this metaphysical indeterminacy is best accounted for by determinable-based approach when compared to their competitor, namely supervaluationism. Al Wilson, who has first argued for the existence of genuine metaphysical indeterminacy in Everettian mechanics and from which Calosi and Wilson start their paper, in Chap. 26, instead, wishes to explore the notion of levels in the many-worlds theory in terms of various notions of fundamentality. He argues that metaphysical ground and concept fundamentality are suitable framework to understand how to combine a deterministic fundamental reality (described by the universal wave function) with an independent emergent reality (the multiple decoherent worlds). Obviously, many other questions are left to be asked (and to be answered) about the topics the articles in this collection have discussed. Some issues are closely related to the ones covered here, and some less so. For instance, some have argued that even if a theory solves the measurement problem it still possible that it is not compatible with scientific realism. Is that correct, and if so, what are the consequences? It has been argued that the emergence of the there-dimensional world in Everettian quantum theory can be accounted in terms of grounding or concept fundamentality. How does this extend to the other approaches? Is there a notion of fundamentality which would fit best within relational or pragmatic approaches? What is the right notion of fundamentality in the primitive ontology framework? It has been suggested that the primitive ontology can be indeterminate. How would that play out? What is the relationship between relational quantum mechanics, pragmatic approaches, and structuralism? Even if these questions have been touched upon in some of the papers, a deeper discussion seems to be necessary, even if not here. I could list many more questions and open problems. In any case, it would have been foolish to even think of trying to provide anything remotely close to a comprehensive set of contributions on topics such as these, which are currently hotly debated. In the literature, one can already find some impressive collections addressing similar issues, and there is certainly room for many more volumes to be written on them. In any case, I am confident that this collection will be of interest for physicists, philosophers of physics, and metaphysicians interested on quantum theories, their different formulations and modifications, their implications for philosophy, and the various ways to provide a naturalized ontology for them. Valia Allori

Contents

Part I

Realism

1

The Unreasonable Effectiveness of Decoherence . . .. . . . . . . . . . . . . . . . . . . . Davide Romano

3

2

Quantum Fundamentalism vs. Scientific Realism . .. . . . . . . . . . . . . . . . . . . . Matthias Egg

19

3

On the Principles That Serve as Guides to the Ontology of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Vera Matarese

33

The Quantum World as a Resource. A Case for the Cohabitation of Two Paradigms . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Laura Felline

49

4

5

Quantum Ontology: Out of This World? . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Travis Norsen

63

6

Why Might an Instrumentalist Endorse Bohmian Mechanics? . . . . . . . Darrell P. Rowbottom

81

Part II 7

Ontology

Beables, Primitive Ontology and Beyond: How Theories Meet the World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Andrea Oldofredi

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8

All Flash, No Substance? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 113 Elizabeth Miller

9

Does the Primitive Ontology of GRW Rest on Shaky Ground? . . . . . . . 127 Cristian Mariani

10 Towards a Structuralist Elimination of Properties .. . . . . . . . . . . . . . . . . . . . 141 Valia Allori xix

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11 Quantum Ontology Without Wave Function . . . . . . . .. . . . . . . . . . . . . . . . . . . . 157 Carlo Rovelli 12 The Relational Ontology of Contemporary Physics . . . . . . . . . . . . . . . . . . . . 163 Francesca Vidotto 13 Explicit Construction of Local Hidden Variables for Any Quantum Theory Up to Any Desired Accuracy .. . . .. . . . . . . . . . . . . . . . . . . . 175 Gerard ’t Hooft Part III

The Wave Function

14 Wave Function Realism and Three Dimensions . . . . .. . . . . . . . . . . . . . . . . . . . 195 Lev Vaidman 15 Reality as a Vector in Hilbert Space . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 211 Sean M. Carroll 16 Platonic Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 225 Jacob A. Barandes 17 cat alive and cat dead Are not Cats! Ontology and Statistics in “Realist” Versions of Quantum Mechanics .. . . . . . . . . . . . . . . 255 Jean Bricmont 18 Cosmic Hylomorphism vs Bohmian Dispositionalism .. . . . . . . . . . . . . . . . . 269 William M. R. Simpson and John M. Pemberton 19 The Governing Conception of the Wavefunction . . .. . . . . . . . . . . . . . . . . . . . 283 Nina Emery 20 Representation and the Quantum State .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 303 Richard Healey Part IV

Indeterminacy

21 Quantum Mechanics Without Indeterminacy . . . . . . .. . . . . . . . . . . . . . . . . . . . 319 David Glick 22 Derivative Metaphysical Indeterminacy and Quantum Physics. . . . . . . 337 Alessandro Torza 23 Explicating Quantum Indeterminacy . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 351 Peter J. Lewis 24 Defending the Situations-Based Approach to Deep Worldly Indeterminacy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 365 George Darby and Martin Pickup

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25 Metaphysical Indeterminacy in the Multiverse . . . . .. . . . . . . . . . . . . . . . . . . . 375 Claudio Calosi and Jessica M. Wilson 26 Fundamentality and Levels in Everettian Quantum Mechanics . . . . . . 397 Alastair Wilson

Part I

Realism

Chapter 1

The Unreasonable Effectiveness of Decoherence Davide Romano

Abstract This paper aims to clarify some conceptual aspects of decoherence that seem largely overlooked in the recent literature. In particular, I want to stress that decoherence theory, in the standard framework, is rather silent with respect to the description of (sub)systems and associated dynamics. Also, the selection of position basis for classical objects is more problematic than usually thought: while, on the one hand, decoherence offers a pragmatic-oriented solution to this problem, on the other hand, this can hardly be seen as a genuine ontological explanation of why the classical world is position-based. This is not to say that decoherence is not useful to the foundations of quantum mechanics; on the contrary, it is a formidable weapon, as it accounts for a realistic description of quantum systems. That powerful description, however, becomes manifest when decoherence theory itself is interpreted in a realist framework of quantum mechanics.

1.1 Introduction: Decoherence Theory Decoherence theory is the best answer to the classical limit problem, i.e. the problem of deriving the classical world (classical systems obeying Newtonian mechanics) at the macroscopic regime from the microscopic quantum world (quantum systems obeying the Schrödinger equation). However, there is no consensus in the literature on what has been really achieved by the results of decoherence. On the one hand, Schlosshauer (2007) and Crull (2015, 2019) argue that decoherence explains the emergence of classical objects by providing a robust description of emergent welllocalized systems in the position basis. On the other hand, Ballentine (2008) and Okon and Sudarsky (2016) argue (even if for different reasons) that the program of decoherence is essentially flawed and does not help in the job to connect quantum with classical mechanics. In this paper I will not take position for one of the two

D. Romano () Centre of Philosophy, University of Lisbon, Alameda da Universidade, Lisbon, Portugal e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_1

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sides; instead, I will focus on three conceptual problems that have been largely overlooked in the debate so far and that emerge every time we attempt to interpret (or un-interpret) decoherence in standard quantum mechanics: (i) the lack of a precise characterization for the status of subsystems (Sect. 1.2); (ii) the lack of an objective dynamics for classical objects from decoherence (Sect. 1.3) and (iii) the selection of position basis as the privileged basis of classical systems (Sect. 1.4). In the following, I will briefly introduce the formalism of decoherence theory.1 Decoherence theory starts when a system | S interacts with a second system | E, generally called “the environment”.2 Assuming, for simplicity, that S is a superposition of two states: | S = c1 | S1  + c2 | S2  and that S- relative states get correlated with E-relative states in the S-E interaction: Hˆ int

|S1  |E → |S1  |E1  Hˆ int

|S2  |E → |S2  |E2  then the interaction eventually leads to the system-environment entangled state:3 Hˆ int

|S |E = (c1 |S1  + c2 |S2 ) |E → c1 |S1  |E1  + c2 |S2  |E2  = |S, E . However, what we are really interested in decoherence theory is the description and behavior of the (initial) systems S, now subsystem of the larger entangled state | S, E , under the action of the “external” environment. We can think of a table (S) scattered by surrounding air molecules (E) or the famous Schrödinger’s cat (S) interacting with an unfortunate chain of events (radioactive atom, Geiger counter, relais, hammer, poison vial: all collectively described by E). Since in quantum mechanics we cannot assign a state vector (or a wave function) to subsystems, we represent the subsystem S by the reduced density matrix:4   ρˆS = T r E ρˆSE ,

1

For a comprehensive presentation of the philosophical and formal aspects of decoherence theory, see e.g. Joos et al. (2003), Schlosshauer (2007, 2019), Zurek (2003) and references therein. 2 The “environment” can be generally thought of as external or internal degrees of freedom with respect to the degrees of freedom representing our system of interest. Spatial degrees of freedom (position coordinates) may be, for example, “the environment” for spinor degrees of freedom of a spin ½ particle (spin-up, spin-down). 3 Technical note: the coefficients of the entangled state superposition will generally be different from those of the initial S state superposition. However, this difference will not be relevant for the present discussion. 4 Strictly speaking, this is a density operator, while the density matrix is the density operator expressed in a particular basis (generally the position basis). However, as this difference will not be relevant, I will just use the term density matrix in both cases.

1 The Unreasonable Effectiveness of Decoherence

5

which is connected to the usual quantum distribution of eigenvalues by the trace rule:     T r ρˆS Oˆ S = Oˆ S . where ρˆS is the S-subsystem density matrix, Oˆ S an observable acting “locally” on the Hilbert space of the subsystem S (we can think of a property belonging only to S, such as the position of electrons on the final screen of a double-slit experiment interacting with   light photons between the silts and the screen).The dynamics of ρˆS x, x  , t will make the reduced density matrix diagonal in a very short time (this process is quasi-instantaneous for decoherence models at macroscopic scale), which, in turn, will make quantum interference between different components impossible to detect in a measurement. For example, when decoherence is induced by scattering of environmental particles on a system S, the evolution of the S-subsystem reduced density matrix (in the long wavelength limit) will be:      2 ρS x, x  , t = ρS x, x  , 0 e−(x−x ) t 

where ρ S (x, x , 0) is the reduced density matrix at the initial time (t = 0),  a constant of the model (the scattering constant) and (x − x ) the spatial separation between two points of the subsystem S. The term (x − x )2 plays thus the role 1 of an exponential rate, and τ = is the characteristic decoherence time (x−x  )2 of the model. As a result, the S-subsystem reduced density matrix is progressively diagonalized at a quadratic exponential rate. After very short time, ρ S will become (approximately) diagonal, and only relative components that describe well-defined states will “survive” the dynamical process (for example: only a well-defined alive cat and a well-defined dead cat):5    |c1 |2 ε    ρˆS =  ε |c2 |2 

(1.1)

where ε on the off-diagonal components represents a “negligible quantity” and |c1 |2 and |c2 |2 are the probabilities to obtain, respectively, the observables’ eigenvalues associated to the eigenstates | S1  and | S2  when a measurement is per-

5 The term ε in the off-diagonal components of the matrix stands for “negligible quantity”: as the diagonalization process is mathematically described by a decreasing (quadratic) exponential, it will reach the zero value only asymptotically.

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formed on the subsystem S of the total entangled state | S, E.6 We finally arrived at the central result of decoherence: open systems, i.e. systems in interaction with the environment, will be effectively described by diagonal reduced density matrices, generally interpreted as improper mixtures of well-localized states. I will analyze and (critically) discuss this important conclusion in the next section.

1.2 The (Lack of) Description for Subsystems 1.2.1 Decoherence Does Not Transform “Pure States into Mixtures of Well-Localized States” Equation (1.1) represents a diagonal reduced density matrix and it is generally called an “improper mixture”: even though the resulting matrix looks like a mixed-state density matrix (a density matrix computed for a mixed state, i.e. a classical sum of pure states), it does not actually represent a genuine mixed state, as in this case all the diagonal components of the matrix are equally real.7 instead, in the case of a mixed-state density matrix, the diagonal components represent classical epistemic probabilities over the real state of the system, and therefore only one component represents the “real state” (a pure state) of the system. One of the principal results of decoherence is thus expressed as follows: Claim decoherence transforms pure states into improper mixtures of well-localized states However: while the definition of an improper mixture is pretty clear, i.e. it is a shorthand to say that a given superposition state looks like a real mixture (a mixedstate) when represented in a density matrix, we may ask: why does the subsystem S should be described by (an improper mixture of) well-localized states? What is it in the formalism of decoherence that justifies this claim? Usually, this claim follows from the consideration that, as the reduced density matrix becomes diagonal, the coherence length (the distance over which quantum interference between different components of the superposition can be detected) shrinks, approaching negligible values in a very short time. As we cannot detect quantum interference between appreciably distant regions in space, it follows that the reduced density matrix represents the system being a (improper) mixture of well-localized states. However, the problem with this argument is that, in the

6

While this is a natural interpretation of the reduced density matrix, it could be interesting to ask whether a reduced density matrix may have a more general significance independently from measurement interactions. 7 As the dynamics of ρ (x, x ,t) is linear, it cannot eliminate any state of the superposition (see, S e.g., Adler (2003)).

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standard context, it is not possible to assign wave functions to subsystems of an entangled state. That is: in decoherence theory, it is not possible to assign a wave function to the system interacting with the environment, as this is a subsystem of the total system-environment entangled state. This is indeed the very first reason why we describe subsystems via reduced density matrix in decoherence. So, the question is: how can we claim that the subsystem S is a well-localized state (or an improper mixture thereof) if we cannot even assign to S a state vector or a wave function? This seems to be physically illegitimate at best, and conceptually wrong. Consider, for example, the Brownian model discussed by Zurek et al. (1993): a system represented by a quantum harmonic oscillator in interaction with a thermal bath of harmonic oscillators at a constant temperature T. After having analyzed the dynamics of ρ S (in particular, the evolution of the linear entropy), the authors argue that the pointer state of the model (the specific state that remains stable in the interaction with the environment) is characterized by position and momentum uncertainties identical to the ground state of a quantum harmonic oscillator. From this mathematical result, it is suggested that the subsystem pointer state is the ground state of the quantum harmonic oscillator, i.e. a Gaussian wave packet. However, while the results concerning the subsystem linear entropy and the position-momentum uncertainties are physically relevant, the conclusion that the subsystem is described by a Gaussian wave packet is definitely an illegitimate step, as it involves a physical representation (assigning a state vector to the subsystem) that goes beyond standard quantum mechanics. To sum up: • the so-called improper mixtures are not mixtures at all: they represent coherent superpositions of non-interfering relative components of the subsystem of a larger entangled state; • the well-localized wave packets resulting from the decoherence are not “wave packets” at all, as in quantum mechanics we cannot assign wave functions (so, wave packets) to subsystems. Concretely: we cannot think of the decoherence process in standard quantum mechanics as a mechanism that separates the initial entangled state into different non-overlapping components and that re-assigns a wave function to these components. This operation cannot be done in the standard context until a measurement is performed: only a measurement interaction with a macroscopic device8 makes the initial (system-environment-measurement device) entangled state collapse into one of the decohered components (eigenstates of the measured observable), with 8

Note that the system-environment interactions in decoherence theory are measurement-like interactions, as there is a coupling between the system’s and environment’s degrees of freedom and the environment relative states get correlated with the system relative states. However, these interactions do not produce collapse of the wave function, as the master equations are linear and do not select one particular component. In order to have collapse of the wave function, the system has to interact—by definition–with a macroscopic measurement device.

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probability given by the usual Born rule. The unique eigenstate of the measured observable selected in the measurement will be eventually assigned a wave function. We are thus led to the following: Counterclaim Decoherence in standard quantum mechanics does not transform pure states into mixtures (of any sort) of well-localized states but makes it impossible to detect quantum interference between distant spatial components of the subsystem if we perform a local measurement on the subsystem. The subsystem is a superposition of non-overlapping components in configuration space.

1.2.2 Undesired Consequences: The Status of Classical Objects in Decoherence Theory The fact that subsystems cannot be generally described by a state vector/wave function is an ontological problem often overlooked in the philosophy of decoherence. This poses a problem not only for the ontology of any realistic microscopic system but also (and, maybe, most importantly, insofar as we are looking for a clear account of the quantum to classical transition) for the status of classical objects. It is standardly accepted that classical objects emerge in decoherence theory as macroscopic systems (systems composed of many degrees of freedom) in a strong interaction with an external (or internal) environment, i.e. when Hint  HS + HE . By definition, classical objects are subsystems maximally entangled with the environment and mathematically described by perfect diagonalized reduced density matrix. Furthermore, they must provide classical numerical values for observables such as position, energy and momentum when “locally” measured via OS  = TrE (OS ρ S ) [more on this in Sect. 1.3]. The problem is that, from the ontological point of view, a classical object is difficult to characterize if on the one hand, the subsystem reduced density matrix is the only tool we have at our disposal and, on the other hand, the ontology of a quantum system usually refers to the wave function and not to the density matrix or, even worse, to the reduced density matrix. It is worth noting indeed that, in this context, the usual debate on the nature of the wave function becomes completely irrelevant for the characterization of the emergent classical objects as the latter are, by definition, macroscopic quantum systems (subsystems) entangled with the environment. Does it mean that classical objects cannot be clearly defined within quantum mechanics? Certainly not. To my view, this is only a sign that the standard interpretation of quantum mechanics cannot conceptually explain what decoherence achieves from the pragmatic/empirical point of view. The empirical predictions of decoherence clearly indicate that subsystems after having interacted with the environment do become independent/autonomous systems. But quantum mechanics, in the standard framework, is unable to provide such an account. And this is not a specific problem taking place at the macroscopic regime, examples can

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be given also at the microscopic one. Think, for example, at the description of atomic orbitals: the explanation of why we do not see a superposition of orbitals but only definite orbitals is again provided by decoherence theory:9 interaction between different orbitals leads to decoherence in the energy basis (in the microscopic regime HS  HE + Hint ). This process selects well-defined separate components in the energy basis: atomic orbitals. Strictly speaking, atomic orbitals should not be described by wave functions, but only by reduced density matrices. However, in practice we do assign a wave function to each atomic orbital and the fact that we are able to assign such a wave function to a state resulting from decoherence is not a sign that decoherence does not work (it works perfectly); instead it is a sign that the standard interpretation cannot do all the job in the interpretation of decoherence. Subsystems selected by decoherence are, practically speaking, systems that become independent from the total entangled state. And so are classical objects. But this fact is simply not accounted for in standard quantum mechanics, where subsystems are described by density matrices even after the decoherence process. Indeed, the idea to assign wave functions to subsystems of entangled states goes beyond standard quantum mechanics, and can be described more precisely by non-standard interpretations of quantum mechanics: the Everett/Many Worlds Interpretation (MWI), the de Broglie—Bohm theory (dBB) and the spontaneous collapse models (e.g. GRW, CSL). This point will be briefly analyzed in the next subsection (while a complete analysis would require a separate work).

1.2.3 Interpreting Decoherence The problem of the characterization of subsystems, which is not solved and in principle unsolvable in the standard interpretation of quantum mechanics,10 is actually solved when decoherence theory is embedded in a realist interpretation of quantum mechanics: MWI, GRW or Bohm’s theory.11

9

See e.g. Crull (2015). is worth noting, however, that in some recent (more refined) interpretations of the quantum formalism this problem does not arise or could not arise, for example: Rovelli’s account of decoherence in relational quantum mechanics (Di Biagio & Rovelli, 2021); Myrvold’s ontology of quantum states in terms of distribution of values of dynamical variables (Myrvold, 2018); Chen’s density matrix realism (Chen, 2018). In all of these interpretations, the problem discussed above requires a careful and distinct analysis, which will be developed in an extended version of the present paper. 11 For a presentation of the different role of decoherence in the realist interpretations of quantum mechanics (MWI, GRW and Bohm’s theory) see also Bacciagaluppi (2020). 10 It

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1.2.3.1 Decoherence and the Realist Interpretations of Quantum Mechanics In GRW theory, the spontaneous collapse process is represented through the multiplication of the ordinary wave function (solution of the Schrödinger equation) by a Gaussian wave packet whose center is random and |ψ|2 - distributed. An important feature of this theory is that the spontaneous collapse rate is amplified for an N-particle system. That is: while the standard collapse frequency for a 1particle system is λ = 10−16s, the collapse frequency for an N-particle system will be = Nλ. For a macroscopic system, i.e. a system composed of a number of particles of the order of the Avogadro’s number (1023), the collapse rate will be very effective and the system will be “always” well-localized (as far as we can reasonably detect at the macroscopic scale and for classical relevant timescales). The significance of decoherence is thus clear in this context: the interaction with the environment is the physical mechanism which largely increases the number of degrees of freedom of a GRW system, making the spontaneous collapse very effective via the amplification mechanism.12 A 1-particle microscopic system S(x) in interaction with the environment E(y) will be part of an entangled state:  SE (x, y). As the number of interactions with the surrounding environment will increase (at least) linearly in time (for collisional models and quantum Brownian motion models), the system will undergo a spontaneous collapse after short time (the precise time depending on the interaction and environment model, but it is reasonable to expect the GRW spontaneous collapse rate to approach the standard decoherence time very quickly). This eventually explains why, in the GRW theory, microscopic “decohered” subsystems (such as atomic orbitals) and ordinary classical systems are well-defined states to which a wave function can be unproblematically assigned. The significance of decoherence theory in GRW is therefore not related to the decoherence effect, i.e. the separation of components in configuration space. Strictly speaking, there is no decoherence process in GRW theory, as the dynamics of the theory is non-linear. Nevertheless, the interaction with the environment supplements the GRW system with additional degrees of freedom, which trigger the amplification mechanism of the collapse rate and make open quantum systems (so, by definition, classical systems) (quasi-)instantaneously and (quasi-)perfectly well-localized.13 In MWI and dBB theory, decoherence effect is the result of the separation of the initial system-environment entangled state into effectively separate components14 in configuration space. As, in the case of GRW theory, the collapse is more likely to happen for systems with a high number of degrees of freedom, analogously, in 12 For a numerical estimate of the increase of the GRW collapse rate due to decoherence see, for example, M. Toroš, S. Donadi and A. Bassi (2016, pp. 10–11). 13 But instantaneously well-localized for relevant timescales at the macroscopic regime and perfectly well-localized with respect to macroscopic localization. 14 Separate components stand here for “components whose overlap is negligible in configuration space”, since the condition of no-overlap is not realistic: tails of Gaussian environmental particles will overlap in any region of space.

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MWI e DBB theory, the separation between components in configuration space is more likely to happen when the configuration space becomes “large enough”. That is: when the number of degrees of freedom of the entangled state increases, it also increases the possibility that different relative environmental states have negligible overlap with each other.15 The separation of different components in configuration space is the well-known branching process in MWI (Wallace, 2012, ch. 3) and the less well-known but equally important process of effective factorization in dBB theory (Bohm & Hiley, 1987, 1993; Dürr et al. 1992; Holland, 1993), i.e. the physical process that permits to assign an effective wave function16 to subsystems.

1.2.3.2 Relation with the Measurement Problem The problem of the subsystem’s description from a larger entangled state (Sect. 1.2.1) is closely related with the measurement problem,17 i.e. the problem of selecting one particular component (the measurement outcome) out of a coherent superposition of macroscopic different states. It is worth noting indeed that in Bohm’s theory and Everett/MWI there is no difference between a measurement process and the decoherence process. In both cases, the description is that of a system interacting with an external system (environment or measurement apparatus) via a suitable interaction Hamiltonian. This process describes an effective factorization and thus the selection of a unique component (effective wave function) in the dBB theory and a branching process of non-overlapping components in the Everett theory. In other words: the measurement process is basically a decoherence process in Bohm’s theory and Everett theory. As decoherence selects specific subsystems in these two theories, a measurement process will select specific measurement outcomes. Instead, as decoherence does not select specific subsystems in standard quantum mechanics, the measurement process (without introducing the collapse of the wave function) will not select specific outcome(s). This can be viewed as a different angle to understand (i) why QM is affected by the measurement problem and (ii) why decoherence does not solve or help solving the measurement problem in standard QM.

15 Even though there is no rigorous formulation, this is not so different from the standard decoherence condition of “orthogonality” of states, that in most cases is reached approximately and asymptotically. For an analysis of the decoherence condition in dBB theory, see Romano (2016a). For a comparative analysis of the decoherence condition in dBB theory and MWI, see Rosaler (2015, 2016). 16 The notion of effective wave function is originally introduced and defined in Bohm and Hiley (1987). Building on Bohm and Hiley’s historical work, I have recently analyzed the connection between the process of effective factorization and the emergence of classical systems in Romano (2016b). 17 Thanks to Craig Callender and one anonymous referee for helping me elaborating this remark.

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1.3 No Dynamics for the Subsystem S – Ehrenfest Theorem A second aspect which is generally overlooked in the presentation of the classical limit via decoherence is the relation between quantum and classical dynamics. A classical object is not only a system with a well-defined position in threedimensional space; it also moves in space according to Newtonian dynamics.18 That is: when a classical system with position x is affected by a classical (gravitational19) potential V(x), it will accelerate according to Newton’s second law of dynamics: F = mx¨

Newton’s 2nd law : an object with position × accelerates due to the force F

where F = − ∇ x V(x) is a classical gravitational force generated by the potential V(x), generated (in turn) by the existence of a physical gravitational field filling up three-dimensional space (at least, the finite region of space over which the gravitational field is non-negligible and well-defined). Newtonian dynamics, i.e. the fact that a classical object evolves in time according to a Newtonian trajectory when affected by classical potentials, is not a secondary feature but a central one for the correct representation of a classical object. So, the question is: how does decoherence theory account for this important feature of classical mechanics? Stating the obvious: without a precise account of Newtonian dynamics in the macroscopic regime, all we have achieved from the quantum to classical transition via decoherence is a (quantum) object well-localized in space mathematically represented in the position basis. This is an excellent starting point, of course, but not the whole story: for an object to be classical, it has to evolve in time according to (approximate)20 Newtonian trajectories. The connection between decoherence theory and the subsystems’ dynamics is given by a generalized form of the Ehrenfest theorem. This can be easily shown in two steps:

18 Or Lagrangian mechanics, or Hamiltonian mechanics. However, as the ontology of classical systems (system dynamics and interaction between systems) is generally built on Newtonian mechanics, I will consider the quantum to classical dynamics transition as the transition from quantum dynamics to Newtonian mechanics. 19 I consider here only the gravitational potential since, strictly speaking, there is no electromagnetic field in non-relativistic quantum mechanics. Classical electrodynamics should emerge from quantum electrodynamics, i.e. a different mathematical and physical framework. It is true that electromagnetic interactions are described also in non-relativistic quantum mechanics, e.g. the textbook presentation of the proton-electron interaction in a Hydrogen atom. However, this kind of analysis is rather phenomenological and relies on semi-classical assumption (particles described by wave functions interacting through classical electromagnetic forces). On the other hand, there is no gravitational interaction either in non-relativistic quantum mechanics, as a real quantization of the gravitational field is only done in quantum gravity. Yes, any realistic description in non-relativistic quantum mechanics is trickier than usually thought. 20 We want to derive Newtonian trajectories approximately and not exactly, as any deviation of the order e.g. of the atomic scale cannot be detected at the macroscopic scale.

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1. Trace rule: we can compute the mean value of any observable acting on the subsystem of a larger entangled state using the reduced density matrix ρ S through the trace rule:     T r ρˆS Oˆ S = Oˆ S where ρˆS is the subsystem reduced density matrix and Oˆ S a Hermitian operator acting “locally” on the subsystem’s Hilbert space. 2. Ehrenfest theorem for open systems: we can use the trace rule defined in step (1) to compute the mean value of the momentum, the acceleration and the “force” operators: Ehrenfest theorem : d m dt xˆ = pˆ

 

d dt pˆ = − ∇V xˆ

 

  where xˆ = T r ρˆS xˆ and pˆ = T r ρˆS pˆ , respectively. When the mean value of the potential can be reasonably approximated with the potential of the mean value of the position operator,21 the second equation above becomes:

d dt

“ Newtonian" dynamics

 



pˆ = − ∇V xˆ ∼ = −∇V xˆ

which is generally taken to describe a quantum system moving along a Newtonian trajectory. From an instrumentalist/pragmatic point of view, this is correct: the mean value of the acceleration operator will be Newtonian-like distributed. However, this formula is not a description of an object moving in space according to a specific Newtonian trajectory. The mean value of an operator is a statistical quantity (e.g. Ballentine (1998, Ch. 2); Bowman (2008, Ch. 5))–a quantity computed after a (reasonably) long sequence of separate measurements on identically prepared systems22 –which is ontologically meaningless for the description of the dynamics of an individual system. The fact that the subsystem of interest does not have a state vector does not help in this situation either, as we can represent the classical system only by means

21 This

approximation turns out to be exact for linear and quadratic potentials (Shankar, 1994) insofar the wave function is well-localized (Wallace, 2012). 22 As these are notions we are already familiar from the basic course in quantum mechanics, I will not enter here in the detail of the difference between a statistical quantity and a quantity describing an individual system. For the interested reader, however, I have analyzed this issue more carefully in another paper (Sakurai, sect. 3).

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of the reduced density matrix. The problem of finding or recovering Newtonian dynamics in the macroscopic regime combines together two conceptual problems already present in standard quantum mechanics: • the definition of the physical state for the subsystem of a larger entangled state; • the definition of the subsystem’s dynamics independently from observables (which implicitly involve the use of ensembles, statistical distributions and measurement operations). To conclude: the Ehrenfest theorem (in the standard or generalized form) does not describe individual systems moving on Newtonian trajectories: it provides, instead, a statistical description which can be consistently applied only to ensembles. No account is provided for trajectories of individual systems. What we obtain from the Ehrenfest theorem is not a classical object moving according to Newtonian mechanics, but a discrete sequence in time of position and momentum eigenvalues which are formally compatible with a Newtonian trajectory. Ontologically speaking, though, Newtonian trajectories are not there: they simply cannot be described (neither exactly nor approximately) in the standard framework of quantum mechanics, even introducing decoherence.

1.4 The Basis Problem: Does Decoherence Really Help? A further important problem of the quantum to classical transition is to explain why the familiar classical world is position-based, i.e. why position is a privileged variable for classical objects, starting from the relative freedom we have in the basis representation of quantum systems. The fact that position is a privileged variable in the classical world comes from the evidence that all classical objects have a welldefined position in space and they are described by continuous trajectories. On the contrary, a quantum system can be generally be described in different bases (e.g. position, momentum, energy), the specific choice being usually taken for pragmatic reasons, such as the eigenbasis of the observable we want to measure in a designed experiment or the energy basis for the particle in a box. Thus, we can frame the problem of the basis as follows: Basis Problem why is position basis a privileged basis in the classical world, starting from the physical equivalence of the basis representation in quantum mechanics? The standard answer given by decoherence is that, when the interaction between the system and the environment is very strong, the only states that remain stable under the action of decoherence are position eigenstates.23 In turn, this leads to

23 Following

Zurek’s terminology, they are generally called pointer states.

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diagonalization of the reduced density matrix in that particular basis, and, therefore, decoherence will select subsystem’s states well-localized in position. Decoherence Claim The classical world is position-based because of decoherence. In particular, the continuous interaction between a system with its surrounding environment selects position as the privileged basis for the representation of that system in the macroscopic regime. This explanation, however, does not consider one particular fact: all decoherence models of the quantum to classical transition describe the interaction between the system and the environment as a function of the position coordinates. That is, if we consider a system S(x) and an external environment E(y), the interaction Hamiltonian (which describes the physical coupling between the two) will be of the form: Hˆ int (x, y). Now: • as the basis “selected by decoherence” is the basis of eigenstates of those ˆ ˆ ˆ ˆ ˆ operators (O) that commute with the total Hamiltonian: H , O = H O− Oˆ Hˆ = 0, • given that in the macroscopic regime the total Hamiltonian can be approximated with the interaction Hamiltonian Hˆ t ot ∼ = Hˆ int , since Hˆ int  Hˆ S + Hˆ E ˆ • and since Hint (x, y) already describes interactions between the system and its surrounding environment in position basis,  • then position basis is selected: Hˆ int , Xˆ = 0 This looks like a consistent physical description, but not really as an ontological argument that explains why the classical world is position-based. In this procedure, we are implicitly assuming that in the classical world “things happen” in position, as we describe the interactions between system and environment already in that particular basis. There is no cogent or specific reason to describe those interactions in position basis if not a very important one: we know how to describe interactions in position basis, we are familiar with this kind of interaction from classical mechanics and we expect classical objects to emerge in position basis.

1.4.1 Example: Quantum Brownian Motion Consider, for example, the quantum Brownian motion model.24 In this model, the system is linearly coupled in position with a collection of quantum harmonic oscillators at constant temperature T (thermal bath). Each environmental particle (harmonic oscillator) interacts independently and only with the system and is

24 I take this specific example as the Brownian model is one of the principal models for the quantum to classical transition. Other important models, such as the collisional model, would be perfectly equivalent for the example.

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mathematically represented by the position-basis Hamiltonian: 1 pˆ + mω2 qˆ 2 Hˆ E = 2m 2 The interaction between the system and the generic (i-th) harmonic oscillator is represented by a bilinear coupling in the position variable: Hˆ int = xˆ ⊗

ci qˆi

i

where xˆ is the system position operator, ci the coupling strength between the system and the environmental particle and qˆi the environmental particle position operator. As the interaction Hamiltonian Hˆ int (x, q) is a function of the system and “environment” position, in the approximation Hˆ t ot ∼ = Hˆ int the master equation will lead to decoherence in position. Furthermore, as states with different momenta will form macroscopic superpositions in position in a short time, these will be also decohered by the model (though at a slower rate with respect to macroscopic superpositions in the position basis). As a consequence, the pointer states of this model will be states well-localized in position and momentum. In particular: minimum-uncertainty Gaussian wave packets.25 In this example, we see in what sense position is already a privileged variable in the model: the environment is described in the position basis, and the interaction with the system is described by a coupling of the position coordinates. One could equivalently describe the same model in energy or momentum eigenstates, probably obtaining different results concerning the selected pointer states. Nevertheless, we do want to express the system-environment interaction in the position basis because what we are looking for in this model is an emergent description of classical objects. Position In-Position Out if we introduce position in the model from the start via Hˆ int (x, y), then it is no surprise that we get position out for the subsystem preferred basis. As it is no surprise that in different regimes (when the condition Hˆ int  Hˆ S + Hˆ E does not hold), other bases will be selected. For example, in the microscopic regime, where the self-Hamiltonian of the system dominates the dynamics: Hˆ S  Hˆ int + Hˆ E , the energy basis will be generally selected, leading to decoherence in that particular basis and the formation of separate energy eigenstates, the atomic orbitals. We thus reach the following: Counterclaim decoherence theory consistently describes/accounts for the selection of position basis in the macroscopic regime, but it does not provide a genuine explanation of why the classical world is position-based. Decoherence selects position basis in the macroscopic regime since the system-environment 25 See

e.g. Schlosshauer (2019, Sect. 4.2).

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interactions are already expressed in position basis. This introduces a positionbasis representation, which becomes particularly relevant in all cases in which: Hˆ int  Hˆ S + Hˆ E . While this scheme/procedure is perfectly fine for pragmatic purposes, this cannot be taken as a genuine ontological explanation of the selection of position basis at the macroscopic regime. In doing this, we would mix up the (sometimes hidden) phenomenological assumptions that physicists put in concrete models to make them work with the genuine ontological explanations coming from the theory. Decoherence theory does not explain why position basis emerges in the classical world, as this is only “explained” by introducing position as a privileged variable for the system-environment interaction.

1.5 Conclusion The way in which decoherence theory describes the emergence of the classical world from quantum mechanics is trickier than usually thought. In particular, I have shown that a precise characterization of the emergent classical objects is lacking in the standard context, as well as the description of Newtonian trajectories for individual systems. The basis problem—the problem to understand why the classical world is position-based—is solved pragmatically, but not ontologically: this leaves the debate open for future research. Finally, I have suggested that decoherence theory itself proves to be philosophically very helpful when interpreted in a realist framework of quantum mechanics. Acknowledgments I want to thank Valia Allori, Mario Hubert, Vera Matarese and Antonio Vassallo for helpful comments on earlier drafts on this paper and Craig Callender, Eddy Chen, Barry Lower and Kerry McKenzie for a nice and useful discussion of the paper in the San Diego philosophy of physics reading group. This work has been supported by the Fundação para a Ciência e a Tecnologia through the fellowship FCT Junior Researcher, hosted by the Centre of Philosophy at the University of Lisbon.

References Adler, S. L. (2003). Why decoherence has not solved the measurement problem: A reply to P. W. Anderson. Studies in History and Philosophy of Modern Physics, 34(1), 135–142. Bacciagaluppi, G. (2020). The role of decoherence in quantum mechanics, Stanford Encyclopedia of Philosophy, substantive revision 2020. https://plato.stanford.edu/entries/qm-decoherence/ Ballentine, L. (1998). Quantum mechanics: A modern development. World Scientific. Ballentine, L. (2008). Classicality without decoherence: a reply to Schlosshauer. Foundations of Physics, 38, 916–922. Bohm, D., & Hiley, B. (1987). An ontological basis for the quantum theory: Non-relativistic particle systems. Physics Report, 144(6), 321–375.

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Bohm, D., & Hiley, B. (1993). The undivided universe: An ontological interpretation of quantum theory. Routledge. Bowman, G. E. (2008). Essential quantum mechanics. Oxford University Press. Chen, E. (2018). Quantum mechanics in a time-asymmetric universe: On the nature of the initial quantum state. The British Journal for the Philosophy of Science, 72, 1155–1183. Crull, E. (2015). Less interpretation and more decoherence in quantum gravity and inflationary cosmology. Foundations of Physics, 45, 1019–1045. Crull, E. (2019). Quantum decoherence. In E. Knox & A. Wilson (Eds.), The Routledge companion to philosophy of physics (1st ed., 2021). Routledge. Di Biagio, A., & Rovelli, C. (2021). Stable facts, relative facts. Foundations of Physics, 51(30). Dürr, D., Goldstein, S., & Zanghì, N. (1992). Quantum equilibrium and the origin of absolute uncertainty. Journal of Statistical Physics, 67, 843–890. Holland, P. R. (1993). The quantum theory of motion: An account of the de Broglie-Bohm causal interpretation of quantum mechanics. Cambridge University Press. Joos, E., Zeh, D., Kiefer, C., Giulini, D., Kupsch, J., & Stamatescu, I. O. (2003). Decoherence and the appearance of a classical world in quantum theory (2nd ed.). Springer. Myrvold, W. (2018). Ontology for collapse theories. In S. Gao (Ed.), The collapse of the wave function (pp. 97–123). Cambridge University Press. Okon, E., & Sudarsky, D. (2016). Less decoherence and more coherence in quantum gravity, inflationary cosmology and elsewhere. Foundations of Physics, 46(7), 852–879. Romano, D. (2016a). Bohmian classical limit in bounded regions. In L. Felline, A. Ledda, F. Paoli, & E. Rossanese (Eds.), New directions in logic and the philosophy of science (SILFS series: 303–317). College Publications. Romano, D. (2016b). The emergence of the classical world from a Bohmian Universe, PhD thesis, University of Lausanne. Rosaler, J. (2015). Is de Broglie–Bohm theory specially equipped to recover classical behavior? Philosophy of Science, 82(5), 1175–1187. Rosaler, J. (2016). Interpretation neutrality in the classical domain of quantum theory. Studies in History and Philosophy of Modern Physics, 53, 54–72. Schlosshauer, M. (2007). Decoherence and the quantum-to-classical transition. Springer. Schlosshauer, M. (2019). Quantum decoherence. Physics Report, 831, 1–57. Shankar, R. (1994). Principles of quantum mechanics. Springer. Toroš, M., Donadi, S., & Bassi, A. (2016). Bohmian mechanics, collapse models and the emergence of classicality. Journal of Physics A: Mathematical and Theoretical, 49(35), 355302. Wallace, D. (2012). The emergent multiverse: Quantum theory according to the Everett Interpretation. Oxford University Press. Zurek, W. (2003). Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics, 75, 715–775. Zurek, W. H., Habib, S., Paz, J. P. (1993). Coherent states via decoherence, Physical Review Letters, 70(9): 1187–1190.

Chapter 2

Quantum Fundamentalism vs. Scientific Realism Matthias Egg

‘As for myself,’ said Éomer, ‘I have little knowledge of these deep matters; but I need it not. This I know, and it is enough, that as my friend Aragorn succoured me and my people, so I will aid him when he calls.’ The Return of the King J.R.R. Tolkien

Abstract Quantum fundamentalism is the view that quantum mechanics (QM) should inform us about fundamental ontology. It is adopted by many who seek to defend scientific realism with respect to QM, and it prompts them to opt for one of the versions (or interpretations) of QM that were developed in response to the measurement problem. I argue that this is a mistake. Not only is realism about QM compatible with neutrality concerning these different versions, but the commitment to any particular one of them is actually in tension with basic tenets of scientific realism. This is demonstrated by a critical assessment of Michael Esfeld’s and Valia Allori’s recently developed versions of quantum fundamentalism.

2.1 Introduction Scientific realism is concerned with empirically successful scientific theories and with what they tell us about reality. No one denies that quantum mechanics (QM) is empirically successful, but there is little agreement as to what it tells us about reality. If we refer to “standard QM” as the kind of predictive apparatus (going back to Dirac and von Neumann) that is employed to generate this empirical success, then it is difficult to pin down what it would even mean to be a realist about standard QM.

M. Egg () University of Bern, Bern, Switzerland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_2

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It is therefore widely assumed that the only way to be a realist about QM is to go beyond the standard account and to opt for one of the versions (or interpretations) of it that were developed in response to the measurement problem (e.g., many-worlds accounts, spontaneous collapse models or pilot-wave theories). This assumption has recently come under attack by authors who suggest that realism about QM can remain neutral with respect to these options by lowering its ambitions concerning the ontological content associated with the realist attitude (Healey, 2020; Saatsi, 2020). While I share these critics’ reluctance to go beyond standard QM, I do not think that the problem lies with the ontological ambitions of scientific realism. In my view, realism should be concerned with ontology,1 but it is a mistake to think of QM as informing us about fundamental ontology. This mistake is what I call quantum fundamentalism. In previous work (Egg, 2021), I have argued that the rejection of quantum fundamentalism allows us to formulate an ontologically robust version of realism about standard QM, which can avoid being committed to any one of the specific ways to go beyond the standard approach. I now want to take a step further and claim that realism about QM is not only compatible with neutrality concerning the non-standard versions of QM, but that a commitment to one of these versions actually contradicts the spirit of scientific realism. Insofar as such a commitment is motivated by quantum fundamentalism, my central thesis can be expressed by saying that the scientific realist should reject quantum fundamentalism. The aim of the present paper is to provide support for this thesis by critically discussing the arguments of those who think that scientific realism in the quantum domain requires (or at least encourages) some form of quantum fundamentalism. Such arguments have rarely been made explicit, because the above-mentioned conjunction of scientific realism with particular responses to the measurement problem has often been taken for granted. Besides, debates on the interpretation of QM have for a long time had little interaction with the debate on scientific realism. However, thanks to recent contributions by Michael Esfeld (2020) and Valia Allori (2018, 2020a,b), we now have two explicitly formulated views on how the foundations of QM might relate to scientific realism. The two views are quite similar to each other in that both Esfeld and Allori endorse a Bohmian version of QM and a view of physical theories that requires them to be formulated in terms of what is known as a primitive ontology (PO). A consequence of this similarity is that my investigation in this paper is not as general as it could be. I suspect that at least a part of my criticism could also apply to other instances of quantum fundamentalism, but I chose to focus on these two authors because they make the connection to scientific realism explicit. My critique of quantum fundamentalism will therefore largely come in the guise of a critique of the PO approach from a scientific realist’s point of view. 1

I admit that there are many formulations of scientific realism (see Chakravartty (2017) for an overview) and not all of them include the ontological component I am concerned with here. However, the authors discussed in the rest of this paper clearly share an ontological understanding of scientific realism in the sense that the realist should take some parts of our best scientific theories to accurately describe elements of reality.

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The investigation will proceed as follows: in Sect. 2.2, I will introduce and critically discuss the distinction between ontology and dynamics, which is a central ingredient of the PO approach. I will then (in Sect. 2.3) work towards the connection with scientific realism by asking what exactly Esfeld means by ‘ontology’ and the related notions of ‘existence’ or ‘being (there)’. Section 2.4 takes a closer look at how his view plays out in the context of QM. Section 2.5 does the same for Allori’s view, including critical assessments of her claim that the PO approach helps scientific realism to deal with familiar antirealist challenges stemming from underdetermination and theory change. A conclusion is drawn in Sect. 2.6.

2.2 Ontology and Dynamics One of the basic ideas of the PO approach is the following piece of methodology: A physical theory should clearly and forthrightly address two fundamental questions: what there is, and what it does. The answer to the first question is provided by the ontology of the theory, and the answer to the second by its dynamics. (Maudlin, 2019, p. xi)

So far, few scientific realists would disagree. Indeed, one might even think that this applies to any theory of empirical science, not just to physical theories. But as I will argue in what follows, a tension with scientific realism arises as soon as PO supporters not only insist that these two questions be addressed, but insist that they be addressed separately. The problem, in other words, is that some PO approaches presuppose a strict separation of (primitive) ontology from dynamics. For example, when Esfeld (2020, p. 90) lists different aspects that are central to the argument of his book, the first item on the list is “the distinction between primitive ontology and dynamical structure”.2 Similarly, Allori (2013, p. 64) postulates that fundamental physical theories “have a dual structure: the primitive variables that specify what matter is, and some other variables that determine its temporal development (its dynamics)”.3 A first thing to note about this separation is that it can hardly be upheld for nonfundamental branches of science. To give a simple example, fluids and solid bodies are arguably part of the ontology of continuum mechanics, but any specification of what differentiates a fluid from a solid depends on how they behave, that is, on the dynamics. Allori acknowledges this by explicitly restricting her considerations to

2

The metaphysical background for this view is a position called “Super-Humeanism” (Esfeld and Deckert, 2018, Section 2.3). In a similar spirit as the present paper, Vera Matarese (2020) has recently questioned Super-Humeanism’s conformity with scientific realism. 3 Another important ingredient of the PO approach is the requirement that the PO is constituted by “entities living in three-dimensional space or in space-time” (Allori, 2013, 60). This will become important in Sects. 2.4 and 2.5, because it excludes the quantum mechanical wave function (which is in general not defined on three-dimensional space, but on some higher dimensional configuration space) from the PO.

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fundamental physical theories. Esfeld, by contrast, also wants to speak about nonfundamental (even non-physical) theories, but maintains that these do not, strictly speaking, have a primitive ontology. What they are ultimately about, according to Esfeld (2020, 72) are specific features of how the PO of fundamental physics (in his view, particles) behaves: for the scientist in the special sciences, it makes handling the theories much easier to pretend that there are intrinsic chemical elements, intrinsic biological species, etc. That this is not so, that these elements or species do not have to be accepted as something primitive, but are functionally defined in terms of the roles of their characteristic features for in the last resort particle motion becomes relevant only when it comes to the relationship between the special sciences and physics.

Thus Allori and Esfeld agree that the strict separation of ontology from dynamics is only applicable in the context of fundamental physical theories. But how is this compatible with the fact that they (along with many other contributors to the PO literature) primarily discuss the PO approach in the context of an obviously nonfundamental theory, namely nonrelativistic quantum particle mechanics? In other words, what is the justification for adopting quantum fundamentalism (as defined in Sect. 2.1)? The most plausible reply seems to be that the ontological lessons learnt in the nonrelativistic case are expected to carry over to more fundamental theories and ultimately to the final, unified theory of physics. However, David Wallace (2020b,a) gives some good reasons to be skeptical about that expectation. In particular, the conceptual differences encountered in the transition from nonrelativistic QM to relativistic quantum field theory are too profound to simply assume ontological continuity between the two theories (not to mention even more fundamental theories, such as quantum gravity). Admittedly, Esfeld and Deckert (2018, chap. 4) argue that a PO of particles can successfully be transferred from QM to quantum field theory, but their model (just like any other PO approach to quantum field theory) has not yet passed the stage of a hopeful research program. While I share Wallace’s misgivings about quantum fundamentalism, I disagree with his conclusion that the scientific realist who rejects quantum fundamentalism should end up endorsing the Everett interpretation. I will discuss my alternative conclusion in Sect. 2.5 below. It is based on my own way of rejecting quantum fundamentalism on realist grounds, which I have developed and defended in a recent paper (Egg, 2021). In the same paper, I have also directly questioned the separation of ontology from dynamics, endorsing Myrvold’s (2018, p. 105) claim that “until we have said something about how the purported ontology acts, we haven’t yet given sense to the claim that it is there at all”. I should add, however, that this claim is somewhat overstated, because the PO framework (and in particular, Esfeld’s rigorous treatment of point particles individuated by changing distance relations) indeed gives sense to the notion of spatiotemporal presence, although that sense differs quite starkly from our ordinary (dynamically infected) notion of “being there”. Be that as it may, what matters for the present discussion is not so much whether a non-dynamical notion of ontology makes sense, but to what extent it aligns with scientific realism. I will now turn to this question.

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2.3 Is Primitive Ontology All There Is? The qualifier ‘primitive’ suggests that the realm of ontology is not exhausted by the PO but also contains a non-primitive part. Yet some formulations of the PO approach seem to imply that the PO is all there is to ontology, for example when Esfeld (2020, p. 61) writes that the dynamical structure (which, as we saw in Sect. 2.2, contains everything that a theory introduces in addition to the PO) “does not call for additional ontological commitments”. In other places, he is more generous, granting that dynamical parameters such as mass and charge are “admitted to the ontology, albeit not as primitive, but as derived notions” (Esfeld, 2020, p. 68). He further elaborates on this kind of derivation by explaining how dynamical properties are not eliminated but located (in the sense of Jackson (1998)) in configurations of the point particles that make up the PO. Scientific realism usually operates with the more generous notion of ontology, because many of the non-primitive parameters are indispensable for the empirical success of science and are therefore the proper targets of a realist’s commitment, which can be expressed by saying that they refer to something real, something that exists or something that is there (to on in ancient Greek, from which the term ontology derives). The more restrictive notion of ontology, according to which only the PO is really there (or in the words of Esfeld (2020, p. 70) “simply exists in the world”), by contrast, has little to do with scientific realism. For a scientific realist, singling out the PO as that which really (or simply) exists would have to be justified by scientific evidence supporting the PO to a larger degree than other theoretical posits, such as the dynamical structure. As we will see in Sect. 2.5, the actual evidential situation is just the opposite: if we have uncontroversial evidence for anything, it is for dynamical (or more generally: non-fundamental) structures, not for any fundamental ontology. And even if we neglect this evidential difference for the sake of the argument, it is hard to see how scientific realism could incorporate a principled restriction of existence claims to an elite class of fundamental entities. After all, when antirealists dispute the existence of electrons and other unobservable entities, a central part of the realist’s response has always been the rejection of any principled restriction of existence claims to an elite class of entities (observable ones in this case). And many of the entities on whose existence scientific realists have insisted in such disputes (molecules, genes, bacteria, for example) are undoubtedly non-fundamental (as, by the way, are the observable entities whose existence is not disputed by either party in the debate). Therefore, scientific realism has no room for the restriction of existence to supposedly fundamental entities.4

4

One might think that this problem can be circumvented by formulating scientific realism in terms of truth rather than existence, but this only works insofar as one disregards the question of what it is in the world that makes some statement true (see Asay (2018) for different realist options in that respect). Now Esfeld is clearly not among those who disregard this question, and his response to it brings us back to the original claim that the only real truth-makers are the elements of the PO. About propositions containing predicates such as ‘mass’ or ‘charge’, he writes:

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2.4 Primitive Ontology with Wave Function Realism Let us now investigate the tension between the PO approach and scientific realism more concretely by focusing on the most important element of non-primitive ontology in QM, the wave function. Here is a beautifully concise statement of why one should want to be a realist about the wave function: Which features of a theory should a scientific realist take to represent the world? Answer: those that are responsible for the theory’s explanatory success. When the theory is quantum mechanics, the wave function is surely one of those features. (French, 2013, p. 76)

This provides anyone who, like Esfeld, seeks to combine scientific realism with the PO approach, with a strong motivation to adopt the more generous of the two notions of existence discussed above. In this way, he can take the wave function to represent something in the world, although it is not part of the PO, but only a dynamical parameter (as Esfeld (2020, p. 81) puts it, “the central dynamical parameter in quantum physics”). Indeed, Esfeld explicitly subscribes to a kind of realism about the wave function, but simultaneously emphasizes the sense in which the wave function does not exist: This view is a realism about the wave-function by contrast to an instrumentalism. The wavefunction is not a sui generis entity that exists in addition to the primitive ontology; but it exists, namely as being located in the particle configuration of the universe and its evolution. (Esfeld, 2020, p. 82)

One might view Esfeld here as trying to work with two rather different notions of existence at the same time. This would confirm a suspicion raised in the previous section, and it was actually my own first response to this passage, but I have in the meantime been convinced (by Michael Esfeld and an anonymous referee) that this is not what he has in mind. He insists that there is only one notion of existence, it is just that some of what exists (e.g., the wave function) depends on other things that exist. If that was the whole story, Esfeld could uphold a robust version of wave function realism, and no conflict with scientific realism would arise. Unfortunately, however, the more restrictive notion of existence (which does not fit well with scientific realism) crops up again in one of the central arguments in Esfeld’s book, which purports to establish that “determinism in science is not opposed to free will” (Esfeld, 2020, Section 2.4). He approaches this issue by critically discussing Peter van Inwagen’s consequence argument, which, in a nutshell, says that if determinism is true, then our present acts are not up to us, because they are consequences of laws of nature and events in the remote past. Esfeld’s rather ingenious response to that argument is to attack the (seemingly

These predicates—as well as all the other ones appearing in the propositions that are true about the world—really apply, and the propositions really are true; there is nothing fictitious about them. But what there is—and hence what makes them true—is nothing over and above the distribution of primitive stuff throughout space and time. (Esfeld, 2014a, 465)

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indubitable) premise that what went on in the remote past is not up to us. Since in his view, dynamical parameters are located in particle movements, there is indeed a sense in which something in the remote past (namely, the values of dynamical parameters pertaining to an early stage of the universe) depends on our choices: “These values depend on what happens later in the universe, including the particle motions that are expressions of free will” (p. 102). What matters for our discussion is how Esfeld defends this proposal against the immediate objection that it involves a kind of backwards causation. It does not, he claims, because “the particle positions and motions before we were born are what they are, independently of what we do” (ibid.). This is compatible with a change in the initial values of dynamical parameters (such as the wave function) due to later particle motions.5 Why does such a change not amount to backwards causation? One possible response would be to adopt the restrictive notion of existence discussed above, according to which the dynamical parameters are not considered as real features of the world at any particular time. The only real features in this sense would be the particle positions, and these are not affected by influences from the future. As already mentioned, this is not what Esfeld has in mind, because he wants to work with only one notion of existence (the one that is in line with scientific realism). Another option (suggested to me by an anonymous referee) is to hold that the wave function does exist as a higher-level entity, but that higher-level causation is not genuine causation. The problem with this response is that causation in all sciences except fundamental physics is higher-level causation and that therefore, to say that higher-level causation is not genuine causation is just as much in tension with scientific realism as to say that higher-order existence is not genuine existence. Relatedly, Esfeld (in personal communication) suggests to view the wave function as just the kind of thing that (unlike the positions and motions of particles) can be influenced by future events. In that case, however, I no longer see what “distinguishes the present proposal from proposals in terms of backwards causation” (Esfeld, 2020, p. 102). At the end of the day, Esfeld’s response to the charge of backwards causation must depend, in some way or other, on the idea that a change of the wave function due to future events is somehow less problematic than a change of particle motions due to future events, and this is hard to square with the scientific realist’s conviction that the wave function is just as much part of objective reality as the particles and their motion. Of course, Esfeld can still pay lip service to realism by saying that the wave function—as located in the complete history of the PO until the end of the universe—is real. But this is a far cry from what the scientific realist means when she says that the wave function—as responsible for successful explanations and predictions at a certain time—is real. 5

As far as I see, justification for this assumption comes from non-relativistic Bohmian mechanics, where a history of particle configurations can be compatible with many different wave functions describing the system’s quantum state at a certain time during that history. It is not clear to me whether this still holds in Esfeld and Deckert’s (2018, chap. 4) Dirac sea model for quantum field theory, where the number of particles is massively increased and the requirement of being able to change the quantum state without changing any particle movements seems much harder to satisfy.

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2.5 Primitive Ontology Without Wave Function Realism In light of the difficulties engendered by working with two distinct notions of existence, it is natural for the PO supporter to abandon the more generous notion and to deny reality to everything that is not part of the PO. In a sense, this is the path chosen by Allori (2018, 2020a,b), who, in particular, rejects wave function realism. Since it was wave function realism and the corresponding generous notion of reality that kept Esfeld’s position in touch with scientific realism, one might expect that Allori’s position is even harder to square with scientific realism. I will argue that this indeed what we find, although Allori is much more outspoken about what she views as an excellent fit between the PO approach and scientific realism. Before we dive into this discussion, it is important to address an ambiguity in the term ‘wave function realism’. Up to now, I have used it in a sense that would more precisely be denoted by ‘scientific realism about the wave function’, motivated by the kind of explanatory success mentioned in the opening quote of Sect. 2.4. The other sense, which is actually more widespread in contemporary philosophy of physics, would better be called ‘wave function ontology’ or ‘wave function fundamentalism’. It is the view, defended most notably by Albert (1996) and Ney (2021), that the wave function is not only a real physical object, but is also part of the fundamental structure of the world (possibly even its only part). In this sense, wave function realism is just as much an instance of quantum fundamentalism as the PO approach, and some of the points of criticism I have raised so far could also be directed against it. Furthermore, much of what Allori has to say against wave function realism is actually a critique of wave function fundamentalism. Since this type of critique belongs to the internal dispute among quantum fundamentalists, I will not address it here (see Egg (2017) for my own take on that dispute). Let us therefore focus on what Allori has to say about wave function realism in the first (non-fundamentalist) sense. Although she ends up arguing against it, it is noteworthy that (just as in the case of Esfeld discussed above) some aspects of wave function realism are still present in her position. For one thing, she acknowledges that the so-called ψ-epistemic views are hard to square with scientific realism and that her own view, which she calls ‘ψ-non-material’ is actually a subtype of a ψ-ontic view (Allori, 2020b, pp. 35–39). Additionally, and again similarly to Esfeld, she seeks to reduce the wave function to its functional role, embracing the slogan “the wavefunction is as the wavefunction does” (Allori, 2020b, p. 40). I am entirely sympathetic to this appeal to functionalism, but would like to put a realistic emphasis onto the first part of the slogan: the wavefunction is. As I have argued elsewhere, functionalism is precisely what enables us to be realists about QM, including (in some instances at least) realism about the wave function (Egg, 2021). I therefore do not think that functionalism by itself supports Allori’s claim that “the wavefunction does not represent matter” (Allori, 2020b, p. 41). Consider the parallel case she uses to introduce functionalism: few of us who readily acknowledge that “a table is as a table does” would thereby be tempted to deny realism about tables, or to say that the term ‘table’ does not refer to a material object. This does not imply,

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of course, that the wave function is a material object (for its function may be quite different from the function of a table). My claim is merely that functionalism about any X is entirely compatible with realism about X and with the view that X is a material object. Moving closer to the traditional debate on scientific realism, Allori (2020a) reminds us that in the face of antirealist criticism, realism was forced to become selective, restricting its commitment to what Kitcher (1993) has called the working posits of a successful theory, while remaining agnostic about its presuppositional posits. This kind of selective realism obviously also constitutes the background for French’s dictum quoted at the beginning of Sect. 2.4. Allori’s account shares the background, but differs radically in the assessment of what belongs to the working posits of QM. Unlike French, she attributes the explanatory and predictive success of QM to the PO, not to the wave function: Quantum theory’s predictions, being encoded in pointer positions, are determined by the PO, not by the wave function. Similarly, the explanations of the phenomena are in terms of the PO, and only indirectly involve the wave function. That is, the PO is reminiscent of the working posits and the wave function is reminiscent of a presuppositional posit. (Allori, 2020a, p. 221)

Given that most standard presentations of QM (in physics textbooks, for example) do not speak about ontology (let alone PO), it is not immediately clear what to make of the claim that QM’s predictions and explanations of phenomena “are in terms of the PO”. One might interpret it as claiming that most QM textbooks— except for the few cases which actually do speak about ontology (e.g., Norsen, 2017)—do not contain any actual successful predictions and explanations. This would be hard to swallow for any scientific realist who seeks to take physics (as it is actually taught and practiced in schools and laboratories around the world) seriously. A more plausible interpretation (suggested by (Allori, 2020a, fn. 2) and (Allori et al., 2008, p. 363)) is that even the standard presentations of QM, when they speak about measurement outcomes, do (at least implicitly) specify a PO, if only a PO consisting of pointer positions and other (macroscopic) means of displaying experimental results. Interpreted in this way, Allori’s claim is more palatable, but still faces two serious problems. First, by attaching the label ‘PO’ to whatever constitutes the physical events that serve to confirm a theory, it guarantees a priori that the PO will be among its working posits. But the resulting version of selective realism would no longer differ in any interesting way from instrumentalism or other antirealist positions, as long as they accept the reality of macroscopic objects.6 Second, by adding macroscopic objects to the list of possible candidates for a PO (alongside point particles, matter fields and flashes, cf. Allori, 2020a, pp. 218–219),

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It might seem that the PO of standard QM also encompasses microscopic entities, since QM textbooks often talk about elementary particles and their properties, for example. However, standard QM describes such entities by a wave function, whose role, according to the proposal under consideration, is “to determine the probability relations between the successive states of [macroscopic] objects” (Allori et al., 2008, 363).

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it leaves us with radically different possibilities of what realism about PO amounts to. This is related to another problem familiar from the scientific realism debate, which we must now investigate in more detail: the underdetermination of theory by evidence. When different theoretical accounts with mutually incompatible ontological commitments are equally well confirmed by the empirical evidence, it gets difficult to justify realism about any particular one of them. Many realists then appeal to non-empirical criteria of theory evaluation in order to arrive at a unique verdict on what to be a realist about. This is what Allori (2020a, sect. 11.8) does as well (see also Esfeld, 2014b), concluding that a pilot-wave theory with a PO of particles is the best option. She admits, however, that the issue remains controversial, as some of the criteria pull in opposite directions. But even if we set this problem aside, her conclusion lacks support, due to her neglect of the ontological option just discussed, which builds on the massive amount of empirical success that (standard) QM achieves without postulating any microscopic PO. Allori (2020a, fn. 2) dismisses this option, because macroscopic entities cannot plausibly be regarded as fundamental. However, to suppose that they need to be so regarded is just quantum fundamentalism at work again. Recall that macroscopic objects such as pointers were (somewhat idiosyncratically) labeled as the PO of standard QM, in order to uphold the claim that the theory predicts and explains phenomena in terms of the PO. As a scientific realist, I agree that some PO (in this minimal sense) must be postulated, but I do not see any justification for the additional requirement that the PO ought to be fundamental. If scientific realism were only applicable to predictions and explanations given in terms of fundamental ontology, no science (except perhaps some future fundamental physics) could claim to teach us much about reality. This is not to say that all is well for realism about QM as soon as a (nonfundamental) ontology of macroscopic objects along with a wave function is postulated. The measurement problem needs to be addressed, because the realist cannot accept the anthropocentric notion of ‘measurement’ to enter into the dynamical description as an unanalyzable primitive (Egg, 2019). Now it is well known that the measurement problem generates a serious case of underdetermination, since there is no consensus as to which one of the proposed solutions is to be preferred (Egg and Saatsi, 2021). However, it can be shown that if one abandons quantum fundamentalism and looks at concrete paradigmatic cases of QM’s explanatory and predictive success, the underdetermination problem disappears, because there turns out to be a striking convergence in the (non-fundamental) ontology postulated by the different proposals, in spite of their radical disagreement about fundamental ontology. In particular, I have argued (Egg, 2021) that in some key experiments of QM (the Stern-Gerlach experiment, quantum tunneling and two-path interference), some aspects of the wave function play an indispensable explanatory role according

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to all proposals (including the PO approaches advocated by Esfeld and Allori).7 These aspects are therefore properly regarded as the working posits of QM, in contrast to any kind of microscopic PO, which is only present in some particular proposals. Allori (2018, p. 72) seeks to deny the wave function its status as a working posit by citing attempts to eliminate the wave function completely (Dowker and Herbauts, 2005; Norsen, 2010). If these attempts were successful, a parallel argument to the one I gave with respect to microscopic PO could be made: since the success of QM can also be achieved without the wave function, it cannot be a working posit. However, the cited proposals are merely demonstrations of the wave function’s eliminability in principle, with no pretension of being applicable in practice to any real-life example like the ones discussed in Egg (2021). Such demonstrations are of interest to the quantum fundamentalist, but not to the scientific realist, who (reminiscent of Éomer in the epigraph of this chapter) is committed to the (nonfundamental) working posits that account for his experiences, and does (for the time being) not care about what may ultimately turn out to be their fundamental nature. I have not yet mentioned what is probably the main motivation to develop selective realism and its distinction between working and presuppositional posits: according to a prominent antirealist argument known as the pessimistic metainduction, many posits of once successful scientific theories later turned out not to represent anything in reality. In response, the selective realist wants to show that this is only the case for presuppositional posits, not for working ones. Allori (2018) claims that a PO of point particles is particularly well suited to ground a selective realist response to the pessimistic meta-induction. Let me conclude this section by raising some doubts about this claim. First, selective realists have traditionally viewed their task as involving an important empirical element: one tries to identify the working posits of any given theory and then looks at the history of science to check whether these posits actually survived when one theory was replaced by another. By contrast, Allori identifies the working posits with the PO, which in turn follows from specific constraints on how physical theories should be formulated (‘theory architecture’, as she calls it). If a theory is successful without respecting these constraints (as is arguably the case for standard QM), it has to be reformulated (as a pilot-wave theory in this case) before one can identify its working posits. To the extent that this determines what the working posits of any physical theory are going to be (namely point particles), ontological continuity between different such theories is guaranteed a priori. Furthermore, it is doubtful whether the statement that “the wave function arguably does not have any classical analog and therefore we have radical discontinuity in the classical-to-quantum theory change” (Allori,

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There are other aspects of the wave function which do not play such a role, an important example being superpositions of macroscopically discernible states of affairs (Everettian “worlds”). That these are not acknowledged in all the proposed solutions of the measurement problem is the reason why (as mentioned in Sect. 2.2) my proposal differs from the Everett interpretation, and is, from the realist point of view, preferable to it.

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2018, p. 71) is really an accurate way of describing the challenge posed by the pessimistic meta-induction. QM (like any other novel theory) taught us surprising things we did not previously know, and it is no embarrassment for the realist that this novelty is reflected in ontological posits that do not have any analog in the preceding theory. A problem for realism would only arise if the novel ontology were supposed to replace the working posits of the preceding theory. If one properly distinguishes wave function realism from wave function fundamentalism (as suggested above), it is clear that the former involves no such supposition. Again, it turns out that many of Allori’s arguments in this context are actually arguments against wave function fundamentalism, not against wave function realism. Finally, Allori’s proposal of ontologically streamlining the classical-to-quantum transition by viewing point particles as the working posits of QM is premised on the claim that point particles were the working posits of classical physics. This is far from obvious. For example, point particles do not play any explanatory role in the paradigmatic applications of Newtonian mechanics (explaining planetary motions, movements of falling bodies, the tides etc.), and even in Allori’s (2018, p. 72) own example (“the transparency of a pair of glasses is explained in terms of the electromagnetic forces acting between the particles composing the glasses, which are such that incoming light rays will pass through them”), one would have to look at the details to judge whether it is really the point particles themselves (rather than their energy levels or some other dynamical parameters) that are doing the explanatory work.

2.6 Conclusion I have argued that the marriage of the PO approach with scientific realism is an unhappy one. Part of the reason for this is the former’s liaison with quantum fundamentalism, which engenders a tendency to regard everything that is not part of the fundamental ontology as somehow less real. Scientific realism advocates exactly the opposite ranking of ontological commitment: since we have much stronger evidence for non-fundamental entities than for fundamental ones, we should be more strongly committed to the former. When it comes to QM, the wave function is the crucial element for achieving the kind of explanatory and predictive success that motivates scientific realism. We saw that even a PO advocate like Esfeld feels the pull of this and accepts a kind of wave function realism. But in the end his commitment to quantum fundamentalism prevails and weakens his realism, pushing his position towards Allori’s explicit antirealism about the wave function. Her arguments reveal another harmful effect of quantum fundamentalism: it prompts its adherents to lose sight of the difference between claims about reality and claims about fundamentality. As a consequence, Allori throws out the (wave function realist) baby with the (wave function fundamentalist) bathwater. This puts pressure on the compatibility of quantum fundamentalism with scientific realism. I conclude that insofar as we are unwilling to give up the latter, we should let go of quantum fundamentalism.

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References Albert, D. Z. (1996). Elementary quantum metaphysics. In J. T. Cushing, A. Fine, & S. Goldstein (Eds.), Bohmian mechanics and quantum theory: An appraisal. Boston studies in the philosophy of science (Vol. 184, pp. 277–284). Kluwer Academic Publishers. Allori, V. (2013). Primitive ontology and the structure of fundamental physical theories. In A. Ney & D. Z. Albert (Eds.), The wave function: Essays in the metaphysics of quantum mechanics (pp. 58–75). Oxford University Press. Allori, V. (2018). Scientific realism and primitive ontology or: The pessimistic induction and the nature of the wave function. Lato Sensu: Revue de la Société de Philosophie des Sciences, 5, 69–76. Allori, V. (2020a). Scientific realism without the wave function. In S. French & J. Saatsi (Eds.), Scientific realism and the quantum (pp. 212–228). Oxford University Press. Allori, V. (2020b). Why scientific realists should reject the second dogma of quantum mechanics. In M. Hemmo & O. Shenker (Eds.), Quantum, probability, logic: The work and influence of Itamar Pitowsky (pp. 19–48). Springer. Allori, V., et al. (2008). On the common structure of bohmian mechanics and the Ghirardi-RiminiWeber theory. British Journal for the Philosophy of Science, 59, 353–389. Asay, J. (2018). Realism and theories of truth. In J. Saatsi (Ed.), The Routledge handbook of scientific realism (pp. 383–393). Routledge. Chakravartty, A. (2017). Scientific realism. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy. Summer 2017. http://plato.stanford.edu/archives/sum2017/entries/scientific-realism/ Dowker, F., & Herbauts, I. (2005). The status of the wave function in dynamical collapse models. Foundations of Physics Letters, 18(6), 499–518. Egg, M. (2017). The physical salience of non-fundamental local beables. Studies in History and Philosophy of Modern Physics, 57, 104–110. Egg, M. (2019). Dissolving the measurement problem is not an option for the realist. Studies in History and Philosophy of Modern Physics, 66, 62–68. Egg, M. (2020b). Quantum theory as a framework and its implications on the measurement problem. In S. French & J. Saatsi (Eds.), Scientific realism and the quantum (pp. 78–102). Oxford University Press. Egg, M. (2021). Quantum ontology without speculation. European Journal for Philosophy of Science, 11, 32, 1–26. Egg, M., & Saatsi, J. (2021). Scientific realism and underdetermination in quantum theory. Philosophy Compass. https://doi.org/10.1111/phc3.12773 Esfeld, M. (2014a). Quantum humeanism, or: physicalism without properties. The Philosophical Quarterly, 64, 453–470. Esfeld, M. (2014b). The primitive ontology of quantum physics: Guidelines for an assessment of the proposals. Studies in History and Philosophy of Modern Physics, 47, 99–106. Esfeld, M. (2020). Science and human freedom. Palgrave Macmillan. Esfeld, M., & D. A. Deckert (2018). A minimalist ontology of the natural world. Routledge. French, S. (2013). Whither wave function realism? In A. Ney & D. Z. Albert (Eds.), The wave function: Essays in the metaphysics of quantum mechanics (pp. 76–90). Oxford University Press. Healey, R. (2020). Pragmatist quantum realism. In S. French & J. Saatsi (Eds.), Scientific realism and the quantum (pp. 123–146). Oxford University Press. Jackson, F. (1998). From metaphysics to ethics: A defence of conceptual analysis. Oxford University Press. Kitcher, P. (1993). The advancement of science. Oxford University Press. Matarese, V. (2020). Super-humeanism and physics: A merry relationship? Synthese. https://doi. org/10.1007/s11229-020-02717-w Maudlin, T. (2019). Philosophy of physics: Quantum theory. Princeton University Press.

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Myrvold, W. (2018). Ontology for collapse theories. In S. Gao (Ed.), Collapse of the wave function (pp. 97–123). Cambridge University Press. Ney, A. (2021). The world in the wave function: A metaphysics for quantum physics. Oxford University Press. Norsen, T. (2010). The theory of (exclusively) local beables. Foundations of Physics, 40(12), 1858– 1884. Norsen, T. (2017). Foundations of quantum mechanics. Springer. Saatsi, J. (2020). Truth vs. progress realism about spin. In S. French & J. Saatsi (Eds.), Scientific realism and the quantum (pp. 35–54). Oxford University Press. Wallace, D. (2020a). Lessons from realistic physics for the metaphysics of quantum theory. Synthese, 197, 4303–4318. https://doi.org/10.1007/s11229-018-1706-y

Chapter 3

On the Principles That Serve as Guides to the Ontology of Quantum Mechanics Vera Matarese

Abstract Is the ontology of non-relativistic quantum mechanics three dimensional (3N) or high dimensional (3ND)? This paper discusses two principles, proposed in North (The structure of a quantum world. The wave function: essays on the metaphysics of quantum mechanics, pp 184–202, 2013) and Emery (Phil Phenomenol Res 95(3):564–591, 2017), that are usually employed to answer this question. The first, the dynamical matching principle (DMP), states that the fundamental structure of the world should match the structure of the dynamical laws of the theory, in this case, the Schrödinger equation. The second, the minimal divergence norm (MDN), states that insofar as we have multiple empirically adequate theories, we ought to choose the one that minimises the difference between what the theory says the world is like and how the world appears. While the former is used to argue in favour of a quantum 3ND ontology and the latter to argue in favour of a quantum 3D ontology, I show that both principles can in fact be used to support either view. This casts doubt on their role and legitimacy as meta-ontological principles that can guide us in the decision between a commitment to a 3ND ontology or 3D ontology. I suggest instead that they are best regarded as useful principles to construct and ‘regiment’ the space of plausible quantum ontologies.

3.1 Introduction Quantum mechanics faces a severe problem of underdetermination, as its ontological interpretations differ vastly, ranging from three-dimensional ontologies to high-dimensional ones. Although several interpretations exist, they can be lumped into two main rival groups: the quantum 3ND ontologies, on the one hand, which regard the wavefunction as a physical 3N-dimensional entity living in a physical

V. Matarese () Institute of Philosophy and Center for Space and Habitability, University of Bern, Bern, Switzerland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_3

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3N-dimensional space,1 and, on the other hand, the quantum 3D ontologies.2,3 This paper discusses two meta-ontological principles that have been advocated in this debate, the dynamical matching principle (DMP) and the minimal divergence norm (MDN). These two principles are usually meant to support and justify, respectively, a quantum high-dimensional ontology and a quantum three-dimensional ontology. Depending on whether we emphasise the view that the fundamental reality of the world should match the structure of our dynamical laws (DMP) or the view that our fundamental ontology should diverge as little as possible from our world of experience (MDN), only one of the two above-mentioned ontologies remains. This is, at least, how the debate is usually characterised: This brings us to a basic disagreement between wave function space and ordinary space views: how much to emphasize the dynamics in figuring out the fundamental nature of the world. Three-space views prioritize our evidence from ordinary experience, claiming that the world appears three-dimensional because it is fundamentally three-dimensional. Wave function space views prioritize inference from the dynamics, claiming that the world is fundamentally high dimensional because the dynamical laws indicate that it is. (North, 2013, p. 196)

Prima facie, this seems to be a problem of priority: which principle should be prioritised? This, I take it, would create a difficult dilemma, as both principles seem to be promising guides for any naturalistic and empirically oriented metaphysics. Yet, it might be a mistake to think that the principles themselves are truly mutually exclusive. One reason to suspect this is found in Chen (2017), where it is argued that the DMP does not uncontroversially favour 3ND ontology. In this paper, I argue for the stronger thesis that each principle could be applied to support either the 3ND quantum ontologies group or the three-dimensional quantum ontologies group (Sects. 3.3 and 3.4). At this point, questions on the validity of these meta-ontological principles naturally arise. What is their significance if they allow opposite conclusions and can be used to justify opposite ontological interpretations? What is their role, and what should it be? My paper warns against invoking such principles to justify a commitment to a 3D or 3ND ontology. To be clear, I am not arguing against the

1

Ontologies of this first group include, for instance, wavefunction monism (the view that all that exists is a wavefunction or wavefield living in a high-dimensional space, while our 3D world is only an illusion; see Emery (2017) and Allori (2010) for a discussion of such a view) and wavefunction fundamentalism (the view that the wavefunction is all that fundamentally exists, while our 3D world exists non-fundamentally; see Emery (2017) for a discussion of this view). 2 Ontologies of the second group include particle ontology, flash ontology, and mass-density ontology. If an ontology is hybrid (for instance, for Bohmian mechanics, Valentini (2009, 2010) postulates a particle ontology in three-dimensional space and a wave-function ontology in a highdimensional space), it counts as a 3N-dimensional ontology. 3 Note that the same quantum theory could feature a 3N-dimensional or three-dimensional ontology depending on its ontological interpretation. For instance, Bohmian mechanics can be interpreted as featuring a high-dimensional (Albert, 1996; Loewer, 1996; Valentini, 2009, 2010) or a threedimensional ontology (Allori, 2013a, 2017; Esfeld, 2014). This means that both groups contain a ‘Bohmian ontology’.

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legitimacy of these principles per se, but against the legitimacy of using these principles to justify a 3D or 3ND ontology. I devote the bulk of this paper to explaining why. The plan of the paper is as follows. First, I briefly review the problem of quantum underdetermination and identify ‘meta-ontological eliminative principles’— principles that support and justify a quantum ontology by ‘ruling out’ other ontologies—as a solution to the problem (Sect. 3.2). Next, I offer an examination of DMP and MDN by focussing on their legitimacy and application as metaontological eliminative principles that can decide between a commitment to the 3ND ontology group or to the 3D ontology group (Sects. 3.3 and 3.4). My conclusion is that neither principle can serve as a meta-ontological eliminative principle that can rule out one group in favour of the other because both can be used to support either group. I suggest instead that they should be used to construct and regiment the space of plausible quantum ontologies.

3.2 Setting Up the Debate: Meta-ontological Principles and Quantum Mechanics Within the philosophy of physics, ontology is normally understood as an inquiry concerned with cataloguing all that physically exists. It addresses questions such as, “Are electromagnetic fields or quantum wave-functions physically real? Are there three-dimensional objects like tables and tigers, or just microscopic particles arranged in table-like or tiger-like patterns?” In this paper, I focus on meta-ontology, which, within the philosophy of physics, usually concerns how we should form ontological commitments. Indeed, a central task of meta-ontology in the philosophy of physics is to define norms that dictate how to establish or identify the ontological commitments of physical theories.4 Some meta-ontological principles are eliminative and function as follows. First, we posit a space of possible ontologies. We then apply such eliminative principles to that space to rule out different ontologies until we arrive at the ultimate, noneliminable one, which conforms to all the principles we used to rule out the alternatives. Such meta-ontological, eliminative principles are essential in cases of underdetermination, such as the one afflicting quantum mechanics, where different ontologies are equally plausible alternatives, and a decision about which is the true one must be made. Since this paper targets the debate contraposing quantum 3ND ontologies and 3D ontologies, these eliminative principles should decide which group is the right one and thus whether the quantum ontology is high-dimensional or

4

There are many similar principles. Here, I consider only two, those that are normally employed to decide between 3D and 3ND. Other meta-ontological principle include the principle according to which we should opt for those ontologies that preserve the ontology from previous theories. For an implementation of this principle, see Allori (2013a).

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three-dimensional. The principles I consider in this case are not supposed to lead to one particular, specific, ontological interpretation, as many different ontologies are available within the 3ND ontology and 3D ontology groups. Instead, they should lead to one group of ontologies by eliminating the other.5 This way, at least, the most threatening underdetermination afflicting quantum metaphysics debates would be resolved. In this regard, the DMP and the MDN appear to be good candidates to serve as such eliminative, meta-ontological principles, for while the former is used in the literature to identify and support a 3ND ontology as the ‘true’ ontology for quantum mechanics, the latter is used to reject it and to justify a commitment to a 3D ontology. Since the task of the eliminative principles in this context is to eliminate one of the two groups, once one principle eliminates one group, the other becomes idle. In this respect, North’s discussion of whether DMP or MDN should have priority is relevant, especially given that it is normally thought that each principle leads to the selection of a different group of ontologies. The desiderata of these eliminative principles can be easily set. The first is that they should be able to eliminate one of the two groups, no more and no fewer. It would be problematic if the principles could not eliminate either view or could not eliminate both. Second, they should not presuppose what they are meant to establish, that is, they should not assume the truth or preferability of either wavefunction ontology or a three dimensional one. Since the DMP and the MDN are used in the literature as meta-ontological eliminative principles that can eliminate either the 3ND or the 3D ontology group, and therefore to ground the legitimacy of an ontological commitment to either 3ND or a 3D quantum ontology, it is important to examine whether they can actually play such a role.

3.3 The Dynamical Matching Principle The first principle I consider, called the ‘dynamical matching principle’ (Chen, 2017), was first spelled out by North (2013) as an inference principle that rules how the ontological commitments of a theory should be drawn: This brings me to a very general principle that guides our physical theorizing, from which the other principles I use all extend: the dynamical laws are about what’s fundamental to a world. [ . . . ] We do not directly observe the fundamental level of reality: we infer it from the dynamics. We posit, at the fundamental level, whatever the dynamical laws presuppose. (North, 2013, p. 186)

5

Before proceeding, let me clarify that, in this paper, I do not deal with the problem of whether these principles in conjunction with other principles can eliminate all the ontological interpretations except for one. I examine only whether these principles can be used to eliminate one of the two groups, as they are normally supposed to do.

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However, it has also been treated as an eliminative principle that eliminates 3D ontologies in favour of 3ND ontologies.6 Indeed, according to the DMP, the structure of the true quantum ontology should be determined by the Schrödinger equation, which governs the temporal development of the quantum wavefunction: DMP: The fundamental structure of the world should match the dynamical structure of quantum mechanics.

According to supporters of quantum 3ND-ontology, since the dynamical law posits a high-dimensional wavefunction living in a 3ND space, three-dimensional ontologies are ruled out. In order to check how this principle should actually be implemented and to proceed on firm grounds, it would be desirable to have a definition of ‘structure’. Neither North (2013) nor Chen (2017) provides a definition; what they seem to mean, however, is that an ontological structure should include both ontological entities and spacetime structure. Moreover, the dynamical structure of a theory is independent of any arbitrary or conventional representational means. I include this requirement as a minimum desideratum for the ontological structure that the DMP aims to select. Second, a definition of ‘matching’ would be desirable. In this regard, one well-known option would be to claim that the structure of our ontologies needs to be isomorphic to our mathematical ‘models’ thereof7: Ever since Newton, physicists have been in the business of constructing mathematical models of aspects of the world, ‘models’ in the sense that they are – or intended to be – isomorphic to the features of the world being studied. (Wallace, 2012, p. 12)

In this respect, the application of the DMP in quantum mechanics is, according to North and the supporters of 3ND ontologies, straightforward. Remember that the Schrödinger equation dictates the temporal evolution of the quantum wavefunction, which is a mapping Ψ : R 3N → C such that (x1 , x2 , x3 , . . . , x3N ) → c From this definition of the wavefunction, it seems clear that it is a high-dimensional entity, specified on a high-dimensional space, and thus we should postulate a highdimensional entity and space in the quantum fundamental ontology. It is important to note that the wave-function high-dimensional space and our three-dimensional

6

For instance, Chen (2017) uses it hoping to find an ultimate conclusion on the debate between 3D-fundamentalism and 3ND-fundamentalism. North (2013) presents it as an inference principle but also uses it to argue for 3ND ontology and against 3D ontologies. Albert (1996) and Ney (2015) implicitly adopt DMP as an indisputable principle to support and justify their view. 7 For a discussion on isomorphism and wavefunction realism, see Halvorson (2018).

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space are related, as each of the coordinates of the former corresponds to a particular degree of freedom of the latter.8 The problem is that there is another entity, the so-called ‘multi-field’,9 which is isomorphic to the structure of the wavefunction and is three dimensional: Ψ : R3 x . . . x R3 → C where R3 x . . . x R3 is an N-fold product.10 In this case, the wavefunction maps each N-tuple of points in three-dimensional space to a complex number:       x1 , y1 , z1 , x2 , y2 , z2 , . . . xN , yN , zN ,

→ c Now, for the sake of our argument, it is important to stress that, by definition, R3N is R3 x . . . x R3 . R3N is simply a symbol which refers to the N-fold product R3 x . . . x R3 , nothing more, as R3N and R3 x . . . x R3 (N-times) are symbols that indicate the very same representation. In other words, the space of the wavefunction expressed in 3ND coordinates (x1 , x2 , x3 , . . . , x3N ) is isomorphic, if not identical,11 to the space expressed in 3D coordinates ((x1 , y1 , z1 ), (x2 , y2 , z2 ), . . . (xN , yN , zN , )). The only difference is the use of some superfluous parentheses and different symbols. Since the wavefunction and the multi-field are isomorphic, the DMP could well

8

To be fair, supporters of quantum 3ND ontology are reluctant to claim that the space where the wavefunction lives is a representational space, a ‘configuration’ space of our three-dimensional space. Moreover, they insist that, strictly speaking, the configuration space is 3M, that is, it does not contain any obvious connection with the 3D world. However, when pushed to solve the problem of the existence of multiple mappings between the two spaces (Monton, 2002) and of the empirical adequacy of their own theory (Maudlin, 2007), not only do they accept that the ‘N’-factored dimensions of the 3ND-space correspond to the degrees of freedom of our threedimensional space, but also that each coordinate of the 3ND space represents a particular particle’s coordinate in our three-dimensional space. Indeed, if the 3ND ontologies did not include a fixed connection between 3ND-space and 3D-space, the wavefunction space could be mapped to our space with different mappings, all equally possible (Maudlin, 2007; Monton, 2002), making the 3ND ontologies empirically unsupported and even empirically unsupportable in principle (Chen, 2017; Maudlin, 2007). Moreover, in order to solve the problem of the connection between the two spaces, they claim that the Hamiltonian potential features degrees of freedom grouped into triples, such that each triple should be ‘naturally’ interpreted as the coordinates of a particle in three-dimensional space (Albert, 1996; Ney, 2015). See Monton (2006) and Allori (2013b) for a discussion of this claim. Notice that this strategy is usually employed to deal with the problem of the connection of the 3ND space with the 3D space within Bohmian mechanics, and that it is not applicable within multi-world interpretations, which demand a separate treatment. 9 For the multi-field view, see Belot (2012), Hubert and Romano (2018), Romano (2020), and Chen (2017). 10 A similar remark can be found in Chen (2017). 11 Given that the multi-mapping problem can be overcome by admitting a natural identification of points in 3ND-space and an N-tuples of points in three-dimensional space (see footnote 8), it could even be claimed that the two spaces are identical. See Chen (2017) for an identity mapping between the two spaces.

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support not only the wavefunction ontology, which is 3ND, but also the multifield ontology, which is three-dimensional. This means that we still have the same problem of underdetermination and that the DMP fails as an eliminative principle, as it fails to eliminate one of the two groups of ontologies. Let me briefly dispel one possible objection to this argument and discuss a possible strategy to overcome the problem of lingering underdetermination. First, the objection: A wavefunction realist may insist that the dimensions of the wavefunction are M, and M just happens to be 3N and that there is no explanation relating to our 3D world for why M is 3N. Maybe, there is no explanation at all.12 This is a fair point. However, even if the dimensions of the space only happen to be 3ND, this is enough to establish the isomorphism between the space of the wavefunction and the space of the multifield. No matter whether the dimensions of the space of the wavefunction are necessarily or contingently 3N, it is a fact, albeit brute and unexplainable, that the wavefunction lives in R3N and that this space is isomorphic to the space where the multi-field is defined. Second, one strategy could in principle break the underdetermination, but it would violate the minimal desiderata discussed above. First, one would have to commit to a strong form of term objectivism,13 according to which there is a fact of the matter regarding which formalism and symbols one should use. Secondly, one would have to choose one of the two definitions of  and its space and ontologically commit to it. The problem is that either we choose the formalism randomly, and so the ontological structure would be the result of a random choice, or we choose it on the basis of whether we prefer a 3ND or a 3D ontology.14 The first option would violate the requirement that the structure of our ontology should be independent of conventional or arbitrary representational choices, while the second option would violate the minimal desideratum that the principle should be applied without presupposing the truth of preferability of the ontological interpretation it aims to justify. Therefore, my conclusion is that the DMP cannot serve as a metaontological eliminative principle to select the 3ND ontologies group and eliminate the 3D ontologies group.

3.4 The Minimal Divergence Norm The minimal divergence norm was first introduced in Emery (2017) and captures the longstanding idea that our perception of reality should play a decisive role in

12 For

a discussion of this objection, see also Allori (2013b). objectivism is discussed in Hicks and Schaffer (2017). 14 Some may suggest that a third option is to choose the ‘simplest’ symbols and infer our ontology from those. However, appealing to simplicity in this case would be groundless, since the difference in symbols does not indicate any difference in their representation. 13 Term

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choosing the kind of ontology to which we should be committed.15 It states that we should choose the ontology that diverges the least from how the world appears. This means that it does not rule out ontologies that diverge significantly, as long as their divergence is the minimum we can choose. MDN: Insofar as we have two or more empirically adequate scientific theories – two theories that both accurately predict the phenomena that we observe – we ought to choose the one that minimizes the difference between the way the theory says the world is and the way the world appears to be.

The MDN has been applied (Emery, 2017) in a way that accords well with the role of meta-ontological eliminative principles. Indeed, the MDN can be applied to eliminate one of the two competing groups – the 3ND ontology group – exactly as we expect it to do qua eliminative principle. Ostensibly, Emery’s application of the MDN, however, is different, as it appears more restrictive. First, her aim is to rule out not the whole 3ND ontology group, but only ‘wavefunction ontology’; second, she does so by comparing it with a particular ontology taken from the 3D ontologies group, namely, mass-density ontology. The ontological debate that I target on the contrary, compares the entire 3ND ontologies group with the entire 3D ontologies group. However, this difference is not significant. First, the wavefunction ontology encompasses all the interpretations belonging to the 3ND ontologies group: all ontologies of this group have a wavefunction ontology. Second, the reason the mass-density ontology is judged to diverge less than the wavefunction ontology encompasses all the interpretations that belong to the 3D ontologies group. Indeed, the debate is settled by taking into consideration the two following related reasons: (1) wavefunction ontology implies a new kind of relation with which we are not familiar, which is the relation between the configuration space and our threedimensional space; (2) mass-density ontology includes three-dimensional objects, which are common in our experience.16 While condition (1) applies to every 3ND ontology, condition (2) applies to every three-dimensional ontology. This means that MDN can be considered a meta-ontological eliminative principle that could serve well in our case, and given (1) and (2), the principle should lead, a fortiori, to the conclusion that 3ND ontologies should be eliminated in favour of 3D ontologies. The problem with the MDN is that it is quite difficult to judge whether it actually leads to this conclusion, given that it does not specify how to quantify the degree of divergence of a theory from how the world appears. Emery anticipates this objection in her paper and proposes that, while this may be a genuine, legitimate worry in 15 This idea has been developed in different forms; for instance, it is behind the principle that whatever is physical must be three-dimensional (Hale, 1988) or that we should reject any theory that cannot locate the ‘manifest image’ of reality (Chen, 2017). Emery attributes some form of MDN to Allori et al. (2008) and to Allori (2013a). Ney also adopts some sort of MDN in her book (Ney, 2021, p. 129), where she uses arguments from intuitions. According to her, a quantum ontology that matches our intuition of what the world is like is desirable. 16 (1) and (2) are intrinsically related. If our ontology is in 3D space, we do not need any ‘new’ relation to connect different dimensional ontologies. Thus, both (1) and (2) concern the dimensions of the space of the ontology.

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some cases, it does not touch on this particular debate. After all, she claims, the application of the principle is straightforward in choosing between 3ND ontologies and 3D ontologies, as it is clear which ones diverge the least: 3D ontologies. On the contrary, I argue that this objection does constitute a problem, especially in cases like this one, in which one must compare ontologies that diverge from reality in more than one salient respect. In these cases, gauging the divergence is not as intuitive as Emery suggests, as we lack a common measure that would be applicable to both criteria. Indeed, one of the ways that supporters of 3ND ontologies justify their view is by applying a sort of MDN, as well, by arguing that 3ND ontologies are the only viable ones that can avoid non-locality (understood here as including both action at a distance and non-separability) in the fundamental ontology.17 Their reasoning can be summarised as follows: we opt for a high-dimensional ontology because we do not want non-locality in the fundamental space, and we do not want non-locality because non-locality would be at odds with how we perceive our three-dimensional macroscopic space.18,19 In light of this reasoning, we would expect them to argue that, given two different ontologies, one with non-locality and three-dimensional space and the other with locality and high-dimensional space, they would prefer the latter, as, overall, it diverges less from how the world appears to us. It is particularly interesting that, on the one hand, non-locality is an undesirable consequence for rejecting wavefunction ontology, and, on the other, wavefunction ontology is an undesirable consequence for rejecting non-locality.20 Which of the two ontological features diverges the least from what the world looks like? I think that it is highly controversial, and so that the MDN does not uncontroversially favour 3D over 3ND. Here, I do not want to argue against the MDN per se. Indeed, Emery gives compelling reasons it should be applied, regardless of its success in the quantum

17 This is, at least, what Albert claimed during a seminar at Rutgers in 2016. Such claims about the undesirability of non-locality in the fundamental ontology can also be found in Ney (2015), Wallace (2012), and Loewer (1996). Ney also motivates her support for a 3ND ontology by appealing to the fact that non-locality is against our intuitions of what the world is like (Ney, 2021). 18 Wallace, for instance, argues that action at a distance correlations flatly contradict our commonsense understanding of how physical objects should interact and our perception of reality. In our daily experience, macroscopic objects do not influence each other instantaneously if they are in different spacetime regions (Wallace, 2012, p. 292). In her recent book, Ney (2021) argues that both action at a distance and non-separability are at odds with our intuitions of what the world is like. 19 It is, of course, controversial that locality matches how the world appears to us and our intuitions of what the world is like, as there are many phenomena that at first appear non-local (for instance, when I switch on the lightbulb of my office from the corridor or when I use the remote control to turn off my TV). However, it is true that we usually know that the true explanation of such seemingly non-local phenomena is local. 20 Like Ney (2021) and Wallace and Timpson (2010), I also endorse the view that wavefunction realism does not imply any form of non-separability. However, this is a controversial issue. See Myrvold (2015) for the opposite view.

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ontology debate. Rather, the issue raised in this paper specifically targets the question of whether the MDN can be properly used as a meta-ontological eliminative principle that rules out the 3ND ontology group and identifies three-dimensional ontology as the true ontology for quantum mechanics. My conclusion on this point is negative. The fact that the two alternative kinds of ontology diverge from our perception of the world with respect to different yet salient criteria makes the MDN inapplicable.21 One option would be to restrict the application of this principle to situations where the ontologies are equal in all respects except for one. All else being equal, if one ontology differs from another, and this difference leads this ontology to feature a more serious divergence from how the world appears, then we ought to discard it.22 However, the MDN would then be inapplicable as a meta-ontological eliminative principle that can decide between the 3ND ontologies and the 3D ontologies groups, as the two groups are heterogenous as they include different ontological interpretations.

3.5 Conclusion: Eliminative Reasoning and the Role of the DMP and the MDN in the Quantum Ontology Debate Meta-ontological principles are crucial in order to specify the ontological commitments of quantum mechanics. As mentioned above, the ontologies that are possible candidates for quantum mechanics vary significantly, as they include both high-dimensional and three-dimensional ontologies. Given this serious underdetermination, one natural strategy is to find meta-ontological eliminative principles which could at least guide us in deciding whether the true quantum ontology is three dimensional or high dimensional. Here, I have examined two important and promising candidates that might serve this function, the DMP and the MDN; indeed, each is usually used to support and justify a commitment to one of the two kinds of ontology. However, I found that both DMP and MDN are ineligible for this role. The DMP, by relying on the notion of isomorphism, leads to unsatisfactory alternatives. The first is lingering underdetermination, which means that the DMP is idle and that it does not fulfil its role as an eliminative principle, since it violates the minimal 21 This is the case at least in the absence of some proposal for how to apply a common measure to both criteria. Some options might be available, such as reflective equilibrium, but this suggestion would need a separate treatment. 22 Emery considers but rejects the possibility that MDN should be applicable only for cases in which the ontologies are equal in all theoretical virtues (simplicity, explanatory power) except for the degree to which they diverge from what the world looks like. Here, I consider the option that the MDN should be applied only in cases where all but one of their ontological features of the ontologies are identical.

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requirement of ruling out at least one of the two kinds of ontologies (3ND or 3D). The second alternative would be to break the underdetermination by adopting a strong form of term objectivism and deciding which set of symbols one should read off the ontology. However, this choice would be either arbitrary (since there is no apparent reason to prefer one use of symbols over the other), or ontologically laden if the choice were made on the basis of the kind of ontological interpretation one preferentially supports. In the case where the set of symbols is chosen arbitrarily, the resulting structure would violate the requirement that it must be independent of any arbitrary representational choice. If it is done on the basis of a preferred ontology, it would violate one of the desiderata of the principles, for one would have to presuppose what the principle aims to prove (whether the wavefunction is a high-dimensional entity or a multi-field). For this reason, I do not think that the DMP can play the role of a meta-ontological eliminative principle that can eliminate the group of 3D ontologies and identify the 3ND ontology as the ‘true’ one for nonrelativistic quantum mechanics. The examination of the MDN was more brief but hopefully no less convincing. The fundamental problem is its vagueness in formulation, more precisely, the lack of specification regarding how we should gauge the divergence between an ontology and how the world appears. I suggested that the MDN cannot be successfully applied in cases where ontologies differ in more than one respect. Both 3ND and 3D ontologies diverge from our perception of the world with respect to two different and ontologically significant aspects: while the former diverge because of the space of the ontology, the latter diverge because they feature non-locality. In both cases, DMP and MDN, it is illegitimate to appeal to these principles to break the underdetermination and justify commitments to a quantum 3ND or 3D ontology. Should we then designate these principles illegitimate? The conclusion I have defended is that they cannot play the role of epistemic guides qua metaontological eliminative principles to decide between the two groups of 3ND ontologies and 3D ontologies. However, I am open to the idea that they may play other roles. For instance, I remained silent on whether they can be used legitimately to rule out some particular ontologies instead of others. Indeed, I suppose that they would, at least in some circumstances.23 Moreover, if used in conjunction with other meta-ontological eliminative principles, they might even help eliminate all but one particular ontology. This is, however, a separate issue which would require a discussion of what these other principles are and how exactly they should be implemented. Rather than examining how the DMP and the MDN could work with regard to particular quantum ontologies and in conjunction with other principles, let me conclude by sketching a meta-ontological role for DMP and MDN that I particularly favour. Returning to the way I described eliminative principles and their role in the

23 For instance, the DMP could rule out a Humean primitive ontology of only particle positions in favour of a multi-field view, as only the latter includes an entity isomorphic to the wave function in its ontology.

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selection of the true ontology, I said that eliminative reasoning normally consists of setting up the space of possible ontologies and then, via eliminative principles, selecting the correct one. The DNM and the MDN cannot play the role I expected them to play in the second step, but what about the first one? We surely need some principles that can guide the construction of the space of possible quantum ontologies. How would one go about constructing this space? On the one hand, there is the thought that the larger the space, the better: the larger the space is, the more likely it would contain the ‘true’ ontology. However, it would be a mistake to think that the size is all that matters.24 Indeed, the space could be large simply because it contains highly implausible, bizarre ontologies or because it contains many slightly different versions of an ontology. If so, its size would not inform us about whether it is likely to contain the true ontology within it. Moreover, if we have a space which is too large, especially if it has many slightly different or bizarre alternatives, we would potentially need many more eliminative principles to identify the right ontology. Lastly, if the size were all that mattered, it would never be possible to consider the space ‘ready’ for an application of eliminative principles, for it would always be possible to find new (and even more bizarre) alternatives. Thus, we need some principles that govern the construction of a large space which contains reasonable, promising, ontologically significant alternatives: a space including not all possible, but plausible, naturalistic ontologies. In this regard, I suggest that the DMP and the MDN can play the role of such principles, since the construction of this space would benefit from considerations on the dynamical laws of quantum mechanics and our perception of the world we live in, among other aspects. First, the MDN can play an important role in regimenting the space of plausible quantum ontologies. In particular, the quantum underdetermination problem consists in the proliferation of too many slightly different ontologies, which not only differ from other ontologies with regard to one ontological aspect but which also are clearly worse off, as they diverge from how the world appears more than similar ontologies. For instance, in addition to wavefunction ontology and a threedimensional ontology, there is a hybrid view (Valentini, 2009, 2010), mentioned in footnote 2, which postulates the fundamental existence of both the wavefunction— as a high-dimensional entity in a 3ND field—and particles in our 3D world. This view differs from the wavefunction ontology by admitting a fundamental non-local three-dimensional ontology in addition to the wavefunction, and it differs from mere three-dimensional ontologies by admitting an unusual, high-dimensional space that involves an unnatural relation to our 3D space in addition to the particle ontology. This view, by keeping a high-dimensional space as well as non-locality, clearly diverges from our perception of the world more than both wavefunction ontology and particle ontology. In light of this, I can safely say that it fares worse than both wavefunction and particle ontologies. Thus, in virtue of the MDN, it should not

24 See

Norton (1994) and McCoy (2021) for a discussion of this point in the context of the use of eliminative reasoning in science.

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be part of the space of plausible ontologies. It would merely add an ontology to the space without making it ‘stronger’ and without increasing the probability that it contains the ‘true’ ontology. The DMP plays an important role in regimenting the space of plausible ontologies by justifying the insertion of 3ND wavefunction ontology into that space. Wavefunction ontology certainly invites the ‘incredulous stare argument’ (Emery, 2017) and could be easily labelled ‘impossible’, ‘too bizarre’, or ‘a metaphysical speculation’. The reason it is and should be received as a naturalistically plausible and promising ontology for quantum mechanics is that it conforms to the DMP, which, in this case is used as an inference principle to draw out the possible ontological commitments of quantum mechanics. Not only does the DMP lend credibility and strength to wavefunction ontology, but it also justifies its inclusion in the space of plausible quantum ontologies. Here, I do not want to claim that the DMP is a necessary condition for an ontology to be included in the space of possibilities; I rather suggest that it can justify its inclusion.25 Moreover, given that, as argued before, the notion of isomorphism can allow different ontologies, the application of the DMP as an inference principle generates not one but many different naturalistic ontologies that deserve to be part of the space of plausible quantum ontologies. In these respects, the DMP plays the role of regimenting, constructing and justifying the space of plausible ontologies. In conclusion, while I have argued that it is illegitimate to justify commitments to 3ND ontology or to 3D ontology by appealing to these principles, since they can be used to support either view, I believe that they may still play an important role in governing the construction, regimentation, and justification of the space of all plausible quantum ontologies. Acknowledgments I am grateful to Valia Allori for inviting me to submit a paper for this volume and for her editorial assistance. I am also thankful to Casey McCoy, Davide Romano and two anonymous reviewers for insightful comments on different parts of this manuscript, and to Jill North for checking my argument regarding her view. Finally, I would like to thank the members of the Theoretical Philosophy Research Colloquium at the Institute of Philosophy of the University of Bern for their feedback. Among them, I owe special thanks to Claus Beisbart, Matthias Egg, and Vincent Lam for sending me written comments.

References Allori, V., Goldstein, S., Tumulka, R., & Zanghì, N. (2008). On the common structure of bohmian mechanics and the Ghirardi–Rimini–Weber theory: Dedicated to Giancarlo Ghirardi on the

25 One might even regard the DMP as playing the role of preventing ontologies that do not include entities isomorphic to the wavefunction from being part of the space of plausible ontologies. However, I do not endorse this role for the DMP, as, to my mind, it would eliminate too many plausible ontologies.

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occasion of his 70th birthday. The British Journal for the Philosophy of Science, 59(3), 353– 389. Allori, V. (2010). Some remarks on wave function monism. Boulder Conference on the History and Philosophy of Science. http://commons.lib.niu.edu/bitstream/handle/10843/16043/ remarks-wave-function-monism.pdf?sequence=3. Accessed 16 Aug 2021. Allori, V. (2013a). Primitive ontology and the structure of fundamental physical theories. In Ney & Albert (Eds.), The wave function: Essays on the metaphysics of quantum mechanics (pp. 58–75). Oxford University Press. Allori, V. (2013b). On the metaphysics of quantum mechanics. In S. Lebihan (Ed.), Precis de la Philosophie de la Physique: 116–151. Vuibert. Allori, V. (2017). A new argument for the nomological interpretation of the wave function: The galilean group and the classical limit of nonrelativistic quantum mechanics. International Studies in the Philosophy of Science, 31(2), 177–188. Albert, D. Z. (1996). Elementary quantum metaphysics. In Bohmian mechanics and quantum theory: An appraisal (pp. 277–284). Springer. Belot, G. (2012). Quantum states for primitive ontologists. European Journal for Philosophy of Science, 2(1), 67–83. Chen, E. K. (2017). Our fundamental physical space: An essay on the metaphysics of the wave function. The Journal of Philosophy, 114(7), 333–365. Emery, N. (2017). Against radical quantum ontologies. Philosophy and Phenomenological Research, 95(3), 564–591. Esfeld, M. (2014). The primitive ontology of quantum physics: Guidelines for an assessment of the proposals. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 47, 99–106. Hale, S. C. (1988). Spacetime and the abstract/concrete distinction. Philosophical Studies, 53(1), 85–102. Halvorson, H. (2018). To be a realist about quantum theory. http://philsci-archive.pitt.edu/14310/ Hicks, M. T., & Schaffer, J. (2017). Derivative properties in fundamental laws. The British Journal for the Philosophy of Science, 68(2), 411–450. Hubert, M., & Romano, D. (2018). The wave-function as a multi-field. European Journal for Philosophy of Science, 8(3), 521–537. Loewer, B. (1996). Humean supervenience. Philosophical Topics, 24(1), 101–127. Maudlin, T. W. (2007). Completeness, supervenience and ontology. Journal of Physics A: Mathematical and Theoretical, 40(12), 3151. McCoy, C. D. (2021). Meta-empirical support for eliminative reasoning. Studies in History and Philosophy of Science Part A, 90, 15–29. Monton, B. (2002). Wave function ontology. Synthese, 130(2), 265–277. Monton, B. (2006). Quantum mechanics and 3 N-dimensional space. Philosophy of Science, 73(5), 778–789. Myrvold, W. C. (2015). What is a Wavefunction? Synthese, 192(10), 3247–3274. Ney, A. (2015). Fundamental physical ontologies and the constraint of empirical coherence: A defense of wave function realism. Synthese, 192(10), 3105–3124. Ney, A. (2021). The world in the wave function: A metaphysics for quantum physics. Oxford University Press. North, J. (2013). The structure of a quantum world. The wave function: essays on the metaphysics of quantum mechanics, 184–202. Norton, J. D. (1994). Science and certainty. Synthese, 99, 3–22. Romano, D. (2020). Multi-field and Bohm’s theory. Synthese. https://doi.org/10.1007/s11229-02002737-6 Valentini, A. (2009). The nature of the wave function in de Broglie’s pilot-wave theory. PIAF 09’ New Perspectives on the Quantum State, Perimeter Institute. PIRSA Number: 09090094. https:/ /www.perimeterinstitute.ca/videos/nature-wave-function-de-broglies-pilot-wave-theory Valentini, A. (2010). De Broglie–Bohm pilot-wave theory: Many worlds in denial. Many worlds, 476–509.

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Wallace, D. (2012). The emergent multiverse: Quantum theory according to the Everett interpretation. Oxford University Press. Wallace, D., & Timpson, C. G. (2010). Quantum mechanics on spacetime I: Spacetime state realism. The British Journal for the Philosophy of Science, 61(4), 697–727.

Chapter 4

The Quantum World as a Resource. A Case for the Cohabitation of Two Paradigms Laura Felline

Abstract In this paper I analyse the contraposition between two families of interpretations of QT. On the one hand, Information-Theoretic Interpretations of QT, the family of interpretations that understand QT to be a theory about information. On the other hand, those interpretations (e.g. Bohmian mechanics or GRW) that provide an analysis of measurement interactions and ‘open the black box’. The main aim of this paper is to undermine the basic assumption that one of the two approaches should prevail over the other and to outline the background for a viable alternative to this assumption. In the first part of the argument I use the three main features of a Kuhnian paradigm-shift (World-change, Kuhn-loss, Conceptual change) to dissect and invalidate the reasons behind the expectation that a solution of the quarrel will determine the ‘victory’ of one position to the other. In the second part of the argument I argue that Evandro Agazzi’s account of scientific objectivity might harmonize the ontologies of the two interpretations in a realist framework.

4.1 Introduction In the last decades, the notion of information has gained a prominent role in quantum physics. The conceptual, formal and methodological tools provided by Information Theory invaded not only the practice, but also the foundations of Quantum Theory (QT). However, while the pragmatic success of Information Theory is hardly debatable, the question of its foundational significance is controversial. In this paper we analyse the contraposition between two groups of interpretations of QT. On the one hand Information-Theoretic Interpretations of QT (IT-QT), the family of interpretations that understand QT to be a theory about information. Specifically, we take into account interpretations where information is intended as a physically defined quantity cashed out in terms of the resources required to transmit messages –

L. Felline () Independent Researcher, Seneghe, Italy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_4

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measured classically by Shannon entropy or in QT by Von Neumann entropy. As a concrete representative of this view, in this paper we take Jeff Bub’s IT-QT, where information is a new kind of physical primitive, whose structure “imposes objective pre-dynamic probabilistic constraints on correlations between events, analogous to the way in which Minkowski space-time imposes kinematic constraints on events.” (Bub, 2018, p. 5). On the other hand (what we call for simplicity) ‘traditional’ interpretations, i.e. those interpretations, like Bohmian mechanics or GRW, that take QT as a theory about the elementary constituents of the physical world and how these elements evolve dynamically over time. In line with the analysis provided in (Felline, forthcoming), we take such interpretations ‘opening the black box’ as providing a mechanistic account of measurement interactions. These two characterizations of the object of QT come with extremely different attitudes towards the main issues in the foundations of QT. The most boasted virtue of IT-QT is that they explain away the conundrums of QT, namely the measurement problem and non-locality; the response from the other side is that they really don’t. Advocates of IT-QT often complain about the dogmatism of the traditional interpretational program and about its obsession over obsolete problems; the others answer that the internal coherence of IT-QT can only be maintained under a plain instrumentalist stance, which makes it useless for the issue of the ontological interpretation of QT. The former charge the latter with dogmatism, viceversa the latter claims that the former’s solutions beg the question. A widely shared implicit assumption in the debate is that a solution of the quarrel will determine the rightness of one position and the wrongness of the other. Such a ‘victory’ is supposed to be grounded on the superior virtues of one approach over the other – for instance, the winning theory is expected to surpass the other in explanatory power. This expectation seems quite natural since the two positions are typically treated as mutually exclusive: if IT-QT are right, and QT is a theory about information, then it is not only useless, but also unscientific to keep trying to account for quantum phenomena in terms of the behaviour of elementary constituents of the physical world. Vice versa, if QT is about elementary entities and processes, then information is ontologically determined by such elementary constituents of the physical world, therefore it is not fundamental, therefore it does not have place in the foundations of QT. The main aim of this paper is to undermine the basic assumption that one of the two approaches ‘wins’ over the other, and to outline a viable alternative to this assumption. The path towards the construction of my proposal starts by an analysis of the foundational debate through analytic tools provided by elements of the Kuhnian doctrine of competing paradigm. More specifically, the three main features of a Kuhnian paradigm-shift (World-change, Kuhn-loss, Conceptual change) will be used to dissect and invalidate the reasons behind the expectation that a solution of the quarrel will determine the ‘victory’ of one position to the other.

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Once this case has been made, then some work will be necessary to make understandable and plausible the claim that both approaches can be truthful and should therefore cohabit as interpretations of QT. In the last part of my proposal, therefore, I will suggest a way to accommodate the coexistence of both approaches to QT: fist I will adopt Kuhn’s late (2000) account of scientific progress as speciation, secondly I will argue that Evandro Agazzi’s account of scientific realism and objectivity and its perspectival character of science might harmonize the two ontologies in a realist framework. One may wonder about the motivation for taking such a path in the interpretation of QT. I think that, if successful, this proposal would provide a key to the understanding of (what I take as) two recent important achievements in the foundations of QT. First of all, after decades of failed attempts in providing a genuinely causal explanation of non-local quantum correlations, IT-QT (and especially Bub’s (2016) interpretation) managed to provide a non-causal explanation of such phenomena that has the great virtue of mirroring actual cutting edge scientific practice (Felline, 2016, 2019). Secondly, without an analysis of measurement interactions as provided by interpretations that ‘open the black box’, QT can’t provide coherent predictions in Winger’s Friend scenarios (Hagar & Hemmo, 2006; Frauchiger & Renner, 2018; Felline, forthcoming). Before we start, few caveats are in order. First of all, the last part of the proposal will not be developed in detail: it is not an aim of this paper to provide a full-fledged account of scientific progress and realism in QT – but the arguments provided will hopefully suffice to show that there are viable alternatives to the realism implicitly assumed in the debate, requiring a monolithic approach to the interpretation of QT. Secondly, the reference to Kuhn’s work is probably going to raise many eyebrows.1 Due to the high evocative power of the Kuhnian terminology, I must clarify that by using such terminology (and in particular the term ‘paradigm’, with which I will refer to both traditional and information-theoretic approaches) I am not subscribing to the whole Kuhnian philosophy of science, but I am rather adopting what Kerry McKenzie and Steven French call the ‘Viking approach’: I take what I need and leave the rest. The reader, therefore, is invited to refrain from thinking of well-known features of Kuhn’s famous 1970’s Structure of Scientific Revolutions, at least if they are not explicitly used here or directly implied by the elements here adopted.

1

I think that part of such mistrust comes from the rejection of the so-called context of inquiry within the ‘Order of Discourse’ of the foundations of physics, and to that extent I consider this possible reaction unjust. Philosophy of science puts at our disposal a rich variety of approaches to epistemology, ontology, methodology etc.., and this variety of approaches contributes to more sophisticated and rich understanding of the practice and of the foundations of science. In this sense, the philosophy of physics tradition is a sort of anomaly, since only a very limited number of approaches is taken seriously.

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4.2 Information Theory as a Resource Theory and the Information-Theoretic Interpretation of QT Information theory and its sub-discipline Quantum Information Theory, study how physical systems can be exploited for communication processing tasks. They are therefore resource theories, i.e. theories that study how something “can be used or consumed, either in order to produce some desirable commodity, or in order to produce other resources for producing desirable commodities, such as machine equipment” (Fritz, 2017, p. 1). Resource theories’ ‘pragmatic’ approach to science is displayed by the questions that drive their investigation: (a) Under which conditions can a resource object x be converted into a resource object y? (b) Can the use of a third resource object z as a catalyst help in achieving the conversion of x in to y? (c) If one tries to convert many copies of x into many copies of y, then how many copies of x does one need on average in order to produce one copy of y? (ibid.)

Resource theories, however, contribute to physics in ways that exceed the straightforward pragmatic interests listed above. As an example, take the line of research originated by Popescu’s and Rorlich’s PR-boxes. The latter are models of ‘fantasy quantum theories’ that replicate some aspects of QT (i.e. that maintain some principles in common with QT), while changing some others. The consequences of these modifications are then studied by manipulation of such toy models. PR-boxes are built from two axioms: relativistic causality, corresponding to nosignalling, and non-locality, neutrally defined in terms of non-local correlations in the sense of Bell’s theorem. Popescu and Rorlich carry an investigation based on the manipulation of toy models defined by these two principles and show that quantum correlations are not the sole non-local correlations that are consistent with such a setup: other, so-called ‘post-quantum’, correlations exist that can be more non-local than quantum correlations, where non-locality is measured as the amount of violation of Bell’s inequalities. This discovery has opened the door to a series of questions about non-locality: if post-quantum correlations don’t violate the ‘no signalling’ principle, why doesn’t our world instantiate them? Why is our world only this much non-local, when it seems that it could be more? Building on Popescu’s and Rohrlich’s result, later works (Brassard et al., 2006; Brunner & Skrzypczyk, 2009) suggest that the explanation of why our world only instantiates quantum non-locality, and no more non-locality than that, lies in another information-processing principle. In order to see how, think that the availability of stronger correlations makes communication tasks easier, i.e. such correlations allow one to solve communication problems with the use of less resources. For instance, let’s say that Alice and Bob have the task of calculating a Boolean function f(x, y), where x is known only by Alice, and y only by Bob. Obviously, in order to solve the problem they must exchange some information. However, such information could be reduced depending on the availability of correlated systems shared by the two. It is, for

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instance, known that, in the calculation of some functions, sharing a pair of entangled systems could reduce the amount of information to be sent from one wing of the experiment to the other (Buhrman et al., 1988); however, for the calculation of some other functions (e.g. the inner product between two numbers) Alice will have to send to Bob the same number of bits as her input, whether or not the experimenters share a quantum entangled pair. Brassard et al. (2006) have therefore shown that correlations that are more nonlocal than quantum correlations would make any communication complexity trivial, i.e. with them, any Boolean function could be calculated by Alice and Bob with the exchange of only one bit of information. Finally, Brunner and Skrzypczyk (2009) have shown that the same happens with post-quantum correlations that are arbitrarily close to classical correlations (i.e. arbitrarily close to the limit imposed by Bell’s inequalities). In short, Brunner and Skrzypczyk show that from such postquantum correlations it is possible to distil2 correlations that are strong enough to make communication complexity trivial. This suggests a possible reason why we have only so much non-locality in the world, as “most computer scientists would consider a world in which communication complexity is trivial to be as surprising as a modern physicist would find the violation of causality.” (Brassard et al., 2006, p. 2) They therefore put forward the conjecture that the explanation of the existent limit in the non-locality of our world lies in a new information-theoretic axiom about the impossibility of trivial communication complexity. Such interwinement between the resource approach and the theoretical aims of fundamental physics is brought to extreme consequences in IT-QT. Such interpretations of QT do not share the pragmatic interests of information theory as a resource theory, but claim that the concepts, properties and laws that are relevant to information-processing tasks should take the place of mechanical concepts, properties and laws in the interpretation of quantum theory as a fundamental physical theory. As a consequence, notwithstanding the partial rejection of the purely pragmatic aims of resource theories, IT-QT inherit a great part of the features that characterize resource theories: methodologies, concepts, styles of representation, but also desiderata and criteria of scientificity. Our case study, Bub’s (2016) Information-Theoretic interpretation of QT can help illustrate this point. Bub articulates his version of the IT-QT approach with the formal background provided by PR-boxes. According to Bub, QT is a theory about information ‘in the physical sense’: a structure of correlations between intrinsically random events which is not Shannon-like, as the classical structure of correlations, but von Neumann-like. Here, information is a new kind of physical primitive, whose structure “imposes objective pre-dynamic probabilistic constraints on correlations between events, analogous to the way in which Minkowski space-time imposes kinematic constraints on events.” (Bub, 2018, p. 5).

2

Distillation is a procedure that begins with a large number of weakly correlated systems and ends up with a smaller number of strongly correlated systems, ‘distilling’ in this way non-locality.

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According to Bub, in the same way as special relativity explains length contraction as a manifestation of the Minkowskian geometry of space-time, QT explains non-local correlations as the manifestation of the structure of information. The explanation of quantum correlations as the manifestation of the fundamental physical structure of information is one of the most impressive results of Bub’s interpretation. However, the issue of the explanatory power of this account is much more convoluted than suggested by the above quote. In Bub’s Information-Theoretic interpretation, not only information is a physical primitive, but its structure is the true and only object of QT. However, this last claim, i.e. that QT is only about information, has been disputed on various grounds. In particular, several arguments based on Wigner’s Friend scenarios (see for instance Hagar & Hemmo, 2006, but also Frauchiger & Renner, 2018) have challenged the claim that black box approaches like IT-QT always provide coherent predictions.3 IT-QT take the determinateness of our experience as a primitive fact, that can’t and should not be explained. This is not only a limitation in itself, but it also has important consequences for Bub’s explanation of non-locality. In order to see why,4 it is useful to take again the analogy with the geometrical explanations of special relativity. In the orthodox interpretation (to which Bub refers) of this theory, length contraction is explained as a manifestation of the geometry of spacetime and as independent of the specific dynamical details of the concerned objects and their properties (Felline, 2011). Yet, if we have to explain the individual occurrence of length contraction of a specific rod within two specific inertial frames, we need to fill the explanation with initial conditions, e.g. the proper length of the rod. This fact, i.e. the rod’s length, can’t be explained within special relativity itself, but it must appeal to the dynamical history that produced an object of such length. This clearly does not undermine the geometrical explanation provided by special relativity, since the latter is not supposed to explain the proper length of the rod, but uniquely the contraction in the switch to different inertial frames. Moreover, the existence of a theory of matter that explains the proper length of a rod in no way undermines the fundamentality of the structure of spacetime. A similar situation applies in the case of Bub’s interpretation and the relative explanation of correlations. Bub’s explanation of non-local correlations makes reference to the noncommutative algebraic structure of information and to the fact that measurement results are determinate. Information Theory (and IT-QT), however, takes such results and their determinateness as a primitive, so it can’t explain them. As the geometrical explanation of length contraction rests on a prior explanation of the proper length of the rod, in the same way Bub’s explanation of non-local correlations rests on a prior explanation of the determinateness of measurement results. Now: on the one hand, this explanation should be provided by QT itself; on the other such an explanation requires an account of what happens

3

We are not going to discuss the details of this debate here, however, for a counterargument see Bub (2018) and (Felline, 2020) for a reaction to Bub. 4 For a more detailed illustration of the argument, see Felline (2019, 2020).

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when you ‘open the box’, i.e. and an analysis of measurement interactions. Since the latter can’t be provided in by a theory only about information, then QT can’t only be about information.

4.3 World Change, Conceptual Change and Kuhn Loss In this section I am going to analyse the relationship between IT-QT (especially Bub’s interpretation) and traditional, mechanistic interpretations of QT, through the lens of the three main features of a Kuhnian paradigm-shift: World-change, Kuhnloss and Conceptual change. To pay tribute to Kuhn I will maintain the term ‘paradigm’ to denote the two approaches. However, as already pointed out in the introduction, the reader should refrain from ascribing to this evocative terminology the whole of 1970’s Kuhnian doctrine about, for instance, scientific revolutions and incommensurability. World-Change Let’s start by noticing that scientists that work in the context of different paradigms experience different worlds. In a sense I am unable to explicate further, the proponents of competing paradigms practice their trades in different worlds. One contains constrained bodies that fall slowly, the other pendulums that repeat their motions again and again. In one, solutions are compounds, in the other mixtures. One is embedded in a flat, the other in a curved, matrix of space. Practicing in different worlds, the two groups of scientists see different things when they look from the same point in the same direction (Kuhn, 1970, 150).

We have already seen that scientists working in resource theories investigate the world not as a reality to be unveiled, nor with the aim of explaining phenomena – but rather as a resource that can be used or consumed. Take for instance, the procedure called superdense coding. In it, Pauli sees two singlet state particles first developing unitarily according to Schrödinger equation, and then undergoing an instantaneous and non-unitary process of collapse resulting into two pure correlated states. In the same procedure, Shannon sees the transmission of two bits of classical information through the delivery of only one single qubit, achieved by the exploitation of a communication channel. In the literature about QIT, world change terminology is ubiquitous. Especially when it comes to the study of entanglement, more often than not introductions to this discipline begin with the invitation to look at entanglement as “a resource for teleporting quantum states and constructing unbreakable codes, a resource that we can extract, purify, distribute and consume” (Popescu & Rorlich, 1998, p. 1). This kind of invitations might be understood as a mere ‘figure of speech’, but a more careful look at the context in which they are made suggests that they are more than that. Technical, formal and methodological training, but also direction of focus, condition pupil scientists to a novel gaze towards the world which is crucial for the establishment of the paradigm. An illustration of the perceived role of world change is the way in which this novel gaze is typically depicted as a cornerstone, a

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crucial breakthrough that allowed the impressive development and success of QIT and entanglement theory. For instance, in the Stanford Encyclopedia of Philosophy about quantum entanglement, Jeff Bub closes his historical review of the birth of entanglement theory exactly with such a cornerstone: But it was not until the 1980s that physicists, computer scientists, and cryptologists began to regard the non-local correlations of entangled quantum states as a new kind of non-classical physical resource that could be exploited, rather than an embarrassment for quantum mechanics to be explained away (Bub, 2020).

IT-QT is the completion of the process of world change, in that it translates the features of quantum states as a resource from its primal pragmatic context of inquiry, aimed at the production of commodities, to a different context, aimed at the explanation of phenomena. Skrzypczyk et al. (2009) investigate ‘what makes quantum mechanics so special’ by studying generalized non-signalling theories, and, more in general, axiomatic reconstructions of QT in terms of informationtheoretic principles are meant to show that the ‘essence’ of quantum theory is to be found in information-processing principles (Felline, 2016). Our elected case study, Bub’s IT-QT, according to which QT as a fundamental theory is about quantum information, and only about information – is a particularly suggestive example of world change. Conceptual Change As a result of perceiving the same reality in different ways, in the passage from one paradigm to another it also happens that some terms change their meaning and their reference. For instance, the distinction between proper and improper mixture originally formulated by Bernard d’Espagnat is a central theoretical element in the foundations of QT. However, at least when it comes to measurements actually performable given the limits of our technology, proper and improper mixtures are operationally indistinguishable, since they yield exactly the same probabilities for measurement results. Due to this (current) operational indistinguishability, proper and improper mixtures are indiscernible with respect to their role in information-processing protocols and, as a result, the distinction vanishes in IT-QT. The distinction is in fact so irrelevant to the practice of QIT that, to the surprise of philosophers and physicists trained in the foundations of physics, more often than not informationtheory working physicists plainly ignore it, even when they apply their work to foundational issues. It is hard to overestimate the consequences of such a conceptual change. For one thing, the distinction between proper and improper mixture is instrumental in the determination of the non-local nature of quantum correlations, which leads to our next point. Kuhn-Loss As we have already said before, one of the most boasted virtues of ITQT is that it explain away the problem of non-locality. The non-pernicious character of non-locality in IT-QT can be formally understood through the operational definition of concepts in this theory, exemplified by the above-illustrated case of the notion of mixture. More concretely, the purported solution to the problem of

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non-locality consists in the fact that that merely performing a local (non-selective)5 operation on a system S cannot convey any information to a physically distinct system. In an EPR experiment, for instance, there is no local measurement (nor series of measurements) that Bob can perform, that allows him to infer the state of Alice and her particle. Since IT-QT interpret QT as a theory only about information, then the impossibility of superluminal information transfer automatically solves the possible problems with Special Relativity (much in the same spirit as the instrumentalist/operational approach to non-locality in standard QT). This result is so fundamental in QIT that it constitutes one of the basic principles of information-theoretic reconstructions of QT, the so-called ‘no superluminal information transfer’ principle. And yet, if one examines such solution from the point of view of the traditional paradigm, she can hardly be impressed. For, first, the impossibility of superluminal signalling has been well-known even before QIT came to play. Secondly, if we investigate the world, as physicists developing traditional interpretations of QT do, as inhabited by entities and processes that yield non-local quantum correlations, then there must be a more fundamental story that explains non-local correlations. But then, we are back to square one: if there is such a story, then we should have an answer to the question ‘is there a local causal account of how quantum correlations come about?’ The other main quantum conundrum that IT-QT claim to explain away is the measurement problem: if QT as a fundamental theory is only about information, then the determinateness of our experiences is a primitive fact of our world, something that must be taken at face value and can’t be explained. The measurement problem therefore, i.e. the problem of explaining the determinateness of our experiences, fades away in this perspective. And yet, again, for an exponent of the traditional paradigm this purported solution of the problem begs the question: from her point of view, interactions between entities and measurement apparatuses yield determinate measurement results, and if QT as a fundamental theory can’t account for such interaction, then this is a problem! Moreover, although there is no consensus yet on any of them, it should be said that the traditional approach displays various still valid accounts (e.g. Bohmian a non-collapse and GRW a collapse) of measurement interactions, which makes even less clear why one should renounce to the whole enterprise from the start. This state of art is described by the phenomenon now called ‘Kuhn loss’: in the translation to a new paradigm it happens that some explanations and achievements of the old paradigm get lost and possibly not even replaced with new explanations and achievements, because the new paradigm lacks the resources of even formulating the problem. These problems might be crucial in the context of

5

Selective measurements operations are obviously not considered here, given that selection changes the ensemble under study and therefore its statistics.

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the old paradigm, and yet, although not really ‘solved’, are perceived in the new paradigm as obsolete and forgettable. The way in which advocates of traditional and IT interpretations of QT seem to talk past each other in the debate about the measurement problem and nonlocality (one side perceived as dogmatic, while the other as begging the question) exemplifies this element of the Kuhnian analysis. Moving on, the passage to a new successful paradigm can’t obviously only bring to an epistemic loss. A successful new paradigm, in fact, contributes to provide explanations outside the reach of the old paradigm. This is for instance the case of Bub’s explanation of non-locality. As we have seen at the end of the previous section, Bub’s interpretation provides a powerful account of the IT-QT explanation of non-local correlations as a manifestation of the fundamental structure of information, tracing the explanation of length contraction in Special Relativity as a manifestation of the fundamental structure of spacetime.6 A successful new paradigm also and most importantly suggests new questions, which leads to novel directions of inquiry. The advocates of QT-IT rightfully claim for their approach the capacity to open new ways of discovery, neglected by older interpretations. This is for instance the case of the impressive results in the study of entanglement and the discovery of post-quantum correlations illustrated in the previous section in the context of PR-boxes.

4.4 Progress, Objectivity and Scientific Realism The analysis above suggests considerations about what constitute progress in the development of QT and about a more pluralistic view of scientific objectivity and realism. Such considerations might prepare the ground for a philosophical framework more in line with the actual state of art in the foundations of QT.

4.4.1 Progress Advocates of IT-QT or of traditional interpretations of QT often argue for the superiority of their approach by appealing to a supposed higher explanatory power of their approach. The analysis provided in the previous section, however, shows that, due to the phenomenon of Kuhn loss, we should, at the least, be very suspicious of such oversimplifications. It provides another reason to reject the idea that the 6

We have seen in the previous section that that, according to (Felline, 2019) Bub’s explanation of correlations must rely on a prior explanation of the determinateness of each correlated measurement result, that can’t be provided by IT-QT. It might be useful to remind at this point that this argument does not go to the detriment of the analysed explanation; it only shows that QT cannot be only about information.

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explanatory capacity of a scientific theory can be measured and scientific theories ordered by score. More believably, the two paradigms address different question and achieve different aims. This conclusion, in turn, undermines the idea that the prevalence of one approach over the other constitute ‘progress’ in the foundations of QT. In his late work (Kuhn, 2000), Kuhn tried to provide both a more refined view of the relationship between paradigms – able to overcome the criticalities of his 1960 notion of incommensurability – and a more articulated view of scientific change, to accommodate an idea of scientific progress with his rejection of a development of science moving towards the cumulation of truth. Accordingly, the main carrier of scientific progress is described as akin to the process of speciation in biological evolutions, where one field develops by splitting in two more specialized subfields. In the context of the present paper, Kuhn’s late view of progress as speciation seems especially promising. Translated in the debate about QT and information theory, one might see the new IT-QT as contributing to the progress of science in a synchronic development of two interpretations of QT, where the new paradigm does not force the old one out, but rather both develop in parallel. This ‘splitting’ process does not guarantee a conceptual, methodological or perspectival homogeneity: world change, conceptual change and Kuhn loss make such that interpretations might not always be completely integrated. However, conceptual work in the foundations of QT can and should aim at explicating apparent idiosyncrasies.7

4.4.2 Objectivity and Scientific Realism So far, the idea of a cohabitation of two interpretations of QT (one that ‘opens the black box’ and provide a mechanistic analysis of measurement interactions, the other, a IT-QT as a resource theory), both with the dignity of ‘fundamental’ interpretations, has not received much attention.8 One reason for this is that a minimal realist stance seems to require that a truthful interpretation individuate the object of quantum theory, as a thing in the world ‘captured’ by the description provided by the theory. However, this assumption is far from being necessary, as we have at our disposal a variety of realist views of science that are less monolithic and might allow a reconciliation of the ontology of traditional interpretations of QT with the ontology of IT-QT.

7

As an example of how conceptual analysis is crucial for the achievement of such a goal, take Timpson (2013) and especially chapter 4. 8 (Koberinski & Muller, 2018) is a notable exception.

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As an example, we take Evandro Agazzi’s account of scientific realism and knowledge (Agazzi, 2014). According to Agazzi, scientific objects (i.e. objects studied and described in scientific theories) are not ultimate primitive entities, ‘things’ in the world, but rather bundles of attributes that we investigate from specific point of views. The elected point of view of an investigation determines the attributes on which to focus and such attributes constitute the object of the theory. In such a perspectival view of science,9 each theory comes with “criteria of protocollarity” permitting “the determination of which propositions are immediately true, that is, the determination of the science’s protocol propositions” (Agazzi, 2014, 87). For instance, protocol propositions in mechanics are about matter, motion and force; while protocol propositions in information theory are about degree of correlations between variables. It does not make sense, in this context, to question which is the quid that possess the properties investigated by different scientific perspectives, because scientific objects do not possess such properties, but are exhausted by them. This reference to objects as constituted of bundles of attributes and selected by criteria of protocollarity, explicates, according to Agazzi, scientific objectivity. If we translate Agazzi’s notion of scientific objectivity to the quantum world, traditional interpretations of QT and IT-QT might share the same quantum ‘thing’, but look at it from two different perspectives. Each perspective, therefore, selects attributes that constitute their objects: traditional interpretations investigate objects constituted of mass, force, spin and their dynamics, while IT-QT investigate objects constituted of properties that are directly relevant to communication protocols (for instance quantum entanglement). In the same procedure of superdense coding, Shannon and Pauli select different attributes of the systems observed to constitute the scientific object of study, and elaborate their accounting theory accordingly. Given Agazzi’s characterization of scientific object, the fact that the two approaches describe different objects does not necessarily imply ontological incompatibility. Particles and waves on the one hand and informational structure on the other are just bundles of different properties belonging to the same domain, selected by different interests.

Bibliography Agazzi, E. (2014). Scientific objectivity and its contexts. Springer. Brassard, G., Buhrman, H., Linden, N., Méthot, A. A., Tapp, A., & Unger, F. (2006). Limit on nonlocality in any world in which communication complexity is not trivial. Physical Review Letters, 96(25), 250401. Brunner, N., & Skrzypczyk, P. (2009). Nonlocality distillation and postquantum theories with trivial communication complexity. Physical Review Letters, 102, 160403. Bub, J. (2016). Bananaworld: Quantum mechanics for primates. Oxford University Press.

9

Cf. Buzzoni (2015), chap. I, § 3.

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Bub, J. (2018). In defense of a “single-world” interpretation of quantum mechanics. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. Bub, J. Quantum entanglement and information. In Edward N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Summer 2020 ed.). https://plato.stanford.edu/archives/sum2020/entries/ qt-entangle/ Buhrman, H., Cleve, R., & Wigderson, A. (1988). Proceedings of the 30th annual ACM symposium on theory of computing (p. 63). ACM. Buzzoni, M. (2015). Science and operationality. In M. Alai, M. Buzzoni, & G. Tarozzi (Eds.), Science between truth and ethical responsability (pp. 27–44). Springer. Felline, L. (2011). Scientific explanation between principle and constructive theories. Philosophy of Science, 78(5), 989–1000. Felline L. (2018). It’s a matter of principle: Scientific explanation in information-theoretic reconstructions of quantum theory. Dialectica, 70, 549–575 (2016). Felline, L. (2019). Quantum theory is not only about information. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. ISSN:1355–2198. Felline, L. (2020). The measurement problem and two dogmas about quantum mechanics. In Hemmo, Meir, Shenker, Orly (a cura di) Quantum, probability, logic: Itamar Pitowsky’s work and influence. Springer. Felline, L. (forthcoming). Mechanistic explanation in physics. In E. Knox & A. Wilson (Eds.), Companion to the philosophy of physics. Routledge. Preprint: http://philsci-archive.pitt.edu/ 16162/ Frauchiger, D., & Renner, R. (2018). Quantum theory cannot consistently describe the use of itself. Nature Communications, 9(1), 1–10. Fritz, T. (2017). Resource convertibility and ordered commutative monoids. Mathematical Structures in Computer Science, 27(6), 850–938. Hagar, A., & Hemmo, M. (2006). Explaining the unobserved—Why quantum mechanics Ain’t only about information. Foundations of Physics, 36(9), 1295–1324. Koberinski, A., & Müller, M. P. (2018). Quantum theory as a principle theory: Insights from an information-theoretic reconstruction. In Physical perspectives on computation, computational perspectives on physics (pp. 257–280). Cambridge University Press. Kuhn, T. (1970). The structure of scientific revolutions (2nd ed.). University of Chicago Press. Kuhn, T. (2000). The road since structure (J. Conant & J. Haugeland, Eds.). University of Chicago Press. Popescu, S., & Rohrlich, D. (1998). The joy of entanglement. In Introduction to quantum computation and information (pp. 29–48). Skrzypczyk, P., Brunner, N., & Popescu, S. (2009). Emergence of quantum correlations from nonlocality swapping. Physical Review Letters, 102(11), 110402. Timpson, C. G. (2013). Quantum information theory and the foundations of quantum mechanics. OUP Oxford.

Chapter 5

Quantum Ontology: Out of This World? Travis Norsen

Abstract Many commentators agree (based on the PBR theorem or other lessrigorous but still-compelling arguments) that the quantum mechanical wave function must represent some physically real thing/things/stuff. Existing proposals for the nature of this thing/things/stuff have tended to reflect the mathematically abstract character of the wave function: it has been suggested that the wave function represents, for example, a physical field of a mysterious and indeed rather incomprehensible character; or maybe a more comprehensibly-physical field that (incomprehensibly) lives not in ordinary physical space but instead in an abstract high-dimensional space; or maybe a non-local beable of a genuinely novel and ineffable sort; or maybe a kind of reified Law of Nature. None of these proposals is fully satisfying, physically and/or philosophically. The goal of the present paper is to advocate, by developing an analogy to two equivalent formulations of classical mechanics, for an under-appreciated alternative possibility—namely that the quantum wave function could represent some thing/things/stuff of a more mundane and more easily comprehensible character.

5.1 A Different World? Imagine a parallel world which, like ours, appears (at least at some sufficiently macroscopic scale) to be populated by massive particles which move on continuous trajectories through the three-dimensional space that the inhabitants of this world ordinarily take themselves to inhabit. The physicists of this world have discovered a theory which successfully accounts for the observed motions of the particles. The theory, however, has a character that most people in our world would regard as unusual, unfamiliar, and indeed downright strange. In particular, the theory seems to suggest that the motion

T. Norsen () Smith College, Northampton, MA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_5

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of particles is in some sense orchestrated by an object S which does not appear to live in the same three-dimensional space that the particles move in. For the case of a single particle of mass m moving in one dimension, this orchestrating field obeys the non-linear partial differential equation −

   ∂S ∂S = E x, ∂t ∂x

(5.1)

where E[x, p] is the expression for the total energy of the particle in terms of its position x and momentum p = mx. ˙ The trajectory X(t) of the particle is then determined by the orchestrating field according to the guidance formula  dX(t) 1 ∂S(x, t)  = . dt m ∂x x=X(t )

(5.2)

Note that for a one-particle system, S is just a function of position (in ordinary physical space) and time and hence can be straightforwardly and unproblematically understood as a physical field. But for a system of N particles, the orchestrating field S = S(x1 , x2 , . . . , xN , t) is a function on the N- (or, in three dimensions, 3N-) dimensional configuration space of the system and evolves in time according to      ∂S ∂S ∂S = E x1 , . . . , xN , − ,..., ∂t ∂x1 ∂xN

(5.3)

where E = E[x1 , . . . , xN , p1 , . . . , pN ] is again the expression for the total energy of the system in terms of the positions and momenta of its constituent particles. The trajectory of the ith particle is then determined by the obvious generalization of Eq. (5.2):  1 ∂S  dXi (t) = . dt mi ∂xi xn =Xn (t ) ∀ n

(5.4)

It will be helpful to develop a bit of intuition for this strange theory, which seems to correctly describe the “classical” physics of the parallel world, so let us consider a few simple examples to see how it works and what it predicts.

5.1.1 A Free Particle Moving in One Dimension Consider to begin with the simplest imaginable situation: an isolated single particle of mass m which is free to move only along the x-direction. The total energy of the particle is given by E[x, p] = p2 /2m, so the orchestrating field S(x, t) in this

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situation satisfies −

1 ∂S = ∂t 2m



∂S ∂x

2 .

(5.5)

If we guess an (additively) separable solution of the form S(x, t) = α(x)+β(t), it is easy to show that solutions take (up to an additional uninteresting additive constant which we henceforth ignore) the form 1 S(x, t) = mvx − mv 2 t 2

(5.6)

where v is an arbitrary constant. The guidance formula, Eq. (5.2), then tells us how the particle will move under the influence of the orchestrating field: dX(t) = v. dt

(5.7)

Thus, v can simply be understood as the constant velocity of the particle, whose position as a function of time will be given by X(t) = X0 + vt

(5.8)

where X0 = X(0) is the particle’s initial position. This is, of course, just the same “inertial” motion we would expect for a similarly isolated particle in our own world.

5.1.2 Simple Harmonic Oscillator As a second example, let’s consider a particle of mass m which is again constrained to move only along the x-axis, but which is now connected to the origin by a (massless) spring of stiffness k and negligible rest length. Its total energy will be given by E[x, p] =

1 p2 + kx 2 2m 2

(5.9)

so evidently the orchestrating field for this particle will obey the following partial differential equation: 1 ∂S = − ∂t 2m



∂S ∂x

2

1 + kx 2 . 2

(5.10)

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We can again search for additively separable solutions of the form S(x, t) = α(x) + β(t). We find, as before, that β(t) = −Et (for some constant E). The differential equation satisfied by α(x), 2  2mE = α  (x) + mkx 2 ,

(5.11)

is a little more complicated, but is only first-order, so it can be solved by separation of variables with the result √ α(x) = ± 2mE · F

k 2E

(5.12)

(x)

(plus an uninteresting additive constant which we again ignore). Here Fγ (x) =

 

1 − γ x 2 dx

√   arcsin γ x 1 2 = x 1 − γx + . √ 2 2 γ

(5.13)

Putting the pieces together, we have that the orchestrating field is given by S(x, t) = −Et ±

√ 2mE · F

k 2E

(x).

(5.14)

How does the particle actually move under the influence of this orchestrating field? Plugging into Eq. (5.2) yields the following first-order differential equation for X(t):   dX 2E k 2 =± X . 1− dt m 2E

(5.15)

This can again be solved by separation of variables with the result  X(t) = A sin

k t +c m

 (5.16)

 2E which describes a sinusoidal oscillation with amplitude A = k , angular √ frequency k/m, and an arbitrary phase c. It is interesting to observe that, again, the motion of the particle is the same as what we would expect in our world.

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5.1.3 Projectile Just like our world, the imaginary parallel world we are considering here contains planets with approximately uniform, downward gravitational fields near their surfaces. In the presence of such a field, a mass-m particle that is launched or thrown, with negligible atmospheric drag, will therefore move under the influence of an orchestrating field S(x, y, t) satisfying −

∂S 1 = ∂t 2m



∂S ∂x

2 +

1 2m p2



∂S ∂y

2 + mgy

(5.17)

p2

since, for such a particle, E[x, y, px , py ] = 2mx + 2my +mgy. Here g is the (constant) magnitude of the downward gravitational field. Using the same mathematical techniques as above, it is straightforward to solve this partial differential equation and show that the orchestrating field will take the form √ 2m 2g S(x, y, t) = −Et + vx x ± (5.18) (Y0 − y)3/2 . 3 where E, vx , and Y0 are constants. (Interested readers will benefit from plugging this expression into the differential equation and finding the relationship between the constants that makes it work!) Having found the orchestrating field, we can now determine the particle trajectory that it produces. The x-component of the velocity is given by 1 ∂S dX = = vx . dt m ∂x

(5.19)

So the particle just moves with constant velocity vx in the horizontal direction. The vertical motion is a little more interesting:   1 ∂S dY = = ∓ 2g Y0 − Y (t). dt m ∂y

(5.20)

The solution, Y (t) = Y0 −

g (t − t0 )2 , 2

(5.21)

corresponds to motion with constant downward acceleration of magnitude g. So— again!—the actual motion of particles, predicted by the strange theory in the parallel world, is the same as the motion we are accustomed to seeing in our own world.

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5.1.4 Two Particles in 1D So far we have considered only situations involving a single particle moving in one or two spatial dimensions. In these cases, the orchestrating field S lives in a space of the same dimensionality as the space the particles move in. That is, the orchestrating field appears to be a genuine physical field that has, at each moment, a well-defined value at each point in physical space. But the really strange thing about this theory is that, in situations involving several particles, the orchestrating field does not seem genuinely physical in this same sense. To see this, we might consider the simple example of two particles, both constrained to move along a single, one-dimensional axis. Suppose particle 1 has a mass m and is electrically neutral, while particle 2 has mass m but has an electric charge q. Then, if there is a uniform electric field of magnitude E parallel to the axis, the total energy will be given by E[x1 , x2 , p1 , p2 ] =

p2 p12 + 2 + qEx2 2m 2m

(5.22)

where x1/2 is the position of particle 1/2 and p1/2 is its momentum. Note that this situation is mathematically identical to the previous example of a single particle—the projectile—moving in 2D. So we can immediately infer that the orchestrating field S(x1 , x2 , t) for our two particle system here will take the form 3/2  S(x1 , x2 , t) = −Et + v1 x1 ± C X20 − x2 .

(5.23)

where E, v1 , C, and X20 are constants. And we can immediately infer that particle 1 will move with constant velocity (v1 ) while particle 2 will move with constant acceleration. The unusual feature here is that the orchestrating field S is a time-dependent function on the so-called “configuration space” of our system. It takes values, that is, not at locations in the (one-dimensional) physical space in which the particles both move, but instead at locations in the abstract (two-dimensional) space each point of which represents a specific possible configuration of the two-particle system, i.e., each point of which represents a pair of locations in physical space. The field S—if it is proper to think of it as a field—is thus very unusual compared to the gravitational and electric fields that we have also mentioned. Unlike the gravitational and electric fields, which unambiguously and unproblematically live in physical space and can hence be understood unambiguously and unproblematically as physical objects, S lives in an abstract, higher-dimensional space and thus seems, somehow, unphysical. One might even say that it seems “out of this world” (and not in the good way usually implied by that phrase). But arguably things are less weird here than they seem, for this unusual theory in our imaginary parallel universe. After all, the orchestrating field S is additively

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separable in its dependence on x1 and x2 . So instead of demanding that there is a single (“out of this world”) field on the abstract configuration space, we could simply break the field into two parts: S = S1 + S2 . In particular, S1 (x1 , t) = −E1 t + v1 x1

(5.24)

could be understood as an unproblematically physical orchestrating field for particle 1, while 3/2  S2 (x2 , t) = −E2 t ± C X20 − x2

(5.25)

could be understood as an unproblematically physical orchestrating field for particle 2. With E1 + E2 = E, this is mathematically equivalent to what we said originally. In this picture, each particle’s motion would be determined by that particle’s own associated orchestrating field according to 1 ∂Si dXi = . dt mi ∂xi

(5.26)

And so, after all, there is nothing particularly weird about the physical picture suggested by the theory, at least in this particular situation. But it turns out that this kind of situation is very special: the (additive) separability of our original S is a consequence of the fact that the two particles do not in any way affect one another in this scenario. Instead they each just independently respond to the external electric field in the same way that they would if the other particle weren’t present. But in a more general kind of case, in which the particles mutually influence one another, the two-particle orchestrating field S(x1 , x2 , t) will not be (additively) separable. It will, instead, be an irreducible function on the (abstract, two-dimensional) configuration space, by which I just mean that the function cannot be broken apart into a function of x1 plus another function of x2 , as in the example we’ve just been discussing. Physicists in the parallel universe describe the multi-particle system, in this kind of situation, as being “intertwined”. It will be helpful, as always, to consider a concrete example. So, suppose again we have two mass-m particles constrained to move in one dimension, but with the two particles connected by a (massless) spring (of negligible rest length). The energy of this system, in terms of the positions and momenta of the p2

p2

two particles, is E[x1, x2 , p1 , p2 ] = 2mx + 2my + 12 k(x1 −x2 )2 . And so the differential equation governing the time-evolution of the orchestrating field S(x1 , x2 , t) is: −

1 ∂S = ∂t 2m



∂S ∂x1

2 +

1 2m

1 + k(x1 − x2 )2 . 2



∂S ∂x2

2

(5.27)

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It is easy enough to show that the orchestrating field which satisfies this partial differential equation takes the form  S(x1 , x2 , t) = − Et ±  ±

mE −

c (x1 + x2 ) 2

c F mk (x1 − x2 ) 2 c

(5.28)

where c and E are constants (E representing the total energy of the two-particle system) and where F is the same function defined above in our discussion of the simple harmonic oscillator. It will no longer be surprising to learn that the trajectories, for the individual particles, implied by Eq. (5.2) with an orchestrating field of this form, correspond exactly to the trajectories we would expect for this same system in our own world. In particular, the two particles oscillate sinusoidally about their average (or “center of mass”) position, which in turn moves inertially. The surprising and interesting thing is rather the additively non-separable character of the orchestrating field in Eq. (5.28). Unlike the earlier example of the two particles which don’t mutually affect one another, the fact that the two particles in this example are connected by a spring results in this non-separable orchestrating field. It results, that is, in what the physicists of our imagined parallel world describe as the “intertwinement” of the two particles in this kind of system. To be clear, though, the troubling and puzzling thing, for the physicists of our parallel universe, about systems exhibiting this sort of intertwinement, is not the motion of the particles. The particles just move the way they move, and the theory’s predictions for that motion perfectly match the motions that are in fact observed. The troubling and puzzling thing has instead to do with understanding the causality of, and the underlying ontology involved in, the motion of the particles. In particular, the theory makes perfectly clear that the motion of the particles is orchestrated by the field S. (The physicists don’t call it the “orchestrating field” for nothing!) But, as we have noted, the orchestrating field simply cannot be understood as an ordinary, physically real field (akin to the familiar and ontologically-unproblematic electric and gravitational fields). It is just not the right kind of mathematical object, depending, as it does, in the general case of N particles moving in the full 3dimensional space, on (time and) 3N spatial coordinates. It is, in short, a field on configuration space (-time) rather than physical space (-time).

5.1.5 Controversy This situation, in which a central mathematical object in a seemingly-correct physical theory does not appear to afford any straightforward physical interpretation, has

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generated a sprawling controversy among the physicists and philosophers of our parallel world. Those of a more scientifically “realist” bent, who agree that our ontological commitments should follow the best established science, have splintered into several opposing camps. One camp contends that the orchestrating field S can be understood as a field in physical space, but a new kind of field (which they call a “many-field”) which assigns values not to individual points in physical space, but rather to N-tuples of points. None of the physicists, however, have found this perspective helpful, and the rest of the philosophers quite reasonably suggest that it sounds more like a description of the original problem (in fancy new terminology) than anything like a solution of the problem. Another camp recommends biting the proverbial bullet: if, they argue, the (seemingly correct) theory involves what looks, mathematically, like a field in a 3N-dimensional space, and if we want to interpret (seemingly correct) theories realistically, then we should accept that the true physical space, on which the fundamental dynamics of the orchestrating field S plays out, is this 3N-dimensional one. One sub-group within this camp takes the 3N-dimensional space (where the physically real orchestrating field lives) to exist in addition to the more familiar 3-dimensional space where the particles (and, presumably, the physicists and philosophers themselves) live. They tend to have a difficult time providing a coherent account of the causality involved in a physically real object (which lives in one space) physically orchestrating the motion of different physically real objects (which live in an entirely different space). Perhaps to avoid such difficult questions, another sub-group claims that everything (including the particles and the physicists and philosophers who study them) is in the 3N-dimensional space. On this view, the true physical reality for (what we ordinarily describe as) an N-particle system involves a single so-called “spectacular particle” moving around in the 3N-dimensional space under the influence of the orchestrating field that lives in that same space. While this sub-camp’s view has an undeniably tidy ontological character, and avoids the troubling questions raised in the previous paragraph about how an object in one space can causally influence objects in an entirely distinct space, it has not made a lot of progress in explaining coherently how a physically-real 3-dimensional world (and/or the illusion thereof) could emerge from the posited fundamental physics. Many philosophers thus quite reasonably note that, despite the clear connection between this interpretation and a seemingly empirically adequate theory, this particular interpretation really cannot be said to account successfully for the observations (which are, in our parallel world, still of an ordinary, three-dimensional world not unlike our own). Another realist camp, recognizing the difficulties involved in interpreting S as a physical field, suggests retaining the particles (in regular physical space) as physically real, but interpreting the orchestrating field S in some other way. What sort of thing is S, exactly, on this view? Often this is left vague, and sometimes this vagueness is characterized as a virtue (on the grounds that only backwards simpletons would demand that novel physical objects, discovered by cutting-edge

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science, should fit neatly into a set of boxes laid out by ancient philosophers). Sometimes it is suggested that we could interpret S not as a physical thing at all, but instead as something like a law. This has some plausibility, especially for those people in the parallel world who have a more Platonist understanding of the nature of laws. But most people have not found this very convincing. After all, the orchestrating field S (at least for most systems within the parallel universe) has a non-trivial time-evolution (already a highly unusual feature for laws), and this timeevolution is governed by a partial differential equation (just like the more familiar and ontologically unproblematic electric, magnetic, and gravitational fields). And it also seems to perform the same fundamental job as those other more familiar fields, namely: making the particles move a certain way. So at the end of the day it is just difficult to make a compelling case that something which (aside from living in the wrong space) looks like a field and acts like a field, is not a field, but is instead a law. And then, of course, in addition to the various realist camps, there exist a number of more-or-less reputable anti-realist camps. One of these, for example, suggests that S could be understood subjectively (i.e., as a description of somebody’s beliefs about how the particles will move, rather than as something physically real that makes them move in a certain way). Another suggests that it’s pointless or indeed meaningless to worry about the ontological status of all these things; the theory seems to work, say the proponents of this view, and that is all that matters. And there are other viewpoints as well. But most sensible physicists feel that all these anti-realist viewpoints fail to provide (and somewhat ridiculously attempt to make a virtue out of failing to provide) any genuine physical explanation of the observations. And so, for quite some time, no real progress has been made. The theory continues to work well and everyone agrees that, in some sense, it must be correct. But nobody in this parallel world can agree about how, exactly, to take the theory seriously.

5.2 Relation to Our World I’m sure it will not surprise anyone to learn that the imaginary parallel world I described in the previous section is not substantially different from our world. In particular, the physics of the imaginary parallel world is identical to the physics of our world. The two worlds, as it turns out, only differ historically and sociologically. Let me explain. The theory that I described in the previous section is just the version of classical mechanics that is known, in our world, as the Hamilton-Jacobi (H-J) formulation. I have presented it in an admittedly simplified way, for the sake of clarity and accessibility. A fuller presentation would flesh out the connections to the more-widely-understood Hamiltonian, Lagrangian, and Newtonian formulations of classical mechanics and would develop a number of technical points that I have

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ignored or breezed past. Interested readers can of course find such details in any Classical Mechanics textbook. For our purposes, though, the important point is just that the H-J theory can be understood as a reformulation of the physical principles articulated (in our actual world) by Newton. One nice way to see the equivalence is to take the time derivative of Eq. (5.2) to see what the H-J theory implies about the acceleration of particles. Recognizing that the right hand side of Eq. (5.2) depends on t two ways, we have that      d 2X 1 ∂ ∂S dX ∂ ∂S a= = . (5.29) + dt 2 m ∂t ∂x dt ∂x ∂x x=X(t ) It is then just a few lines of simple algebra to show that, if dX/dt is given by Eq. (5.2) and if S obeys Eq. (5.1)—which, in our world, is of course called the Hamilton-Jacobi equation—the above simplifies to a=

  1 dV − m dx x=X(t ) 2

(5.30)

p + V (x). Thus, the H-J theory where V (x) is the potential energy: E(x, p) = 2m generates the same particle trajectories as the more intuitive and familiar Newtonian second law, F = ma. What, then, is different about the imaginary parallel world and ours? The point of the parable was to contemplate how things might have developed if, somehow, scientists in that world stumbled onto the Hamilton-Jacobi theory first, without yet appreciating (what, in our world, is known as) Newton’s second law and, in particular, without yet grasping the concept of “force” and the Newtonian principle that applied forces cause particles to accelerate. In our world, with these Newtonian concepts and principles being familiar and well-understood prior to the advent of the Hamilton-Jacobi re-formulation of classical mechanics, there are no deep controversies about how to understand the “orchestrating field” S. It is, somehow, just clear that S need not and should not be taken seriously, as a direct and literal description of some physically real object. Instead, it is somehow just clear that S describes some aspect of physical reality, but in an indirect and abstract way: in particular, S captures something about the way that energy transforms for different possible dynamical evolutions of the system, and energy, in turn, provides a kind of abstract perspective on the forces that particles exert on one another. So, in our world, anybody who suggested that S should be understood as describing a physically-real field living in a (therefore evidently equally real) 3Ndimensional space, and somehow orchestrating the motion of particles in ordinary 3-dimensional physical space from there, or any of the various other things like that, would be laughed off the stage. And quite rightly. The suggestion is patently absurd. It represents an obvious failure to recognize the abstract character of the H-

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J formulation and the function S in particular. We can be realist about classical mechanics, without the need to posit any “out of this world” ontology, by just saying that there are particles, which move around in 3-dimensional space, under the influence of the forces they exert on one another. Of course, even in our world, philosophers argue about things like the ontological status of forces. I am certainly no expert in this area of metaphysics, although I did find much of value in the several papers I read recently (Bigelow et al., 1988; Massin, 2009, 2017; Wilson, 2007). That said, at the end of the day, at least when I am wearing my physicist hat, I think we need to acknowledge that whatever puzzles there might be about how to reconcile Newtonian physics with metaphysics, they are trivial and uninteresting compared to the puzzle that the physicists in our imaginary parallel world took themselves to face. If, to make sense of what we observe, a theory requires us to postulate physicallyreal things which are, in some sense, “out of this world”, it is appropriate for metaphysical alarm bells to go off. That would be a really troubling situation, basically unprecedented in the history of science (at least in our actual world). But Newtonian mechanics just says there are (massive) things, which push and pull on one another in various ways. Some sophistication is of course required to precisely define concepts like “acceleration” and “mass”, and to distinguish them from other related ideas. But we can literally just see and feel that there are things, and we can directly experience the forces that things exert on one another (when one of the things is us). So I think it’s reasonable to say that if metaphysical alarms are sounding in response to this—in response to what amounts to a mere precisification and quantification of phenomena that are amenable to direct perceptual experience (which is how I think Newtonian mechanics can and should be understood)—the problem is with us and our metaphysical expectations. In short, if our metaphysical expectations cannot be reconciled with Newtonian mechanics, it is our metaphysical expectations that should be replaced, not our understanding of Newtonian mechanics. And so, in so far as we correctly recognize the Hamilton-Jacobi theory as an abstract reformulation of Newtonian mechanics, and in particular in so far as we correctly recognize S as simply a highly-indirect way of talking about forces, there should be no metaphysical worries about the H-J theory either. Which, thankfully, in our world, there aren’t. Why, then, was there an irreconcilable controversy about the ontological status of the H-J orchestrating field S in our imagined parallel world? Because, by construction, the people in that world did not have the Newtonian ontology (of particles interacting via forces) to fall back on. So, out of sheer ignorance (or a lack of creativity), they believed that the only way to take their empirically successful theory seriously was to take S as a direct and literal description of some thing/things/stuff. Their various attempts to understand the meaning of the theory thus all involved inappropriately reifying what, from the comfort of our world, is obviously, in fact, a kind of abstraction. To us that looks silly. But perhaps we can and should learn something from their plight.

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5.3 Relation to Quantum Mechanics So far everything I’ve been saying has been intended in the context of classical physics where, at least if you’ve got the Newtonian conceptual foundations in place, there is nothing particularly problematic or controversial. But the discussion so far does relate to quantum mechanics (QM), in several ways. At least in our world, the H-J formulation of classical mechanics is relatively unfamiliar to most people. But it is closely related to quantum mechanics (especially the de Broglie–Bohm “pilot-wave” theory) and drawing out some of the connections may help readers, for whom the H-J theory is currently less familiar than QM, understand how the H-J formulation works. First, the wave function ψ for an N-particle system in QM (for simplicity here, let’s ignore spin) is, just like the “orchestrating field” S in the H-J theory, a function on the 3N-dimensional configuration space. The wave function ψ is, of course, complex, but the complex field ψ can be decomposed into two real fields, R and S, as follows: ψ = ReiS/h¯ . (Thus, R is the modulus, and S the complex phase, of ψ.) If we plug this ansatz into the Schrödinger equation (for, say, a particle of mass m moving, for simplicity, in just one dimension) i h¯

∂ψ h¯ 2 ∂ 2 ψ =− + V (x)ψ ∂t 2m ∂x 2

(5.31)

and demand that the real and imaginary parts match on both sides, we arrive at two new equations that are jointly equivalent to the Schrödinger equation. The first can be understood as relating to the conservation of probability and is not particularly relevant for our purposes. But the second— ∂S 1 − = ∂t 2m



∂S ∂x

2 + V (x) + Q

(5.32)

—just is the Hamilton-Jacobi equation, but with a new term, the so-called “quantum potential”, ∂2R

Q=−

h¯ 2 ∂x 2 . 2m R

(5.33)

So already in standard quantum mechanics there is a close connection between the Schrödinger equation and the H-J equation, with a slightly-modified (“quantum”) Hamilton-Jacobi equation describing the time-evolution of the phase of the wave function ψ. And there is an even closer connection in the de Broglie–Bohm pilot-wave theory, in which the role of the wave function is to guide (or “pilot”) the motion of actual particles. Indeed, the Bohmian “guidance formula” is nothing but Eq. (5.2),

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but with, of course, S being a solution of the quantum H-J equation (with the additional term, Q). There is also a sense in which the H-J formulation of classical mechanics captures the classical limit of quantum mechanics. This is, again, particularly clear in the de Broglie–Bohm pilot-wave theory where what has been said above implies that, if the “quantum potential” Q is negligible (which occurs, for example, when a particle’s guiding wave has a local plane-wave structure) the quantum H-J equation becomes identical to the (classical) H-J equation and the Bohmian particle will move along a classical trajectory. Finally, note that thinking of the orchestrating field S (in the context of the classical H-J theory) as being closely associated with the complex phase of the QM wave function perhaps clarifies a few technical details from our earlier examples. For instance, the additively-separable character of S corresponds to the more standard multiplicative-separability of stationary, energy-eigenstate solutions of the time-dependent Schrödinger equation. And, of course, the failure of S to be (additively) separable in the spatial coordinates for our example of two particles connected by a spring—the situation that I said the physicists in the parallel world described as “intertwinement”—is of course analogous to the failure of (multiplicative) separability of the wave function in the situations that physicists in our world describe as involving “entanglement”. So, perhaps, those brief comments will help connect some of the technical details of our earlier discussion to some more-familiar ideas from QM. But there is also a second—more important and undoubtedly alreadyanticipated—relation between quantum mechanics and the H-J formulation of classical mechanics that I want to raise for discussion. Namely: the (imaginary, parallel world) metaphysical controversies that I described in Sect. 5.1 bear an amazing similarity to the real controversies, in our actual world, about how to understand quantum mechanics generally and the wave function specifically.

5.4 Proposal All of the different viewpoints I described previously, for understanding the role and ontological status of the orchestrating field S in the context of the classical physics of our imaginary parallel world, correspond exactly to viewpoints that have been put forward (in our actual world) as possible ways to understand the role and ontological status of the QM wave function (Albert, 1996; Bohr, 1928; Caves et al., 2002; Dürr et al., 1997; Hubert and Romano, 2018; Maudlin, 2019; Ney, 2021). I think it should be admitted that the corresponding viewpoints in the two worlds are equally problematic. That is, none of the standard, extant proposals for understanding the role and ontological status of the QM wave function, ψ, in our actual world, have the kind of compelling, convincing character that generates consensus. That is why, in our actual world just like in the imaginary parallel world, the debates have raged, without any apparent progress, across decades.

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So I have a simple proposal. Let’s learn a lesson from the imaginary parallel world. In particular, let’s recognize the possibility that—especially when there are some unusual historical/sociological circumstances involving anti-realist philosophies of science, as has indeed been the case in our world (Beller, 2001; Cushing, 1994; Freire, 2015)—physicists may stumble into what should be regarded as an abstract re-formulation of some other, more straightforward and ontologically unproblematic theory, prematurely. . . and hence fail to recognize their theory for what it really is. . . and hence make the mistake of trying to understand, as a direct and literal description of something exotic, something that should instead be understood as an indirect and abstract description of something more mundane. Specifically, let’s leave room in our thinking for the possibility—just the mere possibility!—that Schrödinger’s equation captures something fundamental and true about physical reality, but that it has a status like the status of the Hamilton-Jacobi equation in classical mechanics. Let’s leave room, that is, for the possibility that the quantum mechanical wave function ψ might not describe, in a literal and direct way, something weird and exotic and “out of this world”, but may instead indirectly describe something more mundane, just as the orchestrating field S in H-J theory can be understood as indirectly describing forces that particles exert on one another. To be clear, I am not suggesting that there is a perfect parallel between QM and H-J theory. In particular, I do not mean to suggest that we can or should re-interpret the QM wave function in terms of forces that particles exert on one another. It is, of course, controversial whether particles even exist in the context of QM. (They do, according to the pilot-wave theory, and, as discussed above, the connection between QM and H-J is especially close if we adopt the pilot-wave theory of QM, but it certainly wouldn’t be appropriate here to presuppose the truth of the pilot-wave picture.) And even if the correct ontology for QM does involve particles (as in the pilot-wave picture), there are several reasons why I don’t think it is likely that the right way to round out the ontology will involve “forces”. (What are these reasons? One is that even in the pilot-wave picture, the dynamics for the particles is fundamentally very non-Newtonian. In particular, the dynamics is first-order in time rather than second-order (Goldstein and Struyve, 2015). Another is just the qualitative evidence, for example from the 2-slit experiment with individual electrons, for something like “guiding waves.” Bell summarized this evidence perfectly when he wrote: “Is it not clear from the smallness of the scintillation on the screen that we have to do with a particle? And is it not clear, from the diffraction and interference patterns, that the motion of the particle is directed by a wave? De Broglie showed in detail how the motion of a particle, passing through just one of two holes in a screen, could be influenced by waves propagating through both holes. And so influenced that the particle does not go where the waves cancel out, but is attracted to where they cooperate. This idea seems to me so natural and so simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored” (Bell, 2004). Indeed. But note that for such guiding waves to, e.g., “propagat[e] through both holes” of a 2slit screen, the waves in question must propagate in ordinary 3-dimensional physical space, not an abstract high-dimensional configuration space. To me this suggests that

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the QM wave function might be an abstract/indirect description of, say, a number of guiding-waves in 3-space. . . which is of course very different from the forces I am suggesting are the ontological referent of S in H-J classical mechanics.) So, again, my proposal is not that ψ in QM should be understood in precisely the same way that I think S in H-J should be understood. The proposal is much more general—and hence, unfortunately, much more vague. It is simply that we not forget the logical possibility that quantum mechanical wave functions could be indirect, abstract descriptions of something metaphysically unproblematic (compared to, for example, fields on high-dimensional spaces, Platonic laws, etc.)—something as simple and mundane and ontologically unproblematic as forces are in the context of Newtonian mechanics, but whose exact nature, unfortunately, I cannot at present claim to know. The only concrete possible example I can provide is my own TELB (“Theory of Exclusively Local Beables”) (Norsen, 2010). In this theory, the QM wave function should be understood as an indirect description of a set of interacting fields that live in ordinary physical space. The theory thus raises exactly as many difficult metaphysical questions (about the status of the wave function) as classical mechanics does (about the status of the H-J function S)—namely, none. Unfortunately, though, TELB is so ugly, complicated, and contrived that I don’t think it can be taken too seriously, as anything but a proof of concept for the general proposal I’m making here. But still, that is something. We know it is possible, in principle, to understand quantum mechanics in a way that doesn’t require relinquishing the proper, realist attitude toward scientific theories, but in a way that also doesn’t require postulating any weird, “out of this world” ontology. But to find a genuinely compelling quantum theory of this sort—to find a theory that would allow the community to move beyond its seemingly endless philosophical debates between partisans of different, but equally problematic, viewpoints—we need more people to start looking for the right kind of thing. Maybe the parable about the parallel world can provide the needed motivation?

References Albert, D. Z. (1996). Elementary quantum metaphysics. In J. T. Cushing, A. Fine, & S. Goldstein (Eds.), Bohmian mechanics and quantum theory: An appraisal (pp. 277–84). Springer Netherlands. Bell, J. S. Speakable and unspeakable in quantum theory (2nd ed., p. 191). Cambridge. Beller, M. (2001). Quantum dialogue: The making of a revolution. University of Chicago Press. Bigelow, J., Ellis, B., & Pargetter, R. (1988). Forces. Philosophy of Science, 55(4), 614–630. Bohr, N. (1928). The quantum postulate and the recent development of atomic theory. Nature, 121(Supplement), 580. Caves, C. M., Fuchs, C. A., & Schack, R. (2002). Quantum probabilities as bayesian probabilities. Physical Review A, 65, 022305. Cushing, J. T. (1994). Quantum mechanics: Historical contingency and the Copenhagen hegemony. University of Chicago Press.

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Dürr, D., Goldstein, S., & Zanghì, N. (1997). Bohmian mechanics and the meaning of the wave function. In R. S. Cohen, M. Horne, & J. Stachel (Eds.), Experimental metaphysics – Quantum mechanical studies for Abner Shimony, Volume One. Boston studies in the philosophy of science (Vol. 193, pp. 25–38). Kluwer. Freire, O. (2015). The quantum dissidents: Rebuilding the foundations of quantum mechanics. Springer. Goldstein, S., & Struyve, W. (2015). On quantum potential dynamics. Journal of Physics A: Mathematical and Theoretical, 48, 025303. Hubert, M., & Romano, D. (2018). The wave-function as a multi-field. European Journal for Philosophy of Science, 8, 521–537. Massin, O. (2009). The metaphysics of forces. Dialectica, 63(4), 555–589. Massin, O. (2017). The composition of forces. British Journal of Philosophical Science, 68, 805– 846. Maudlin, T. (2019). Philosophy of physics: Quantum theory. Princeton foundations of contemporary philosophy (Vol. 33). Princeton University Press. Ney, A. (2021). The world in the wave function: A metaphysics for quantum physics. Oxford University Press. Norsen, T. (2010). The theory of (exclusively) local beables. Foundations of Physics, 40, 1858. Wilson, J. (2007). Newtonian forces. British Journal of Philosophical Science, 58, 173–205.

Chapter 6

Why Might an Instrumentalist Endorse Bohmian Mechanics? Darrell P. Rowbottom

A Briton wants emotion in his science, something to raise enthusiasm, something with human interest. He is not content with dry catalogues. —George F. Fitzgerald (1896: 441)

Abstract I show that Bohmian mechanics may be highly valued by anti-realists, such as cognitive instrumentalists, although it is typically understood to be a realist interpretation of quantum mechanics. I argue that it possesses significant theoretical virtues – and is a vehicle for fostering non-factive understanding concerning how many phenomena interrelate – even if it fails to be approximately true (or true to any significant degree in what it says about unobservable entities). I employ several examples from the history of science to illuminate and support this view.

6.1 Introduction When I first encountered quantum mechanics, I was alarmed. The picture I’d built up as a secondary school student was of a clockwork world where positions and momenta were continuously well-defined and where future states were uniquely determined by past states. But now I was being taught a manifestly incompatible view, where probabilities were needed not merely due to our ignorance or the consideration of ensembles, where entities sometimes behaved like waves and other times behaved like particles, where the act of observation could cause striking changes, and so on. Quantum mechanics lacked external consistency – or

D. P. Rowbottom () Lingnan University, Tuen Mun, N.T, Hong Kong e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_6

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consistency with the other physical theories I knew – as Kuhn (1977, ch. 13) might have put it.1 Why did I care about this? I was a (naïve) scientific realist. I believed the physics I’d learned previously was largely true. Hence my concern wasn’t directly with inconsistency, but with what this showed; two inconsistent stories often can’t both be approximately true (or are strongly contradictory).2 In short, what I took myself to know about the way the world worked was in jeopardy. And that was extraordinarily intellectually uncomfortable. I didn’t realise it at the time, but my reaction heralded that I was destined for a career in philosophy, rather than science, perhaps because I was not ready to be indoctrinated – in Kuhn’s (1963) sense – any further.3 My peers found quantum mechanics harder than many other areas of physics but were typically happy to ‘shut up and calculate’; and when I raised my worries with lecturers, they were somewhat dismissive because they had similar leanings. I became a dissident student, appalled by what I presumptuously, in my callow state, took to be a refusal to do physics properly and to consider various possible interpretations.4 I began to read exclusively about the interpretation of quantum mechanics. I started with D’Espagnat (1989). And subsequently encountering Bohm’s ontological interpretation of quantum theory, in Squires (1994), was a revelation. Dissident student discovered dissident scientist, wrote an angry dissertation about how he’d been taught false things, stormed off into philosophy, and resolved to argue for scientific realism (with a methodological component). All didn’t go to plan, however, because I gradually became convinced that the arguments against scientific realism are stronger than those for it (and was spurred on by my vain attempts to determine how confirmation values could be absolute and objective); and by the time I reached Oxford as a postdoc, I had strong antirealist leanings. While extemporising in a lecture there, I came up with the idea

1

I mean quantum mechanics as I was taught it, which included an incoherent mixture of elements of Bohr’s and Heisenberg’s interpretations of quantum theory. As Wallace (2019: 306) notes: ‘philosophers’ version of “orthodoxy” does physicists too much credit in providing a selfconsistent realist account of QM that just lacks a satisfactory account of exactly when collapse happens, even as it does them too little credit in failing to recognise the unsuitability of the orthodox version of orthodoxy to physical practice.’ 2 I introduce the notions of weak and strong contradiction, which are useful in discussing several aspects of scientific realism, in Rowbottom (2021). In summary: (1) p strongly contradicts q precisely when ≈p contradicts ≈q; (2) p weakly contradicts q when p contradicts q but ≈p does not contradict ≈q. 3 The following passage from Popper (1970: 52) resonates with me as a result: ‘In my view the “normal” scientist, as Kuhn describes him, is a person one ought to be sorry for ... [is one who] has learned a technique which can be applied without asking for the reason why (especially in quantum mechanics)’. I do, however, think that using indoctrination to some extent may be good for science from a social epistemological perspective; see Rowbottom (2011a). 4 As I earlier intimated, what passed for ‘the Copenhagen interpretation’ didn’t correspond to the views of Bohr or Heisenberg, either in isolation or in any reasonable combination. It wasn’t the Bohr-Heisenberg view accurately presented by Faye (2019).

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for a position inspired by logical positivism, which I fleshed out and outlined in Rowbottom (2011b); and this led to my recent monograph, proposing a novel instrumentalism of a ‘cognitive’ variety (Rowbottom, 2019). I’ll say a little more about that later. I recount this story because I believe many would be surprised, in the light of it, that Bohmian mechanics remains my preferred approach to quantum theory (or non-relativistic QM). Physicists of an anti-realist persuasion might be particularly surprised. For example, Leggett (2005: 871) opines that Bohmian mechanics involves ‘little more than verbal window dressing of the basic [measurement] paradox’ and echoes earlier thoughts of Heisenberg (1955: 18), that: ‘[Because] Bohm’s interpretation cannot be refuted by experiment . . . From the fundamentally “positivistic” (it would perhaps be better to say “purely physical”) standpoint . . . [It is not a counter-proposal to] the Copenhagen interpretation but . . . its exact repetition in a different language.’ It is especially pertinent that this attitude among physicists may originate from Bohr’s work on quantum theory. As Allori (2015: 316) notes: ‘a good portion of physicists ended up with some sort of anti-realist position because they took Bohr’s proposal as a reductio ad absurdum for scientific realism in the quantum framework.’5 Some philosophers of physics might also be somewhat surprised by my story because, as Wallace (2020b) points out, Bohmian mechanics is normally labelled a ‘realist’ interpretative strategy. Wallace (2020b: 79) continues by noting that: The debate over realist-vs-non-realist strategies has been fairly cursory in recent discussions and has largely turned on general disagreements about the legitimate aims of science . . . In the bulk of philosophy of physics (in particular in its more metaphysically inclined corners), indeed, the non-realist strategies are set aside almost without comment. Meanwhile, comparative assessment of the realist strategies has tended to turn on relatively detailed, and fairly metaphysical, concerns with those strategies.

This state of affairs is intellectually unsatisfactory on several counts. First, being an anti-realist about quantum theory doesn’t entail preferring one of the ‘nonrealist strategies’ normally identified; in short, those strategies aren’t exhaustive. Second, on a related note, one may prefer one of the interpretations labelled ‘realist’, on principled grounds, without believing it to be approximately true. Third, and finally, comparative assessments of theories might involve epistemological and practical concerns, rather than metaphysical ones (at least in any realist, especially revisionary, sense of ‘metaphysics’).6 I will expand on all these points in this paper. I will begin, in the next section, by showing how and why an instrumentalist (or an anti-realist of another variety) might prefer an ontologically rich theory (or interpretation), involving extensive claims 5

There are exceptions. For example, Bowman (2002: 313) holds that Bohmian mechanics ‘possesses intrinsic heuristic value, arising from calculational tools and physical insights that are unavailable in “standard” quantum mechanics’. I agree in spirit, but don’t think any value is intrinsic. 6 I might also have again taken issue with the very idea that science has aims, but I will just point to Resnik (1983) and Rowbottom (2014, Forthcoming) on this issue.

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about unobservable entities, to a sparser or more austere counterpart. In doing so, I will point to pertinent work in the history of philosophy of science and in the history of physics. Before I continue, however, I’d like to address two issues to prepare the stage. The first concerns what, exactly, I intend to consider Bohmian mechanics to be an interpretation of. This is a pressing consideration of in light of Wallace’s argument that quantum mechanics is ‘a framework theory covering a huge number of particular theories’ (2020a: 4303), and his resultant position that what we should expect from an interpretation thereof is ‘a set of instructions which tells us, for any given quantum theory, how to understand that theory.’ (2020b: 85). Wallace may be right about what quantum mechanics is (although, as I will explain shortly, this depends on what one takes a theory to be). However, having an interpretation of such broad scope is of questionable import, except from the point of view of economy (and other things that follow from that, ceteris paribus, such as memorability). A patchwork of interpretations could do the trick and might be superior to any all-encompassing alternative. Moreover, more than one interpretation might be adequate in any given context. (Strictly speaking, one could have a set of instructions of the form: in contexts of type C1 , employ interpretations I1 or I2 ; in contexts of type C2 , employ interpretation I3 ; and so forth. But this is not what Wallace wants, as far as I can discern, and he contends that a many-worlds view may be employed in all pertinent contexts.) This said, I will restrict my attention, as is typical in such discussions, to the interpretation of the part of quantum mechanics governing what Wallace (2020a: 4303) calls ‘nonrelativistically moving point particles interacting by long-range forces’. This element is significant in physics education because it stands in direct contrast to Newtonian mechanics, which is invariably taught long beforehand. This brings me to my second point, which concerns what counts as a theory rather than an interpretation thereof. I shall not be able to say much about this, beyond noting that philosophers often talk past one another due to having different implicit views on the matter. As I alluded to above, even what counts as theory is unsettled; ‘theory’ is used with gay abandon by most philosophers of science, although there is no agreed referent thereof; see, for example, French (2020). Indeed, I hesitate to agree with Wallace (2020a) about quantum mechanics being a ‘framework theory’ because I think of some of the things he calls ‘theories’ as ‘models’, and others as ‘reformulations’; and Wallace (2020a) doesn’t spell out exactly what it takes for something to be a theory, by his lights. What he does say is highly controversial, against the background of the history of philosophy of science.7 But this essay cannot adequately deal with this issue, so suffice it to 7

Wallace (2020b: 81) suggests that something is not a theory (although it might be a theoretical framework) if ‘by itself it makes no predictions and explains no phenomena; [if] by itself it cannot be tested or falsified’. However, there is a long-standing (confirmational holist) view in the philosophy of science that most theories are like this. Duhem (1954), for instance, is best known for his proclamation that there are no crucial experiments in physics because physical theories can’t be tested in isolation.

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say that I will take an interpretation, for present purposes, to be an account of happenings on the (currently) unobservable ‘level’ which is explicitly associated with a specific set of putative relations between observable entities. This is a rough and inadequate definition, partly for reasons I allude to in the previous footnote, but it serves to point, at least, to what I have in mind. Many other accounts of interpretations are compatible with what I argue below being true.

6.2 Instrumentalism and Representations of the Unobservable Wallace (2020b) hits the nail on the head when he notes that the interpretations of quantum mechanics which are typically labelled ‘realist’ are simply representational. Thus, put concisely, the questions I will address are as follows. Why would an anti-realist (such as an instrumentalist) prefer a representational theory to a nonrepresentational alternative with equal predictive power, when the representations in question concern unobservable entities? Why, that’s to say, would someone advocate a representational theory concerning a domain that they thought we could only have highly limited knowledge about? Let me begin by giving a direct but condensed answer. Representations of unobservable entities – things, systems, processes, and states of affairs – can play important epistemic roles even when they aren’t accurate (and even when there is nothing there to represent). First, they may make it easy for us to spot connections between phenomena, and thereby enable us to predict more readily how said phenomena interrelate. Second, they may also serve as vehicles for understanding how phenomena interrelate, or for giving us an orientation and a sense of familiarity, when employing a theory in a class of contexts. Third and finally, such representations may serve as heuristics for the development of new theories and models. Or so some philosophers of science and physicists have, as I will shortly show, argued. In particular, the notion that ‘understanding why’ is non-factive is explored in several recent monographs, including Elgin (2017), De Regt (2017), and Rowbottom (2019). And many pertinent papers have appeared in recent years. For instance, Reutlinger et al. (2018) provide an account of how toy (or extremely idealized) models can provide understanding, and Rancourt (2017) argues that introducing more falsehoods (or making models less representationally accurate) often promotes understanding. There is not an entirely united front among those who are inclined towards such views; for example, Elgin thinks understanding why is quasi-factive, whereas I do not. There is also some opposition to the very idea that understanding why is not factive.8 Yet the view is respectable, and I will summarise the case I make

8

See, for instance, Hills (2016) and Bird (Forthcoming).

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for it in Rowbottom (2019), while showing how it bears on Bohm’s interpretation, in the final section of this piece. Before I get to that, however, I should like to emphasise that some instrumentalists have advocated avoiding unobservable posits in science to some extent. Mach is a notable example, and his opposition to atomic theory is well known. But Mach allowed that sometimes it is useful to posit such unobservable entities. Consider the following passage: Even when the sound has reached so high a pitch and the vibrations have become so small that the previous means of observation are not of avail, we still advantageously imagine the sounding rod to perform vibrations . . . . [T]his is exactly what we do when we imagine a moving body which has just disappeared behind a pillar, or a comet at the moment invisible, as continuing its motion and retaining its previously observed properties . . . . We fill out the gaps in experience by the ideas that experience suggests. (Mach, 1893: 588)

In brief, Mach’s (1911: 49) view was that ‘[w]hat we represent to ourselves behind the appearances exists only in our understanding, and has for us only the value of a memoria technica or formula’. But one can disagree on two counts without renouncing instrumentalism, as I argue in detail in Rowbottom (2019). First, one may think that unobservable things probably do exist and that some of what we represent ‘behind the appearances’ may be correct, but just that it’s extremely difficult to know when we are right (because, for example, there are considerable limits to what we’ve conceived of at any point in time).9 Second, one may hold that achieving a sense of understanding is important, above and beyond saving the phenomena in an economical way, in science. Mach might even have been convinced of this with a little argument, in so far as he held that a significant task of science is to give us an ‘orientation’; see Rowbottom (2019: 113–116). It is also worth remembering the heated controversy, around the turn of the twentieth century, about how physics should be done. Duhem is especially notable for his critique of ‘English’ science – which is better dubbed ‘wrangler science’ (after the Cambridge wranglers who held so much influence in British science of the time) – and for his criticisms of the way these British scientists employed (typically abstract, but occasionally concrete) representational models. Poincaré had similar thoughts. Yet this wasn’t a battle concerning scientific realism because wrangler scientists didn’t take their models to accurately represent anything. In the words of Heilbron (1977: 41–43): [T]he representations were not meant or taken literally . . . . The same physicist might on different occasions use different and even conflicting pictures of the same phenomena . . . piecemeal analogies or provisional illustrative models.

These models weren’t considered to be optional extras. They were thought to be necessary for the reasons I mentioned above. So the kind of anti-realist view I’m bringing into focus in this paper – a view akin to cognitive instrumentalism – was exemplified in a key period of historic science. As Heilbron (1977: 42) puts it: 9

I have in mind here the argument from unconceived alternatives, presented by Stanford (2006) and extended by Rowbottom (2019: ch.3).

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[T]he principal English pedagogues of physics considered a theory incomplete without an accompanying model or analogy, ideally elaborated to the last detail. Such pictures, they believed, fixed ideas, trained the imagination, and suggested further applications of the theory . . . . This pedagogical bias was built into the training of British physicists.

I cannot recount the pertinent history covered in Rowbottom (2019: ch. 4) here. But I would emphasise that several leading physicists of the day discussed how models aid understanding. Kelvin (1884: 270), for instance, wrote: ‘If I can make a mechanical model, I understand it. As long as I cannot make a model all the way through I cannot understand . . . . I want to understand light as well as I can without introducing things that we understand even less of.’ Some physicists also wrote about the significance of analogies with mundane observable scenarios in discussing unobservable things or processes. Consider this charming passage from Fitzgerald (1888: 168–169): All reasoning about nature is . . . in part necessarily reasoning from analogy . . . . Notwithstanding the danger of our mistaking analogies for likenesses there is a great advantage in studying analogies . . . . [I]f the forms of energy were as familiar a conception as eggs and money, people would find it as easy to reason about its [sic] transformation as they are [sic] about the number of eggs the old woman brought to the market and sold at one dozen at 3 a penny and so forth.

This view was far from new. For example, Kepler wrote explicitly of his love for analogies – even suggesting that they hold all the secrets of nature – and used them extensively in his work.10 And even in modern psychology, one finds Dunbar (2002: 159) concluding, on the basis of his observations of scientists, that ‘[M]undane analogies are the workhorse of the scientific mind’. In summary, the (instrumentalist) view of representations that I hold has a significant precedent in the history of physics (and, as I will hint at below, of chemistry). In closing this piece, I will illustrate some of the functions of representational models from this perspective, with the aid of scientific examples. I will also show how Bohm’s interpretation performs some of those functions.

6.3 Functions of Representations of the Unobservable In this final section, I will show how representational models can be valuable in four distinct respects despite failing to be accurate: (1) in highlighting or revealing connections between phenomena; (2) in promoting understanding of how phenomena interrelate; (3) as heuristics for theory (or model) construction; and (4) in illustrating or revealing connections between theories (or laws or models). I will argue that Bohm’s interpretation is valuable in each respect only after providing less controversial examples of models performing these tasks.

10 For

more detail, see Gentner et al. (1997).

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6.3.1 Highlighting or Revealing Connections Between Phenomena Representational models can be particularly important in emphasising connections between phenomena. In Rowbottom (2019: 8–14), I provide an extended discussion of how the simple pendulum – which involves numerous idealizations, including that the motion is only two-dimensional, that no frictional forces are acting, that the gravitational field is uniform, and that the angle of swing is very small (such that the angle is approximately equal to the sine thereof) – is useful in this regard. (The simple pendulum is an impossible entity. It would swing perpetually.) For example, the model makes it evident how changes in pendulum length and changes in gravitational field strength tend to affect swing frequency; it’s clear that in summer a pendulum clock might run slower due to expansion in the rod bearing the bob (in the absence of some compensatory mechanism like a mercury-filled bob), and that it would also run slower on the moon than it would on Earth, ceteris paribus, for example. So how does Bohmian mechanics perform a similar function? As a precursor, note that I will follow Bohm (1952) in construing this in a ‘second order’ (Goldstein, 2021) fashion. So construed, Bohmian mechanics is similar to ‘Newtonian’ classical mechanics in its focus on forces acting on particles with definite positions. It merely introduces a new quantum variety of force, which is normally discussed in terms of ‘a quantum potential’; the associated guiding equation concerns how particles react to such forces. This has several striking consequences when it comes to making it evident how phenomena interrelate in ways not anticipated in classical mechanics (or indeed in special or general relativity). Most obviously, the picture is explicitly non-local, such that occurrences (like ‘measurements’) in one part of the universe might instantaneously affect states of affairs in other distant parts. For instance, getting a pointer reading of +1/2 after measuring the spin of one of an entangled pair of photons entails getting a pointer reading of −1/2 in a subsequent measurement of the spin of the other photon on the same axis. (Measurement breaks the entanglement.) Moreover, on a related note, the Bohmian view makes it clear how changing experimental setups can be expected to alter outcomes (in virtue, so the story goes, of changing the quantum potential). It is evident, for example, that placing a detector behind one slit, in a two slits experimental scenario, will prevent an interference pattern of the kind expected without the presence of such a detector.

6.3.2 Promoting Understanding of How Phenomena Interrelate Rather more controversially, representational models may also serve as vehicles for providing understanding of how phenomena interrelate even when they are highly inaccurate. It will be difficult for me to detail my account of understanding –

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developed in Rowbottom (2019: ch. 5) – in the present context. But I will give the briefest of outlines. Imagine you have identified several potential explanantia for some known explanandum. They are potential in the sense that none are known to be true or approximately so; and let’s also imagine, for simplicity’s sake, that all fail to be approximately true to the same extent. Isn’t there nonetheless a sense in which some of those explanantia may be better than others? I answer in the affirmative and propose that each might provide different degrees of understanding why. (Of course, thinking we could infer to the best explanation also involves answering in the affirmative. You might think of my position as being that we should prefer, but not infer to, the best available explanation in any given context.11) As we saw above, this view fits well with what the wrangler scientists thought. But what kind of understanding is at play here? Poincaré (1905: 182) saw it as follows: [Some hypotheses have] only a metaphorical sense. The scientist should no more banish them than a poet banishes metaphor; but he ought to know what they are worth. They may be useful to give satisfaction to the mind, and they will do no harm as long as they are only indifferent hypotheses. (Poincaré, 1905: 182)

Poincaré (1905: 171) also wrote that such hypotheses/models may help ‘by concrete images, to fix the ideas.’ And to cut a long story short, I believe that there is a connection between giving ‘satisfaction to the mind’, which is indicative of strongly mentally grasping, and fixing the ideas. ‘Concrete images’ – or visualizable models – often help with such grasping. So I think an important aspect of science is ‘empirical understanding’ (Rowbottom, 2019: ch.5), which involves subjective understanding of a neat (and normally not even approximately true) story that saves the phenomena (or more precisely, saves some specific class of phenomena to a desirable extent). Empirical understanding is useful above and beyond being able to predict successfully for the following reasons. First, it makes it easier for a subject to remember a model. Second, perhaps most importantly, it makes it easier for a subject to use said model. Third – in a more double-edged fashion – it bolsters the user’s confidence in the model as a means towards desired ends. Consider again the example of the simple pendulum and the factors that affect its period of swing. Due to its simplicity, the model provides a pellucid means by which to grasp some aspects of how many real pendulums behave. For instance, it makes it clear why changing the mass of the bob shouldn’t be expected to have any significant effect on swing period. Relatedly, it can also be mentally manipulated easily – in line with Wilkenfeld’s (2013) view of understanding – such that it’s easy to appreciate why increasing (or decreasing) the gravitational force would (tend to) decrease (or increase) the period, and so forth. Bohmian mechanics is excellent in this regard because it gives a visualizable or imaginable picture of some microprocesses, such as the two slits experiment (and similar scenarios). It also provides a sense of familiarity in so far as it connects well with the pictures in Newtonian mechanics and statistical mechanics; the system 11 I also accept that understanding can sometimes be achieved without anything resembling an explanation being present, e.g., following Lipton (2009), via a model such as an orrery.

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is deterministic, and the probabilities (and hence uncertainty relations) are present ultimately due to ignorance about the positions of particles. Measurement involves disrupting the system, such that one doesn’t always measure the way things were beforehand. And the path to the classical limit is conceptually easy to navigate; when the quantum potential ceases to be, then so does any non-locality, and the classical force laws just fall out. As Cushing (1994: 52) puts it: ‘Conceptually, a continuous passage from the microdomain to the macroworld is possible.’ Bowman (2002: 317) concurs, in his Bohmian treatment of the role that wave packets play in the simple harmonic oscillator: ‘By providing a valuable link with familiar classical physics, the Bohmian approach arguably can provide a better understanding of why quantum systems behave as they do.’ There is one other aspect of Bohmian mechanics that is preferable to the normal story one finds in physics textbooks yet is rarely commented on; spin is construed to be a feature of the way that the wavefunction acts on particles, rather than a rather mysterious ‘intrinsic angular momentum’.12 As Norsen (2014: 346) puts it: ‘so-called “contextual properties” (like the individual spin components in the pilotwave theory) are not [intrinsic categorical] properties at all.’ Naturally, however, some other parts of the Bohmian picture are less than ideal. For instance, as Cushing (1994: 47) notes, ‘we have no understanding of the physical origin of the highly nonlocal quantum potential’. A mitigating factor, he goes on to note, is that mysterious (instant) action at a distance is familiar from the Newtonian account of gravitational attraction. In any event, I am not arguing that Bohmian mechanics should be preferred for thinking about all, rather than many, aspects of quantum mechanics. And if one isn’t committed to the Bohmian picture being accurate, it is easy to use a different picture in some quantum mechanical applications.

6.3.3 As Heuristics for Theory (or Model) Construction Models may also serve as important heuristics. An especially interesting historical example concerns Hofmann’s introduction of concrete/material ‘ball and stick’ models of molecules in 1865, which came to be known as glyptic formulae. The short story – see Rowbottom (2019: 43–48) for a more detailed discussion – is that the spatial relationships and connections (‘bonds’) in the model were not intended to be taken literally. As Meinel (2004: 246–247) explains: Convinced as he was that symbolic notations in chemistry were purely formal tools that did not immediately correspond to reality, this approach explicitly avoided the question of truth . . . [Hofmann’s glyptic formulae] were not meant to represent the physical arrangement of the atoms. They rather supplied a pattern according to which the chemical operations of elimination and substitution could be classified and analogies found.

12 See Rowbottom (2019) for an argument that discourse concerning this cannot be taken literally. What would the distance involved in the moment be?

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To cut a long story short, however, it is plausible that these models and others developed subsequently – where, for instance, carbon bonds were instead arranged tetrahedrally to make the models easier to handle – affected the direction of chemistry. Most strikingly, the fact that double bonds did not allow rotation inspired Paternò (1869) to posit isomers of halogenated ethanes.13 Overall, the change was remarkably rapid. In 1866, Kolbe wrote that ‘graphic representations are . . . dangerous because they leave too much scope for the imagination . . . . It is impossible, and will ever remain so, to arrive at a notion of the spatial arrangement of atoms’ (Rocke, 1993: 314). But by 1874, when Le Bel and van’t Hoff both presented theories involving spatial relations playing active roles, ‘[s]tereochemists . . . believed they had access to the physical appearance of the molecule’ (Ramberg, 2003: 328). Similarly but less dramatically, Bohm’s theory has inspired further theoretical work, mainly focused on preserving a pilot-wave type picture in relativistic contexts; see Struyve (2011) for a summary of several attempts to employ it in quantum field theory. It is important to note that many of these approaches depart from the nonrelativistic Bohmian view in significant ways; for instance, some take fields rather than particles as fundamental in their ontology, and some of those that take particles as fundamental are not deterministic.

6.3.4 Illustrating or Revealing Connections Between Theories (or Laws or Models) A final way in which models may be valuable is in illustrating connections between theories or laws (and thereby promoting understanding of how these interrelate and might be conjoined or simultaneously applied). Take the planetary model of the solar system normally associated with Rutherford – although he was not the first to propose it14 – as a simple case in point. This not only highlights the similarities in the force laws governing gravitational and electrostatic attraction, but also makes it apparent how both kinds of force could be present simultaneously in a ‘solar system’ type arrangement. Some such highlighting isn’t of much import in this case. For example, the similarity between the force laws may be self-evident, especially if Coulomb’s law is thought of in terms of the Coulomb constant. Yet even then, the model might cause a user to reflect on why 4π is present when Coulomb’s law is

13 In fact, Crum Brown presented the familiar graphical way of depicting molecules just 1 year before Hofmann first used a glyptic formula. But the glyptic model more obviously suggests the idea that carbon-carbon double bonds (or connections) might restrict motion in a way that single bonds do not. 14 Forerunners include Colding, Fecher, and Weber. See Kragh (2012: 3–5) and Rowbottom (2019: 90).

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expressed in terms of the permittivity of free space, and thereby come to consider why the formula for the surface area of a sphere is present.15 Bohm’s theory also isn’t especially remarkable in this regard, although it does, for reasons I’ve already covered, fit nicely with Newtonian classical mechanics (in so far as classical behaviour falls out in the absence of quantum forces). Thus, it shows how quantum mechanics and classical mechanics may connect. Inter alia, as I have previously hinted at, it also provides an interesting means by which to reflect on the role of probabilities in statistical mechanics. As Callender (2007) puts it: ‘quantum mechanics is to Bohmian mechanics roughly as statistical mechanics is to classical mechanics’.

6.4 Conclusion I have shown why an anti-realist, such as a cognitive instrumentalist, might prefer Bohmian mechanics to alternatives in several contexts. It may be construed as a pleasing fiction, which provides an excellent way to think about, and anticipate, many phenomena falling under the remit of quantum mechanics. It is also fertile, in so far as it prompts further representational model/theory construction. Acknowledgements My work on this paper was supported by a GRF Grant from Hong Kong’s RGC (‘Scientific Progress: Foundational Issues’) and the Center for Foundations of Science, University of Pittsburgh (via a Visiting Fellowship). I am also grateful to David Wallace and two anonymous referees for comments on an earlier version.

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15 The

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Reutlinger, A., Hangleiter, D., & Hartmann, S. (2018). Understanding (with) toy models. British Journal for the Philosophy of Science, 69, 1069–1099. Rocke, A. J. (1993). The quiet revolution: Hermann Kolbe and the science of organic chemistry. University of California Press. Rowbottom, D. P. (2011a). Kuhn vs. Popper on criticism and dogmatism in science: A resolution at the group level. Studies in history and philosophy of science (Part A), 42, 117–124. Rowbottom, D. P. (2011b). The instrumentalist’s new clothes. Philosophy of Science, 78, 1200– 1211. Rowbottom, D. P. (2014). Aimless science. Synthese, 191, 1211–1221. Rowbottom, D. P. (2019). The instrument of science: Scientific anti-realism revitalised. Routledge. Rowbottom, D. P. (2021). A methodological argument against scientific realism. Synthese, 198, 2153–2167. Rowbottom, D. P. (Forthcoming). Scientific progress. Cambridge University Press. Squires, E. J. (1994). The mystery of the quantum world (2nd ed.). CRC Press. Stanford, P. K. (2006). Exceeding our grasp: Science, history, and the problem of unconceived alternatives. Oxford University Press. Struyve, W. (2011). Pilot-wave approaches to quantum field theory. Journal of Physics: Conference Series, 306, 012047. Wallace, D. (2019). What is orthodox quantum mechanics? In A. Cordero (Ed.), Philosophers look at quantum mechanics (pp. 285–312). Springer. Wallace, D. (2020a). Lessons from realistic physics for the metaphysics of quantum theory. Synthese, 197, 4303–4318. Wallace, D. (2020b). On the plurality of quantum theories: Quantum theory as a framework, and its implications for the quantum measurement problem. In S. French & J. Saatsi (Eds.), Scientific realism and the quantum (pp. 78–102). Oxford University Press. Wilkenfeld, D. A. (2013). Understanding as representation manipulability. Synthese, 190, 997– 1016.

Part II

Ontology

Chapter 7

Beables, Primitive Ontology and Beyond: How Theories Meet the World Andrea Oldofredi

Abstract Bohm and Bell’s approaches to the foundations of quantum mechanics share notable features with the contemporary Primitive Ontology perspective and Esfeld & Deckert minimalist ontology. For instance, all these programs consider ontological clarity a necessary condition to be met by every theoretical framework, promote scientific realism also in the quantum domain and strengthen the explanatory power of quantum theory. However, these approaches remarkably diverge from one another, since they employ different metaphysical principles leading to conflicting Weltanschaaungen. The principal aim of this essay is to spell out the relations as well as the main differences existing among such perspectives, which unfortunately remain often unnoticed in literature. Indeed, it is not uncommon to see Bell’s views conflated with the PO programme, and the latter with Esfeld and Deckert’s proposal. It will be our task to clear up this confusion. Keywords Quantum mechanics · David Bohm · John Bell · Local beables · Primitive ontology

7.1 Introduction Quantum Mechanics (QM) is one of the most efficient descriptions of the inherent structure of matter and its behavior. As is known, however, this theory represents elementary objects in a way that drastically changed our conception of reality, speaking about intrinsically indeterminate systems, non-local interactions, waveparticle dualities, etc.. Indeed, it is still a pivotal philosophical challenge to understand what scientific image of the world it provides—i.e. to comprehend its ontology, and how the macroscopic realm can be explained from it. Referring to this, it is worth noting that quantum theory is affected by several conceptual and technical conundra; hence, we cannot consider it the final word about ontological

A. Oldofredi () Section de Philosophie, Université de Lausanne, Lausanne, Switzerland © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_7

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matters, notwithstanding its empirical success. In particular, scholars agree that QM contains physically ill-defined notions within its axioms, and inconsistencies among its dynamical laws—resulting in the (in)famous quantum measurement problem. Against this background, the Primitive Ontology (PO) approach has been advanced to overcome these issues. Today, in fact, we have several PO theories that resolve the quantum puzzles and that can be considered serious alternatives to the standard formulation of QM, as for instance Bohmian Mechanics (BM), dynamical reduction models as the Ghirardi-Rimini-Weber (GRW) theories, etc.. Although the expression “primitive ontology” has been introduced in Dürr et al. (1992), the roots of this programme can be found in Bohm and Bell’s foundational works. Following their steps, the modern formulation of the PO perspective requires that well-defined physical theories explain the macroscopic regime through a more fundamental, microscopic picture, postulating a clear primitive ontology—i.e. specifying a set of theoretical entities representing real objects moving in physical space—and a consistent dynamics. Moreover, elaborating on the notion of primitive ontology, Esfeld and Deckert (2017) extended and modified this concept proposing an atomistic ontology that is taken to be valid at every energy/length scale between the classical regime and the quantum field theoretical level. Bohm and Bell’s approaches share notable features with the modern PO perspectives; for instance, they consider ontological clarity a necessary condition to be met by every theoretical framework, promote scientific realism also in the quantum domain and strengthen the explanatory power of quantum theory.1 However, these positions remarkably diverge from one another, since they employ different metaphysical principles leading to conflicting Weltanschaaungen. The principal aim of this essay is to spell out the relations and the main differences existing among such perspectives, which unfortunately remain often unnoticed in literature. Indeed, it is not uncommon to see Bell’s methodological approach conflated with the PO programme, and the latter with Esfeld and Deckert’s views. It will be our task to clear up this confusion. The paper is organized as follows: Sect. 7.2 analyzes Bohm’s reflections on the nature of theoretical entities and physical laws, taking into account also his metaphysical infinitism. Section 7.3 focuses on Bell’s theory of local beables. In Sect. 7.4 the modern formulation of the PO programme is presented underlying the main metaphysical and methodological differences w.r.t. Bohm and Bell’s approaches. Section 7.5 introduces Esfeld and Deckert’s minimalist ontology and illustrates how this perspective diverges under important respects with all the previous proposals. Finally, Sect. 7.6 concludes the paper.

1

I assume that the reader has some familiarity with this literature.

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7.2 David Bohm, Pluralism and Infinitism Ontological clarity is an essential feature of physical theories according to Bohm. On his view the problems of the standard interpretation of QM come from its obscure metaphysical content. Bohm was deeply unsatisfied by the lack of explanation of measurement results, since quantum theory excludes a precise characterization of individual systems as well as an accurate description of their dynamical evolution “without proving that such a renunciation is necessary” (Bohm 1952, 168). Furthermore, he argued that the empirical robustness of QM and its contingent mathematical structure are not sufficient to exclude a priori other ontologically clearer formulations. To tame these issues, Bohm joined the foundational debate proposing an alternative interpretation of QM where quantum particles have definite positions and velocities, and are dynamically guided by a real ψ-field, challenging the metaphysical indeterminacy of the standard formulation. Hence, postulating a dual ontology of particles and fields, he showed that an individual description of quantum systems following continuous trajectories in space was not only mathematically possible, but also physically consistent. Moreover, this theory is able to explain measurement outcomes without invoking mysterious collapses of ψ, strengthening the explanatory power of quantum theory. In addition, Bohm extended his approach to electromagnetism employing fields—different from ψ—as fundamental entities, showing that he was open to implement various ontologies in different theories (cf. Appendix A of Bohm (1952)). The common trait between such theories is their metaphysical clarity, since Bohm always specified which variables represent matter and how they dynamically behave. In addition, he claimed that QM—as any other theory—has a limited validity and that “at distances of the order of 10−13cm or smaller and for times of the order of this distance divided by the velocity of light or smaller, present theories become so inadequate that it is generally believed that they are probably not applicable” (Bohm 1952, footnote 6). At these regimes he expected that new ontologies and new theories will be discovered. More generally, Bohm thought that physical theories have limited domains of application at precise length/energy scales, and remarked that what is measured at a particular level—thereby what exists at that scale— depends on the theory at hand: our epistemology is determined to a large extent by the existing theory. It is therefore not wise to specify the possible forms of future theories in terms of purely epistemological limitations deduced from existing theories (Bohm 1952, 188).

In his view this is a direct objection against the empiricist basis of QM, which follows the positivistic principle of not accepting entities that cannot be currently observed. According to Bohm, this is a poor working hypothesis, since the history of physics showed the fruitfulness to assume the existence of certain items before their empirical discovery, and the atomic theory is a pivotal example. On the other hand, he rejected another perspective, later called “mechanistic philosophy”, for which reality can be fully explained starting from a fixed set of entities, and a

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restricted set of laws—something close to what philosophers call foundationalism.2 He warns us not to expect such a knowledge “because there are almost certainly more elements in existence than we possibly can be aware of at any particular stage of scientific development. Any specified element, however, can in principle ultimately be discovered, but never all of them” (Bohm 1952, 189). This is an hint of the metaphysical infinitism endorsed in Causality and Chance in Modern Physics to which we now turn. In Bohm (1957) we find detailed objections against mechanistic philosophy. Firstly, history of physics disconfirms the basic tenets of this view, since the revolutions that occurred from Newton to this day radically changed the entities and the laws of our theories. Moreover, future frameworks will be as revolutionary as QM was w.r.t. classical physics. Secondly, the assumptions concerning the final character of any particular ontology are neither necessary, nor provable, because future theories may demonstrate its limited validity.3 Finally, foundationalism contravenes the scientific method, since the latter imposes that every object and law must be continuously subjected to verification. This process of testing may end up in contradiction with new discoveries or new domains of science. Looking at how physics evolved, says Bohm, such contradictions not only systematically appeared, but also led to a deeper comprehension of the world. Contrary to such a view, he proposes a form of infinitism. In essence, Bohm stated that physical sciences and experimental data push us to a conception of nature composed by an infinity of different entities, which do not depend ontologically on a fixed set of absolutely fundamental objects (Bohm 1957, 91). According to this view of science, physical theories do not always lead us closer to a fundamental ground, but instead show the infinite complexity of nature. Furthermore, he believes that empirical data cannot a priori provide any justification to metaphysical restrictions concerning a particular set of entities to be chosen as absolutely ontologically independent. On the contrary, conforming to Bohm’s infinitism, scientific practice always discloses new entities, laws and phenomena which contribute to our continuous, never-ending process of understanding the structure of reality. However, although Bohm denied the existence of a fundamental level, he firmly believed that every theory must be ontologically unambiguous in its domain of application. Therefore, theoretical frameworks must provide a clear ontology to be applied at the relevant energy/length scale, meaning that the entities forming the basic ontology of a given theory can be considered relatively fundamental. Referring

2 In Sect. 7.5 we will see that Esfeld and Deckert (2017) proposed a view similar to the mechanistic philosophy described by Bohm. It should be noted, however, that according to these authors physical laws can change in time in order to accommodate the discovery of new phenomena. 3 Referring to this Bohm writes: “Newton’s laws of motion, regarded as absolute and final for over two hundred years, were eventually found to have a limited domain of validity, these limits having finally been expressed with the aid of the quantum theory and the theory of relativity” (Bohm 1957, 90).

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to this, he stated that [a]ny given set of qualities and properties of matter and categories of laws that are expressed in terms of these qualities and properties is in general applicable only within limited contexts, over limited ranges of conditions and to limited degrees of approximation, these limits being subject to better and better determination with the aid of further scientific research (Bohm 1957, 91).

Thus, Bohm claimed that a well-defined physical theory should provide a clear ontological picture for the domain in which it is a reliable description of physical phenomena. Nonetheless, its ontology can be substantially modified with the progress of scientific research. This certainly exemplifies Bohm’s scientific pluralism and his heterodox metaphysical views w.r.t. to the dominant paradigm towards the interpretation of QM (cf. van Strien 2019). As we can see, the seeds of the PO programme—i.e. the requirements of ontological clarity and explanatory robustness—have been sowed. They will germinate in Bell’s theory of local beables which we are going to discuss.

7.3 Local and Non-local Beables It is well-known that Bell was disappointed by the lack of ontological clarity in QM, which in his opinion was the source of its inconsistency and redundancy. Inconsistency because the dynamical laws of the theory contradict each other;4 redundancy because some central notions contained in its axioms, e.g. observation and observable, should be derived from more fundamental concepts, the beables of the theory: [t]he concept of ‘observable’ lends itself to very precise mathematics when identified with ‘self-adjoint operator’. But physically, it is a rather woolly concept. It is not easy to identify precisely which physical processes are to be given the status of ‘observations’ and which are to be relegated to the limbo between one observation and another. So it could be hoped that some increase in precision might be possible by concentration on the beables, which can be described in ‘classical terms’, because they are there (Bell 1987, 52).

According to Bell, a well-defined physical theory T must postulate a clear ontology, or in his jargon, a set of local beables. These are the theoretical entities of T referring to real objects ascribed to bounded regions of space-time, i.e. they correspond to the elements of reality that the theory postulates to exist independently on any observation. It is worth noting that the beables cannot be derived from other more fundamental notions of T ; hence, they establish what is fundamental in the context of T.

4

These laws are the Schrödinger equation and the collapse postulate.

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Moreover, together with dynamical laws,5 the beables explain the physical phenomena lying within the domain of application of the theory under consideration, connecting its formal structure to the macroscopic world accessible to us: [t]he beables must include the settings of switches and knobs on experimental equipment, the currents in coils, and the reading of instruments. ‘Observables’ must be made, somehow, out of beables. The theory of local beables should contain, and give precise physical meaning to, the algebra of local observables (ibid.).

But how do we select a particular set of beables for a theory? In answering this question Bell’s pluralist attitude becomes manifest, since he claimed that the only necessary requirements for an ontology is to state explicitly which variables represent matter, and to provide unambiguous explanation of macroscopic observations.6 These liberal requirements do not impose any strict limitation to the selection of a certain ontology. Such a freedom of choice is justified because (1) there are always several adequate options to account for the macroscopic data that a certain theory must explain, (2) physical theories have a provisional character. Therefore, according to Bell, there is no one-to-one relation between the beables of a theory and the physical world, meaning that we cannot definitively establish whether a certain ontology is the correct, ultimate description of reality. To this regard, Bell used “the term ‘beable’ rather than some more committed term like ‘being’ or ‘beer’ to recall the essentially tentative nature of any physical theory. Such a theory is at best a candidate for the description of nature. Terms like ‘being’, ‘beer’, ‘existent’, etc., would seem to me lacking in humility. In fact ‘beable’ is short for ‘maybe-able’ ” (Bell 1987, 174). Indeed, there are various examples of beables implemented in different classical and quantum theories, as for instance electromagnetic fields, point particles, matter density fields, strings etc..7 Bell himself, although he was a supporter of the pilotwave theory (Bell 1987, Chapter 17), proposed a rival picture for non-relativistic QM, namely a GRW theory with a flash ontology (ibid., Chapter 22). Moreover, he extended Bohm’s approach to Quantum Field Theory (QFT) implementing an ontology of fermion number density (ibid., Chapter 19). Such beables have no classical analogues, implying an ontological discontinuity between the classical and the quantum regime; however, such a discontinuity was tolerated by him. Similarly, Bell’s approach allows that a single theory can postulate the existence of different

5

Remarkably, according to Bell in the context of a given theory T there is a distinction between physical and non-physical entities: the former denote the beables of the theory, the latter are the mathematical structures needed to formulate dynamical laws for the variables representing matter in space; cf. (Bell, 1987, Chapter 7). 6 To this regard Bell writes that “what is essential is to be able to define the positions of things, including the positions of pointers or (the modern equivalent) of ink of computer output” (Bell 1987, 175). Cf. also (Bell, 1987, Chapter 5). 7 Maudlin (2016) underlines that a similar choice is available also for the specification of the spacetime structure of a given theory.

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kinds of objects, i.e. it can provide a multi-category ontology, as for instance in classical electromagnetism, where we find both point particles and fields. Thus, the theory of local beables does not require that reality must be reduced to a unique category of physical entities. These facts certainly denote his liberal views about the ontology of physical theories, in close analogy with Bohm’s pluralist perspective. Finally, it is interesting to note the Bell admitted the existence of non-local beables.8 This fact shows that according to him, the general category of what exists is divided into two subsets: local and non-local beables. The former must be defined in 3dimensional space and associated with bounded portion of spacetime; while the latter can be either defined in high-dimensional spaces or in the usual 3D space. For instance, he considered the state vector defined in configuration space as a proper beable in his essay Beables for quantum field theory. Another important example is given by the multi-field interpretation of the ψ field in Bohm’s theory. According to this approach, given a N-particle system one projects the ψ values defined on configuration space into multi-field values in 3-dimensional space, i.e. one associates a particular field value not with individual points (as one would do with usual fields), but with N-tuples of points. In this manner, the multi field determines the motion of a configuration of particles in physical space. This new object is a beable because it is considered a real physical item living in 3-dimensional space, and it is non-local since its value is specified for a configuration of N points and not for a single point.9 In sum, from our discussion we can say that Bohm and Bell’s10 reflections concerning the ontology of physical theories led to the following ideas: • the beables of a theory T are fundamental in the context of the framework under consideration, i.e. they may be non-fundamental in another, deeper theory; • the beables must explain all the physical phenomena lying within the domain of application of T ; • several ontologies can be proposed to explain the same set of physical phenomena; • it is possible to implement different beables within the same theoretical framework postulating a multi-category ontology; • beables can be non-local and defined either in 3D space or in higher-dimensional spaces; • finally, it is possible to implement different beables at different energy/length scales (scientific pluralism).

8

Cf. Bell (1987, 53). For details on this particular perspective cf. Hubert and Romano (2018). Clearly, such type of beables have been admitted also by Bohm himself since he explicitly considered the wave function a real physical field, as said in the previous section. Another recent example of non-local beables can be found in Smolin (2015). 10 It is worth stressing that Bell did not explicitly endorsed a metaphysical infinitism as Bohm did. Nonetheless, the similarities between their views are evident. 9

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Since the PO programme evolved from the work of these physicists, let us then see what has been kept and what has been left behind.

7.4 The Primitive Ontology Approach The PO approach is a normative perspective about the construction of physical theories since it provides a set of requirements that any theoretical framework should met to be considered empirically and metaphysically consistent. Indeed, according to the proponents of this view, well-defined theories share a common architecture deriving from these constraints.11 Following Bell,12 also in this programme the mathematical structure of a theory T is divided into two subcategories. Firstly, there are “primitive” variables provided with a physical meaning; they represent matter and refer to real objects precisely localized and moving in 3-dimensional space (or in space-time), which is generally considered a real substance as well. These are the PO of the theory, the fundamental entities postulated by T which constitute the building blocks of macroscopic reality. Remarkably, the PO defines the observable quantities of T — i.e. the properties of physical systems—and its symmetries, since it establishes which entities remain invariant under symmetry transformations.13 On the other hand, T contains mathematical structures that are responsible for the dynamical evolution of the primitive ontology, without representing material objects—the nonprimitive (or nomological) variables (cf. footnote 5). An example is given by the wave function in QM: in every PO theory ψ is not considered a physical substance, but rather an essential mathematical tool useful to formulate empirically adequate laws for the dynamical history of the primitive ontology, similarly to the parameters of mass and charge.14 Furthermore, the PO has explanatory power since the macroscopic reality is metaphysically dependent on the primitive ontology of a particular theory and

11 Here

I follow the modern presentation of this perspective contained in Allori (2013) and Allori (2015). NB: other proponents of the PO approach as for instance R. Tumulka, S. Goldstein or N. Zanghì may have different opinions concerning this programme and its main goals. For details about the common structure of PO theories see Allori et al. (2008). 12 For the sake of conceptual accuracy, it should be stressed that the PO approach not only is rooted in Bohm and Bell’s works, but it is also inspired by Einstein, de Broglie and Schrödinger’s reflections on and critiques to the ontology of QM. These physicists, indeed, explicitly rejected the instrumental philosophy of standard quantum theory and tried to provide an unambiguous metaphysical picture for this theoretical framework. Given the space limit, here I cannot elaborate of their influence on the PO approach. However, proponents of this perspective in several occasions—as for instance Nino Zanghì in many conversations and Valia Allori (private communication)—emphasized their influence. 13 For technical details see Allori et al. (2008). More on this later. 14 Arguments against wave function realism can be found in Allori (2013), Allori (2015), and Allori (2021).

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its dynamical laws. This dependence relation can be intuitively characterized as mereological composition: the entities of a PO interact and form nuclei and atoms, which in turn aggregate into molecules and more complex biological organisms, arriving eventually to the macroscopic realm.15 For instance, according to BM, GRWm or GRWf, the macroscopic regime is literally composed by particles, matter density fields or flashes respectively, meaning that such PO theories explain the existence of macroscopic objects in terms of the motion and interaction in space of their different mereological primitives. Thus, the properties and the behavior of macroscopic objects depend on—i.e. are reduced to—the properties and interactions of their fundamental constituents. Clear examples of these explanations are macroscopic measurement outcomes. For instance, in BM measurements of spin are explained via—and reduced to—particles’ positions and their dynamical evolution as shown in Bell (1987, Chapter 17) and Dürr et al. (2004b). This example is generalizable (1) to every other PO theory and (2) to every observable, since these frameworks provide rigorous descriptions of the physical processes taking place in measurements situations, and then also detailed explanations of the obtained results.16 All these are remarkable virtues, because observation constitutes the only connection between theory and experience. To conclude this brief presentation, it is worth stressing that the PO approach puts very few constraints on the selection of the primitive variables: the latter must be microscopic, since they must explain the macroscopic regime, and located in 3-dimensional space.17 Referring to this, indeed, Allori states that there is no rule to determine the primitive ontology of a theory. Rather, it is just a matter of understanding how the theory was introduced, it has developed, and how its explanatory scheme works (Allori 2013, 65).

The definition of the primitive variables of a given theoretical framework depends on the metaphysical assumptions used to construct it, and may vary from a theory to another, as witnessed by the several PO theories available. However, the primitive ontology is never chosen a posteriori, i.e. it is not read off from the mathematics of a theory. Rather, the PO itself provides an interpretation of the formalism: in the process of theory construction, a scientist selects a particular PO and will use the appropriate mathematical structures to implement her choice. Let us now turn to the differences between the PO programme w.r.t. the previous approaches. To this regard, Allori (2013, 2015, 2021) stress two related crucial

15 The exact rules of mereological composition as well as the dependence relation based on them have not yet been investigated in detail in the context of PO. This will be the subject for future research. 16 For technical details cf. Dürr et al. (2004b) and Goldstein et al. (2012). 17 The reader may refer to Allori (2015, Sections 4 and 5) for arguments in favor of the requirement of “microscopicality” and 3-dimensionality.

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points:18 firstly, not every local beable is necessarily a PO. For instance, electromagnetic fields in classical electrodynamics can be interpreted as nomological variables instead of primitive ones given their role in the theory: not only there is asymmetry between particles and fields—i.e. the particles can generate fields, but not vice versa—but also the latter act as mediators of particles’ interactions. These facts provide indications that fields are not fundamental entities. Consequently, E and H are not part of the PO, i.e. they do not represent matter in space, but describe how particles move. Thus, they are part of the explanatory machinery of the theory. Hence, some local beables can be nomological, while PO must be exclusively material. Secondly, if every local beable of a given theory T is considered part of the PO, then T may lose symmetry properties. A simple example is provided by the symmetry of time reversal in electromagnetism: if E and H would be real physical entities, one would expect that, under time-reversal, they would still represent possible ways in which fields may be. However, the transformations E(t) → E(−t) and H(t) → H(−t) are not solutions of the Maxwell’s equations. On the contrary, particles’ trajectories are invariant under time reversal. Thus, if E and H are considered part of the PO, then classical electromagnetism would lose the symmetry of time-reversal. Hence, Allori concludes the PO preserves the symmetries of the theory, while sometimes local beables do not.19 In addition, one can also find other important differences between these perspectives. Firstly, a given PO theory T defines a set of entities which are considered fundamental tout court, since they are considered the building blocks of nature. According to the PO perspective, indeed, our macroscopic realm is completely ontologically reduced to the primitive variables of T and its dynamical laws, which are considered the fundamental elements of reality.20 On the contrary, Bohm stated that a particular ontology should be valid and reliable only within the domain of application of a given theory. Similarly, Bell stressed that we may have two different sets of local beables at different energy/length scales, implying that a certain theory T postulate an ontology which is theory dependent and only relatively fundamental. Secondly, the PO approach does not admit different ontologies at diverse scales.21 Indeed, if either GRWm or GRWf were the correct description of the

18 To

my knowledge Allori is the first that pointed clearly out the differences between the PO and local beables. In literature, however, such differences are often neglected and these approaches are frequently conflated. See for instance Esfeld (2014) or Tumulka (2016). 19 This is another elegant argument against the reality of fields. More details are given in Allori (2021, Sections 4.2 and 4.3). 20 To this regard, Allori writes that in the PO approach “macroscopic objects are thought to be fundamentally composed of the microscopic entities the PO specifies. As such, the PO approach is (ontologically) reductionist, at least to the extent that it allows to make sense of claims like the PO being “the building blocks of everything else”, and of the idea that macroscopic regularities are obtained entirely from the microscopic trajectories of the PO” (Allori 2018, 71–72). 21 If BM would be the correct description of reality at a fundamental level, we would have ontological continuity between the classical and the quantum scale. Referring to this, not only

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world, the non-fundamental particle-like character of classical mechanics would emerge and explained in terms of the fundamental GRW PO. However, since we do not have direct access to the fine-grained nature of the primitive variables, we can consider the flashes/matter fields as if they were particles for explanatory purposes, recovering the appearance of classical corpuscles: [a]t the level of microphysics we may have flashes or a continuous field, but at some mesoscopic level they produce trajectories as if they are produced by particles. So, even if the microscopic PO is not one of particles, there is a mesoscopic scale in which they behave as if they are in the sense that from that level up to the macroscopic level the explanation is the same as if they were particles (Allori 2018, 73).

In this case, then, we would know that at the fundamental level the PO is not corpuscular, even though we would keep talking of particles for explanatory aims. Clearly, Bohm and Bell pluralist attitude seems to be lost in favor of a foundationalist perspective. Consequently, infinitism is abandoned as well. Thirdly, contrary to the multiple-category ontologies endorsed by Bohm and Bell, PO theories explain our manifest image of the world postulating exclusively a one-category ontology, i.e. the macroscopic regime is reduced to a single type of physical objects and their evolution. However, this reduction may be noneliminative: for instance, as state above, in the case a GWR-type theory would postulate the correct PO, particles would be still considered an effective description of reality from a mesoscopic to a macroscopic scale. Thus, for all explanatory purposes a classical particle ontology can be maintained. Finally, as we have seen in the previous sections, Bohm and Bell conceived the existence of non-local beables, whereas in the PO perspective these cannot be considered material entities. Thus, they cannot be part of the primitive ontology of a theory, and must be included within its nomological variables. In sum, if for Bell and Bohm what physically exists can be defined in terms of local and/or non-local beables, the PO can only be a subset of the local beables. Thus, the PO programme imposes stricter criteria to determine the fundamental elements of a theory w.r.t. the preceding approaches. In this section we saw that the PO approach diverges from the views of its pioneers, since it employs different metaphysical assumptions and methodological principles that lead to diverse pictures of reality. Thus, we should be aware of these dissimilarities not to conflate these approaches. To conclude the essay, let us now consider the most recent development of the PO programme.

Allori is inclined to postulate particles as the fundamental PO, but also she argues that the ontological discontinuity between classical mechanics and GRW theories is an argument against the latter (cf. Allori 2018). Clearly, I cannot speak for other proponents of this programme.

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7.5 Beyond the PO: Minimalism and Fundamental Ontology Starting from the modern formulation of the PO programme, Esfeld and Deckert radicalized the notion of primitive ontology eliminating the theoretical dependence of the PO: “to our mind, it is inappropriate to speak of the ontology of this or that physical theory. Ontology is about what there is” (Esfeld and Deckert 2017, 12–13). Hence, they move from the notion of “primitive ontology of a theory T ” to ontology tout court, developing their metaphysical proposal independently of any particular theoretical framework. To achieve this result, the authors introduce a background independent atomistic ontology which seeks to recover the predictions of classical mechanics, QM and QFT via the definition of (1) different dynamical laws to be applied at the relevant energy/length scales, and (2) appropriate typicality measures.22 More specifically, they formulate a relationalist ontology of propertyless, finite and permanent matter points uniquely individuated by their mutual distance relations—the only property of these points is position. The change of these relations, which is taken as a primitive fact, constitutes the dynamical aspect of the proposal. Referring to this, the authors suggest a general dynamics of matter points in terms of the first derivative of a given configuration  with respect to time t: vt (t ) =

d t , ∀t ∈ R. dt

Clearly, to describe actual modifications of distance relations at various length scales, one has to provide the correct laws of motion. Here parameters as masses, charges, fields, potentials etc. enter into the scene: they are useful mathematical tools used to implement the evolution of matter points, consequently they are not part of the PO but have an essential explanatory function. An explicit example is provided by the velocity field appearing in the guidance equation of BM (cf. Esfeld and Deckert 2017, equation 3.6), as well as those of classical mechanics and QFT (cf. equations 3.3 and 4.4 respectively). Indeed, following this metaphysical project, it is always the dynamics that changes, while the ontology of matter points remains unaltered in every theory change from the classical regime to QFT—and possibly beyond it in more fundamental theories as e.g. quantum gravity. Such an ontology, thus, is primitive in a new sense: it is the fundamental ontology of our physical world. These matter points are (1) ontologically independent from any other entity, and (2) constitute the mereological basis of every object in the universe at every scale. Hence, contrary to Bohm and Bell approaches, Esfeld and Deckert’s project excludes the possibility to have different ontologies at different energy/length scales. Here the PO is chosen a priori in virtue of the guiding metaphysical principle of ontological parsimony. This ontology, in fact, is the simplest possibility to explain our macroscopic reality

22 For technical and metaphysical details of this proposal

see Esfeld and Deckert (2017, Chapter 2).

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providing the minimal ingredients to recover spacetime and matter in motion (once the correct dynamical equations are given).23 This proposal captures the idea of fundamentality expressed in terms of wellfoundedness (cf. Tahko 2018): a dependence chain is said to be well-founded if and only if it terminates with entities which do not ontologically depend on other items. In the case of Esfeld and Deckert’s project, matter points close the ontological dependence chain, and constitute the ultimate ingredients of reality. Well-foundedness, in turn, is closely related to metaphysical foundationalism, which can be defined in full generality by saying that every non-fundamental entity is dependent on some fundamental item—in this case matter points—that fully account for its being and reality. Hence, Esfeld and Deckert’s project expresses the idea that reality has an ultimate foundation of matter points to which everything else is ontologically reduced—the notion of a relative fundamentality as conceived by Bohm and Bell’s is no longer maintained.24 Let me conclude this section underlying other important differences w.r.t. the PO programme. Firstly, according to Esfeld and Deckert, space-time and its geometry emerge from the configuration of matter points, contrary to the substantivalist perspective of the PO approach, where the primitive ontology of a theory is interpreted as a decoration of space.25 Secondly, while the PO perspective allows the postulation of several primitive ontologies, these authors accept the existence only of particles since in their opinion (1) almost all the evidence coming from the most advanced experimental research is given in corpuscular terms, and (2) atomism can be retained not only at the classical level, but also in QM and QFT in virtue of the existence of BM and Bohmian QFTs. Thus, they exclude every non-particle primitive ontology. Interestingly, if the PO is a subset of the local beables, then such an atomistic ontology is a subset of the possible POs; hence, Esfeld and Deckert’s project imposes stricter criteria to determine what is real w.r.t. the PO approach. Thirdly, these authors postulate that the number of matter points remain constant in time. This is an important difference w.r.t. the PO programme where theories with a variable number of entities have been given. For instance, the number of particles may not be constant in time, as we can see in particular Bohmian QFTs. This fact leads to significant differences when QFT and its phenomenology is taken into account.26

23 Esfeld and Deckert argue that parsimony has been a guiding principle in both philosophy and physics in their historical evolution. Indeed, given a class of theories explaining the very same set of physical phenomena, we prefer the one(s) able to explain them with the lowest number of entities and laws. 24 It is obvious that this project rejects Bohm’s infinitism and his objections against a foundational metaphysics. 25 This expression is used explicitly in Allori et al. (2008). 26 In Dürr et al. (2004a) it is postulated an ontology with a variable particle number. For a discussion of the various Bohmian QFTs with a particle ontology cf. Oldofredi (2018).

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7.6 Conclusion Starting from Bohm’s infinitism and Bell’s theory of local beables, we arrived to the PO programme and its most recent development, i.e. Esfeld and Deckert’s minimalist ontology. Although the initial steps of this journey are in open contradiction w.r.t. the latest works proposed within the PO community, there is still some confusion in literature, where it is not uncommon to conflate Bell’s methodological approach with the PO programme, and the latter with Esfeld and Deckert’s views. Thus, in this essay I underlined the main differences among these projects. In particular, we saw that Bell’s methodological approach shows important similarities with Bohm’s ideas, sharing the belief about the provisional character of physical theories and a pluralist attitude towards their ontology. Then the PO programme has been presented emphasizing some important differences w.r.t. Bell’s and Bohm’s views. Elaborating on Allori’s works, I have pointed out new divergences between these two schools of thought. Finally, I considered Esfeld and Deckert’s minimalist ontology, showing (1) the radical difference of this project w.r.t. Bohm and Bell’s views, and (2) its subtle dissimilarities with the tenets of the PO programme. In conclusion, I hope to have clarified at least some of the differences existing among such views, and to have shown to the reader that these approaches employ divergent metaphysical and methodological principles leading to contrasting pictures of reality. Acknowledgments I warmly thank the Editor of this volume, Valia Allori, for her kind invitation to contribute to this book. I would like to thank Olga Sarno for helpful comments on previous versions of this manuscript and the reviewers for their positive and constructive feedback. This work is financially supported by the Swiss National Science Foundation (Grant No. 105212175971).

References Allori, V. (2013). Primitive ontology and the structure of fundamental physical theories. In D. Z. Albert & A. Ney (Eds.), The wave function: Essays on the metaphysics of quantum mechanics (Chapter 2, pp. 58–75). Oxford University Press. Allori, V. (2015). Primitive ontology in a nutshell. International Journal of Quantum Foundations, 1, 107–122. Allori, V. (2018). Scientific realism and primitive ontology or: The pessimistic meta-induction and the nature of the wave function. Latosensu, 5(1), 69–76. Allori, V. (2021). Primitive beable is not local ontology: On the relation between primitive ontology and local beables. In E. Okon & C. Romero (Eds.), Special issue of Critica: The meptaphysical foundations of physics. Universidad Nacional Autónoma de México. Allori, V., Goldstein, S., Tumulka, R., & Zanghì, N. (2008). On the common structure of Bohmian mechanics and the Ghirardi-Rimini-Weber theory. British Journal for the Philosophy of Science, 59(3), 353–389. Bell, J. S. (1987). Speakable and unspeakable in quantum mechanics. Cambridge University Press. Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of “hidden” variables. I, II. Physical Review, 85(2), 166–193.

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Bohm, D. (1957). Causality and chance in modern physics. Routledge. Dürr, D., Goldstein, S., Tumulka, R., & Zanghì, N. (2004a). Bohmian mechanics and quantum field theory. Physical Review Letters, 93, 090402. Dürr, D., Goldstein, S., & Zanghì, N. (1992). Quantum equilibrium and the origin of absolute uncertainty. Journal of Statistical Physics, 67, 843–907. Dürr, D., Goldstein, S., & Zanghì, N. (2004b). Quantum equilibrium and the role of operators as observables in quantum theory. Journal of Statistical Physics, 116, 959–1055. Esfeld, M. (2014). The primitive ontology of quantum physics: guidelines for an assessment of the proposals. Studies in History and Philosophy of Modern Physics, 47, 99–106. Esfeld, M., & Deckert, D. A. (2017). A minimalist ontology of the natural world. Routledge. Goldstein, S., Tumulka, R., & Zanghì, N. (2012). The quantum formalism and the GRW formalism. Journal of Statistical Physics, 149(1), 142–201. Hubert, M., & Romano, D. (2018). The wave function as a multi-field. European Journal for Philosophy of Science, 8(3), 521–537. Maudlin, T. (2016). Local beables and the foundations of physics. In M. Bell & S. Gao (Eds.), Quantum nonlocality and reality. 50 years of Bell’s theorem (Chapter 19, pp. 317–330). Cambridge University Press. Oldofredi, A. (2018). Particles creation and annihilation: Two Bohmian approaches. Latosensu, 5(1), 77–85. Smolin, L. (2015). Non-local beables. International Journal of Quantum Foundations, 1, 100–106. Tahko, T. E. (2018). Fundamentality. Stanford Encyclopedia of Philosophy. Tumulka, R. (2016). Paradoxes and primitive ontology in collapse theories of quantum mechanics. In S. Gao (Ed.), Collapse of the wave function: Models, ontology, origin, and implications (pp. 134–153). Cambridge University Press. van Strien, M. (2019). Pluralism and anarchism in quantum physics: Paul Feyerabend’s writings on quantum physics in relation to his general philosophy of science. Studies in History and Philosophy of Science Part A, 80, 72–81.

Chapter 8

All Flash, No Substance? Elizabeth Miller

Abstract The GRW dynamics propose a novel, relevantly “observer”-independent replacement for orthodox “measurement”-induced collapse. Yet the tails problem shows that this dynamical innovation is not enough: a principled alternative to the orthodox account demands some corresponding ontological advancement as well. In fact, there are three rival fundamental ontologies on offer for the GRW dynamics. Debate about the relative merits of these candidates is a microcosm of broader disagreement about the role of ontology in our physical theorizing. According to imprimitivists, the GRW dynamics directly describe (only) some (element’s) undulation in an unfamiliar high-dimensional physical field. Primitivists resist this GRW0 proposal on the grounds that it fails to secure comprehensible contact with our data about macroscopic objects in ordinary low-dimensional space-time. They expect an adequate fundamental ontology to include at least some spatiotemporally localized entities—intuitively, concrete constituents of our familiar macroscopic landscape. The most compelling case goes by way of distributional basing: minimally, primitivists expect a theory’s predictions immediately about spatiotemporal distributions of fundamental entities to provide a supervenience base for data about configurations of macroscopic objects. But while the background intuition is familiar, the distributional model is surprisingly subtle. Lack of clarity about its details generates serious confusion for both sides of our debate.

While textbook quantum theory provides a remarkably successful recipe for predicting the outcomes of experiments, its standard interpretation rests on gerrymandered dynamics in which “observers” making “measurements” play a starring role. This is the price orthodox quantum mechanics pays to solve—or, more accurately, evade—

E. Miller () Brown University, Providence, RI, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_8

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the measurement problem (§1).1 GRW’s dynamical innovation promises a more principled solution, but a reincarnation of the measurement problem persists for GRW (§2). On one view, the tails problem proves that our initial innovation is not enough: we need some ontology to go along with our new dynamics.2 In fact, there are three candidates on offer: GRW0, GRWm, and GRWf propose three rival fundamental ontologies for the GRW dynamics. Debate about their relative merits is a microcosm of broader disagreement about the role of ontology in our physical theorizing. One point of dispute is whether an adequate fundamental ontology must or should include some “local beables”—physical entities localized “at definite places and times in the real world” (§3).3 Local beables play a starring role in our data. Since our antecedent vision of the world features familiar macroscopic objects in space-time, any comprehensible theory about our world will be at least indirectly about these. Minimally, predictions directly about the features and behaviors of its fundamental ontology will provide a supervenience base for facts about “the arrangement of things in ordinary 3-dimensional space”.4 But Bell imposes a further continuity constraint: we must recognize, implicit in these predictions, a familiar “image of our physical world”. Since this image features objects in space-time, primitivists expect our base itself to include at least some fundamental local beables (or flobs)—intuitively, concrete constituents of our macroscopic landscape. On the other side are imprimitivists, who maintain that a theory can do just as well, if not better, without any flobs at all. In our microcosm, imprimitivists champion GRW0, which takes the GRW dynamics to directly describe (only) some (element’s) undulation in a highdimensional physical field. Primitivists resist GRW0 on the grounds that it fails

1

I am deeply grateful to Ned Hall for encouraging me to develop these ideas and for sharing many invaluable insights in conversation about them. For further discussion, I am grateful to David Albert and Nina Emery, as well as to the editor of and referees for this volume. My thanks also to David Baker, Michael Della Rocca, Heather Demarest, Barry Loewer, Tim Maudlin, Mark Maxwell, Michaela McSweeney, Alyssa Ney, Jill North, Zeynep Soysal, Scott Sturgeon, Elanor Taylor, and Alastair Wilson; to members of my philosophy of physics seminars at Brown and Yale; and to audiences at the Metro Area Philosophy of Science Association (at NYU), the University of Vermont, and USC. 2 For Maudlin, an ontology is “a collection of items taken to be physically real” by a theory, which can include familiar macroscopic objects featuring explicitly in our data but only implicitly in the theory’s own predictions (2013, 143). A theory is stated “in terms of” some subset of its ontology so understood. I follow, among others, Allori (2013) and Emery (2017) in describing this subset as the theory’s fundamental ontology; for simplicity, though, I drop the ‘fundamental’ where possible. Some instead use the label ‘primitive ontology’ for this subset, but since others reserve that for fundamental elements that also meet some further condition, I avoid this term here; cf. Dürr et al. (1992), Maudlin (2013). Still my own label ‘primitivism’ alludes to discussions of primitive ontology—as well as to discussions of “spatial primitivism” elsewhere in metaphysics; cf. Chalmers (2012) 325ff. 3 Bell (1987) 45. 4 Bell (1987) 44.

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to secure adequate contact with our data (§4). The most compelling resistance goes by way of prior commitment to distributional basing, which—in Bell’s words— links a “piece of matter” from our data to some “galaxy” of constituents within our theory.5 Interestingly, though, recent arguments for imprimitivism begin from the premise that distributional basing is a non-starter: even for primitivists, (i) a minimal supervenience base must include something besides an occurrent distribution of flobs. Imprimitivists focus on showing that (ii) once we add requisite functional or dynamical factors to the base, distributional inputs are superfluous: GRW0 can do just as well without any flobs. Such a focus obscures two concerns. One is that imprimitivists may prove too much: if (i) and (ii) are true, we risk undermining some motivation (reviewed in §§1–2) for turning to ontology in the first place. The other, and my primary, concern is that imprimitivists lack an adequate defense of (i). In advancing (i), they conflate distributional supervenience with an implausible claim of geometric sufficiency. But imprimitivists are not alone in their mistake. A lack of clarity about the details of distributional basing generates serious confusion on both sides of our dispute (§5). When it comes to recovering Bell’s “image” of our data, it is not enough for primitivists to know it when they see it: before we can hope for any further progress, we need careful investigation into their—familiar yet surprisingly subtle—model of contact between macroscopic objects and “galaxies” of localized constituents. My aim is to highlight the distributional model’s subtlety—and, in light of that, to suggest a path forward in disagreement about quantum ontology.

8.1 The GRW Dynamics Start with Schrödinger’s hapless cat, “penned up in a steel chamber”: [I]n a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of one hour one of the atoms decays . . . ; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed.6

Schrödinger seals the chamber at t0 , expecting one of two outcomes after a tense hour. The textbook recipe promises a cat-flask system with one of two wavefunctions, |ψ dead  or |ψ alive  , when he opens the door at t. In the interim, it predicts something even more “diabolical”. Between t0 and t, the theory ascribes neither |ψ dead  nor |ψ alive  to Schrödinger’s system; instead, it evolves the system’s initial state to √1 | ψdead  + √1 | ψalive  just before t. On one hand, it is not clear 2

what

5 6

√1 2

| ψdead  +

2

√1 2

| ψalive  amounts to physically. On the other, it seems clear

Bell (1987) 45. Trimmer (1980) 328, translating Schrödinger (1935).

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enough what it does not: anything we actually find in our world. Schrödinger himself suggests “the living and the dead cat (pardon the expression) mixed or smeared out in equal parts”, and, in essence, the orthodox interpretation agrees: Schrödinger’s mathematical evolution indicates a physical “indeterminacy” only “resolved” by observation. More generally, since the textbook recipe frequently yields pre-measurement functions in superpositions of relevant eigenstates, its standard interpretation says that systems themselves frequently evolve away from familiar physical states. Nevertheless, it does not follow that we should expect to observe any unfamiliar happenings. Given pre-measurement state √1 | ψdead  + √1 | ψalive  , the recipe 2 2  2   promises one of two familiar eigenstates, each with prior probability  √1  = 12 , 2 at t. Orthodox quantum mechanics tailors its physical dynamics to underwrite this guarantee: when Schrödinger opens the chamber, his system’s character transforms into either a dead state or a live one. By design, systems evolve away from familiar states only when, and indeed because, we are not looking—luckily, though, our data only concerns what the world is like when we are. An alternative from Ghirardi, Rimini, and Weber targets this “measurement”induced collapse, proposing a novel stochastic mechanism in its stead.7 The resulting unified GRW dynamics yield, as a theorem, something close to the orthodox story, thereby cohering with the same empirical data. Crucially, though, there is nothing special about observers or measurements per se. Instead, the GRW proposal starts with a division of the world into units (particles), each assigned some probability per unit time of undergoing a hit event. A hit on a particle transforms the wavefunction representing its position: roughly, we can think of a hit centered around some point in space-time as prompting our particle to jump there. A larger system’s probability of undergoing a hit-induced transformation is then a function of the number of particles interacting within it: a hit on one particle transforms the wavefunction of the whole. Schrödinger’s possible outcomes differ with respect to the positions of macroscopically many particles: a system in state |ψ dead  has flask particles strewn across region Rdead of the chamber floor, rather than arranged in Ralive . While any single particle will undergo a hit, on average, only once every 108 years, we can expect a hit somewhere in our flask—on the order of 1023 particles—every 10−8 seconds.8 The matched coefficients in √1 | ψdead  + √1 | ψalive  indicate that 2 2 the next hit is just as likely to be centered in Rdead as in Ralive , amplifying one of two components. Wavefunctions still spread out between hits, but Schrödinger’s system is large enough—and so hits frequent enough—that its spreadings are imperceptibly brief. Thanks to our first hit, one of |ψ dead  and |ψ alive  stands amplified at t, and subsequent hits are highly likely to reiterate that choice.

7 8

Girardhi et al. (1986); cf. Albert (1992) 80–116, Albert and Loewer (1996), and Lewis (2006). Maudlin (2011) 226–8.

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8.2 The Tails Problem and Fundamental Ontology GRW’s insight is that anything recognizable as an orthodox “measurement” shares key physical characteristics with Schrödinger’s case. Whenever these characteristics are present, stochastic hits naturally rein in smeared-out wavefunctions. Importantly, though, these hits do not transform such wavefunctions into strict eigenstates. Mathematically, a hit on a particle multiplies its wavefunction by a Gaussian, amplifying some component over the rest. But thanks to non-zero “tails” on our multiplying curve, such amplification does not eliminate the others. In Schrödinger’s case, a hit takes √1 | ψdead  + √1 | ψalive  to some function retaining multiple components, 2√ √ 2 say 0.999 | ψdead  + 0.001 | ψalive  , at t. If our pre-hit function was troubling because it was not a plain eigenstate, this tailed post-hit function should be just as bad. The natural fix, curtailing the tails, is not a live option: our measurement problem is back with a vengeance, just reincarnated in a subtler form.9 Strictly, the tails problem proves only that GRW’s initial innovation is not enough. We find our tailed function troubling because we think it amounts to neither a definitely dead nor definitely live cat. Yet this conclusion rests on a further assumption about the relationship between representation and reality: a system has a definite “value” for some given “observable” only if its wavefunction is in a strict eigenstate of the mathematical operator associated with that observable. We might avert the tails problem by revising this link, providing a more sophisticated way of extracting physical outcomes from Specifically, we √ √ formal representations. need a sort of sophistication that counts 0.999 | ψdead  + 0.001 | ψalive  as close enough to |ψ dead  . One option is stipulation: we could propose a mapping between almost eigenstates and physical outcomes.10 To provide a principled alternative to the orthodox account, however, GRW needs to motivate our close enough verdict. A different approach starts by getting serious about quantum ontology.11 Maybe, as Schrödinger suggests, mathematical evolution to √1 | ψdead  + √1 | ψalive  2 2 represents some worldly spreading out or other, but what are the entities doing this spreading, what sort of physical processes are involved, and how does any of this bear on our data? The eigenstate-eigenvalue link—like any stipulative revision to it—attempts to map directly from mathematical representations to the recognizable macroscopic outcomes at issue in our data. In doing so, it allegedly skips a crucial step, obscuring the mediating role of ontology in our theorizing. Compare a classical case: a candidate theory assigns formal state C to the current contents of my office. It also issues predictions about those contents later, assigning mathematically distinct

9

In brief, “tailless” collapses threaten the conservation of energy, since they reduce uncertainty about position without any compensatory increase in uncertainty about (velocity and so) energy; cf. Albert (1992) 78. 10 Albert and Loewer (1996) and Lewis (2006) discuss this strategy. 11 I learned the term ‘serious ontology’ from John Heil; cf. Heil (2012).

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C* to them at t. I leave my office, planning to return at t. Should I expect to find the furniture rearranged when I do? To answer, it is not enough to know that our theory’s predictions have some bearing on the contents of my office. We need to know what its mathematical states directly represent, and how these underwriting physical happenings relate to our observable outcomes. In fact, C depicts an arrangement of particles in my office. C* depicts another, differing with respect to, say, the relative positions of three constituents. Equipped with this ontological insight, we can opine: I should not expect anything new when I return. Our distinct mathematical states represent different physical happenings, but so slight a difference at the level of particles will not show up in our macroscopic data. In this sense, C* is close enough to C: we can recognize Bell’s image of my desk√in both. To dissolve √ the tails problem, we need to issue the same sort of verdict: 0.999 | ψdead  + 0.001 | ψalive  is close enough to |ψ dead , since they depict happenings with the same (unfortunate) import for our data. For this, we need some new and improved—or at least some, reasonably clear—ontology for the GRW dynamics.

8.3 GRW0 and Primitivism According to Bell, orthodox disregard for ontology proves especially troubling once we note that the wavefunction for my desk, Schrödinger’s flask, or any other macroscopic system “lives in a much bigger space” than the one we ordinarily take ourselves to inhabit.12 Perhaps, then, any complete interpretive backstory for the textbook recipe will include some physically high-dimensional ingredient(s). Regardless, Bell and likeminded primitivists expect it to include at least some flobs in “ordinary 3-dimensional space” as well. Since science promises us deeper, even revisionary, insight into our world, characters in the scientific story may be antecedently unfamiliar to us. Still, they cannot be so unfamiliar that we fail to recognize the story as about our world—Bell’s familiar “arrangement of things”— at all. GRW0 proposes a thoroughly non-local fundamental ontology for the GRW dynamics. On the GRW0 proposal, these dynamics depict (only) some (element’s) undulation in a high-dimensional field, a physical counterpart to the mathematical space in which the universal wavefunction resides. Ordinary space-time is not a straightforward subspace of this realm: concrete happenings are aspects of, projections from, or patterns in some fundamental affairs “outside” of our familiar milieu. GRW0 furnishes a complete supervenience basis for our data: we can find some mapping between various high-dimensional affairs and familiar macroscopic outcomes.13 According to primitivists, however, this is not enough: GRW0 does

12 Bell 13 It

(1987) 44. is “informationally” complete in the sense of Maudlin (2007a).

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not afford the right sort of contact with our data: at the very least, some alternative including flobs can—and in√fact does—do better. √ Before we ask whether 0.999 | ψdead  + 0.001 | ψalive  and |ψ dead  amount to the same outcome, we need to recognize both as concerning fundamental happenings that bear, somehow or other, on outcomes here in the lab. But why should these particular undulations tell us about the contents of Schrödinger’s chamber here, rather than about, say, the furniture in my office, or some distant configuration of elm trees? Not because the undulations are here too: GRW0 has no fundamental happenings here—or, indeed, anywhere in space-time. Minimally, we need to say which high-dimensional happenings bear on which pieces of data, but we started down the road of ontology looking for something better than stipulation. Rather than turning back, primitivists urge us to keep going. GRW0 shows that not just any ontological ingredients will do: we need at least some flobs in the mix. But not just any localized additions will do either. Consider (fictional) GRW0+ , which supplements GRW0’s high-dimensional happenings with arbitrary flob o. Like GRW0, GRW0+ furnishes at least one supervenience basis for our data. Unlike GRW0, it furnishes at least one basis also comprising a localized ingredient(s). Yet o’s addition does not secure any relevant advantage over GRW0. Intuitively, o is a mere dangler: while all relevant facts supervene on a base including it, they do so only because they already supervene on GRW0’s high-dimensional happenings alone. Primitivists expect their flobs to play a more essential, distinctive role in securing contact with our data. Whatever else it may comprise, that data includes facts about the gross configuration of macroscopic objects within, or the gross distribution of matter across, space-time. Primitivists expect such facts to be fixed entirely by their theory’s predictions just about some underwriting flobs. Our theory may bring other— perhaps non-local, even non-spatiotemporal—ingredients along for the ride. Still, these play a comparatively indirect role in “determining” our data: in Maudlin’s terms, their effects are “screened off” by our “[p]rimary” mediating flobs.14 Intuitively, other fundamental ingredients can causally or physically influence these flobs, but they alone directly ground our data. For this intuitive distinction to have any bite, primitivists must circumscribe those predictions relevantly about their candidate flobs. Otherwise, GRW0+ could count as screening off all our data by ascribing some complicated but nominally “local” state—merely represented, in our theory, by the universal wavefunction—to o alone. A “Democritean” fix blocks this move: the relevant theoretical predictions about primary local beables ascribe (only) spatiotemporal positions to otherwise qualitatively indistinguishable elements.15 To make contact with familiar facts

14 According to Maudlin, “if we imagine keeping the behavior of the Primary Ontology fixed but altering the behavior of the Secondary Ontology, the data would remain the same” (2013, 144). 15 Oppenheim and Putnam (1958) 16 n. 18; cf. Ney (2013). Alternatively, we can think of these predictions as ascribing fundamental states of “occupation” to points or small regions of space-time itself.

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about my office, our theory need not explicitly mention desks. Still, it should describe at least something here in Roffice , where we antecedently take my desk to be. Whatever features this something may or may not turn out to have on our scientific story, it at least should be distinguishable from what the theory describes over there in Rempty , where we find no macroscopic objects at all. Minimally, then, our theory should depict some elements in Roffice without corresponding counterparts in Rempty . We can extend this reasoning to smaller scales. Consider the edge of my desk here: some materially occupied subregions of Roffice abut comparatively vacant neighbors. This contrast at the macroscopic level should show up, somehow or other, as some difference in corresponding contents at the level of fundamental ontology. What results is a primitivist paradigm of distributional basing. Rather than stipulating some link between data about the macroscopic configuration of my office and discontinuous fundamental happenings in GRW0, primitivists hope to draw on antecedently familiar continuity between configurations of macroscopic objects and underwriting distributions of concrete constituents. Primitivists can disagree amongst themselves about the nature of this continuity, and so about further metaphysical or epistemological constraints on the base.16 Minimally, though, they expect data about macroscopic configurations to supervene on the theory’s distribution of flobs in space-time.

8.4 GRWm and GRWf Primitivists are spoiled for choice. There are two alternatives to GRW0, each proposing some flobs for the GRW dynamics. But this choice also generates controversy, since even for primitivists, not just any localized additions guarantee a relevant advantage over GRW0. Maudlin argues that GRWm does not secure the right sort of contact with our data: it does not provide even a minimal distributional supervenience base for “the arrangement of things in ordinary 3-dimensional space”. Behind both Maudlin’s original argument and Albert’s criticism of it is common confusion about what exactly the distributional model demands. According to GWRm, reality is not ultimately particulate. Instead, each “particle” from our initial presentation of the dynamics gets associated with some portion of continuous mass density. Our dynamics now depict evolving distributions of mass density in ordinary space over time: smooth temporal evolution of a particle’s wave function corresponds to a spreading out of mass density. What we originally described as a particle undergoing a hit centered within a region involves consolidation of associated mass density there. As before, a hit on one part of the systems transforms the √ wavefunction for the whole: Schrödinger’s system √ in state 0.999 | ψdead  + 0.001 | ψalive  has much more of its mass density in Rdead than in Ralive . Given our transformed coefficients, subsequent hits are likely to

16 For

discussion of candidate constraints, Allori (2013), Ney (2013), and Emery (2017).

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consolidate mass density in Rdead , leaving Ralive ’s shadow to spread and thin. What about the lingering |ψ alive ? Our Gaussian tails guarantee some thin smear of mass density in Ralive , but, allegedly, that smear does not show up at the level of our data: our tailed function is close enough to |ψ dead  to yield a familiar outcome. Maudlin illustrates his doubts about this last step with the case of a pointer on a dial. If we like, we can imagine the pointer’s possible orientations, A and B, representing dead and alive states of Schrödinger’s system, but for now just consider the pointer itself: its wavefunction spreads from an √ initial ready state, splitting briefly into √1 | in RA  + √1 | in RB  , before yielding 0.999 | in RA  + 2 2 √ 0.001 | in RB  at t. Then: On the assumption that . . . [R√B ’s] small (and ever √ shrinking) mass density can be safely neglected, the post-hit state [ 0.999 | in RA  + 0.001 | in RB  ] would be as satisfactory in accounting for our beliefs about the outcome as it is in the case of predictive certainty [|in RA  ]. But on what basis, exactly, can the small mass density be neglected? After all, that mass density is something, and it has the same shape and . . . behavior and dispositions to behave as it would have had if the hit had left it with the lion’s share of mass density.17

Our data includes one pointer at A, with no macroscopic counterpart at B, but GRWm depicts two portions of mass density. According to Maudlin, these portions match in all relevant respects, even at t. Indeed, the actual distribution of mass density across B matches, in all √ both RA and R√ such respects, what we would find in the case of 0.001 | in RA  + 0.999 | in RB  . Of course, RB ’s actual mass density is supposed to be too thin to amount to a macroscopic object, but on what “basis” can we discount it? In principle, we might introduce some concentration threshold: only RA contains any recognizable distribution of portions with thickness greater than ρ at t. If we stipulate some value for ρ, however, we risk undermining our motivation for introducing flobs in the first place: GRW0 can just as well stipulate some mapping between its discontinuous fundamental happenings and macroscopic outcomes. Albert and likeminded imprimitivists share Maudlin’s risk assessment, but they embrace this consequence for GRW0. They take Maudlin’s case to point to a deeper problem with primitivism itself. According to Albert, familiar objects are distinguished, in part, by characteristic behaviors and dispositions. As a result, everyone, even primitivists, must accept: (i) an adequate basis for our data will specify more than the occurrent distribution of fundamental ontology. Albert takes Maudlin’s reference to “behavior and dispositions to behave” to mark his own explicit acceptance of (i). On Albert’s diagnosis, a confused fixation with distributional factors gets in the way: Maudlin implicitly expects “whatever is shaped like” a pointer to be a pointer, and so mistakenly assumes that his similarly shaped portions of mass density have the same dispositions at t.18 Thanks to the GRW

17 Maudlin 18 Albert

(2010) 135; his emphasis. (2015) 151.

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dynamics, only Maudlin’s thicker portion is actually disposed to consolidate—its thinner counterpart is likely to spread and fade. According to Albert, the same sort of confusion lying behind Maudlin’s error in this case is also what motivates primitivists’ attachment to fundamental local beables in the first place. Primitivists fail to appreciate (i) because they overemphasize distributional considerations. If we take (i) to heart, though, we find that GRW0 can do just as well as its rivals. That is: (ii) once we have functional information in the base, distributional facts are, at best, superfluous.19 It is precisely because of (ii), however, that primitivists should not, and Maudlin plausibly does not, grant (i) to begin with: primitivists hope to secure an advantage over GRW0 by providing a distributional basis. As Maudlin observes, even if we need further factors to distinguish those distributions that amount to genuine pointers from mere pointer-like objects, such functionalism is beside the point in his case: pointer or not, GRWm depicts “something” macroscopic in RB . Muddying the waters throughout Albert and Maudlin’s exchange are their differing uses of the term ‘disposition’. Albert and his fellow imprimitivists are concerned with causal or functional dispositions. Yet Maudlin instead seems to have in mind the “arrangement, order; [or] relative position of the parts or elements of a whole”.20 In comparing “dispositions” in his sense, then, Maudlin is quite plausibly affirming his commitment to distributional basing. After all, he rejects GRWm because he thinks it fails to mark any requisite difference in the arrangements of flobs across RA and RB . Maudlin’s own response is to swap GRWm’s continuous mass density for GRWf’s discrete event ontology. The physical correlate of a hit centered around some point in space-time is now a “flash” there. Our matched weights in √1 | in RA  + √1 | in RB  signify that RA and RB are equally likely to host the 2 2 √ √ next flash. In fact, a hit in RA takes us to 0.999 | in RA  + 0.001 | in RB  at t. Our transformed coefficients indicate that the next flash is highly likely (probability .999) to be in RA as well. The GRW dynamics ensure that further hits are likely to√reiterate this choice. √ Crucially, no shadows haunt GRWf: where GRWm looks at 0.999 | in RA  + 0.001 | in RB  and posits a thin but sure smear of mass density in RB , GRWf posits only some probability of a flash—and then a cascade of flashes—there. But probabilities of flashes are neither flashes nor pointers nor shadows √ of these. Our tailed function is close enough to an eigenstate: √ 0.999 | in RA  + 0.001 | in RB  depicts a galaxy in RA —and none in RB .

19 Cf.

Rubenstein (forthcoming). online (2020); thanks to Maudlin (p.c.) for confirming. Compare Maudlin’s characterization of the Humean mosaic as “determined by nothing more than the values of the individual pixels plus their spatial disposition relative to one another . . . ” (2007b, 51), and his claim that (for our primitivists) “the disposition of the local ontology screens off the quantum state from the data” (2013, 148). 20 OED

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8.5 Massy Gluts and Flashy Gaps Since GRWf has “no ‘low density’ or ‘low intensity’ flash-sequences to be ignored or discounted or argued away”, Maudlin takes it to avoid the “interpretive problems” that stymie GRWm.21 Yet GRWf’s comparative sparseness also ensures that, sometimes, GRWf has mere probabilities where GRWm has sure non-shadowy mass. On the GRWf underdetermine occurrent distributions √ √ account, wavefunctions of flashes. Given 0.999 | in RA  + 0.001 | in RB  at t, we can expect another flash in RA very shortly. Still, its actual occurrence is determined by GRW’s stochastic dynamics. In the interim, however brief, we have no flashes at all in RA . Indeed, if flashes are truly instantaneous, then GRWf promises many entirely empty moments, amid some barely flashy accompaniment. GRWf virtually guarantees some unfolding cascade in RA , but it does not promise a multi-flash galaxy at any moment. Where GRWm has some sure, thick mass density in RA , GRWf offers—if we are lucky—a single flash. Flashy gaps generate a challenge for GRWf analogous to the challenge shadowy mass density generates for GRWm. One potential objection to GRWm starts from √1 | in RA  + √1 | in RB  , before our initial hit in Maudlin’s case: we have two 2 2 exactly matched portions of mass density at time t-. But this cannot be a basis on which to prefer GRWf, because it is in the same boat: √1 | in RA  + √1 | in RB  2 2 depicts two equally √ of flashes at t-. In fact, the situation for √ vacant distributions GRWf is worse: 0.999 | in RA  + 0.001 | in RB  at t likely depicts two vacant distributions after our hit as well. Since Maudlin takes GRWf to avoid the “interpretive problems” that plague GRWm, the image of our data need not comprise some single pointer-shaped distribution of flobs at t. Our data concerns humanly detectable happenings, so perhaps what we need is a recognizable distribution across some suitably extended temporal interval. One option is to link RA ’s momentary state to some broader distribution of flobs. Another is to deny that there are any momentary macroscopic states at all: our data concerns configurations spread over space and time. Either way, once we shift our attention to an extended interval, we can find a difference in the decorations of RA and RB . But we also find at least some difference between GRWm’s portions of mass density: though √1 | in RA  + √1 | in RB  depicts a 2 2 momentary match, the contents of RB soon spread and fade. For Maudlin, of course, this is not the right sort of difference to underwrite our macroscopic data: thick or thin, GRWm still predicts “something” in RB , even after our hit. His remark might seem to suggest that GRWf, unlike GRWm, satisfies the following condition: a region of space is macroscopically vacant only if there are no flobs in it during some relevant interval. Even in the classical case, though, we learn that “empty” space is not devoid of occupants. In a large enough region, a single flash, or even some sparse scattering of flashes, does not suffice

21 Maudlin

(2010) 138–9.

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for anything macroscopic at all. Instead, perhaps the claim is that for any relevant region of space-time R, R is macroscopically vacant only if there is no recognizably shaped distribution of flobs there. Even after our first hit, GRWm retains two recognizably pointer-shaped distributions, only one of which is thick enough to be a genuine macroscopic object. According to Maudlin, GRWm needs some basis on which to discount some recognizably shaped distributions as macroscopically inconsequential. In contrast, perhaps any recognizably pointer-shaped galaxy of flashes amounts to something macroscopic for GRWf. While our choice between continuous and discrete ontologies can obscure this point, GRWf’s appearance of an advantage arises from ambiguity in such talk of shape. Trace the outline of Maudlin’s distribution of mass density in RA over some interval. Now imagine filling in this outline with flashes. How many flashes does it take to yield a recognizably pointer-shaped distribution? Perhaps any distribution “consistent” with our outline—even just one flash within this boundary—counts as pointer-shaped in some minimal sense. But not every distribution that counts as pointer-shaped in this sense does—or should—amount to a genuine macroscopic object. Perhaps Maudlin has some more robust notion in mind: a relevantly pointershaped distribution of flashes is not only minimally consistent with our outline but also sufficiently clustered to show up in our data. But then GRWf and GRWm are in the same boat: both must distinguish some special, sufficiently concentrated candidates from among all minimally pointer-shaped ones. Perhaps this result offers Albert a way of shoring up his case for (i): GRWf and GRWm are in the same boat, because neither has any advantage over GRW0. On Albert’s diagnosis, implicit commitment to geometric sufficiency motivates primitivists’ attachment to local beables. As Maudlin himself points out, however, GRWm violates this condition. Since GRWf is in the same boat, it does too: On what basis can either one discount insufficiently concentrated distributions? According to Albert, the right response is to give up on the expectation of distributional basing: even primitivists must agree that we need more than the occurrent distribution of fundamental ontology in the base. But then GRW0 can do just as well without any localized fundamental ontology at all. Alternatively, maybe GRWf and GRWm are in the same boat because both offer a relevant advantage over GRW0. According to Maudlin, GRWf does better than GRW0. If GRWm is in the same boat, then—despite his own suspicions to the contrary—it does too. To defend this alternative, primitivists must separate distributional basing from any implausible claim of geometric sufficiency. Recall our primitivists’ motivation for introducing fundamental local beables in the first place: roughly, they hope to link macroscopic configurations to arrangements of localized constituents (§3). Our data includes a single macroscopic pointer, at A, over some relevant interval. The tip of our pointer extends through small region RA1 , while some neighbor R is (comparatively) empty. Primitivists expect our theory to predict at least some flobs in RA1 without any counterparts in R. Still, more flobs in RA1 is consistent with some—sparser—flobs in R. In Maudlin’s case, though, GRWm does mark this sort of distributional difference, even between his similarly shaped portions of mass density. To make the

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parallel with GRWf vivid, imagine GRWm’s mass density packaged in thin layers: mass density gets “painted on” a region in a series of discrete strokes. We can have some difference in distributions—whether massy “strokes” or flashes—across RA and RB , without any difference in “shape” per se. Neither GRWf nor GRWm counts every minimally pointer-shaped distribution as a macroscopic object—but nor should they. The key question for primitivists is whether this result demotes both to the level of stipulation or, alternatively, whether both preserve continuity that secures their alleged advantage over GRW0. Primitivists expect at least minimal (“global”) supervenience between our data and spatiotemporal distributions of flobs.22 Intuitively, they expect a special link between our data about one part of the world and some distribution in that part. To make progress, though, primitivists need to clarify what exactly, beyond minimal supervenience, their model demands. GRWf’s sparseness underscores this need. For whatever exactly distributional continuity comes to, the distribution of flobs just within a region does not, in general, metaphysically suffice for its macroscopic state on the GRWf account. For small regions, this result is unsurprising: it is just the spatial analogue of our earlier point about temporal sparseness. While there is a pointer in RA through some interval, subregion RA* may well contain no flashes at all. So, if RA* counts as macroscopically occupied at all, it is because of some distribution extending beyond its bounds. But now consider larger RA : this hosts the broader distribution D that, intuitively, makes up our pointer at A. Perhaps within our actual world, any flashy duplicate of D amounts to something macroscopic. Yet consider another (near) world w, which hosts some duplicate of D in much flashier surroundings. Does this—indeed, should this—duplicate amount to any macroscopic object? Surely, it depends: not if even “empty” space is densely decorated in w. In that case, after all, D is not even recognizably pointer-shaped in anything but our minimal sense. Plausibly, in fact, any relevant further sense is essentially contrastive: what it takes to be sufficiently dense, in any given case, may itself depend on some more global distribution of flobs.

References Albert, D. (1992). Quantum mechanics and experience. Harvard UP. Albert, D. (2015). After physics. Harvard UP. Albert, D., & Loewer, B. (1996). Tails of Schrödinger’s cat. In R. Clifton (Ed.), Perspectives on quantum reality: Non-relativistic, relativistic, and field-theoretic (pp. 81–92). Kluwer. Albert, D., & Ney, A. (2013). The wave function: Essays on the metaphysics of quantum mechanics. Oxford UP.

22 Cf. Kim (1987). A referee asks how distributional supervenience relates to Humean supervenience (HS). There is much more to say, but prima facie, primitivists can remain neutral on HS, which claims supervenience for all facts about the world, including causal and nomological facts, not just facts about macroscopic configurations. Cf. Maudlin (2007b), Maudlin (2015), Miller (2018).

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Allori, V. (2013). Primitive ontologies and the structure of fundamental physical theories. In Albert & Ney (eds.) pp. 58–75. Bell, J. (1987). Are there quantum jumps? In C. Kilmster (Ed.), Schrödinger: Centenary celebration of a polymath (pp. 41–52). Cambridge UP. Chalmers, D. (2012). Constructing the world. Oxford UP. Dürr, D., Goldstein, S., & Zanghì, N. (1992). Quantum equilibrium and the origin of absolute uncertainty. Journal of Statistical Physics, 67, 843–907. Emery, N. (2017). Against radical quantum ontologies. Philosophical and Phenomenological Research, 95(3), 564–591. Ghirardi, D., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34, 470–491. Heil, J. (2012). The universe as we find it. Oxford UP. Kim, J. (1987). ‘Strong’ and ‘global’ supervenience revisited. Philosophy and Phenomenological Research, 48, 315–326. Lewis, P. (2006). GRW: A case study in quantum ontology. Philosophy Compass, 1(2), 224–244. Maudlin, T. (2007a). Completeness, supervenience and ontology. Journal of Physics A: Mathematical and Theoretical, 40, 3151–3171. Maudlin, T. (2007b). Why be humean? In The metaphysics within physics (pp. 51–77). Oxford UP. Maudlin, T. (2010). Can the world be only Wavefunction? In S. Saunders et al. (Eds.), Many worlds?: Everett, quantum theory, and reality (pp. 121–143). Oxford UP. Maudlin, T. (2011). Quantum non-locality and relativity (3rd ed.). Wiley-Blackwell. Maudlin, T. (2013). The nature of the quantum state. In Albert & Ney (eds.), pp. 126–53. Maudlin, T. (2015). The universal and the local in quantum theory. Topoi, 34, 349–358. Miller, E. (2018). Local qualities. Oxford Studies in Metaphysics, 11, 224–242. Ney, A. (2013). Ontological reduction and the wave function ontology. In Albert & Ney (eds.), pp. 168–83. OED online. (2020). Disposition, n. https://www.oed.com/view/Entry/55123?redirectedFrom= disposition. Accessed 19 Sept. Oppenheim, P., & Putnam, H. (1958). Unity of science as a working hypothesis. In H. Feigl et al. (Eds.), Minnesota studies in the philosophy of science (Vol. 2, pp. 3–36). Minnesota UP. Rubenstein, E. (forthcoming). Grounded shadows, groundless ghosts. The British Journal for the Philosophy of Science. Schrödinger, E. (1935). Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaften, 23, 807–812. Trimmer, J. (1980). The present situation in quantum mechanics: A translation of Schrödinger’s “cat paradox” paper. Proceedings of the American Philosophical Society, 124, 323–338.

Chapter 9

Does the Primitive Ontology of GRW Rest on Shaky Ground? Cristian Mariani

It’s all about gelatin. An electron can be here and there and that’s it. Angelo Bassi, 2020, Interview for the NY Times

Abstract The notion of Primitive Ontology (PO) has recently received a great deal of attention in the quantum foundations literature. The PO is the fundamental ontology posited by a certain theory, what is out there in the world which makes the predictions of theory true. Can we make sense of the idea that the PO is indeterminate? And if we do, would this idea be explanatory useful in some way, or would it simply lead us too far from the very reasons we had to posit a PO in the first place? As I will show in this paper, these issues become of crucial importance when it comes to understanding what the ontology of the Mass Density approach to GRW (GRWM ) ultimately looks like. Proponents of the PO are never explicit in claiming that the determinacy is a requirement for the notion, yet arguably this is entailed by one of the criteria for a suitable PO, namely its being always well defined in every point in 3D space. I will argue that this requirement is however not satisfied in GRWM . The conclusion I will draw is that the notion of indeterminate PO should be taken seriously, for it is suggested by one the major interpretations of quantum mechanics. Keywords GRW · Mass density · Primitive ontology · Quantum mechanics · Quantum indeterminacy

C. Mariani () Institut Néel (CNRS), Grenoble, France Universitat de Barcelona, Barcelona, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_9

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9.1 Introduction Some of the major issues in both contemporary philosophy of science and naturalistic metaphysics come from the conceptual puzzles arising from quantum mechanics (QM). And in particular, they come from the fact that this theory seems to challenge many of our classical intuitions, thus making it hard to be reconciled with the everyday picture of the world. Experiments have revealed that microscopic particles sometimes behave as if they can be located in two places at the same time, that the precise values of certain pairs of quantities cannot jointly be assigned, that certain properties are not possessed independently of observation, or that instantaneous interactions at a distance are allowed. A long standing, still lively tradition has taken all of this to suggest, if not entail, that the standard reductionist program should be abandoned, and even that novel forms of realism have to be developed if we wish to understand the quantum world. It is against this background that we can understand the reasons why the notion of Primitive Ontology (PO) has been advocated (Allori et al., 2008), and why it is recently gaining more and more attention. The inspiration for this approach comes from Bell’s (1987) reflection on the idea of be-ables, a term coined to be clearly distinguished from the notion of observ-ables (p. 52). ‘Observables’ must be made, somehow, out of beables. The theory of local beables should contain, and give precise physical meaning to, the algebra of local observables. (p. 52)

As the term itself indicates, the be-ables of a theory should represent what exists in the world, the ontology: The beables of the theory are those entities in it which are, at least tentatively, to be taken seriously, as corresponding to something real. (p. 234)

While introducing this notion, Bell expresses full agreement with Bohr’s idea that every experience and physical evidence must be accounted for in classical terms, and claims that “it is the ambition of the theory of local beables to bring these ‘classical terms’ into the equations” (p. 52). So it is quite natural to suppose that according to Bell whatever beables a certain theory has, these must be described in classical terms. The PO program aims at developing Bell’s insights into a fullfledged approach to the ontology of physical theories, be them quantum or not. Here is Allori (2016), summing up nicely the guiding ideas behind the view: According to this approach, any satisfactory fundamental physical theory, if taken from a realist point of view, contains a metaphysical hypothesis about what constitutes physical objects, the PO, which lives in three-dimensional space or space-time and constitutes the building blocks of everything else. In the formalism of the theory, the variables representing the PO are called the primitive variables. In addition, there are other variables necessary to implement the dynamics for the primitive variables: these non-primitive variables could be interpreted as law-like in character. Once the primitive and the non-primitive variables are specified, one can construct an explanatory scheme based on the one that is already in use in the classical framework. (p. 177)

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According to Allori, a suitable PO has to be (i) microscopic, (ii) well localized in 3D space, and (iii) appropriately distinguished from the dynamical variables of the theory. The criterion (ii) plays a crucial role, for it is meant to avoid problematic states of superposition at the level of the fundamental ontology. Once again, this idea traces back to Bell’s claim (Bell, 1987) that the “beables are those which are definitely associated with particular space-time regions” (p. 234, emphasis mine). If met, the requirements (i)–(iii) would ground a classical reductive explanation of the behaviour of the macroscopic ontology as determined by the behaviour of the PO. The overall goal of this approach is then to show that such a classical explanatory scheme, which is not dissimilar from the one we can find in classical physics, can be obtained in most of the live interpretations of QM (Allori, 2013; Allori et al., 2008). And if this is true, as proponents of this approach claim, then arguably we would have little reasons to completely revise the way we think about physics and its place in our understanding of nature. How far this program can go, and especially how classical, in Bell’s sense, can the PO be, is highly debated. The two major lines of criticisms seem to point in opposite directions. On the one hand, it has been argued that the notion of PO is too general and empty to be useful when applied to concrete cases, especially if we consider that every theory that has been proposed as an example of the PO scheme is not a fundamental theory (e.g. Wallace 2018, p. 20). On the other hand, it has been stressed that some of the criteria for a suitable PO are too strict, and may not be satisfied in QM (e.g. Myrvold 2018, p. 115).1 In this paper I will be focusing on this latter line of criticism, and especially on whether the criteria according to which the PO is “definitely associated with particular space-time regions”, to use Bell words, is indeed satisfied within the context of the Mass Density approach to GRW (Ghirardi et al. 1995, GRWM ). Proponents of the PO approach have argued that GRWM provides a good exemplification of their view, and yet, as I will show, the PO in this theory cannot always be ascribed a definite localisation in 3D space. Once the desiderata of definiteness for the PO is put into question, it becomes legitimate to ask whether the notion we are left with does indeed serve the purposes for which it was developed. The very idea of an indeterminate PO, as I will be calling it, may seem to contradict our intuitions. The reason is probably that the PO is the fundamental ontology according to a given theory, and while we may be tempted to accept indeterminacy at some derivative level of reality (Mariani, 2021), it is hard to entertain the thought that the world is indeterminate at its fundamental level. However, I will suggest that if this idea is supported by one of the major interpretations of QM, and if it proves to be both consistent and explanatory useful, then a good naturalistic attitude should prevent us from ruling it out just based on our intuitions.

1

I note that in this context quantum non-locality also poses a very serious threat since, as Bell himself claimed, “it may well be that there just are no local beables in the most serious theories” (Bell 1987, p. 235). Here I will not be concerned with this issue.

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Roadmap In Sect. 9.2 I analyse how the Mass Density ontology of GRWM has been interpreted within the PO framework. In Sect. 9.3 I argue that a crucial feature of the Mass Density is that the PO is not always well localised in 3D space. In Sect. 9.4 I suggest the notion of indeterminate PO as a way to understand the ontology of GRWM .

9.2 The PO Approach to GRW The guiding idea behind the spontaneous collapse models such as GRW (Ghirardi et al., 1986), is to modify the Schrödinger’s dynamical equation of standard QM by adding a stochastic and non-linear element to it. This allows for an explanation of the wave function collapse within the dynamics itself, and provides what Ghirardi Rimini and Weber called a “unified dynamics for microscopic and macroscopic phenomena” (Ghirardi et al., 1986). In theories like GRW, the collapse is an objective, physical mechanics, and contrary to standard QM, we need no obscure reference to observers, measurements, or experimental apparata in order to explain it. Given the dynamics of the theory, collapses happen spontaneously and randomly with a certain probability rate per unit time.2 The rate is such that for microscopic systems (like nucleons) the collapse of the wave function is incredibly rare, whereas for macroscopic objects made of a large number of mutually entangled particles, the collapse is practically certain to occur in a very short time. In this way, the theory allows to explain why microscopic objects can show quantum behaviour (such as interference pattern in a double slit experiment), and at the same time why at the macroscopic scale this behaviour has no effect. A major problem with any spontaneous collapse model is that the dynamical evolution never evolves into eigenstates of the relevant operators, but only very close to them. When a GRW collapse occurs, the wave function gets multiplied by a Gaussian that localizes the system with a certain accuracy. And although a large part of the post-collapse state is localized in a small portion of space, the system is also spread infinitely in both sides of the tails of the Gaussian. This is known as tails problem, and it is among the most discussed issues in the literature on GRW (for an overview, see Lewis (2003), and McQueen (2015)). The main strategy to solve this problem is to change the standard way of ascribing properties to physical systems starting from the quantum state, namely the Eigenstate-Eigenvalue Link (EEL). Several revisions to the EEL have been proposed in order to explain the definiteness of experimental outcome in GRW (Albert and Loewer 1996; Monton 2004; Lewis 2016, inter alia), and despite the differences between them, the general 2

This is achieved by introducing two constants for the spontaneous localization, one for its accuracy in space (α = 10−5 cm), and one for its frequency in time (λ = 10−16 s−1 ). These values for α and λ were proposed in Ghirardi et al. (1986), but I report that during the years different values have been proposed (e.g. Adler 2003), some of which have been empirically falsified. For a recent discussion, see Toroš and Bassi (2018).

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idea is to allow for property ascription even when the relevant observable is not in an eigenstate, but appropriately close to it. The major drawback of this strategy is that it seems to introduce a certain degree of vagueness and arbitrariness at the level of the ontology (Lewis 2016; Tumulka 2018). An alternative option to solve the tails problem, proposed in the context of the PO approach, consists in postulating additional ontology over and above the wave function.3 Moreover, this strategy looks especially motivated when we consider in more details one of the most developed versions of the theory, namely the one proposed in Ghirardi et al. (1995) and Bassi and Ghirardi (2004), and later called Mass Density GRW (GRWM for short).4 In every GRW-type of theory, the collapse is defined by picking a preferred basis on the Hilbert space on which it occurs. The crucial conceptual amendment of GRWM with respect to previous versions of the theory concerns the introduction of a new operator M(r) for the Mass Density, which serves as the preferred basis of collapse, and which is defined in Ghirardi et al. (1995) as follows: M(r) =

mk Nk (r)

(9.1)

k

Where k are the particles of a given type, r stands for a given spacetime point, and N is the operator describing the number of particles, which is in turn defined as: N(r) = a † (r)a(r)

(9.2)

In Ghirardi et al. (1986), the eigenbasis of N(r) was the preferred basis in the Hilbert space on which collapses occur, whereas in GRWM the selected basis is M(r). An important consequence of this amendment is that it provides a way to indicate unambiguously what the theory is about, its be-ables, by defining a Mass Density Function M(r, t) in 3D space.5 Consider a physical system S of N particles with corresponding Hilbert space H(S) of 3N dimensions. We then define M(r, t) as follows: M(r, t) = ψ(t)|M(r)|ψ(t)

3

(9.3)

See Tumulka (2018) for an extensive review of the reasons why a PO helps solving the conceptual problems of GRW, tails problem included. 4 Proponents of the PO have also individuated another version of GRW as a good exemplification of their view, namely the theory developed by Tumulka (2006) and known as GRW Flash. Since my focus in this paper is on GRWM only, I will not discuss this other option any further. 5 I note en passant that the reasons behind the choice made in Ghirardi et al. (1995) of a Mass Density operator also have to do with certain technical issues (see Ghirardi (2011) for a review) which are largely independent from the philosophical problem I am discussing here. This is important insofar as it reminds us that whatever inclination one has towards the PO approach, it is still a fact that in the most developed version of GRW mass is going to play a crucial role.

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|ψ(t) is the normalized vector6 describing S at time t, and M(r) is the mass density operator defined in Eq. (1) above. The mass density function defined in (3) provides a way to map H(S) onto the space of 3D functions r, at a given time t. If we now suppose that the physical system S is the whole universe (and therefore that H(S) is its corresponding Hilbert space), it follows that (3) gives the average, continuous distribution of mass throughout the 3D space. The conceptual move made by proponents of the PO approach is to posit an ontology, the Mass Density, that always has definite values in every point in 3D space, and which is fully represented by the Mass Density function M(r, t).7 This view was first suggested in Goldstein (1998), and then discussed in more details in Allori et al. (2008), Egg and Esfeld (2015), and Tumulka (2018). Ghirardi himself has expressed sympathy towards this approach in many of his writings (e.g. Ghirardi 2007; Ghirardi 2011). It is important to note that, on this view, although the ontology is determinate in every point in 3D space, the location of microscopic objects is not definite (Egg and Esfeld 2015, p. 2), since their mass is literally smeared out in physical space (again due the tails of the Gaussian). This is however not problematic as far as the large portion of the mass of a certain object is located within a small region (Egg and Esfeld, 2015). Microscopic objects are derivative entities which are grounded on the mass density distribution. And if there is any indeterminacy to them, this does not affect the PO itself. What really matters is that the fundamental ontology, which is given by the distribution of mass throughout space, is not itself indeterminate. And indeed, the idea that the mass density distribution is, to use Glick’s words, “perfectly determinate” (Glick 2017, p. 205) has been advocated many times in the literature (Allori, 2013; Chen, 2020; Egg and Esfeld, 2015; Glick, 2017; Tumulka, 2018). In the next section I argue that this supposition is however unmotivated, and that we have good reasons to believe that the mass density (so the PO) may not be always well localised in 3D space, and may therefore be indeterminate.

9.3 Accessible and Non-accessible Mass The Mass Density function M(r, t) is a many to one mapping, as Ghirardi et al. (1995) immediately notice. To see this, consider a large number of particles N and two regions A and B both of spherical shape and of the same size, and then compare the following two states |ψ ⊕  and |ψ ⊗ : 1  |ψ ⊕  = √ |ψNA  + |ψNB  2

6

(9.4)

Notice that the Stratonovich equation of any GRW-type of theory does not actually generate normalized vectors. I am going to set this complication aside here. 7 This also means that there no hidden variables here.

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A B |ψ ⊗  = |φN/2  ⊗ |φN/2 

(9.5)

Eq. (9.4) expresses a linear superposition of equal amplitudes of the states |ψNA  and A  |ψNB . Eq. (5), on the other hand, expresses the tensor product of the states |φN/2 B  describing the physical situation of N/2 particles in region A and N/2 and |φN/2 particles in region B. Now notice that the states |ψ ⊕  and |ψ ⊗  give rise to the same mass density function M(r, t) for each region A and B. Consider for example region A: ⊕ ⊕ M⊕ (r,t) = ψt |M(r)|ψt  ≈

1 A Nm ψN |M(r)|ψNA  ≈ 2 2

(9.6)

Nm 2

(9.7)

⊗ ⊗ A A M⊗ (r,t) = φt |M(r)|φt  ≈ φN/2 |M(r)|φN/2  ≈

The same goes for region B. Although the functions generated by (9.6) and (9.7) are the same, it is important to discriminate between the states that originate them. For instance, Monton (2004, pp. 14–15) imagines a particle traveling between regions A and B, and ask what we should expect to happen in both cases. In the case of |ψ ⊕ , the particle would become entangled with the mass in both regions, and would therefore be deflected upwards or downwards with equal probability. In the case of |ψ ⊗  instead, since both regions have the same mass density, the particle would proceed its trajectory undeflected. To explain the difference between the two states, Ghirardi et al. (1995) define a criterion for individuating what are the states that give rise to accessible mass distributions (like |ψ ⊗ ), and what are the states that do not (|ψ ⊕ ). Their method is simply to define the ratio between the mean expectation value for a given outcome and the variance. We first define the variance V(r, t) for the mass density operator M(r) as follows: V(r, t) = ψ(t)| [M(r) − ψ(t)|M(r)|ψ(t)]2 |ψ(t)

(9.8)

Given V(r, t), we can define the ratio: R2 (r, t) = V(r, t)/M2 (r, t)

(9.9)

Now, if R turns out to be much smaller than 1, this suggests that the corresponding mass density can be considered accessible. If instead R is close to 1, the corresponding mass is defined as non-accessible. Thus, we can now state the following Criterion of Accessibility (CAM) for any Mass Density state: CAM — M(r, t) is accessible iff R(r, t)  1.

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Given CAM, it can be shown that in the above example the mass corresponding to the state |ψ ⊗  is accessible because the value of R is much smaller than 1: R⊗ (r, t)  1

(9.10)

Contrariwise, for |ψ ⊕  the value of R is close to 1, and therefore the corresponding mass is non-accessible. R⊕ (r, t) ≈ 1

(9.11)

According to Bassi and Ghirardi (2004), the Criterion of Accessibility, along with the distinction between accessible and non-accessible mass, is what explains why, as we should have expected all along, macroscopic superposition states like |ψ ⊕  are not empirically accessible. Now let us ask: what is the ontological status of the non-accessible portion of mass? Recall that, on the PO approach, the ontology of GRWM is fully represented by the Mass Density function M(r, t). However, as I have just shown, there can be different states corresponding to the same M(r, t), not all of which describe well localised mass density configurations. We seem to have two options here: either we reject states like |ψ ⊕  as representing something real, or we provide an explanation of the difference between the states |ψ ⊕  and |ψ ⊗  in terms of the PO.8 The first option is to simply stipulate that there is no ontology corresponding to non-accessible Mass. As a matter of fact, this option is suggested by Ghirardi and collaborators in the very same paper where the argument I gave above is presented (Ghirardi et al., 1995), where instead of “not accessible” it is used the adjective “not objective” to refer to the mass corresponding to states like |ψ ⊕ . This option looks however highly problematic. Given that the Criterion of Accessibility is purely operational (recall that it is given by the variance), by claiming that states like |ψ ⊕  are not objective or real it would follow that what exists according to the theory, its ontology, depends on what observers can and cannot do.9 And in effect Tumulka— one of the major defenders of the PO approach for GRWM —is explicit in rejecting this option: [. . . ] the PO does provide a picture of reality that conforms with our everyday intuition. All this is independent of whether the PO is observable (accessible) or not. Bassi and Ghirardi sometimes sound as if they did not take the matter density seriously when it is

8

There is actually a third option, which is to accept that there is more to the quantum state than just the PO. The reason why I do not consider this view is because it would entail that the PO is redundant, since we would still need the quantum state to play an ontological role (beyond just determining the dynamics). But since the whole point of positing the PO is to avoid that, this option should not be adopted. 9 A similar argument can be found in McQueen (2015), Tumulka (2018), Monton (2004), inter alia. See Mariani (2022) for a more extensive critique of this approach to non-accessible mass, and for a review of the various options.

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not accessible; I submit that the PO should always be taken seriously. (Tumulka 2018, p. 142, italics mine)

Tumulka suggests that the second option is preferable, and so we have to accept the existence of non-accessible mass and give an ontological explanation of the difference between |ψ ⊕  and |ψ ⊗  in terms of the PO. The most immediate way to do so would be to simply assume that, even if not accessible, in the state |ψ ⊕  the mass is spread evenly over the two regions, half in A and half in B (precisely as it happens for the state |ψ ⊗ ). But then the natural worry is that we are left with no explanation whatsoever as to why, in the case of |ψ ⊕ , the test particle we send through regions A and B is deflected, whereas in the case of |ψ ⊗  it is not. As Myrvold (2018) nicely puts it: “[s]omething that you might be inclined to call a “mass”, if it doesn’t act like a mass, is not a mass” (p. 114). And even if we were to call it “mass”, the problem is still that if there is PO corresponding to states like |ψ ⊕ , and if we want to explain the physical difference with respect to the state |ψ ⊗ , we simply have to accept that in the former case the “mass” is not associated to a definite region of 3D space.10 We cannot simply stipulate that it is, for if the configuration of the PO is the same, we would have no explanation whatsoever for the difference between the two states. The explanation has to be based in the PO itself.11 Thus, the question before us is this: is there a plausible way to take nonaccessible mass states seriously, and give an ontological explanation to them, while still endorsing the PO approach?

9.4 The Indeterminate Primitive Ontology I suspect that at this point the reader may be inclined to wonder why not simply take the arguments in the previous section as a straightforward objection to the PO approach. The thought would go something like this: if the ontology of GRWM allows for states of non-accessible mass, and if these states are not well localised in 3D space (as the view requires them to be), then the PO approach fails to apply to GRWM . This is a fair point, and yet I believe it generates from a confusion on the

10 Also

note that, for this reason, the ontology of GRWM is crucially different from the case of a classical field in which, despite not being well localised, the ontology is definitely associated to any spacetime point. I thank two anonymous reviewers of this journals for inviting me to elaborate on this point. 11 As Tumulka claims in the very same context: [The problem] concerns whether GRW theories provide a picture of reality that conforms with our everyday intuition. Such a worry cannot be answered by pointing out what an observer can or cannot measure. Instead, I think, the answer can only lie in what the ontology is like, not in what observers see of it. (Tumulka 2018, 142)

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very role that the PO is supposed to play as something we need to postulate over and above the wave function. In a somewhat general way, the need to posit a PO comes from the simple thought that any physical theory should describe something in the world, the stuff in 3D space, and that everything else should be reduced to, and explained by the behaviour of the PO. Moreover, this approach is motivated by the idea that the quantum state or the wave function are not the right candidates for a satisfactory ontology. If both these goals are achieved in GRWM , then why bother if the PO turns out to be indeterminate in the sense of lacking definite properties? Perhaps, once we realise that the general reductive explanatory scheme proposed by the PO approach is indeed satisfied, and that an ontology beyond the wave function is provided, a more interesting question to address is what motivated in the first place the claim that the PO must be definite and well localised in every point in 3D space. As I anticipated in the introduction to this paper, I think that the justification for this claim goes something like this: (i) the PO of a theory is, by definition, its fundamental ontology; (ii) the fundamental ontology cannot be indeterminate; therefore, the PO cannot be indeterminate. This very simple two-premise argument not only explains why proponents of the PO assume that the determinacy is a requirement, but also suggests why, for instance, Myrvold (2018) takes the existence of non-accessible mass as a good reason to reject the PO and endorse realism towards the quantum state in the context of GRWM . As it should be clear enough by now, the point I am trying to make is that premise (ii) of the above argument is unwarranted, for we have no reasons to believe that fundamental indeterminacy is incoherent. I will shortly come to this. First though, since I think it is very instructive to realise that the truth of this premise is assumed by both defenders and detractors of the PO approach, let me spend a few words on Myrvold’s view. Some of the attempts to provide an understanding of the ontology of GRW refer, sometimes explicitly, to the notions of indefiniteness, vagueness, or fuzzyness. In most of the cases, these notions are meant to indicate that the fundamental entities described by this theory may objectively lack definite values for their properties.12 Myrvold’s Distributional Ontology (Myrvold, 2018) is a clear example: In classical physics, dynamical quantities always possess precise values. In quantum theory, there is always some imprecision [. . . ] But the full reality is that associated with each dynamical variable is a distribution of values. This distribution, though formally like a probability distribution, is to be thought of not as a probability distribution over a precise but unknown possessed value but as reflecting a physical, ontological, lack of determinacy about what the value is. (p. 118)

Myrvold argues for this view mainly based on Ghirardi’s argument about nonaccessible mass which I also gave in Sect. 9.3. However, he also explicitly takes this argument as an objection to the PO program, and then defends a view according

12 For reasons of space, I cannot discuss other views in this vicinity. A notable example is Monton’s Mass Density Simpliciter view, which, as discussed in McQueen (2015) and Mariani (2022), also seems to allow for indeterminacy in the ontology.

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to which ultimately the mass density (along with every other physical property) is grounded on the quantum state. So in this case too, any indeterminacy appears at some derivative level (the mass density, for instance), but does not affect the fundamental level (the quantum state). However, if one were convinced that the quantum state is not the right kind of entity to be a candidate for the PO, and that we need to posit additional ontology beyond it, then presumably Myrvold’s conclusion would hardly be taken to follow from the premises.13 Once again, for such conclusion to be justified, we also need the assumption that the PO must be determinate, if it exists at all. If instead, as I am suggesting in this paper, we give up on this idea, there is no need to reject the PO approach and endorse the view that the quantum state is more fundamental than the mass density distribution. I have shown that there are good naturalistic reasons for taking seriously the idea that the PO may be indeterminate in GRW. Moreover, I have individuated what seems to be the cause of the scepticism towards this idea, and which is also probably why it has not been developed so far, namely the thought according to which the fundamental ontology cannot be indeterminate. As a matter of fact though, some fairly recent developments in the metaphysics of physics seem to go in the very opposite direction, suggesting that we can indeed make sense of this very idea. In particular, many authors (Darby 2014; Calosi and Wilson 2018; Torza 2017; for an overview see Calosi and Mariani 2021) have suggested that QM, by violating the supposition that objects always have definite values for their properties, may provide an instance of what philosophers call ontological indeterminacy. For several decades this notion was not even considered to be consistent (notably, Lewis 1986). Quite recently however, it has been shown that we can indeed provide clear accounts of what it means for something to be objectively indeterminate (Barnes and Williams 2011; Wilson 2013, inter alia), along with well defined criteria for distinguishing determinate from indeterminate states of affairs. As it happens sometimes, a good conceptual analysis may be useful in providing a more refined picture of what physics tells us, even though this might entail a departure from our classical presuppositions about the world. For all these reasons, I take it that the main lesson we learn from GRW is that we should start to seriously entertain the possibility that the world is fundamentally indeterminate.14

13 Myrvold

himself recognizes this, and goes on to defend the viability of quantum state realism. develop this idea further here. However, a good working hypothesis seems to me that the approach to indeterminacy developed in Calosi and Wilson (2018) may be used in the context of GRWM , for it would allow to distinguish between indeterminate and determinate states of affairs, thus providing an explanation to the distinction between accessible and non-accessible mass that is based on the PO itself rather than on the quantum state. 14 I cannot

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9.5 Conclusions The core motivation for adopting the Primitive Ontology approach is to provide a classical reductive explanation of the behaviour of macroscopic objects as determined by the behaviour of the microscopic, fundamental ontology. Contrary to what proponents of this approach seem to suggest, however, all of this is independent from whether the PO is indeterminate or not. To make this point, in this paper I have been focusing on GRWM , which has been taken by proponents of the PO approach as one of the best exemplifications of their view. The main claim of this paper is that, since it is suggested by one of the major interpretations of QM, the notion of indeterminate PO should be taken seriously from a naturalistic point of view. Acknowledgments For useful comments on previous versions of this paper, I thank Valia Allori, Claudio Calosi, Vincent Lam, Maria Maffei, Giuliano Torrengo, and an anonymous reviewer for this journal. I acknowledge the generous support of the Foundational Questions Institute Fund (Grant number FQXi-IAF19-05 and FQXi-IAF19-01).

References Adler, S. L. (2003). Why decoherence has not solved the measurement problem: A response to P.W. Anderson. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 34(1), 135–142. Albert, D., & Loewer, B. (1996). Tails of Schrödinger’s cat. In R. Clifton (Ed.), Perspectives on quantum reality (pp. 81–91). Kluwer. Allori, V. (2013). On the metaphysics of quantum mechanics. In S. Le Bihan (Ed.), Precis de la Philosophie de la Physique: d’aujourd’hui ‘a demain. Vuibert. Allori, V. (2016). Primitive ontology and the classical world. In R. Kastner, J. Jeknic-Dugic, & G. Jaroszkiewicz (Eds.), Quantum structural studies: Classical emergence from the quantum level (pp. 175–199). World Scientific. Allori, V., Goldstein, S., Tumulka, R., & Zanghì, N. (2008). On the common structure of Bohmian mechanics and the Ghirardi-Rimini-Weber theory. British Journal for the Philosophy of Science, 59, 353–389. Barnes, E., & Williams, R. (2011). A theory of metaphysical indeterminacy. In K. Bennett & D. Zimmerman (Eds.), Oxford studies in metaphysics (Vol. 6). Oxford University Press. Bassi, A. & Ghirardi G. C. (2004). Dynamical reduction models. Physics Reports, 379, 257. Bell, J. S. (1987). Speakable and unspeakable in quantum mechanics (2nd ed., 2004). Cambridge University Press. Calosi, C. & Mariani, C. (2021). Quantum Indeterminacy. Philosophy Compass, 16(4), e12731. Calosi, C. & Wilson, J. (2018). Quantum metaphysical indeterminacy. Philosophical Studies, 176, 1–29. Chen, E. K. (2020). Fundamental Nomic Vagueness. arXiv:2006.05298 [physics.hist-ph]. https:// arxiv.org/abs/2006.05298 Darby, G. (2014). Vague objects in quantum mechanics? In K. Akiba & A. Abasnezhad (Eds.), Vague objects and vague identity. New essays on ontic vagueness (pp. 69–108). Springer. Egg, M., & Esfeld, M. (2015). Primitive ontology and quantum state in the GRW matter density theory. Synthese, 192, 3229–3245.

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Ghirardi, G. C. (2007). Some reflections inspired by my research activity in quantum mechanics. Journal of Physics A, 40, 2891. Ghirardi, G. C. (2011). Collapse theories. Stanford encyclopedia of philosophy. Published online by Stanford University at http://plato.stanford.edu/entries/qm-collapse/ Ghirardi, G. C., Grassi, R., & Benatti F. (1995). Describing the macroscopic world: Closing the circle within the dynamical reduction program. Foundations of Physics, 25, 5–38. Ghirardi, G. C., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic physics. Physical Review, D34, 470–491. Glick, D. (2017). Against quantum indeterminacy. Thought, 6(3), 204–213. Goldstein, S. (1998). Quantum theory without observers. Physics Today, Part I, (3), 42–46; Part II, (4), 38–42. Lewis, D. (1986). On the plurality of worlds. Blackwell. Lewis, P. J. (2003). Quantum mechanics and ordinary language: The fuzzy-link. Philosophy of Science, 70, 1437–1446. Lewis, P. J. (2016). Quantum ontology. A guide to the metaphysics of quantum mechanics. Oxford University Press. Mariani, C. (2022). Non-Accessible Mass and the Ontology of GRW. Studies in History and Philosophy of Science, 91, 270–279. Mariani, C. (2021). Emergent Quantum Indeterminacy. Ratio, 34(3), 183–192. McQueen, K. J. (2015). Four tails problems for dynamical collapse theories. Studies in History and Philosophy of Modern Physics, 49, 10–18. Monton, B. (2004). The problem of ontology for spontaneous collapse theories. Studies in History and Philosophy Modern Physics, 35, 407–421. Myrvold, W. C. (2018). Ontology for collapse theories. In S. Gao (Ed.), Collapse of the wave function. Models, ontology, origin, and implications (pp. 97–123). Cambridge University Press. Toroš, M., & Bassi, A. (2018). Bounds on quantum collapse models from matter-wave interferometry: Calculational details. Journal of Physics A: Mathematical and Theoretical, 51(11), 115302. Torza, A. (2017). Quantum metaphysical indeterminacy and worldly incompleteness. Synthese, 197(10), 4251. Tumulka, R. (2006). A relativistic version of the Ghirardi–Rimini–Weber model. Journal of Statistical Physics, 125(4), 821–840. Tumulka, R. (2018). Paradoxes and primitive ontology in collapse theories of quantum mechanics. In S. Gao (Ed.), Collapse of the wave function. Models, ontology, origin, and implications (pp. 134–153). Cambridge University Press. Wallace, D. (2018). On the plurality of quantum theories: Quantum theory as a framework, and its implications for the quantum measurement problem. Phil.Sci Archive. [Preprint]. Wilson, J. (2013). A determinable-based account of metaphysical indeterminacy. Inquiry, 56(4), 359–385.

Chapter 10

Towards a Structuralist Elimination of Properties Valia Allori

Abstract Scientific realists investigate the ontology of the world and explain the observed phenomena by using our best fundamental physical theories. These theories describe the behavior of fundamental objects in terms of their fundamental properties, which determine their behavior. This paper is the natural companion of another paper in which I propose an alternative to this traditional account of metaphysics, according to which fundamental objects have no other fundamental property than the one needed to specify their nature. In that paper I argue that my view fares better than the traditional metaphysics both in the classical and in the quantum domain. In this paper I compare my view to structuralism. After discussing that my proposal shares many motivations with structuralism, I argue in which ways I think mine is superior.

10.1 Introduction In discussions about ontology, traditionally one talks about objects and their properties: objects live in three-dimensional space and change in time according to suitable laws of nature. In naturalized metaphysics, following the tradition of scientific realism, the nature of these objects and laws is given by our best fundamental physical theories. In a classical world one has point-like particles (or particles and fields, if we include electromagnetic phenomena). Their temporal evolution is determined by laws of nature like the Newtonian law of gravitation or Maxwell’s electromagnetic equations. Particles come into families: electrons, protons, and so on, identified by their intrinsic properties of mass and charge. A distinctive feature of this understanding is that different families move differently under the same circumstances because their properties are different. This traditional approach is therefore an object-oriented metaphysics grounded on properties. In

V. Allori () Department of Philosophy, Northern Illinois University, Dekalb, IL, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_10

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a previous article, I have proposed instead a thin object-oriented metaphysics grounded on structure based on the idea that fundamental objects have no intrinsic properties other than the ones necessary to identify their nature. That is, assuming the world is made of particles, they merely have the property of being localized in space, but they have no additional properties such as mass or charge. Conversely, laws are effectively instantiated multiple times. Therefore, an electron in a magnetic field turning the opposite way of a proton is not explained in term of their opposite charges but in terms of them following distinct ‘effective’ laws. In my previous paper I have argued that this approach is better than the traditional view. In this article, which is a continuation of my previous work, I first review my approach in Sect. 10.2, then I compare it with (ontic) structuralism. In this approach structure is fundamental, and symmetries play a crucial role. As such, it constitutes a structureoriented metaphysics based on symmetries. In Sect. 10.3, I argue that my approach and structuralism share many motivations. However, I conclude that my view is to be preferred as it is less problematical in several respects.

10.2 A New Take on Naturalized Metaphysics Scientific realists assume that scientific theories are approximately true and use them to investigate questions about the nature of reality. Thus, they engage in naturalized metaphysics: they look at scientific theories to individuate the nature of fundamental entities, their properties, and the laws governing their motion.

10.2.1 The Traditional View: Thick Object-Oriented Metaphysics Grounded on Properties Metaphysicians disagree about the nature of the objects that exist in the world (particles, fields, or else). They also disagree about the nature of laws governing their behavior (supervenient over the objects, or not), about the nature of their properties (dispositional or categorical properties), as well as the nature of space (fundamental or emergent). However, most share the common understanding that objects behave differently due to their properties. I have called this the traditional view.1 It can be tracked back to Aristotle, and one of its modern proponents is David Lewis: at the fundamental level, the world is a ‘mosaic’ of “local, particular matters of fact.”2 For instance, he would say that in a world governed by classical electromagnetism there are point-particles obeying Newton’s law, and electromagnetic fields, whose intensity changes from point to point as dictated by Maxwell’s laws. Fundamental 1 2

Allori (forthcoming). Lewis (1986).

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properties are introduced to account for the objects’ different behavior under the same circumstances: two particles accelerate differently in an electromagnetic field because they have different charges. In this sense objects are thick: they are ‘dressed up’ with properties. Consequently, there are families of objects: ‘electrons’ are particles with negative charge, ‘protons’ are particles with positive charge, and so on. There is one fundamental law, and many fundamental properties: there are many families (electrons, protons, and so on) of the same kind of fundamental entities (particles).

10.2.2 Objections The traditional approach seems the natural approach at least until one considers the quantum domain. What are the quantum objects, and what are their properties? As for the first question, it is not clear what the ontology of the theory is. One possibility, dubbed wavefunction realism,3 is to think of quantum mechanics as a theory about the wavefunction, as it is the main mathematical object on the theory. However, the wavefunction, unlike electromagnetic fields which are in three-dimensional space, represents a wave in configuration space, which is a high-dimensional space. Accordingly, wavefunction realists argue that the threedimensional objects of our experience are emergent or derivative.4 An alternative to wavefunction realism is the primitive ontology approach,5 according to which the ontology is some three-dimensional entity instead. In contrast with wavefunction realism, this approach preserves most of the classical explanatory schema in terms of compositionality and reductionism: macroscopic objects are composed of the three-dimensional microscopic entities represented by the primitive ontology. The main challenge to this approach is that it is unclear what wavefunction is. There are various proposals,6 the best of which is, I think, to understand the wavefunction structurally.7

3

See Albert and Ney (2013), and then most notably Albert (1996, 2015), Lewis (2004, 2005, 2006, 2013), Ney (2012, 2013, 2015, 2017, 2021), North (2013). 4 In one approach, macroscopic objects can be functionally defined in terms of their role (Albert, 2015). In another approach three-dimensional objects exists as derivative when considering symmetry properties as fundamental facts about the world (Ney, 2021). 5 Dürr et al. (1992), Allori et al. (2008), Allori (2013a, b, 2015a, b, 2019a, b) and references therein. 6 Some have argued it is a property of matter (Monton, 2002; Lewis, 2013; Solé, 2013; Esfeld et al., 2014; Suàrez, 2015). Another approach is to take the wavefunction as a law (see Goldstein & Zanghì, 2013; Allori, 2018a, b, and references therein for a discussion), which seems particularly fitting to the Humean account of laws (Esfeld, 2014; Callender, 2015; Miller, 2014; Bhogal & Perry, 2017). For antirealism about the wavefunction see Healey (2012). 7 This view has been defended in Allori (2021b). Another structuralist perspective has also been defended by Lewis (2020), who writes: “the wave function describes the structure instantiated by whatever fundamental entities there may be in ordinary three-dimensional space: particles, fields,

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Moreover, new intrinsic properties such as spin have been introduced in quantum theory to account for new experimental results. However, they are very different from classical properties, as they are contextual: their value depends on the experiment performed to measure them. How does the traditional view fare in the quantum domain? As I have previously argued,8 the traditional account is not a good fit for wavefunction realism. In fact, this view requires a substantial revision of the traditional view’s explanation based on properties, even without considering spin. Property talk is fictional: useful, but not fundamental. In contrast, being based on a three-dimensional reductionist explanatory schema, the traditional view may still be viable for the primitive ontology approach. However, there are other, more general, reasons to prefer an alternative view, which I briefly summarize in the next subsection.

10.2.3 Fundamental Entities Without Fundamental Properties: Thin Objects-Oriented Metaphysics Grounded on Structure The basic idea is that there are fundamental objects, but they have no property other than the one uniquely characterizing their nature. Thus, objects are ‘thin.’ For instance, the only property possessed by particles is their location. The only property possessed by fields is the set of their intensity values. In contrast with the traditional account, therefore, there are no different families of the fundamental entity that constitute matter, and all fundamental entities are identical, as far as (nonspatiotemporal) properties are concerned. In this framework the observed different behavior of particles appearing to belong to different families is accounted for in terms of laws of nature: fundamental entities behave differently in the same situation because they are governed by different effective laws. There is one effective law for what appears to be a different family in the traditional approach. In their mathematical formulation, laws include constants, like the gravitational constant, and parameters describing properties, like masses and chargers. In the traditional approach the constants are part of the definition of the law, while the parameters are part of the definition of matter: constants remain identical independently of the object the law applies to, while masses do not. In my view instead, the parameters are part of the definition of the law too. Thus, a single law ‘splits’ into a number of effective laws, characterized by the relevant parameters, one for each particle family of the traditional view.9 Thus, the main idea is the opposite of the traditional

flashes, mass density, or something else entirely. A structure is not in itself an object, but rather a way that objects relate to each other.” 8 Allori (forthcoming). 9 For example, in case of a gravitational field, there is one effective law for the ‘electron,’ Eff law 1 = Hr 21 , where H1 = Gme M, one for the ‘proton,’ Eff law 2 = Hr 22 , where

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view: there one would minimize the number of laws to maximize the number of properties while here we minimize the number of properties, allowing for as many effective laws of nature as needed to make the theory empirically adequate. The laws and the effective laws are naturally seen as a structure, a network of relations, grounded on thin objects. The nature of the objects and the form laws have an interesting connection with symmetry properties of the theory, which in my approach are symmetries of the objects, not the structure. First, if one cares about theories having symmetries, one should not allow certain entities to be though as fundamental. For instance, the wavefunction thought of as a fundamental object makes nonrelativistic quantum mechanics no longer Galilei invariant.10 Moreover, the form of the law of nature is constrained by symmetry considerations: for instance, in the pilot-wave theory, the law for the particles is chosen as the simplest equation which allows for the theory to be Galilei and time-reversal invariant.11 To summarize, in my view we have thin objects, individuated by their only natural property, namely by the property that uniquely characterizes their nature. Then the law of nature determines how they evolve in time. For the theory to be empirically adequate, the law has to break down into a set of effective laws, each of which applies to a subset of the set of fundamental objects. The laws and the effective laws are naturally seen as a structure, a network of relations, grounded on thin objects. Moreover, symmetries constrain the possible nature of the fundamental objects, and the form of the laws.

10.2.4 Advantages and Replies to Objections As I argued in my companion paper, this view extends nicely to the quantum domain, regardless of whether one endorses the primitive ontology approach or wavefunction realism. In fact, in the primitive ontology approach the primitive ontology represents objects, while the wavefunction relates the location of the objects at different times. Moreover, since in wavefunction realism property talk is completely fictional, the derived three-dimensional objects are thin: they have no fundamental property. Wavefunction realism is doubly structural: the structure of the wavefunction allows for the derivation of three-dimensional objects, and it explains the objects’ behavior by connecting their locations at different times. H2 = Gmp M, and one for the ‘neutron,’ Eff law 3 = Hr 23 , where H3 = Gmn M (where G is the gravitational constant, me , mp , and mn are respectively the mass of the electron, proton and neutron as traditionally intended, while M is the reference mass, and r is the distance between the reference particle and the one under examination). See Allori (forthcoming) for more details. 10 Allori (2018a, b). See also Allori (2015c, 2019b, 2021a) for an argument that electromagnetic fields cannot be thought as fundamental objects otherwise they either would transform at odds with their nature, or we should stop thinking of classical electrodynamics as time-reversal invariant.. 11 Dürr et al. (1992).

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Aside from extending to the quantum domain, my approach also by-passes the debate over the nature of properties, as it focuses on laws. Also, it is more parsimonious than the traditional view, given that it has an ontology of objects and laws, rather than objects, laws and properties. In this sense, even if it is compatible with other views, my approach may be seen as a natural extension of Humeanism, since it provides the best combination of simplicity and strength.12 Moreover, it is more explanatory than the traditional view, as it has less things to account for, like for example the values of the masses of the fundamental particles. In addition, in the framework of classical electrodynamics, my account makes sense of certain asymmetries between particles and fields, like the fact that particles have properties and fields do not. Another consideration is that we observe objects that move, not their properties: we see positions that change; we do not see masses, charges, or spin. Furthermore, the problematic contextuality of spin in quantum mechanics goes away, because there is no spin property. Here are some possible objections.13 First, this view seems unnecessarily radical, as properties are essential to explanation. However, this is not so: properties add a mysterious ontological category, while their explanatory role can be taken over by the laws. In addition, one may think that the way in which an object ‘pairs up’ with its effective law is mysterious. Nonetheless, this is merely using a different primitive: while in the traditional view it is a primitive fact that positive charge will result in ‘going left’ in a given magnetic field, here it is a primitive fact that that effective laws act as they do on the various objects.

10.3 Structuralism I have used the word ‘structure’ to describe my view. However, the dominant metaphysical approach invoking the notion of structure is structural realism (structuralism). In this section I compare these two frameworks. Structuralism is a realist approach partly motivated by the development of quantum theory. Nonetheless, an early version of structuralism was proposed in response to the pessimistic metainduction argument. This goes after the ‘no-miracle’ argument for realism: the best explanation for the success of our best theories is their truth, otherwise their success would be a miracle. The pessimistic meta-induction argument states that success is no indication of truth, as some past false theory were successful. In response, if one can show that the entities that are retained in moving from one theory to the next are the ones that are responsible for the empirical success of the theory, then the previous argument is blocked. Structuralists notice that what carries over in

12 See 13 For

Esfeld (2014) for a similar argument for his super Humeanism. See also Hall (2015). more details of this view, its objections and motivations, see Allori (forthcoming).

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theory change is the mathematical structure of the theory.14 Arguably, one could limit structuralism to epistemology: there are objects and structure, but we can only know about structure. However, other types of structuralism are ontic: they maintain that there is only structure. In this paper I focus on ontic approaches.15 In particular, I discuss two such views. One is radical structuralism, or eliminativism16 which claims that there are no objects, only relational structures. Another is the socalled moderate structuralism according to which, contrary to eliminativism, both objects and relations among them exist but, contrary to the traditional view, objects cannot exist independently on the structure they are related by.17 Objects are only characterized by the relations in which they stand, and these relational properties constitute the way the objects can be.18 While initially the view stated that objects do not possess any intrinsic property, in reply to criticisms it was later amended to include them.19

10.3.1 Arguments from Quantum Mechanics and Other Motivations One underlying principle that supports structuralism is that we are not justified in postulating the existence of something whose existence cannot be known. This coherence between epistemology and metaphysics leads the structuralist to consider all cases of underdetermination as evidence for their views. This type of argument has been put forward first in quantum theory, but later applied to relativity as well. The idea is that quantum statistics suggest that it is underdetermined whether quantum objects are individuals (i.e., have intrinsic properties) or not, from which eliminativists conclude there is only structure.20 Similar arguments are based on symmetries and the corresponding invariances,21 and on the existence of multiple empirically equivalent mathematical formulations of the same theory.22 14 Worrall

(1989). whenever I write ‘structuralism’ in the rest of the paper, I mean ‘ontic structuralism.’ 16 See Ladyman (1998), French and Ladyman (2003), Ladyman and Ross (2007), French (2010, 2014) and references therein. 17 This view is defended most notably defended by Esfeld (2004) and refined in Esfeld and Lam (2008, 2010, 2012). 18 This view is inspired by Heil (2003) and Strawson (2008), who argue that the intrinsic properties of an object are the ways that object can be. See also Armstrong (1989). 19 Esfeld and Lam (2010). 20 French (2014) and references therein. See Saunders (2006), French and Krause (2006), Muller and Saunders (2008), Ladyman and Bigaj (2010) for further discussion. 21 Permutation invariance in many-particle quantum mechanics (Muller, 2009), gauge diffeomorphism invariance in general theory of relativity (Rickles, 2006, Esfeld & Lam, 2008). 22 See Bain (2006, 2009) in the context of the general theory of relativity; however, see Cao (2003), Pooley (2006). 15 Therefore,

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Another argument for structuralism focuses on entanglement. In quantum mechanics, the sub-systems composing an entangled system have no individual wavefunction, and thus they have no intrinsic properties. Therefore, structuralists think they are best understood as relata of a common, entanglement structure.23 Another argument from quantum mechanics comes from the failure of Humean supervenience, given that the wavefunction is nonlocal, being in configuration space.24 Structuralists argue that the nonlocal relations being instantiated by individuals has to be abandoned toward an ontology of structural relations.25 Structuralism is also motivated by ontological parsimony and simplicity of description: why postulate two categories (objects and relations), if one (relations) is enough? This is connected to another important aspect of structuralism: the relation between laws and properties.26 Part of the traditional debate is between the defenders of categorical properties and the defenders of dispositional properties, without any consensus. Structuralists avoid this dilemma by assuming that laws supervene on the fundamental relations, and fundamental properties are emergent. French calls it ‘reverse engineering:’ while the Humeans take properties as fundamental and laws as emergent, the structuralist takes laws as primary and recovers properties from them.

10.3.2 Objections The strongest charge against eliminativism is that it is unintelligible: relations cannot exist without relata.27 However, one could reply assuming these relations are abstract universals, and we are wrong when we think that they need relata because they do when they are physically instantiated.28 Alternatively, moderate structuralism has been proposed to respond to this charge, as the relata exist.29 In addition, many have complained that the relationship between structure and laws is insufficiently detailed.30 Nonetheless, I believe the structuralist can

23 For instance, a singlet state of two entangled spin ½ sub-systems is in a definite spin state, namely 0, but neither of the sub-systems has a definite spin state on its own. As such, it is argued that these sub-systems are best understood as relata of the fundamental entanglement relation they stand in, in this case: ‘has opposite spin to.’ For more on this argument, see Esfeld (2004). 24 See Teller (1986) for this argument. For discussion, see for instance Maudlin (2007), Ladyman and Ross (2007), Esfeld (2009), French (2014). 25 Ladyman (1998), French and Ladyman (2003), Esfeld (2004). 26 French (2014), and Esfeld (2014). 27 See for instance Busch (2003), Cao (2003), Chakravartty (2003), Morganti (2004), Psillos (2006). 28 Esfeld and Lam (2008). 29 Esfeld (2004). 30 Chakravartty (2007).

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overcome this objection taking structure as primitive, and then define laws as ‘whatever determines the behavior of such structure.’ Moreover, structuralism seems to be unable to account for intrinsic properties like masses and charges.31 This charge is address by using symmetries, to recover properties using suitable group representations.32 Then, people have objected that the structuralism is unable to account for causation.33 However, one could simply maintain that in physics there is no room for causation,34 or propose an account of causal structure.35 Finally, it has been maintained that structuralism does not defeat the pessimistic meta-induction argument.36 In particular, it was argued that structuralism fails in the classical-to-quantum theory change: structuralists think that the wavefunction is the structure, but there is no classical analog for it, so how can structure be preserved?37 This problem is not mitigated in moderate structuralism, given that objects and structure are ontologically at the same level.

10.3.3 Comparison between the Thin Objects View and Structuralism The first similarity between my view and structuralism is that they both rely on the notion of structure, which is a network of relations: without relata, in the case of eliminativism, definitive of objects, in the case of moderate structuralism, and between thin objects in my case. Both approaches also work well in the quantum domain, in contrast with the traditional view. At least partially, structuralism was motivated from the desire of understanding quantum theory. My account less so, but if we can make sense of the view that there are no genuine quantum properties like spin, then it also seems natural to extend this attitude into the classical framework and assume that what we called intrinsic properties (mass, charge) are part of the law. Moreover, both approaches tend to consider the wavefunction as playing a secondary, perhaps instrumentalist role, even in different ways. French (2013) has argued that since it is underdetermined what the wavefunction is. However, this underdetermination can be broken by conceiving the wavefunction as being “constituted by the laws and the associated symmetry principles.” Thus, since

31 Ainsworth

(2010). (1998) and references therein; see also Muller (2009). See Esfeld and Lam (2010) for criticisms. 33 Psillos (2006). 34 Russell (1912), Ladyman (2008). 35 Esfeld (2009), Esfeld and Sachse (2011), chapter 2. 36 Chakravartty (2004). 37 Allori (2019b). 32 See Castellani

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the wavefunction ‘comes out’ from the symmetries, it is defined in terms of the fundamental structure. In my view, instead, the wavefunction is part of the definition of the law. As such, in both approaches not only the wavefunction is not a material object but also it may not even be real. That is, unless one has a realist conception of laws, which however is not necessary in this approach. Indeed, both views do not have to commit to a particular account on the nature of laws. They are compatible with Humeanism: laws can still be imagined as the axioms and posits of our best theories. But primitivism or necessitarianism are also not a priori excluded. Also, they both use simplicity as a guide to metaphysics. Structuralism eliminates objects in favor of structure, while I eliminate properties. However, these attitudes come to the same suspicion for properties, and the corresponding desire to have laws and symmetries do the modal, explanatory work. Different is the case of moderate structuralism that does not get rid of properties: the structure is constituted by the relational properties and by the intrinsic properties.38 A final point in common is the importance both approaches give to symmetries, even if in different ways. In my framework, the symmetries of the theory help identify the physical objects and constrain the form of the laws. In this way, thin objects together with symmetries generate the structure, namely the laws and the properties. In structuralism, however, symmetries are ontological prior to objects, so that the structure together with the symmetries generate the laws, the objects and the properties. This leads directly to the main difference between the two views: my approach is still object-oriented, even if in my case objects are thin. So, in eliminativism we have structure, which through symmetries grounds thick-objects and their properties, while in my view we have thin objects which through symmetries ground the structural relations between thick-objects and their properties. In this sense, my approach is more in line with the traditional object-oriented metaphysics than eliminativism. Instead, both my view and moderate structuralism may be thought of as object-oriented. However, they are very different. In moderate structuralism, objects and structure are ontologically on the same footing: structure is a network of relational properties between objects, which are defined in terms of them. Objects may have intrinsic properties like mass or charge, but mainly they are defined in terms of the relations, which are the ways the object could be. In my view this is not so: there are spatiotemporal relations, but intrinsic properties and relational properties which account for the motion of objects (nomological relations) do not exist. In my view thin objects are ‘interconnected’ with one another by laws, not properties. Laws do not define objects and objects do not define laws.

38 Esfeld

and Lam (2010).

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10.3.4 Advantages of the Thin Objects View Over Structuralism To summarize, my view is structuralist: we have thin objects without intrinsic properties, and we have structures, intended as the nomological network needed for the objects to move. It is not an eliminativist view, given that there are objects. It is not a variety of moderate structuralism, as object and structure (in this case, the laws) are not ontologically on the same level: objects constitute matter, and laws either supervene (if Humeanism is true) or not on them, but do not define the objects. Nonetheless, my approach can account for underdetermination and entanglement. In fact, in the example of quantum mechanics, the way the wavefunction evolves in time is irrelevant, as long as the law such wavefunction defines for matter remains the same. Since different wavefunctions may give rise to the same behavior for the objects, the wavefunction evolution is underdetermined. Nonetheless, in this context it does not pose a problem, since the wavefunction is not material. Rather, the underdetermination regarding objects is naturally broken, since the nature of the fundamental objects is postulated as the ontology that provides the simplest and most unifying explanation, constrained by symmetries. This understanding of the wavefunction is also helpful in accounting for entanglement. While the sub-systems of an entangled system do not possess their own wavefunction, in this framework this does not entail individuals described by the subsystems do not exist: the wavefunction does not describe matter, the thin objects do, and the dependence captured by entanglement is understood in terms of laws. As discussed, eliminativism suffers from the problem of intelligibility: it makes no sense to say that relations exist without relata. Moderate structuralism postulates relata to avoid this problem, but immediately runs into another problem, namely properties. Indeed, this seems to go against one of the original motivations for structuralism against the traditional view, namely getting rid of properties. Instead, by adopting my approach one avoids the intelligibility objection against eliminativism because there are relata, the thin objects. Moreover, one avoids the mystery charge against properties in moderate structuralism, as instead of having properties and laws, one merely has laws. As far as the pessimistic meta induction argument goes, in my approach if the nature of the object is preserved, then the problem is solved, otherwise it is not, but at least the wavefunction with its non-classical nature is not involved. To conclude, I have argued that both my approach and structuralism share many motivations but overall, mine has less objections. So, if one is already convinced by the arguments put forward by the structuralists and that radical changes are needed, then one should endorse my view because it is less problematical.

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10.4 Conclusion In this paper, I have briefly described a view that provides an object-oriented structuralist alternative to the traditional object-oriented metaphysics. In my account there is only one kind of fundamental entity, it has no other fundamental property over and above its spatio-temporal ones. Then, there are structural relations between the fundamental objects, which can be seen as ‘effective laws’ and which are able to account for what we usually regard as different families of fundamental entities (like protons and electrons). I have argued that this account, in contrast with the traditional view, nicely extends to quantum physics, and captures the main ideas and motivations of structuralism, without falling pray of many of its objections.

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Chapter 11

Quantum Ontology Without Wave Function Carlo Rovelli

Abstract The relational interpretation offers an understanding of the theory rooted in realism (as opposed to instrumentalism) where agents or knowledge play no role, while at the same not interpreting realistically the quantum state ψ.

11.1 Realism Without Wave Function The discussion on how to make sense of quantum theory is plagued by a widespread confusion generated by mixing up two distinct questions: 1. Do we get more clarity by interpreting quantum theory “realistically” namely as an account of what happens in the world; or do we get more clarity by interpreting it as the account of the knowledge of a subject? 2. What precisely is the wave function, or more in general, the quantum state ψ? Do we get clarity by interpreting ψ “realistically” as the best picture of the way a system is, the “beable” of the theory? Or do we get more clarity by interpreting ψ as a calculation devise, an account of past interactions, a bookkeeping of the probability for something to happen? These are two distinct questions. I think that mixing them up is the source of much of the confusion about quantum theory. In fact, we can get clarity about quantum theory by interpreting the theory realistically, without interpreting the wave function realistically.

C. Rovelli () Aix Marseille University, Université de Toulon, CNRS, CPT, Marseille, France Perimeter Institute, Waterloo, ON, Canada The Rotman Institute of Philosophy and the Department of Philosophy of Western Ontario University, London, ON, Canada © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_11

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This is not an artificial option, it is the natural one: the one we adopt for classical mechanics. The classical mechanics of—say—a non-relativistic particle admits a realistic interpretation based on a simple ontology: there is a particle with a certain mass somewhere in space, that moves in a way described by the values of its physical i variables. In the semi-classical limit, the wave function ψ(x, t) ∼ e h¯ S(x,t ) of a particle is (the exponential of) the Hamilton-Jacobi function S(x, t). The HamiltonJacobi function is a solution of the Hamilton-Jacobi equation, which is the classical limit of the Schrödinger equation, confirming the fact that the wave function and the Hamilton-Jacobi function refer to the same quantity. Now, nobody in their right mind would interpret the Hamilton-Jacobi function S(x, t) realistically. The Hamilton-Jacobi function is a calculation devise, certainly not a “realistic” picture of the particle. The same natural option is available for quantum theory: interpreting the theory realistically, without interpreting the wave function realistically. How?

11.2 Ontology In order to be a realist, without being a realist about the wave function, we have to be realist about something else. In other words we have to adopt an ontology which is not based on the wave function. In both classical and quantum mechanics the ontology can be taken to be defined by systems and properties, where properties are described by the values of the physical variables associated to a system. The difference between the classical and the quantum case is the following: In classical mechanics: (a) physical systems have properties at any time t, and (b) these properties pertain to the systems itself. In quantum mechanics: (a) physical systems have properties only when they interact with other physical systems and (b) these properties are relative to the systems in interaction. This is the realistic ontology of the relational interpretation of quantum theory (Laudisa and Rovelli, 2021; Rovelli, 1996). The wave function is taken only to be a calculation devise to compute the probability distribution of the values taken by the relative properties of a system at interactions. The main idea is that the very meaning of the (contingent, variable) properties of a system, which are described by values of its physical variables, is not something attached to a single system at any time, but rather a description of the way in which the system affects other systems, or manifest itself to other physical systems, at (discrete) interactions. Not only this reading of quantum theory is possible, but it is in fact how quantum theory originated (Rovelli, 2017). Quantum theory was completed in Göttingen in

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1925—including the calculation of the spectrum of hydrogen (Pauli, 1926)—by Max Born and his young collaborators (Born and Jordan, 1925; Born et al., 1926), and independently by Dirac (1925), before 1926s Schrödinger introduction of ψ. Hence quantum mechanics can be formulated without ψ, like the dynamics of a non relativistic classical particle can be formulated without Hamilton-Jacobi function S. The original Göttingen formulation of quantum mechanics is still philosophically the clearest. The breakthrough was of course Heisenberg’s (1925), but I think that the theory was mostly the brainchild of Max Born. It is extremely elegant. In a sense, it can be condensed by simply saying that a quantum theory has the same variables as a classical theory, with the same physical interpretation, and obeying the same equations, plus one: xp − px = i h¯ . The theory is not about wave functions or quantum states: it is about physical variables taking values at interactions, precisely as the classical theory. Its equations describe (probabilistic) regularities that these values happen to satisfy. The technical step that makes this interpretation possible is the realisation that the fact that a variable takes a value relative to one system has no bearing on the probabilities for values taken with respect to other systems. This resolves the apparent contradiction between wave function collapse and unitary evolution. Both are relative to physical systems. The first pertains to the interaction, the second connects separated interactions. Schrödinger’s cat sees the poison’s bottle open, or not. But this has no bearing on the fact that, in an actual interaction with her, an observer outside the box can still observe quantum interference effects that would not exists if the poison’s bottle was open, nor if it was closed. Wigner’s friend can measure a single given value of the spin, and nevertheless Wigner can measure a quantum interference effect between the two values. No contradiction emerges, if we avoid postulating absolute, nonrelational facts between events. This is a way of interpreting quantum theory that avoids many worlds, hidden variables, physical collapses and other similar hypostases, and at the same time avoids radical instrumentalism. Of course there is a price to pay for interpreting quantum physics in this way, like there is a price to pay in any interpretation of quantum mechanics. The price is accepting the idea that variables take value only at interactions and the value they take is relative to the systems involved in the interaction. Accepting to pay this price and interpreting quantum theory as a realistic account of the relative value that physical variables take in physical interactions between system is the relational interpretation of quantum mechanics (Laudisa and Rovelli, 2021; Rovelli, 1996). Below I detail what are the reasons, the implications and the advantages of this perspective.

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11.3 Contextuality Niels Bohr identified the core novelty of quantum physics –contextuality– when he wrote (Bohr, 1998) The description of a system cannot be separated from the measuring instruments that interact with it.

But this formulation is misleading because it seems to make “measuring instruments”, or special “observers” necessary. A century of successes of quantum theory later, we are confident that quantum theory applies to everything in the universe, whether or not there is a measuring apparatus or a scientist observing. Bohr’s contextuality observation needs therefore to be generalised and the measurer removed. This can be done by reformulating it as follows: The description of a system cannot be separated from the other physical systems that interact with it.

Abandon the notion of a real wave function that mirrors reality and take this statement seriously, and we have a way to making sense of quantum theory. The variables of a quantum system do not denote how the system “is”. They denote how the system “affects” another system. The consequences of this shift are two. First, variables have in general no value outside interactions (because they only have meaning at interactions). Second, the value that a variable takes are labeled by the systems involved in the interaction. They are not properties of one single system. To put it bluntly: whether Schrödinger’s cat’s life is on or off has a value for the cat, but not a value for the world external to the box, which is not interacting with the cat. This means that in spite of the fact that there is a value of this variable relative to the cat, quantum interference between its two values can still show up in future interactions with systems outside the box. The apparent contradiction between the two postulates of quantum theory (unitary evolution and state projection) is resolved because they apply to different quantities. The second is only relevant for quantities relative to the systems in interaction. The first is relevant for the other systems. This relativity of value attribution becomes invisible in the macroscopic world, because decoherence suppresses quantum interference, which is the only hint we have that betrays the relativity of value attribution (Di Biagio and Rovelli, 2021).

11.4 Relations What is then the ontology of quantum mechanics, at the light of the relational interpretation?

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In a naturalistic perspective, we want our ontology to be compatible with our current best science. From the perspective of the relational interpretation, quantum mechanics allows us to maintain an ontology in which the world can be organised in distinct physical systems, but blocks us from assuming that the contingent (variable) properties of these systems are independent. All contingent (variable) properties are relational, namely they are relations: they refer to two systems, not one. We already use relational thinking widely in many sciences (biology, psychology, economics, for instance, are more about relations than about autonomous entities) as well in physics (velocity, position, potentials, are relational notions). Quantum mechanics indicates that fundamental physics does not provide a substratum of substance with properties: for properties, it is relations all the way down. The world is woven by relationships that go all the way down to the smallest physical entities. The word can be described as a collection of facts (the particle is here, the particle is there, the cat is alive, the cat is dead). A fact is accounted for by a variable of a physical system having a value with respect to another system. For instance, a fact is a particle having a certain position in the moment in which it hits a screen. Facts are relative (Bong et al., 2019; Brukner, 2018; Frauchiger and Renner, 2018), because they always can be thought as the interaction between two entities (the particle, the screen). Facts happen at interactions. More precisely, properties described by facts are relative. Fact appear to be stable when decoherence allows us to forget their labeling (a macroscopic stone has a position by itself, because no practically detectable quantum interference betrays the relativity its position). The happening of the world is the happening of discrete quantum events: discrete interactions where the variables of individual systems take value. This is a discrete relational ontology.

References Bohr, N. (1998). The philosophical writings of Niels Bohr (Vol. IV, p. 111). Oxbow Press. Bong, K. W., Utreras-Alarcón, A., Ghafari, F., Liang, Y. C., Tischler, N., Cavalcanti, E. G., Pryde, G. J., & Wiseman, H. M. (2019). Testing the reality of Wigner’s friend’s observations. arXiv:1907.05607. Born, M., & Jordan, P. (1925). Zur Quantenmechanik. Zeitschrift für Physik, 34, 854–888. Born, M., Jordan, P., & Heisenberg, W. (1926). Zur Quantenmechanik II. Zeitschrift für Physik, 35, 557–615. ˇ (2018). A No-Go theorem for observer independent facts. Entropy, 20, 350. Brukner, C. Di Biagio, A., & Rovelli, C. (2021). Stable facts, relative facts. Foundations of Physics, 51, 30, arXiv:2006.15543. Dirac, P. A. M. (1925). The fundamental equations of quantum mechanics. Proceedings of the Royal Society of London, 109, 645–653. Frauchiger, D., & Renner, R. (2018). Quantum theory cannot consistently describe the use of itself. Nature Communications, 9, arXiv:1604.07422. Heisenberg, W. (1925). Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Zeitschrift für Physik, 33, 879–893. Laudisa, F., & Rovelli, C. (2021). Relational quantum mechanics. In The Stanford encyclopedia of philosophy.

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Pauli, W. (1926). Uber das Wasserstospektrum vom Standpunkt der neuen Quantenmechanik [On the hydrogen spectrum from the standpoint of the new quantum mechanics]. Zeitschrift für Physik, 36, 336–363. Rovelli, C. (1996). Relational quantum mechanics. International Journal of Theoretical Physics, 35, 1637 (1996), arXiv:9609002 [quant-ph]. Rovelli, C. (2017). Space is blue and birds fly through it. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376, arXiv:1712.02894.

Chapter 12

The Relational Ontology of Contemporary Physics Francesca Vidotto

Abstract Quantum theory can be understood as pointing to an ontology of relations. I observe that this reading of quantum mechanics is supported by the ubiquity of relationality in contemporary fundamental physics, including in classical mechanics, gauge theories, general relativity, quantum field theory, and the tentative theories of quantum gravity.

12.1 Introduction The image of the world offered by quantum mechanics leaves us with a dizziness. It is hard to make sense of the discreteness and the indetermination revealed by the theory. Different reactions to this vertigo lead to different ways of understanding what the theory tells us about reality, namely different ‘interpretations’ of the theory. Two opposite attitudes are possible. One is to try to bend discreteness and indetermination into an underlying hypothetical continuous and determined reality. For instance, the Many-Worlds interpretation assigns ontological value to the quantum states, which are continuous and always determined; while the DeBroglieBohm, or pilot-wave, interpretation assigns ontological value to a commuting algebra of preferred variables such as positions of particles, which are also assumed to be continuous and always determined. All these are interpretations based on an ontology of objects, or entities, with properties. The other possible attitude is to take discreteness and indetermination at their face value, and study their consequences. The relational interpretation of quantum mechanics (Rovelli, 1996) starts from this second position. It does not make use of an ontology of entities that have always properties, but rather an ontology of

F. Vidotto () Department of Physics and Astronomy, Department of Philosophy, Rotman Institute of Philosophy, The University of Western Ontario, London, ON, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_12

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relations, where the properties (of relata) are only determined at discrete interaction times and are always relative to both interacting systems.1 This is a rather radical metaphysical position: it places relations, rather than objects, or substances, at the center of the metaphysical conception. Articulating such a position has its difficulties: How to think of relations before relata? How to preserve the objectivity of our representations, if properties turn out to be so relative? What grants the commensurability between perspectives? There are answers to these questions (French & Krause, 2006; Ladyman & Ross, 2007; Rovelli, 2018), but they involve a radical rethinking of the conceptual basis of all our representations of reality. So, why should we venture into this arduous journey, when more pacifying readings of quantum theory, compatible with a more naive realism, are available? After all, different interpretations offer coherent frameworks for understanding the content of quantum mechanics and interpreting reality around us. Internal coherence is a necessary condition for a consistent interpretation of quantum mechanics, but is insufficient in helping us choosing between different interpretations. One possibility to settle this problem is to delegate the answer to the future: some interpretations may turn out to more fruitful. This is for instance how the debate on wether or not it is better to consider the Earth to be the center of the Universe (a non empirical question!) was settled: one option turned out to be definitely more fruitful. It has been argued that quantum gravity might be easier within one interpretation. Or perhaps a future theory superseding quantum mechanics will require one particular interpretation (Smolin, 2019; Valentini, 2021). In all these cases, however, the new results against which to evaluate current interpretations are not yet available. Here I want, instead, to investigate a different strategy for evaluating interpretations of quantum theory: their coherence with the conceptual frameworks of other physical theories that best capture our recent advances in understanding the physical world. I argue below that the relationality that characterizes the relational interpretation of quantum mechanics is in fact not so unconventional after all. Rather, it characterizes modern physics. My aim is to provide in this way a more solid foundations to the idea that relationality is central to quantum mechanics: through the analysis of how relationality is present, perhaps in a transversal way, in virtually all aspects of contemporary physics. In fact, I shall argue that the relationality at the base of quantum mechanics is already present in classical mechanics: by putting this classical relationality in evidence, we better situate the emergence of the more subtle relationality of the quantum case. To this end, we need to look at classical theory from a modern perspective, in particular using the language of symmetries and gauge theories. This allows us to create a natural bridge with quantum mechanics in its relativistic 1

The relationality of relational quantum mechanics has been compared with ontic structuralism by Candiotto (2017) and Dorato (2020). The metaphysical implications of relational quantum mechanics, and the association with the more general structuralist framework, are still requiring an in-depth investigation. In this respect, of particular interest are the positions developed by Esfeld (2004) and French and Ladyman (2003).

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version. When we talk about interpretations of quantum mechanics, it is misleading to restrict us to the non relativistic domain: we must consider the compatibility with quantum field theory, with Yang Mills theories and with gravity. Relationality offers a key to do so. I also discuss the specific connection between quantum theory and the relativistic theory of the gravitational field. The problematic nature of this connection can be solved by using a common language: that of totally constrained systems. This can serve as common ground for understanding the foundational problems of quantum mechanics, gravity, and the role of symmetries/gauge, within a common conceptual framework. If these steps are carried out carefully, then the image of a fundamental conceptual structure for understanding reality at our present level of knowledge opens up: that of covariant quantum fields. This is what quantum gravity is about: a quantum description of the gravitational field must follow from a covariant description of quantum fields in full generality. Physics forces us towards engaging metaphysically in a specific direction: everything that exists is quantum, everything that exists is covariant. I argue below that this is clarified by seeing that everything quantum is relational, and everything covariant is relational.

12.2 Relationality in Quantum Mechanics Taking relations as fundamental in quantum mechanics implies a change in the ontology that does not prevent to be realist. The relational interpretation is a realistic one: when a self-adjoint operator, which codes the physical properties of a system, assumes a certain eigenvalue, this corresponds to an element of reality. Rovelli refers to these elements of reality of relational quantum mechanics as facts (see Rovelli’s article in this volume), or events. Like the events in relativity, quantum events are about physical systems in interaction. We may label these systems as observer or “observed”, but subjectivity, agents, mind, idealism, phenomenology, etcetera, play no role in this interpretation.

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These facts actually have a clear correspondence with the ontology of classical mechanics. In the quantum formalism, the observables correspond by definition to the measurable quantities of classical physics. On the other hand, the relational interpretation is characterized by the fact that these facts are understood as intrinsically relational: they are real, but their actualization as real is always related to both systems interacting when the value of a measurable quantity is determined. A fact can be true or actualized with respect to a system (which acts with the abstract role of observer or measuring apparatus) and also not be true with respect to another. Reality and relationality are therefore inextricably linked. We attribute existence to a system from its possessing certain properties: location, speed, energy, angular momentum, area, volume. . . In quantum mechanics, we realize that it is not It is possible to speak of any of these properties except in a relational manner. Each property is determined by a relationship between systems. When this relationship does not materialize, the property is not determined. In a Galilean system, in order to define the speed of an object we must have a reference system with respect to which the speed is measured; different reference systems associate a different speed to the same object. If no reference system is defined, the object does not have any “speed” property. There can not be a notion of “speed”, for instance, associated to the universe as a whole. In relational quantum mechanics this principle is the foundation of the ontology of the theory: the elements of reality, the facts are aspects of a relationship, and take place in interactions. The ontological priority of the interactions invests the whole structure of what we call real.2 In particular, interactions determine our notion of locality. In contemporary physics we emphasize the fact that interactions are local. But the notion of interaction is more primitive than that of localization. Nonetheless, as we shall see, it is precisely the locality of the interactions that saves us from some apparent paradoxes of quantum mechanics. The prototype of these paradoxes is the EPR one (Martin-Dussaud et al., 2019; Smerlak & Rovelli, 2007). Two spatially separated systems (observers) A and B interact with (measure) two entangled particles, one each. This determines a fact relative to A and a fact relative to B. A paradox arises only from the assumption that what is an element of reality for A is also an element of reality for B, and vice versa. A and B may have an element of reality in common only when a local interaction occurs between them. We cannot consider a fact for A, or a fact for B, as absolute. We can eventually introduce a third observer C, with respect to which there will be some element that regards the comparison of the two, but only provided that this is interacting (hence locally interacting) with both A and B.

2

Notice that in an ontology of relations it is still possible to refer to relata in a meaningful way. For instance here we have used the notion of systems and we will talk about objects such as particles and fields. All of them, it is argued, have a relational nature. But, as a structuralist would say, it does not follow from logical principles that they cannot be objects of predication (Saunders, 2003).

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12.3 The Relationality of Symmetries In modern physics, interactions are largely encoded in symmetries. The symmetries of a system determine the possible interactions such system can have with another system. Symmetries therefore capture the potential of interactions among systems. The apparent arbitrariness that often appear in the definition of the symmetries of a system reflect different possibilities for the system to couple to another system. For instance, general relativity formulated in a tetrad formalism has an additional local symmetry with respect to Einstein’s metric formulation, which captures the possibility of coupling the theory to oriented local objects such as fermions. In particular, from this perspective, gauge symmetries do not represent redundant superstructures. They do not just express an indeterminism that needs to be eliminated to get a deterministic theory, or a redundancy in any other sense. The apparent arbitrariness has its origin in the relationship between gauge and relationality. A gauge transformation is s mathematical redundancy only when we consider systems in isolation. The coupling of the system with other systems can well be given by (gauge invariant) interactions that couple gauge degrees of freedom of one system with gauge degrees of freedom of the other. Together, new physical degrees of freedom are born in this coupling. For instance, the Maxwell potential is redundant in the dynamics of the electromagnetic field alone, but is needed in coupling the field to some charged fields such as an electron. The Maxwell potential is not a just a redundant mathematical addition to reality: it is the handle through which the electromagnetic field couples to electrons. Notice that what is relevant, what captures the essence of physical reality, is the coupling between systems, not what we identify as the system. Two systems coupled to each other cancel their respective gauge redundancies: by coupling a gauge-dependent quantity of one system to a gauge-dependent quantity of the other system, we give rise to a give-independent physical interaction. This observation leads to an important distinction regarding observable quantities. We refer to gauge-dependent quantities as partial observables (Rovelli, 2002), and gauge-independent quantities as complete observables in the sense of Dirac. Both kinds of quantities are associated with operators whose eigenvalues corresponds to elements of reality. Both are associated to relative facts. In this sense, partial observables and complete observables have the same ontological status. The difference, on the other hand, is clear cut: partial observables can be measured, but cannot be predicted by dynamic equations alone, while gaugeindependent observables can be measured and also predicted (Rovelli, 2013). In this sense, as Dirac noted, only the latter lead to a determinism in the theory. The indeterminacy of the evolution of the former simply reflects the fact that their value depends on the dynamical evolution of another system whose equations of motion are not considered. For instance, the Einstein equations do not determine uniquely the evolution of the metric tensor because a measurement of this tensor is always relative to a specific (say, material) reference frame, whose equations of motion are in general not included in the Einstein’s equations alone.

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12.4 Relationality in Quantum Field Theory A striking example of relationality is provided by the notion of particle in quantum field theory. While some presentation of quantum field theory rely heavily of the notion of particle taken as fundamental (Weinberg, 1995), it is also very well known that the number of particles present in a given quantum field theory state depends on the reference system. On a generic curved spacetime, in particular, there is no unique notion of number of particles. Physically, different particle detectors count different numbers of particles. Mathematically, in the absence of global Poincaré invariance there is no natural Fock structure in the (nevertheless well defined) Hilbert space of states. The existence of particle can be true with respect to one system but not with respect to another. Different detectors probe different bases in the same Hilbert space. When a detector measure a certain number of particles, we say that the existence of these particles is an element of reality. But the point above makes clear that this is a relational reality: it is the number of particles with respect to the interaction with that detector.

12.5 Relationality in General Relativity The relational nature of space and time has been longly debated. General relativity, while defining space and time as manifestation of the gravitational field, has a structure that is deeply relational (Vidotto, 2016). Dynamical objects are not localized with respect to a fixed background but only with respect to one another. Notice how the collection of dynamical objects includes the gravitational field itself. The very structure of spacetime is built upon contiguity relations, namely the fact of spacetime regions to be “next to one another”. But in the case of the gravitational field, saying that different regions are contiguous one another through their boundaries means exactly that these regions are interacting. Alternatively, when we couple general relativity to the matter of a material reference system, the components of the gravitational field with respect to the directions defined by this system are gauge-invariant quantities of the coupled system; but they are gauge-dependent quantities of the gravitational field, measured with respect to a given external frame. In this case, a prototypical example of a partial observable is time: a quantity that we routinely determine (looking at a clock) but we can not predict from the dynamics of the system.

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12.6 Relationality in Quantum Gravity The relational aspect of spatio-temporal localization that characterizes general relativity and the relational aspect of quantum mechanics that is emphasized by its relational interpretation combine surprisingly well precisely thanks to the fact that interactions are local. This combination provides a solid conceptual structure for quantum gravity (Vidotto, 2016). In fact, locality is a main discovery of in XX century modern physics: interactions at distance of the Newton’s kind don’t seem to be part of our world. They are only approximate descriptions of reality. In the particles’ standard model, as well as in general relativity, things can interact only when they “touch”: all interactions are local. This means that objects in interactions should be in the same place: interaction require localization and localization requires interaction. To be in interaction correspond to be adjacent in spacetime and vice versa: the two reduce to one another. In other words, the fact that interaction are local means that they require spacetime contiguity. But the contrary is also true: the only way to ascertain that two objects are contiguous is by means of having them interact. Therefore we can identify the Heisenberg cut that defines the separation with respect to which (relative) facts are realized in quantum theory, with the boundary of spacetime regions that define the (relative) localization in general relativity.

By bringing the two perspectives together, we obtain the boundary formulation of quantum gravity (Oeckl, 2003, 2008): the theory describes processes and their interactions. The manner a process affects another is described by the Hilbert state associated to its boundary. The probabilities of one or another outcome are given by the transition amplitudes associated to the bulk, and obtained from the matrix elements of the projector on the solutions of the Wheeler-DeWitt equation. Let us make this more concrete. Consider a process such as the scattering of some particles at CERN. If we want to take into account the gravitational filed, we need to include it as part of the system. In doing quantum gravity, the gravitational field (or spacetime) is part of the system. Distance and time measurements are field measurements like the others in general relativity: they are part of the boundary data of the problem. Thinking in terms of functional integrals, we have to sum over all possible histories, but also all possible geometries associated to a given finite spacetime region. In the computation of a transition amplitude, we need to give the boundary data of the process that are for instance the position of a particle at an initial and a final time. We use rods and clocks to define them. But those measure geometrical informations that are just value of the gravitational field. Everything we have to give is the value of the fields on the boundary. This includes the gravitational fields from which we

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can say how much time have passed and the distance between the initial and the final point. Geometrical and temporal data are encoded in the boundary state, because this is also the state of the gravitational field, which is the state of spacetime.

This structural identification is in fact much deeper. As noticed, the most remarkable aspect of quantum theory is that the boundary between processes can be moved at wish. Final total amplitudes are not affected by displacing the boundary between “observed system” and “observing system”. The same is true for spacetime: boundaries are arbitrarily drawn in spacetime. The physical theory is therefore a description of how arbitrary partitions of nature affect one another. Because of locality and because of gravity, these partitions are at the same time subsystems split and partitions of spacetime. A spacetime is a process, a state is what happens at its boundary (Rovelli & Vidotto, 2014). This clarifies that in quantum gravity a process is a spacetime region. Relational quantum mechanics describes systems in interaction. What defines the system and when is it interacting? For spacetime, a process is simply a region of spacetime. Spacetime is a quantum mechanical process once we do quantum gravity. Vice versa, this now helps us to understand how to do quantum gravity. Notice that from this perspective quantum gravitational processes are defined locally, without any need to invoke asymptotic regions. Summarizing:

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12.7 Conclusion: The Relational Nature of Contemporary Physics The debate on the interpretation of quantum mechanics is far from having reached a consensus. Addressing it is unavoidable in order to answer the question of “what does exist?” as far as current physics tells us. But considering this a question related to Quantum Mechanics alone deprives ourselves of some fundamental conceptual inputs, that come from the core of the picture of the world revealed to contemporary physics. I have described the lesson of quantum mechanics from the perspective of relational quantum mechanics. General relativity, quantum field theory and quantum gravity, are compatible and they support such point of view. Gauge theories and quantum fields theories have a deep relational core: gauge degrees of freedom are handle for interactions to other systems. Even the particles of quantum field theory, that in an ontology of objects we would be tempted to call fundamental objects, are in fact relative, not absolute, entities. Locality reveals a deep structural analogy between the relations on which quantum mechanics is based and those on which spacetime is based. Quantum gravity makes this connection completely explicit. In quantum gravity a process is not in a spacetime region: a process is a spacetime region. Analogously, a state is not somewhere in space: it is the description of the way two processes interact, or two spacetime regions pass information to one another. Viceversa, a spacetime region is a process: it is like a Feynman sum of everything can happen between its boundaries. The resulting relational ontology, compatible with quantum mechanics as well as with the rest of our current physical theories, is a minimalistic one. There is no necessity to attribute an ontological role to states nor some mysterious hidden variables: only facts, or events, are part of the ontology. It is also a “lighter” ontology: facts are sparse and relative. This means for instance that particles only exists in interactions, not in between, and exists only with respect to the system they are in relation with, not with respect to the rest of the universe. One may ask: what happens between two interactions? In between, there are other interactions of the field: these interactions are what gives sense to the expression “in between”. We can distinguish a particle that appears here and then there, being some interaction made by the field: what does define the identity of the particle and its story? Only regularities in the interactions. In fact we may think, if we wish, that there is no particle, only correlated interactions (Hume, 1896). These correlations are such that I measure the field here now and later on there, I obtain correlated values. This is what we mean by saying that there is the same particle. There are just manifestations of a field. A field exists through its interactions. This stance weakens usual realism, but makes it compatible with our current empirical knowledge and spare us pernicious paradoxes. The relational realism, it should be stresses, is not in any form relativist: going relational does not weaken

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the reality of the world. If there are only interactions that are intrinsically relational, there is no absolute reality with respect to which the relational events are “less real”. Relationalism should not be confused here with a form of subjectivism, which can lead to solipsism. The relations we considered are among any physical systems in interactions, not subjects or agents that require conscious agency. Conscious agents are a peculiar case among the different systems. Systems can acquire and store information about one another: here information should be understood as physical correlations, without a necessary epistemic connotation. This leads us to think of relations in a completely physical way, discarding a possible reading of the restriction to the relations as only epistemically motivated (as, for instance, in epistemic structural realism). An interpretation of relations that restricts them to be only epistemic would require the assumption of a hypothetical non-relational underlying substance, not accessible to our knowledge: such a move seems circular and redundant, not adding any clarity to our understanding of the world. In particular, for the sake of philosophy of science, it appears as a useless epicycle. On the other hand, embracing a relational perspective, we may be able to leave a monolithic reality for a richer kaleidoscopic one. One in which it is required an epistemology where the notion of objectivity is pluralistic and perspectival (Barad, 2007; Massimi, 2022).

References Barad, K. (2007). Meeting the universe halfway. Duke University Press. Candiotto, L. (2017). The reality of relations, Giornale di Metafisica, 2017(2), 537–551. Dorato, M. (2020). Bohr meets Rovelli: a dispositionalist account of the quantum limits of knowledge. Quantum Studies: Mathematics and Foundations, 7, 233–245. Esfeld, M. (2004). Quantum entanglement and a metaphysics of relations. Studies in History and Philosophy of Modern Physics, 35, 601–617. French, S., & Ladyman, J. (2003). Remodelling structural realism: Quantum physics and the metaphysics of structure. Synthese, 136, 31–56. French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical and formal analysis. Oxford University Press. Hume, D. (1896). A treatise of human nature. Ladyman, J., & Ross, D. (2007). Every thing must go: Metaphysics naturalised. Oxford University Press. Martin-Dussaud, P., Rovelli, C., & Zalamea, F. (2019). The notion of locality in relational quantum mechanics. Foundations of Physics, 49(2), 96–106. Massimi, M. (2022). Perspectival realism. Oxford University Press. Oeckl, R. (2003). A ‘General boundary’ formulation for quantum mechanics and quantum gravity. Physics Letters B, 575, 318–324. Oeckl, R. (2008). General boundary quantum field theory: Foundations and probability interpretation Advances in Theoretical and Mathematical Physics, 12(2), 319–352. Rovelli, C. (1996). Relational quantum mechanics. International Journal of Theoretical Physics, 35, 1637. Rovelli, C. (2002). Partial observables. Physical Review, D65:124013. Rovelli, C. (2013). Why gauge? Foundations of Physics, 44(2014):91–104.

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Rovelli, C. (2018). Space is blue and birds fly through it. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376, 20170312. Rovelli, C., & Vidotto, F. (2014). Covariant loop quantum gravity: An elementary introduction to quantum gravity and Spinfoam theory. Cambridge University Press. Saunders, S. (2003). Structural realism, again. Synthese, 136(1), 127–133. Springer. Smerlak, M., & Rovelli, C. (2007). Relational EPR. Foundations of Physics, 37, 427–445. Smolin, L. (2019). The search for what lies beyond the quantum. In Einstein’s unfinished revolution. Penguin Press. Valentini, A. (2021). Quantum gravity and quantum probability Preprint arXiv: 2104.07966. Vidotto, F. (2016). Atomism and relationalism as guiding principles for quantum gravity. PoS, FFP14, 222 Weinberg, S. (1995). The quantum theory of fields. Cambridge University Press

Chapter 13

Explicit Construction of Local Hidden Variables for Any Quantum Theory Up to Any Desired Accuracy Version 4 Gerard ’t Hooft

Abstract The machinery of quantum mechanics is fully capable of describing a single realistic world. Here we discuss the converse: in spite of appearances, and indeed numerous claims to the contrary, any quantum mechanical model can be mimicked, up to any finite accuracy, by a completely classical system of equations. An implication of this observation is that Bell’s theorem is not applicable in the cases considered. This is explained by scrutinising Bell’s assumptions concerning causality, retrocausality, statistical (in-)dependence, and his fear of ‘conspiracy’ (there is no conspiracy in the language used to describe the deterministic models). The most crucial mechanism for the counter intuitive Bell/CHSH violation is the fact that, regardless the settings chosen by Alice and Bob, the initial state of the system should be a realistic one. The potential importance of our construction in model building is discussed.

13.1 Introduction Quantum mechanics is usually perceived as being a revolutionary new theory for the interactions and dynamics of tiny particles and the forces between them. Here we expand on our earlier proposal (’t Hooft, 2016) to look at quantum mechanics in a somewhat different way. The fundamental interactions could be entirely deterministic, taking place in a world where all laws are absolute, without requiring statistics to understand what happens. The infinite linear vector space called Hilbert space, is then nothing more than a mathematical utensil, allowing us to perform unitary transformations. The set of all possible states is re-arranged

G. ’t Hooft () Faculty of Science, Department of Physics, Institute for Theoretical Physics, Utrecht, The Netherlands e-mail: [email protected] http://www.staff.science.uu.nl/~hooft101 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_13

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into states that look like superpositions of the original ontological, or realist,1 states. As soon as we loose control of these original realist states, we can, instead, interpret the absolute values squared of the superposition coefficients as Born probabilities. Quantum mechanics as we are familiar with today, is then arrived at. This paper is about regaining control. The ‘original, realist states’ then are what is referred to as ‘hidden variables’. Let there be given some quantum model, for instance the Standard Model of the fundamental particles and their interactions. Can we then identify these hidden variables? Often, hidden variables are looked upon as rather ugly attempts to regain a dated interpretation of quantum theory. In contrast, this theory can be far more beautiful and elegant than the original theory of quantum mechanics as displayed in the Copenhagen frame of axioms. This is what we intend to show in this paper. We start with a generic ontological theory. This may seem to be hard to visualise, until we realise that demanding locality severely restricts the possibilities, see Sect. 13.6. One can almost derive what the hidden variables are: Local Hidden Variables (LHV). It was thought that local hidden variables should be easy to refute, but this, we now claim, is a mistake. The use of quantum mechanics as a procedure for vector analysis of a classical system, is merely a mathematical trick, which does not change the equations; therefore it does not introduce ‘conspiracy’ either in the classical or in the quantum description. In this paper, we first explain the use of quantum mechanics to describe deterministic systems in their full generality (Sect. 13.2). Basically, a system is deterministic if its evolution law can be regarded as an element of a large permutation group. We explain how it can be mapped on the set of unitary transformation matrices in Hilbert space. There are various advantages of using such representations in pure mathematics. The Schrödinger equation appears naturally, reflecting what happens if one uses the fact that unitary matrices form a continuum while permutations are discrete. In short: any deterministic theory can be written as a quantum theory, in the sense that one may use the concept of Hilbert space for doing statistics, and a Schrödinger equation describes the evolution law. Next, in Sect. 13.3, we show the converse of that: how any quantum system can be linked to a deterministic model. The observation we use is that, in any classical or quantum system, one may add physical degrees of freedom that move periodically in a compact space. This adds new energy levels to the system that we assume to be invisible, in particular if the detection devices used have a limited time resolution. We call these ‘fast fluctuating variables’. Our new twist is that this compact space may be assumed to be sufficiently tiny, so that these variables return to their previous values with very high frequencies. Consequently, in our artificial vector spaces, their energy spectrum is discrete with wide separations between the energy levels. These separations are so large that, under normal physical conditions, only the very lowest energy state will be occupied. This state is a single wave function, which happens

1

Everywhere in this paper, the words ‘ontological’ and ‘realist’ are used interchangeably. They emphasise that we avoid ‘statistical’ or ‘uncertain’ expressions.

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to be a constant: Our fast variables are all in uniform probabilistic distributions. This makes them invisible in practice, just as the fast vacuum fluctuations due to the fields of extremely heavy virtual particles. Nevertheless, we now have quantum states that may interact with the slow particles non-trivially. If now we construct the vector space representation of such a model, we shall find that any quantum model one wishes to investigate can emerge from this. Some readers noted that our model is non-local. This was due to an error in the notation that could easily be corrected. In Sect. 13.4, we explain the situation. Locality is not a problem, but special relativity is. This is due to the fact that we need to restrict ourselves to finite models, forcing us to work with lattice theories. Making such theories relativistically invariant is notoriously difficult. It will presumably involve general relativity, and this is far beyond what we can handle presently. Of course, the reader may wonder how it can happen in such a model that Bell’s theorem (Bell, 1964, 1982, 1987; Bell & Gao, 2016) is disobeyed. How could this happen? This we describe in Sect. 13.5. Actually, we have to deal with two questions: one, what was wrong or misunderstood in Bell’s classical argument, and two, what is the origin of the apparent clash between totally natural intuitions on the one hand, and the actual quantum calculation—as well as the real experiment— on the other. We think both questions can be answered, but we keep the answers brief. Thus we summarise what, according to this author, the principal weaknesses are in Bell’s argument, which is not the mathematical calculations but the general assumptions, in particular those connected with causality and ‘free will’. The question why our intuition does not agree with the result of the experiments is actually more interesting. It all comes from an important footnote in our theory: the realistic degrees of freedom depend on the basis chosen; one can constrain the choice chosen by nature by imposing more demands. Of our various options for the cause of our discrepancy with Bell, the last explanation given in Sect. 13.5, ‘option # 4’, appears to be the salient one. It explains why and how the ‘free will’ of Alice and Bob to change their settings, enters into the argument: they do not have the free will to choose their settings in some quantum superimposed state, since the initial state of the universe is a realistic one. There are numerous examples of quite counter intuitive consequences of the original Copenhagen interpretation. All these can be traced to the same features. We mention a few in Sect. 13.6: the reason why quantum mechanics is weird, arises from the fact that deterministic underlying theories do not work in the way expected. The deterministic variables do randomise when generating their ‘quantum effects’, but this goes along mathematical paths that differ from our intuitive ideas. Their behavior looks as if ‘conspiracies’ take place, while these variables do nothing else than obeying physical laws. We claim that this paper is not just a philosophical one, but to the contrary, it may well show novel guidelines towards building models, see Sect. 13.7. Our conclusions are in Sect. 13.8.

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13.2 The Generic Realistic System Let us briefly review the precedure described in Ref. ’t Hooft (2016). In a deterministic theory, we identify the set of states that can be realised. In general, the number of different possible states, N, will be gigantically large, but the principle will always be the same. The dynamics is defined by the evolution law over a smal lapse of time, δt. This is nothing but a single element of the permutation group SN : |k(t + δt) = U (δt)|k(t) ;

U (δt) ∈ SN .

(13.2.1)

It maps the set of states onto itself. This set should be regarded as discrete, although it will often be useful to consider one of several possible continuum limits, so as to simplify notations and calculations. It will be useful to regard our system as bounded, by imagining it to be surrounded by a wall. Later, the wall will be removed, enabling us, for instance, to set up plane wave expansions. We start with choosing the time variable as being discrete, t = k δt, where k is integer and 0 ≤ k < N. N is finite. This implies that there is a number T ≤ N such that |k(t + T ) = |k(t) ;

|k(t + t1 ) = |k(t) if 0 < t1 < T . (13.2.2)

Thus, our system is periodic in time (though the period T rapidly tends to infinity as we choose our wall to be further away). In general T  N, so that, consequently, there will be many states |k that are not in the periodic set (13.2.2) at all. Starting with such a state gives us another periodic set with period T  . Continuing this way, we find that the completely generic finite deterministic system consists of a large number of periodic sets. Each periodic set (r) can now be subject to a discrete Fourier transform, to write its elements as superpositions of energy eigenstates |n, r , n = 0, · · · Tr − 1 , Tr −1 1 e−iEn,r t |n, r , |k(t), r = √ T r n=0

En,r =

2πn + δEr , (13.2.3) Tr δt

where δEr is, as yet, an arbitrary normalisation of the energy that may well be different for different sets r. Its physical interpretation is that the energy may depend in an arbitrary way on the quantum number r, which obeys a conservation law. Note, that the concept of energy E used here is the quantum mechanical one, not necessarily the classical Hamiltonian. The above describes the formal, complete solution of all deterministic systems. In a given set r, the energy spectrum (13.2.3) forms an equidistant sequence. If we take the limit where δt vanishes while Tr δt is kept fixed, this spectrum ranges to infinity. At given r, the distance between adjacent energy eigenvalues, E = En,r − En−1,r , is determined by the period, E = 2π/(Tr δt). They are strictly equidistant, regardless how complicated the interactions may be.

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Since the energy eigenvalues do not depend on the basis chosen, the equidistant energy level sequences make one wonder how well such a theory can represent the real world, where such exactly equidistant sequences seem not to be observed. The answer to this will be that there are equidistant sequences, but the energy separations are too wide to be noticeable in physics. We’ll see how this can happen.

13.3 A Deterministic Theory for Every Quantum Model In Ref. ’t Hooft (2021), it is found how a deterministic model can be constructed that mimics any given quantum system. Here we briefly repeat the derivation, by first formulating the deterministic model, and then showing how to link its dynamical evolution law to the given Schrödinger equation. We shall then explain why the result is counter intuitive. For simplicity our quantum system will be described as a one-particle theory, but generalisation to more complex structures such as quantum field theories will be straightforward. The classical model contains N primary states, |i, i = 1, · · · , N, which we declare all to be realistic. These will later be declared to correspond to the basis elements of the quantum model, but we’ll come to that. In addition to these primary states, there will be a number,2 M, of fast fluctuating, periodic variables ϕi (t) , 0 ≤ ϕi < 2π, where i = 1, · · · , M. Anticipating demands that will be needed later (in arguments concerning locality, see Sect. 13.4), we shall choose M to be the number of points in 3-space, regardless the other properties of the system under investigation: M = #( x).

(13.3.1)

What these fast variables actually are, is to a large extent immaterial, as long as they move so fast that all positions are taken more frequently than the largest quantum frequencies in the system we wish to describe. For instance, we may suggest that the variables might represent excessively heavy virtual particles, situated at the space points i ∈ M. In view of the above, the evolution law of ϕi is written as ϕi (t) = ϕi (0) + ωi t mod 2π ,

t = k δt ,

(13.3.2)

where k is integer while δt is very small (though not infinitesimally small). The periods Ti are given by integers Li , such that Ti = Li δt , and ωi = 2π/Ti = 2π/(δt Li ).  This means that the variables ϕi are arranged to sit on an M-dimensional lattice with M i=1 Li points, and periodic boundary conditions in all M directions. 2

Our notation often changes in different publications.

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The values for the ωi are large compared to the physical time parameters, so δt and Li δt are all small, but L2i δt will be assumed to be relatively large. We choose the Li large but not exactly equal. It will make things easier if we assume them to be relative primes (so that the ϕi will mix properly). The primary states |i, also called ‘slow variables’ in Ref. ’t Hooft (2021), undergo deterministic transitions whenever the variables ϕi reach some given points in their lattice. These transitions are all elements of the permutation group SN . For simplicity, we limit ourselves to pairwise permutations between two states at the time. In general, we assume for each pair (i, j ) a number ni,j of given, fixed points in the 2 dimensional sub-lattice spanned by ϕi and ϕj , such that, if ϕi and ϕj arrive simultaneously at one of the points ni,j , the transition |i ↔ |j  takes place. The ϕ variables just continue obeying (13.3.2). Let us call these special points on the ϕ-lattice  crossing points. When locality is considered, we choose i and j to be neighbors in 3-space. So-far, our model is just like any other realistic, deterministic model. But now comes the most important constraint: the ϕi variables go so fast that we cannot detect their values directly. Therefore, we assume them to be in a totally even probabilistic distribution in the entire space spanned by them all. In contrast, the primary variable hops from one state to others at a slow pace, since it happens relatively infrequently that the ϕ variables arrive simultaneously at one of the crossing pints. When we compute what happens classically, we see that the distribution of the fast variables stays even, but how do the slow states |i behave? This is most easily found out by using the quantum mechanical notation, as described in the previous section. It was also discussed in Ref. ’t Hooft (2021). Simply re-write the time evolution described above as an equation for a deterministic, real wave function ψi (ϕ,  t). The dominant part of the Hamiltonian takes care of Eq. (13.3.2). Then, we have perturbation parts for each pair (i, j ), taking the form Hijint =

π [i,j ] (s) (s) σy δ(ki = ki ) δ(kj = kj ) , 2

(13.3.3)

i,j,s

where ki(s) and kj(s) indicate the crossing point(s) on the (i, j ) sub-lattice. Here, the number δt was set equal to one for simplicity, and ‘int’ stands for ‘interaction’. The pairwise permutation between the states |i and |j  is written as a Pauli matrix, [i,j ]

− iσy

=

0 −1 , 1 0

(13.3.4)

We regard the total Hamiltonian describing the transitions between the fast and the slow variables, over small time steps δt or L δt, as small perturbations. We have to assume that the fast variables are in their lowest energy state. Since their excited energy levels are far separated form the lowest value (zero) and energy is exactly conserved, this situation is stable and therefore persists in time. Keeping them in the

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lowest energy state guarantees that the statistical distribution on the ϕ lattice stays perfectly even as time proceeds.3 Since we consider small steps in time, perturbation theory tells us that this is controlled by the expectation value of the Hamiltonian Hijint for the zero energy states of the ϕ variables. This replaces the Kronecker deltas in Eq. (13.3.3) by the factor 1 . Li Lj

(13.3.5)

This is very important. The expression in Eq. (13.3.3), with the factor π/2 included, describes a completely deterministic exchange of states, as it was designed to do. But the factor (13.3.5) turns this into a Hamiltonian that generates superpositions. Thus, from here on, we are dealing with real quantum mechanics. The outcome of our calculation in perturbation theory is, that the probability distribution of the primary state, can be written as Pi (ϕ,  t) = Pi (t) = ψi (t)2 ,

(13.3.6)

where ψi (t) form a real-valued wave function. This wave function slowly varies in time, as we already explained. The ϕi run along their respective circles so quickly  2 that the even distribution of the probabilities, given by the norm N |ψ i | , does i=1 not show any significant dependence on the variables ϕi or on time itself. This takes care of unitarity in the space of the slow variables. The Hamiltonian we obtain is

π ni,j [i,j ] Hslow = σy , (13.3.7) 2Li Lj i,j

where ni,j is the number of crossing points on the (2-dimensional) lattice spanned by the values for ϕi and ϕj .4 This is a purely imaginary, antisymmetric matrix. Such Hamiltonians keep the wave function real, and have no diagonal parts. By choosing the numbers ni,j , Li , and Lj we can generate almost any such antisymmetric, imaginary-valued Hamiltonian. If the desired Hamiltonian has terms with irrational ratios in them, the limits n, L → ∞ have to be taken. So-far, the wave function here came out to be real. Usually, in quantum mechanics, we have complex valued wave functions. This is easy to arrange here. Complex numbers are pairs of real numbers, so having complex numbers means that there is an extra, somewhat hidden, binary degree of freedom, called ‘c-bit’ in The matrix elements φ |Er  may be assumed to have exactly the same amplitude everywhere on the hypertorus of the φ lattice. 4 It is here that we prefer to assume L and L to be relative primes, so as to ensure that any i j crossing point will be passed equally often. 3

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Ref. ’t Hooft (2021), such that this c-bit takes the value 0 for the real part of the wave function, and 1 for the imaginary part. It was easier for us to start with the real numbers only. The c-bit doubles the total number of slow states. Comparing the quantum calculation with what happens classically, we can see what exactly is happening. Quantum mechanically, a wave function ψi (t) obeys a Schrödinger equation. Classically, the system undergoes transitions from the states |i to all other states |j  in fairly rapid successions. Our classical theory now tells us what is causing these transitions: they depend on the exact values of the ϕ variables, which however are difficult to follow since they go fast. Only if measurements are done so fast that we know exactly where the ϕ variables are, the evolution will turn out to be deterministic. But as long as we cannot follow the fast variables, we have to deal with the quantum expressions. The calculation of the effective Hamiltonian for the slow states may require higher order corrections if the numbers n, L, . . . are not very large. This has not been done yet, partly because it is the principle that counts here. In higher order corrections, one may have to deal with some of the higher energy states of the ϕ variables as virtual states; the emergence of virtual ϕ states at higher energy levels then betrays deviations from the strictly even distribution of probabilities for ϕ.  Much like the effects of heavy, virtual particles, these higher order effects may represent virtual particles that cause non-local interactions and other complications. This would not lead to any violation of quantum mechanics itself, in spite of the deterministic underlying theory.

13.4 Special Features. Locality There may seem to be some arbitrariness in the above construction. Why do transitions between the states |i and |j  only involve the fast variables ϕi and ϕj ? Why not the others? Also, the question may arise: where do we have to choose the crossing points, and how does the theory depend on this choice? And then: why do we not unfold the M-dimensional lattice  of all ϕ-variables to form just one line of consecutive states, with length Ltot = i Li ? All these questions have to do with locality. We may wish to avoid theories where the properties of a particle at site x depend on things happening at site y far away from x . In particular, in second-quantised theories this may be important. Signals should not go faster than light. Classically, such conditions are easily seen to be obeyed if the crossing points (i, j ) can be associated with one single position in real space (or space-time). Opening up the ϕ lattice would generate difficulties. It would force us to re-arrange the energy levels of the ϕ variables to form much larger sequences with much smaller energy gaps; we would not be able anymore to keep the higher energy levels out of the discussion on how real quantum mechanics can be obtained. Thus we see that demanding locality narrows down much of the arbitrariness of our models, though some arbitrariness may seem to remain, such as the choice

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of the locations of the crossing points on the lattices. However, for these points, another remark may be made. We can handle the effects of the crossing points by rearranging the field components in terms of eigenstates of the permutation operator considered. We could consider the wave functions ψi and ψj and decompose them along the lines of the eigenstates, ψ± =

√1 (ψi 2

± ψj ) ,

(13.4.1)

We then find that, at a crossing point, ψ+ is not affected at all, while ψ− gets a minus sign. but this minus sign could be chosen differently, which would amount to displacing the crossing point. Thus we obtain a symmetry of the system, implying that the exact location of the crossing point is, to some extent, physically immaterial. Thus, we do emphasise that this theory is local, provided that the original quantum Hamiltonian is local: int Htot =

Hijint ,

(13.4.2)

i,j

where the points x associated with i and j must be neighbours only. A reader thought to have found a counter example in the two-particle system. If the interaction Hamiltonian is written as a potential function V (xi , xj ), of course both the quantum and the classical theory are non-local. If states with indefinite particle numbers are considered, the only known local theory is quantum field theory (QFT), where the Hamiltonian contains kinetic parts exactly of the form (13.4.2). Indeed, the Standard Model is a QFT with all desired locality properties. Since we frequently use a lattice formulation, we do have to indicate more precisely what we mean by locality in a lattice theory. There, the best approximation to locality for the Hamiltonian isa Hamiltonian that is the sum over the entire lattice of a Hamiltonian density, H = x H( x ), where H( x ) must obey [H( x ), H( y )] = 0

if

| x − y| > a ,

(13.4.3)

where a is the lattice mesh size. The limit a ↓ 0 then reproduces locality (and causality) as it usually is formulated in QFT. Our classical formalism will be local in the same sense. It implies that all of the fast variables ϕi are only allowed to affect the Hamiltonian density component(s) that sense the same operators at the coordinate xi . Thus, contradicting some claims to the contrary, we have no problem with locality. In contrast, we do have a problem with special relativity. The Hamiltonian that we start from may easily be chosen to be the one of the Standard Model, which is Lorentz covariant. However, its lattice version is in general not Lorentz covariant. Although this disease does not seem to be serious, as Lorentz invariance

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will naturally be recovered in the continuum limit, it will have to be a topic for discussion for the time being.5

13.5 About Bell’s Theorem Bell (1964, 1982, 1987) explicitly emphasised that hidden variables in general, and local hidden variables in particular, should be incompatible with quantum mechanics. To start with, there seem to be just three options. First, there may be fatal flaws in the present paper. I am sure that quite a few readers will be quite comfortable with this possibility and not look further. The author would be eager to know where they disagree with our approach; there is no fatal flaw. Small inaccuracies in the formalism would not invalidate the fundamental principle stating that the quantum formalism can be applied just as much in deterministic theories as in wave mechanics with uncertainty relations. All that was done here is turn this observation around: there is no reason to draw a sharp dividing line between quantum theories that have a deterministic foundation, such as here, and theories where this should be fundamentally impossible: we claim that deterministic theories are a dense subset of all quantum mechanical theories. Secondly, there could be a flaw in Bell’s arguments. Now his derivations are clear, and it is generally agreed that, given his assumptions, his derivations are correct. The technical part of his arguments seems to be flawless. It has been argued that different number tsystems, non-commuting numbers, or other concoctions should be used, but that would be hard to defend since, whether a calculation is correct or not, should not depend on mathematical procedures and number systems employed. There is a third explanation, which seems to be somewhat more plausible than the other two: Bell did make assumptions, both concerning the nature of the hidden variables, and the nature of physical law. Bell’s assumptions are notoriously controversial. For a discussion, see the papers contained in Bell and Gao (2016). An important assumption made by Bell is “statistical independence”: the idea that due to irreproducible changes in the background, any unwanted correlation between the photons and the settings chosen should disappear. In our models, scrambling the data would require changes in the basis elements of the cellular automaton, such that the entangled photons cannot stay in the same entangled state. The Schrödinger equation derived here remains the same, but not its realistic basis elements. Did Bell think of fast fluctuating hidden variables? It appears that he would have answered this himself with yes, but his assumptions seem not to include this possibility in-depth. Usually, authors discussing Bell’s theorem seem to have extremely conventional classical systems in mind, but most of them would have argued that having local hidden variables move very fast would not affect their results.

5

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Bell did make other assumptions that this author views as suspect, see for instance (Vervoort, 2013). He discusses at length the notions of causality and “retrocausality”: the settings chosen by Alice and Bob cannot affect the polarisations of the photons they observe, since “these photons were there earlier”. Can’t they? In any case, this is not enough to argue that the settings chosen by Alice and Bob must be statistically independent of the polarisations of the photons. It is a standard calculation to find out that quantum theory does generate statistical correlations (’t Hooft, 2016). Classical theories can do the same thing. Bell derives his theorem by assuming the absence of such correlations. It is well-known that there are statistical correlations everywhere. For instance, if there is a laptop somewhere to be found in the universe, then the odds are very high that there are more laptops in its vicinity, even space-like separated ones. In other parts of the universe there are no laptops around at all. This is an example of a nonlocal correlation function. The correct way to distinguish forward from backward causality is not about statistics but about effects that would be connected by laws of nature. In QFT, a sound definition of causality can only take one form, where we cannot distinguish forward from backward: Observable operators must commute when space-like separated

(see Eq. (13.4.3) for the Hamoltonian density). In quantum field theory, statistical correlations are expressed in the propagators connecting space-time points. They are non-vanishing all over space and time. The commutator of two operators vanishes completely outside the light cone.6 Bell knew about this formulation of causality (“No Bell telephone”) but found it to be not enough. Indeed, if only this form of causality would be allowed he would not have been able to prove his theorem. It is important to realise that the equations of motion are needed if one desires to distinguish forward from backward causality. Cause and effect are statistically correlated. Thus, our option #3 is that, by way of the ‘butterfly effect’, even minor fluctuations in a photon wave function in the past could generate a correlation with settings Bob and Alice decide about much later as well as much earlier. Do Alice and Bob have no ‘free will’ then? Not in a deterministic world, it seems. Many experienced scientists have difficulties with that (Conway and Kochen, 2009). Physicists desire more solid arguments for the emergence of statistical correlations between the settings chosen by Alice and Bob, and the fluctuations of photons that existed at earlier times, in such a way that every hint of conspiracy can be avoided. The notion of causality, and the demand that there should be no ‘retro-causality’, may seem to be quite plausible intuitively. It is intuitively false to allow for theories where photons in different parts of the universe ‘conspire’ with one another. Let us

6

As stated earlier, special relativity is notoriously hard to obtain on a finite lattice, and at first sight, there may not be a lightcone at all. However, the cellular automaton usually does have a speed limit for signals, even if nearest neighbors can communicate directly.

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now announce that there exists a much more powerful explanation for the apparent ‘conspiracy’ in Bell’s set-up. This, option #4, goes as follows: Our classical description of any quantum system comes at a price: we first must choose for an orthonormal basis (for the slow states |i). Then we add the fast variables, in such a way that the entire system evolves with probability distributions as described by the given Schrödinger equation. But how does this compare with the system we get if all states would have been transformed to a different basis? This happens when Alice or Bob change their settings. In that case, the Schrödinger equation looks different, and the probability distributions are calculated in a different way. It is inevitable that the realistic states, including the initial states, in both cases are chosen different. In Bell’s set-up, it has been tacitly assumed that only one set of realistic states is accepted to be the ‘physically correct one’. No, the physically correct realistic states inevitably depend on the basis chosen. The automaton allows to choose the basis, but it will have to be modified. This produces precisely the ‘loophole’ needed to avoid Bell’s conclusions. If we now work with superpositions, such as states |a = α1 |i + α2 |j  and |b = β1 |i + β2 |j , then the Born probabilities do not apply to the inner products a|b.7 In the new basis, the realistic states differ from what they were before. It is here that we are not allowed to assume the photons to stay in the same entangled state. This is because the ontological theory eventually predicts only ones and zeros as outcomes. If we start with realistic photons in our initial states (photons that can only be in states either with probability one or with probability zero) then a change in Alice’s and Bob’s settings later in time, will force us to perform a basis transformation first.

We are then forced to choose how to change our realistic photons in the initial state. What is nice about this explanation is that we only have to reformulate our basis of states, which looks as a ‘conspiracy’, but it isn’t that. If Alice and Bob’s actions also fit in the deterministic equations, then there is no further need to change the basis. The deterministic model is a totally natural one. This implies that indeed Alice and Bob have no free will, but they do not have to fear a ‘conspiracy plot’. Our ‘change of basis’ will be typically the basis transition caused by a symmetry such as a rotation. But this means that: Symmetry transformations such as rotations may transform realistic states into superpositions!

This would actually be an argument against our construction. However, there are ways to avoid the need for such symmetry structures: we may postulate that basis elements are chosen as realistic states, only if these among themselves obey the same symmetry principles as the quantum system we are attempting to describe. In the case of photons, the obvious choice is to take the vector potential fields located at well-defined space-time points. Rotating the polarisation of a photon then requires

This is because, if only the states |i and |j  have an ontological meaning, then the probabilities for having a state |a or a state |b are meaningless.

7

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rotating the vector potential field, and this will now be an ontological transformation (the equal-time commutators for vector potential fields vanish). However, in this case the photon states themselves are not realistic: photons are energy packets of these vector fields and these do not commute. Following this line of argument, Bell was not allowed to assume the presence of a photon with a given polarisation anywhere. Only the fields are real. Taking this new constraint into consideration, it is possible to construct a model that exactly reproduces the standard quantum mechanical predictions for Bell’s experiment. Bell would not have accepted such models because every setting chosen by Alice and Bob would require different microscopic configurations everywhere, leading to the absence of ‘free will’ for Alice and Bob. In this special case, there would still be an other issue: due to lack of gauge invariance, not the vector potential but the electric and magnetic fields are welldefined, see Sect. 13.7.

13.6 Note Added: Quantum Weirdness It may seem odd that quantum mechanics, with all its remarkable and paradoxical properties, can be explained at all using fairly mundane classical, deterministic degrees of freedom. Many investigators were confident that such simple explanations should be impossible. And yet, here we are: our model has two kinds of dynamical variables, the slow ones, |i, which we proclaim to be ontologically observable, and in addition fast moving variables that vaguely resemble heat baths (which they are not). The reason why, nevertheless, this system can accurately mimic quantum mechanics is that it only does so after we choose a preferred basis: the set of basis states represented by the |i. All we did is to find extra variables that force the probabilities Pi = i|i to be such that the states |i accurately obey a Schrödinger equation. What is not allowed is to change the basis for these states |i half-way an experiment. We should keep in mind that Born’s rule for the probabilities does not apply when one superposition is compared with another superposition; it only applies if a superposition is compared to realistic states. Practically all examples of counter intuitive quantum features such as the EPR-Bell experiments, the GHZ state (Greenberger et al., 2007), and similar constructions, require as a crucial step the freedom of one or more of the observers, be it Alice, Bob, Cecile, Dave, . . . , to change their basis of states, for instance by rotating their polarisation devices. In our set-up, the fast variables change completely after any such change, regardless how small. The original state that these set-ups start from is entangled some way, and any change of basis states gives the initial particles a completely different probabilistic distribution—as if some signal went back to the past to instruct this initial state. Of course, the actual configuration of classical variables in our set-up does not allow for such a signal, which means that, in our models, the observers do not have the ‘free will’ to rotate their detectors without any modifications of the (quantum or classical) states in the past.

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13.7 Model Building One reader suggested that our theory can be compared with Bohm’s pilot theory (Bohm, 1952). A big difference is that Bohm’s theory appears to hinge on an essential statistical component: the pilot wave generates a distribution of particle positions. We claim that our approach also applies to particles that do not spread at all into statistical wave forms. Most notably, if it is our fast variable ϕi that one would be tempted to compare with pilot waves, the contrast is that our ϕ variables might be interpreted as representing the most massive virtual particles in some construction of unified theory of elementary particles. So-far, neither special relativity, nor general relativity entered in the discussion. We also ignored theories with global or local gauge symmetries. Taking these various very special symmetries of nature into account may however be tremendously important. Special relativity requires the replacement of simple-minded quantum mechanics by quantum field theory, QFT. Keeping QFT discrete would be tantamount to putting the theory on a lattice. In the classical case one then obtains a cellular automaton (Fredkin, 2003; ’t Hooft, 2016; Wolfram, 2002, 2020; Zuse, 1969). Turning this in a quantum theory requires procedures as described in this paper (’t Hooft, 2019, 2020; Wetterich, 2020). Now QFT is a quantum theory just as any other one, except that it may possess Lorentz invariance as a symmetry. Can we take Lorentz invariance into account? Lorentz invariance requires locality in quantum fields, but locality does not impose any difficulty at all. Our fast fluctuating variables, ϕi , may very well be strictly localised in points of space and time.8 The problem with the Lorentz group, however, is that it is not compact. By applying Lorentz transformations successively, one can reach ever stronger Lorentz boosts. Where is the end? Is there an end? We may attempt to construct a theory with Lorentz invariance containing fast fluctuating variables: when strongly boosted, any particle system may then be transformed into a highly energetic one. These can also play the role of fast variables. Our problem may be the converse: our variables ϕi may move very fast but they were postulated only to come in a finite number of states. A Lorentz invariant theory will necessarily have an infinite number of states, as dictated by the infinite Lorentz group. Is there a way to ‘compactify’ the Lorentz group? There is a mathematical trick that is very useful in QFT: the Wick rotation (Symanzik, 1969). This transforms real time into imaginary time, and, miraculously, turns the Lorentz group into the group of rotations in a 4 dimensional, Euclidean space. This group is compact, so that our problem disappears. But now we have no unitary evolution law anymore. Instead, our system now describes interactions in a stochastic system at equilibrium.

8

In a previous version of this paper, the fast variables ϕi were associated directly to the states |ψi  of the slow variable. This would work for a one-particle case, but should not be done in more complex quantum systems, particularly if we wish to recover locality. The index i for the variables ϕi must be associated to coordinates x . I thank the referee for spotting this inaccuracy.

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There is a physical argument telling us that there will be a limit to the Lorentz boosts: gravity. In ordinary physical circumstances, there is only one state with vanishing energy: the vacuum. All other states contain a positive amount of energy. If we Lorentz-boost such a state, we get particles with a large amount of energy as well as momentum. These energies and momenta generate gravitational fields, and with those, curvature in space and time. Consequently, The Lorentz boosts, which lie at the center of special relativity theory, can be terminated by the theory of general relativity. This is caused by the fact that there will be limits to the amount of space-time curvature that we may be able to handle in our system. The present paper suggests that we should not worry too much about quantum mechanics when this problem arises, but we should try to formulate classical evolution laws where these features are taken into account. This is also very difficult, but at least it may seem to be a more manageable problem. Physics is about making accurate models of the real world around us. This paper encourages us to continue doing this. The most advanced model we have today is the Standard Model. It consists of a small set of fundamental fields, which interact in ways where we need a few dozen of freely adjustable parameters, the constants of nature. A fundamental problem today is to speculate what the physical origin of these numbers might be. What one can notice from the results of this paper is, that we cannot have just any set of real numbers for these interaction parameters. Finiteness of the lattice of fast fluctuating parameters would suggest that, if only we could guess exactly what the fast moving variables are, we should be able to derive all interactions in terms of simple, rational coefficients. Thus, a prudent prediction might be made: All interaction parameters for the fundamental particles should become calculable in terms of simple, rational coefficients.

Needless to say that we are unable to pin down more precisely how to predict these values today, but we can say, for instance, that a smooth space-time dependence of the interaction parameters, as is being speculated about by some investigators, is not possible in the framework of this paper.

13.8 Concluding Remarks Every cellular automaton allows for a description in terms of a quantum Hamiltonian, which reproduces the evolution of all cells with infinite perfection. Conversely, every quantum system can be approximated by an Hamiltonian derived from a cellular automaton. Since there appears to be a continuum of distinct quantum field theories but only a denumerable number of cellular autumaton rules, this latter mapping cannot always be perfect, but, if a limit on the time resolution is allowed, the deviations (in terms of energy eigenvalues) can be made arbitrarily small. In particular, we showed how, in principle, to construct a cellular automaton to match the Standard Model of the elementary particles.

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This inspires us to formulate our physical theory for the interpretation of quantum mechanics: Our universe is a cellular automaton.

A special future of this hypothesis is that it can be phrased with infinite perfection, and this entails that the universe may have begun with a single, fundamental state, and will continue to be in realistic states forever. This actually allows one to counter the persistent critical objection that the cellular automaton does not seem to describe the evolution of superimposed states correctly: This does not matter; the universe never is in a quantum superposition of CA states!

This sounds like a fundamental departure from Copenhagen, but it is easy to defend. The reason why nevertheless, interference patterns and superposition phenomena are observed, is that such phenomena are only observed if an experiment is repeated many times. What happens when an experiment is repeated many times is described correctly by our Hamiltonian. This now leaves a lot to be done: Now do the calculation and find the rules for the world’s automaton! Of course, this is extremely difficult. We still haven’t quite understood symmertries such as Lorentz invariance, and, certainly, gravity has not been understood. A good feature of our theory is that, now, we only need to investigate realistic theories for gravity, which may be conceptually much easier to do than keeping everything ‘quantum’. Why not try? The author thanks A. Schwarz and C. Wetterich for interesting discussions. We also had constructive, though sometimes fierce, discussions in weblogs with Ron Maimon, Mitchell Porter, Alan Rominger, Manuel Morales, Lubo˘s Motl, and others. Among them were, and no doubt still are, pertinacious nay-sayers.

References Bell J. S. (1964). On the Einstein–Podolsky–Rosen Paradox. Physics, 1, 195. Bell J. S. (1982). On the impossible pilot wave. Foundations of Physics, 12, 989. Bell, J. S. (1987). Speakable and unspeakable in quantum mechanics. Cambridge: Cambridge University Press. Bell, M., & Gao, S. (Eds.) (2016). Quantum nonlocality and reality: 50 years of Bell’s theorem. Cambridge: Cambridge University Press. Bohm, D. (1952). A suggested interpretation of the Quantum theory in terms of “Hidden Variables, I”. Physical Review, 85, 166–179; id., II. Physical Review, 85, 180–193. Conway, J., & Kochen, S. (2009). The strong free Will theorem. Notices of the AMS, 56(2), 226. arXiv:0807.3286 [quant-ph]. Fredkin, E. (2003). An introduction to digital philosophy. International Journal of Theoretical Physics, 42(2), 189–247. Greenberger, D. M., Horne, M. A., & Zeilinger, A. (2007). Going beyond Bell’s theorem. arXiv:0712.0921, Bibcode:2007arXiv0712.0921G. Symanzik, K. (1969). Euclidean quantum field theory, Conf.Proc.C 680812 (1968) 152–226, Varenna 1968, Proceedings Of The Physics School On Local Quantum Theory, New York 1969, 152–226, Contribution to: 45th International School of Physics ‘Enrico Fermi’: Local Quantum Theory [Scuola Internazionale di Fisica ‘Enrico Fermi’: Teoria Quantistica Locale].

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’t Hooft G. (2016). The cellular automaton interpretation of quantum mechanics. Fundamental Theories of Physics (vol. 185). Springer International Publishing. eBook ISBN: 978-3-31941285-6. Hardcover ISBN: 978-3-319-41284-9, Series ISSN: 0168-1222, Edition Number 1. ’t Hooft, G. (2019). the ontology conservation law as an alternative to the many world interpretation of quantum mechanics. In Presented at the Conference PAFT2019, on ‘Current Problems in Theoretical Physics’, Session on ‘Foundations of Quantum Theory’, Vietri sul Mare, Salerno, April 13–17. Arxiv:1904.12364. ’t Hooft, G. (2020). Deterministic quantum mechanics: The mathematical equations. Submit to the Article Collection: “Towards a Local Realist View of the Quantum Phenomenon”. Frontiers Research Topic, A. Casado et al. (Eds.). Arxiv:2005.06374. ’t Hooft, G. (2021). Fast vacuum fluctuations and the emergence of quantum mechanics. arxiv:2010.02019. Wetterich, C. (2020). Probabilistic cellular automata for interacting fermionic quantum field theories, e-Print: 2007.06366 [quant-ph]. Wolfram, S. (2002). A new kind of science, Champaign, IL: Wolfram Media. ISBN 1-57955-008-8. OCLC 47831356. Wolfram, S. (2020). A project to find the fundamental theory of physics. Wolfram Media (2020). ISBN 978-1-57955-035-6 (hardback) 031-8 (ebook). Vervoort, L. (2013). Bell’s theorem: Two neglected solutions. Foundations of Physics, 43, 769– 791. ArXiv 1203.6587v2. Zuse K. (1969). Rechnender Raum. Braunschweig: Friedrich Vieweg & Sohn; Transl: Calculating space, MIT Technical Translation AZT-70-164-GEMIT, Massachusetts Institute of Technology (Project MAC), Cambridge, Mass. 02139 (1969). A. German & H. Zenil (eds).

Part III

The Wave Function

Chapter 14

Wave Function Realism and Three Dimensions Lev Vaidman

Abstract It is argued that our experience of life in three-dimensional space can be explained by an ontological picture of quantum mechanics consisting solely of the wave function of the universe formally defined in the configuration space. Our experience supervenes on a part of the universal wave function which is defined in three dimensions, while the other parts (defined in configuration space) explain physical properties of objects. A deterministic universe without action at a distance requires the acceptance of the existence of parallel worlds similar to our world.

14.1 Introduction The foundations of quantum mechanics are still far from consensus and the meaning of its basic concept, the wave function, continues to be under heated debate. I consider the quantum state, the wave function of the universe, to be the only ontology of quantum theory (Vaidman, 2016, 2019): “All is Ψ ”. In parallel, we witness an extensive discussion of the term wave function realism (Ney & Albert, 2013). In this paper I want to clarify my approach and put it in the context of the current discussion. There are many different meanings of realism. My experiences are real, but the word “real” does not have the same meaning as in the expression “wave function realism”. Our “real” experiences supervene on the physical ontological reality. In my semantics, ontology describes substance, matter. I separate it from nomological entities (like the Hamiltonian) which specify how the ontological description of the universe changes in time. In physics, realism is frequently considered as local realism, which has two aspects: separability and local causality. Separability: the combined complete local descriptions of all space points provide the complete description of reality. Local

L. Vaidman () Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_14

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causality: objects can influence other objects only when they are physically close together. The locality (or nonlocality) of quantum physics is the main question to be discussed here. Once, the realistic picture of the world was the following: There is a threedimensional space (3D). In this space there are local macroscopic objects: people, animals, stones, trees, etc. interacting locally among themselves. These interactions explain the time evolution: changes in the form and position of the objects in space. A cat has to reach a mouse to eat it. Although the development of classical physics, in an attempt to reach a deeper understanding, encountered difficulties - Newton worried that the gravitational interaction between objects is apparently nonlocal - at the end of the nineteenth century classical physics seemed to be very close to reaching a satisfactory picture: particle-field realism. The fundamental ontology consists of point particles moving in space affected locally by fields. All objects are made of atoms (stable configurations of nuclei and electrons) which have fundamental interactions among themselves by local creation of fields which propagate in space and then locally affect other particles. Not only familiar macroscopic objects move and interact in 3D, but also the microscopic objects move and interact in the same space. However, the success of classical physics was illusory. The stability of atoms and other objects, together with many other phenomena, had no explanation by classical physics and it was replaced by quantum physics. Quantum mechanics explains the stability of atoms, existence of rigid bodies and (apart from gravity) all our observations. It has an extraordinary success: there is no discrepancy between what can be calculated and what is measured. In some cases the agreement is up to ten decimal numbers. The quantum solution was achieved by introducing ontology which is very different from the ontology of classical physics: there are no particles moving on trajectories in 3D. The particle realism of classical physics is replaced by the wave function realism. The wave function is defined in the configuration space of N particles (in a simple case of nonrelativistic quantum mechanics). The route to explain our experience based on this ontology (Albert 2013, Ney 2021) is not simple and this is apparently the reason why the approach encounters skepticism (Maudlin 2010, Wallace 2020): how one can see in an abstract quantum state, a complex valued wave function in the configuration space, the familiar objects in 3D? I suggest accepting the fundamental role of 3D from the beginning. Macroscopic objects, as well as microscopic objects, interact in 3D. The role of 3D was not questioned in classical physics since there was no need for the configuration space to provide complete ontological description. Quantum mechanics needs the configuration space, but only for quantum effects rarely seen in everyday life. Macroscopic objects reside and interact in the familiar three dimensional space.

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14.2 The Single World Universe Consider a naive understanding of the textbook (von Neumann) view (Von Neumann, 2018). Everything, including measuring devices, is described by the wave function. The wave function evolves locally and unitarily, until it evolves toward a superposition of macroscopically different states of a macroscopic object, when it collapses non-locally to a wave in which all macroscopic objects are well localised. A similar picture is given by the Pearle-Ghirardi-Rimini-Weber collapse theory (commonly known as GRW) Pearle (1976), Ghirardi et al. (1986) in which a concrete physical (but ad hoc) proposal replaces a vague postulate of “well localised macroscopic objects”. In this (frequently collapsing) wave function we can see a realistic picture of the world. Local macroscopic objects: people, animals, stones, trees, etc. interact among themselves locally in 3D, changing their form and position in space. We do not know the precise expressions for wave functions of macroscopic objects, they consist of too many (>1020 ) particles. In classical physics we would have a similar difficulty, but we can gain understanding by considering simpler, smaller systems because there is no conceptual difference in the behaviour of microscopic and macroscopic systems: particles, as macroscopic objects, move in 3D and interact locally (directly, or through the creation of and interaction with local fields) between themselves. This move, however, is not available in quantum physics. The wave function of a microscopic system does not collapse to a well localised state, so analysis of behaviour of microscopic systems does not provide proper understanding of the behaviour of macroscopic systems. Macroscopic objects, due to their large mass and moment of inertia, can have a well defined position, orientation, and other variables describing their macroscopic properties changing slowly enough to explain the time evolution of our perceived world. The way to express this is to describe the world wave function as a product of quantum states of all macroscopic objects |Ψobject j  times the state of the remaining particles |Φrest  which do not belong to any macroscopic object. |Ψworld =



|Ψobject j |Φrest .

(14.1)

j

The wave function of every object is a product of wave functions of collective variables describing the macroscopic properties of the object, times entangled wave functions of its microscopic constituents: |Ψobject  =



|ψ(An )|ψ(b1 , b2 , . . . , bNb )|ψ(c1 , c2 , . . . , cNc ) . . . |ψ(o1 ,2 , . . . , oNo ).

n

(14.2) Examples of macroscopic variables An are: the center of mass of the object, center of charge of the object, variables describing orientation of the object, electric and

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magnetic dipoles, quadrupoles, etc. These are the variables which appear in the Hamiltonian of the interaction with other macroscopic objects, so these are the variables describing how we perceive the object. These variables describe the object in 3D. Entanglement appears in the wave function of microscopic constituents of macroscopic objects described by variables bj , cj , . . . oj . Start with an atom. It is a composite system of a nucleus and electrons. A stable atom, say, an atom in its ground state, exhibits a highly entangled state of its electrons. A piece of a solid body consists of ions and electrons (at this level ions can be considered as elementary units). The electrons are in a complex entangled state which explains rigidity and other properties of the object. Ions (depending on the temperature) might be (or might not be) in a product state. We might have entangled states of microscopic systems even if they are not responsible for the rigidity of a macroscopic object. A sealed can with a gas definitely has entangled states of gas molecules due to collisions between them. What we observe is the total action (pressure) of a large number of molecules together, which is essentially independent of the entanglement. Possible decoherence with external systems does not result in measurable differences. Another way to see the 3D reality in the wave function of a world is to draw a “cloud” of expectation value of mass density in 3D, or cloud of atom density, etc. The geometric structures of places where these densities are significantly larger than the background (say, due to air) provide the familiar 3D pictures of objects. The mass (or matter) density is sometimes considered as the “primitive ontology” (Allori et al., 2014). I do not see an advantage in defining this new ontology: the 3D cloud is the property of the already defined ontology, the wave function. Moreover, in some cases mass density might not be useful. If I only observe the 3D distribution of mass density, I will not be able to distinguish my body from the water in a swimming pool. The 3D picture of the density of organic molecules will distinguish me from the surrounding water. Thus, the wave function ontology describes our observed reality also in cases when the mass density ontology does not. The true complete story of the world must include fields: this is how Newtonian nonlocal gravitational interaction becomes local: a massive object creates a gravitational field, the field propagates in space and affects the motion of other local objects. Moreover, the complete picture must also include creation and annihilation of particles. Modern research (especially the deadlock of quantum gravity) suggests that we should go even further. It seems that Wallace (2020) considers this as the main reason why wave function realism is not the correct picture. However, I feel that the road to an exact precise and complete story will not bring new conceptual philosophical difficulties and, on the other hand, it also will not provide the solution to the quantum foundations controversies. Moreover, it seems that we can simplify our consideration by neglecting relativistic effects and approximate the interaction between particles by forces described by potentials depending on the relative distances. This simplifies tremendously the analysis of the interaction between particles by removing the necessity of the separate treatment of fields created by the particles. Thus, I can follow Albert (1996) by modeling the world as a

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(large and constant) number N of quantum particles. For simplicity, I do not discuss the important issues related to the wave function of identical particles (fermions, bosons). The state of the world of N classical particles is fully specified by the position and velocity of each particle in 3D. Mathematically, we can represent it as the position and velocity of a single point in the configuration space of 3N dimensions. Moving to this representation seems very strange and not useful. Motion of this point fully represents evolution of all objects made out of classical particles, but in a very indirect way: the configuration space does not look like “space” as it is defined in Wikipedia or Britannica: Space, a boundless, three-dimensional extent in which objects and events occur and have relative position and direction.

Albert does not suggest using configuration space in case of classical particles, however, he argues that we have to do it for the case of quantum particles. A quantum particle is not described by a point moving in 3D, but by a wave function changing in time in 3D. N noninteracting particles can be described by N wave functions in 3D, but the interaction between particles invariably leads to entanglement between the particles and thus N 3D spaces are not enough to describe the state of N quantum particles. The wave function in the configuration space is. This is the reason why Albert considers the configuration space of N particles as the fundamental space. Ney (2021) finds support for this picture because it avoids the non-separability following from entanglement if we consider the particles separately. However, avoiding separability by moving to another space does not seem helpful. It is the non-separability between objects, the non-separability between spatial regions of 3D, which is problematic. Even if the nonseparability persists in the highdimensional configuration space, it does not represent a serious weakness. We do not expect properties of familiar 3D space to be present in an abstract configuration space. Albert (2013) argues that the connection to the perceived 3D comes from the dynamics defined by the Hamiltonian of our world. This program is similar to the approach which starts with an abstract Hilbert space of the universe and attempts to derive the emergence of the three dimensional picture we observe. The key element of the programs of deriving 3D is locality of interactions in 3D. In my view, this fundamental feature justifies the postulate of 3D. Familiar macroscopic objects are certainly present in 3D and we should try to find their ontological three-dimensional representation. Entanglement is invariably present among particles, and it prevents their description in a set of 3D spaces, but macroscopic objects we perceive are not entangled, so they do “have relative position and direction” in 3D. Every set of entangled constituents of a macroscopic object does require a description in the configuration space, but the sets of constituents of different macroscopic objects are separate, and there is no entanglement with other objects. Macroscopic objects consist of sets of entangled microscopic objects. There can be a hierarchy of sets. Sets of entangled quarks make protons and neutrons. Sets

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of entangled nucleons make atomic nuclei. Sets of entangled nuclei and electrons make atoms. . . At some level, we get sets of systems which are not entangled with anything else. The collective variables of every such set of systems have well defined positions and directions in 3D. These positions are not exact, as they are described by well localised wave functions which cannot be localised too well to allow well localized conjugate momenta of these variables, necessary for avoiding fast changes of the positions and orientations. For everyday macroscopic objects, this constraint is not expected to be seen. The Heisenberg uncertainty for position and momentum of a person allows his localization to be smaller than 10−10 meter during all his life. Our perception of macroscopic objects supervenes on the wave functions of the collective variables of microscopic constituents of these objects, the wave functions in 3D. The complete description of a macroscopic object involves entangled states of its constituents defined in their configuration space. A more detailed description involves entangled states of even smaller systems, the set of which makes the microscopic systems described above. And so on. At the top of the hierarchy are the wave functions of macroscopic objects in 3D. Thus, it seems legitimate to view this picture as a wave function realism in 3D. Albert’s wave function realism in 3N dimensional configuration space is a more fundamental description, but clearly it is also not the fundamental description, there are several levels of more fundamental theories. There is a long-winded way of recognising the familiar three dimensional objects from the wave function in the configuration space. This process heavily relies on the Hamiltonian of the world. By contrast, the role of the Hamiltonian in my picture is to explain changes in our experience, but not the experience itself. Imagine that Mephistopheles changes the Hamiltonian of the world when Dr. Faust says to the Moment flying: “Linger a while - thou art so fair!” such that the current wave function of the world becomes an eigenstate of the Hamiltonian. A minute later, Mephistopheles switches the Hamiltonian to be as it was before. I postulate that we all will have one minute of unchanging (beautiful) experience. We will not remember it, so it is not clear what is the operational meaning of this statement, but it provides a consistent definition which is apparently absent in Albert’s picture. This demonstrates that there is no contradiction between the two approaches, the two pictures are similar, but built up in a different way. Three dimensional reality is a derived property of Albert’s approach while it is a fundamental basis of my approach.

14.3 The Many-Worlds Universe Although I asserted above that the wave function realism describing N quantum particles can be upgraded to a more precise and complete quantum theory of relativistic fields without conceptual changes, I did not mean that this is true for a collapsing wave function. The collapse of the wave function includes genuine

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randomness and action at a distance, and because of this it does not fit with the standard picture of physical science. Consider a particle in an equal amplitude superposition of two spatially separated wave packets. Our decision to measure or not to measure the presence of the particle in one location affects the wave function at another location immediately (for von Neumann collapse) or in a very short time (in the GRW model) irrespective of the distance between the two wave packets. Indeed, if we decide not to measure the particle in one location, the wave function at another location does not change. The probability of the GRW jump of the wave function of an isolated single particle is very small. If however, we perform a measurement which entangles the presence (absence) of the particle in one location with two macroscopically distinct positions of a pointer consisting of a macroscopic number of particles, in a very short time the GRW hit of one of the particles of the pointer will cause a random change of the amplitude of the second wave packet from √1 to a number close to 0 or 1. 2 Even if the wave function is not the ontology (contrary to the approach taken here), this example demonstrates an action at a distance. Measurement in one location changes the situation in another location: without the first measurement, there is a genuine uncertainty about the result of the measurement in this location. Immediately after the first measurement, the result of the measurement in the remote location is deterministic. Removing collapse makes quantum theory sensible from the physics point of view. The theory becomes deterministic (Vaidman, 2014) as a default required from a scientific theory (Earman, 1986), and it avoids an action at a distance. Why then was the collapse invented to be a part of quantum theory? Quantum theory without collapse apparently contradicts our empirical evidence that a quantum measurement ends up with a single outcome. There is no contradiction here, and it seems that Schrödinger (Allori et al., 2011) and maybe other fathers of quantum theory understood this, but the inescapable consequence of quantum theory without collapse - the existence of parallel worlds - was and still is difficult to accept. The wave function of the universe is not given by (14.1). It is a superposition of the wave functions of the form (14.1): |Ψuniverse =

αi |Ψworldi .

(14.3)

i

Since a world i, and thus the quantum state of the world |Ψworldi  is not rigorously defined, the decomposition (14.3) is not rigorously defined too. One property of the decomposition is specified. The worlds are macroscopically different ensuring the mutual orthogonality of various terms. Note that one of the terms in (14.3) might correspond to an unstructured microscopic systems, i.e. it might have no macroscopic objects. Even this world, at least formally, fulfills the definition: there are no macroscopic objects in a superposition of macroscopically different states. You, the reader of this paper, live in a particular world corresponding to one of the terms, |Ψworldj  of the decomposition (14.3). If you adopt the Copenhagen or a physical collapse interpretation, you assume that this term is all that there is,

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|Ψuniverse = |Ψworldj . This assumption clearly simplifies the task of finding the correspondence between our experience and the formalism of the physical theory on which our experience supervenes, but such a physical theory is hard to accept. A much nicer physical theory tells us that the ontology of the universe is described by the superposition (14.3). The many-worlds interpretation (MWI) restores the 3D picture of the world with the single outcomes of quantum measurements we experience by postulating that we and other macroscopic objects exist only within a single world. Our experience supervenes on the wave function of one of the worlds Ψworldi exactly in the same way as it was described in the section about the single-world universe. The (only) ontology in quantum theory is the universal wave function (14.3), but what is relevant for our experience is one term of the universal wave function |Ψworldi  corresponding to the world we live in. Both Ψuniverse and Ψworldi are defined in configuration space, but Ψworldi can also be written in the form of the product of wave functions of sets of particles corresponding to macroscopic objects and the wave function of the set of remaining particles, see (14.1). Each wave function of the set of particles corresponding to a macroscopic object can be written as a product of wave functions of various variables of the set, including collective variables defined in 3D, see (14.2). These wave functions are well localized and they describe familiar macroscopic objects. The interaction of macroscopic objects one with the other is fully specified by their descriptions in 3D. This is the way to answer Maudlin’s worry (Maudlin, 2010) that the wave function of the universe defined in 3N configuration space is not appropriate for describing our familiar objects in 3D. The explanations of properties of macroscopic objects like conductivity, rigidity etc. are based on the analysis of their microscopic ingredients, including entanglement of microscopic systems, which requires the configuration space. We also need the configuration space for the description of microscopic systems in quantum information tasks like teleportation, secure communication, etc. However, the configuration space is not needed for the explanation of the macroscopic behavior of macroscopic objects. A popular view is that we need decoherence with the environment (Bacciagaluppi, 2020) to explain why the existence of parallel worlds does not alter our experience in a particular world. I, however, fail to see the relevance of the environment. An observer living in world j has the same experience with or without the presence of parallel worlds i = j . The quantum states of other worlds are macroscopically different and, therefore, it is not feasible to expect interference between the worlds. In principle, such interference can happen when world j splits into several worlds and at least two of them will appear from the splitting of some other world i, but due to macroscopic differences between worlds i and j we have no technological means to arrange such an experiment and the probability that this happens without our intervention is negligible. Let us analyze an example. A Geiger counter is placed near a weak radioactive source such that it clicks on average once in ten seconds. A runner waits for the first click after 12 AM to start running on a 100 meter circle. There will be numerous worlds differing by the observationally distinguishable times the runner starts

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running. There is no precise definition for observationally distinguishable, although we do have a bound: the worlds should correspond to orthogonal states. Until now we introduced only one splitting, so the terms in the superposition will not interfere due to unitarity. The runners will have an overlap in space, since there will be time differences of the take off between the worlds, but even if the centers of mass of the runners will be exactly in the same place, corresponding quantum states will be very different. To observe an interference, we need a very special situation which splits these states into pairs of identical states. The decoherence due to the environment is irrelevant for the suppression of the interference, because macroscopic objects cannot (without unrealistic super-technology) exhibit the interference anyway. The program of deriving the emergent classical world from the universal wave function (Wallace, 2012), i.e. the derivation of the decomposition (14.3), is conceptually close to Albert’s approach: start with fundamental 3N configuration space and argue that the Hamiltonian describing fundamental interactions leads to the 3D structure. The emergence program might be difficult: it is hard to start with the wave function of the universe and recognize the world we see around. Try to find the superposition of the runner from the complex entangled states of particles smeared around the running circle. It is also not clear how helpful the emergence program is. We do not know much about the wave function of the universe which includes all the worlds. But a more modest task is not problematic. We can reconstruct the relevant parts of the universal wave function to explain our world. We need to accept the existence of other parts (corresponding to parallel worlds) for having a good (simple, deterministic and without action at a distance) physical theory. I presented here my preferred concept of a world Vaidman (2002) in which all macroscopic objects are well localized in 3D and thus the wave function of macroscopic variables of these objects is defined in 3D. Note that there is a legitimate alternative to the concept of a world in the MWI, closer to the original proposal of Everett III (1957) which can be understood as a subjective world of an observer. Only he, and all objects he is in contact with, are well localized. Measuring devices (e.g. Schrödinger’s cat) which are not in contact with the observer are in a superposition of macroscopically different states after remote measurements (the meaning of this is clear, even if the semantics is forbidden according to my approach). In this alternative, the configuration space is needed not only for constituents of macroscopic objects, but also for macroscopic objects which are not in contact with the observer. The same argument for the necessity of the configuration space is even stronger if we consider the wave function of the universe which includes all the worlds. Still, the 3D space is important as it is the space of the fundamental interactions.

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14.4 Our World We see a single world. It is not difficult to imagine that there are other worlds like ours in remote galaxies, but that there are parallel worlds here, in the same place, is counter-intuitive. We see our one world and, naturally, are looking for a theory which will tell us how our world evolves. The MWI tells that our world evolves into multiple worlds, it definitely happens at every quantum measurement, but maybe it happens much more often (Albrecht & Phillips, 2014). We experience only a single world at any moment of time, so it is understandable that we are reluctant to accept existence of our copies. What is even harder to accept is that the natural question: “What will be my world in the future?” makes no sense. Understanding that our world often splits should change the paradigm of a world. We should accept that the picture of an evolving world is incorrect: there is no concept of our world line evolving towards the future: the world line becomes a tree which does not correspond to our (single) experience. There is nothing in the MWI which points to the connection of a world in the past to a particular world in the future. However, we can follow our world line backwards in time. Every world has a history as a single world at every moment in the past. (We disregard here in principle possible, but not feasible, situations of merging worlds in experiments of super technology such as Wigner’s friend experiment (Wigner, 1995). This past world line is what we have in our memories now and this is what led to the paradigm of the world evolving forward in time. This provides the possibility to consider a forward evolving world. We can consider the time reversal of the well defined backward evolving world as the time evolution of our world. This world does not follow the laws of physics which are relevant to all worlds together. This “evolution” is not unitary and it includes an action at a distance. It is identical to the evolution of the textbook (collapsing wave function) world. This is the world which allows description of essential features of macroscopic variables (in particular their interaction with other macroscopic objects) by wave functions in 3D in product with entangled states of microscopic constituents of these macroscopic objects required for explanation of the rigidity, the conductivity and other properties of the objects, see (14.2). Let us look more carefully at the wave function of our world. Apart from the wave functions of collective variables of macroscopic objects ψ(An ) represented by wave functions in 3D, the entangled wave functions in configuration space ψ(b1 , b2 , . . . , bNb ) of the constituents of macroscopic objects described in (14.2), there is an (in general entangled) wave function of microscopic systems which are not entangled with macroscopic objects, signified by Φrest in (14.1). We do not experience directly these microscopic systems and usually there is no need to discuss their state. We do not directly experience also details of states of many microscopic constituents of macroscopic objects. Thus, it is natural to describe our world, as humans did a long time ago, by specifying only states of macroscopic objects. However, today, at the time of the quantum information technology revolution, it is sometimes important to describe quantum entangled

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states of microscopic systems in various experiments: single particle interferometry, teleportation, quantum key distribution. These particles are not well localized in 3D and, moreover, we need at least small parts of configuration space to describe their entanglement. I argued above that decoherence is not important in the MWI, but the lack of decoherence of microscopic particles is important. Then, between preparation and measurements, microscopic systems are described by the coherent superposition of macroscopically different states, including wave packets at remote locations. Between measurements, these superpositions either remain approximately constant or evolve in a known unitary way. The results of all measurements specify completely these states. So, one can choose to consider these results as an ontology. This does not seem to be an attractive option for ontology. Anyway, here we discuss the wave function as an ontology. The world wave function is not a complete ontology (the wave function of the universe is), but to explain our experience in our world, the world wave function is the relevant one. Our world is the world in the past which for every moment includes both preparation before this moment and postselection to our particular world after this moment. So, the complete description at every moment has, in addition to the standard forward evolving wave function, the backward evolving wave function specified by the measurement after this moment, see Aharonov and Vaidman (1990). This description with two wave functions evolving forward and backward in time is relevant only for microscopic systems, because only for microscopic systems can these wave functions be macroscopically different. Within a world, by definition, there are no superpositions of macroscopically different wave functions of macroscopic objects. Usually, we do not experience directly microscopic objects, so the two wave function description is important only for understanding quantum experiments. I find it useful because it provides a new consistent (and surprising) picture of the reality of pre and post-selected quantum systems. Microscopic objects can be assigned positions in 3D, but in contrast to the classical behavior of macroscopic objects, they might not necessarily follow classical continuous trajectories: they can leave (weak) traces simultaneously in several places (McQueen & Vaidman, 2020) and these traces have a complex structure (Dziewior et al., 2019). In a single world universe, the connection between experience and the wave function is very natural: there is one experience and one wave function. In the manyworlds universe, the situation is more subtle. If the wave functions of worlds are different only in their distant locations, such that we have only one wave function of particles corresponding to the observer, we have only one observer with a particular experience. But consider a situation in which I performed a quantum measurement and it is arranged that I am moved slowly in a closed chamber to different locations according to the results of the experiment (Vaidman, 1998). After the experiment, there are several Levs, each aware of parts of the wave function corresponding to all Levs and each having the experience of being in a chamber. My wave function corresponds to my experience, but in which chamber am I? Another postulate about the probability of self-location (that it is proportional to the measure of existence

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of the world with the corresponding result of the experiment) is needed (see more details in Vaidman 2020).

14.5 Summary: Three Dimensional Aspects of Universal and World Wave Functions Wave function realism is a thesis that the only ontology of Nature is a pure quantum state without additional or alternative primitive ontology (Allori, 2017) (e.g. beables Bell 1995) in 3D. The wave function of the universe by itself has an intimate connection to the 3D space, although formally it is not defined in 3D. The source of the connection is that the fundamental interaction takes place in 3D which leads to the following explanation why we experience life in 3D. Our experience supervenes only on one of the terms of the superposition (14.3), our world wave function. In my semantics, in every world all macroscopic objects are well localised in 3D and every world wave function describes full 3D space including remote galaxies. (I do not enter cosmological issues of the size of the universe.) In a world wave function every macroscopic object is described by a well localised wave function in 3D in a product state with a (usually entangled) state of their constituents, states of other macroscopic objects, and (possibly entangled) states of microscopic systems which do not form what we might describe as a macroscopic object. Our experience supervenes only on the part of the world wave function in 3D near us, so the same “we” live in multiple worlds which differ by locations of remote macroscopic objects. We can split our worlds locally by performing quantum experiments. Do it right now with the help of the Tel Aviv WorldSplitter (2012)! There is no meaning to ask in which world will we be after the splitting, but we can ask what was our world in the past. During the whole history of our world (at least not too close to the Big Bang) our world had macroscopic objects well localised in 3D. In our world it might be of interest to assign locations in 3D to some microscopic objects, e.g. a single photon passing through an interferometer. Note, that when its forward and backward evolving wave functions are different, the evolution of these locations might not behave in a classical way (McQueen & Vaidman, 2020). I suggest a direct connection between our experience and our world wave function, recognising our three-dimensional picture in the world wave function by, for example, drawing a three-dimensional map of the density of the wave function of human tissue cells. This is instead of arriving at our experience through operators, e.g., awareness operators (Page, 2003). Although the universal wave function is not defined in 3D, it has a very important property in 3D: there is no action at a distance. Disturbances cannot propagate with superluminal velocity. If we consider the time evolution of two universal wave functions which differ at a particular time only in a localised 3D region, they might

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differ in the future only in the regions that the light starting at that time from the localised region can reach. The universal wave function is nonlocal in 3D in the sense that it is nonseparable, so I have to explain exactly what is the meaning of local differences. If a system C in a remote location is entangled with a system A near me, I can, by a local swap operation, change the entanglement of the remote system C to another system in my location. However, the complete local description (given by local density matrix) in remote locations (the locations the light cannot reach) cannot be changed. The wave function of a particular world has different 3D properties. Remember the way we consider the time evolution of a world. At a particular time, based on our records of events (results of quantum measurements in the past), we reconstruct the forward evolving (collapsing) wave function. To formulate action at a distance we consider two situations in which we reconstruct world wave functions in a particular time in the past which are different only in a localised 3D region. Since world wave functions evolve in a non-deterministic way, there (most probably) will be local differences in remote locations. But if the difference is a change of a setup which specifies the measurement in the local region on a particle entangled with a system in a remote location, we can be certain that there will be differences between the two world wave functions in the regions the light cannot reach. Within a particular world, there is an action at a distance in 3D. On the other hand, in the world wave function we have separability in the 3D of the part of the wave function describing macroscopic properties of macroscopic objects. There is no entanglement between macroscopic objects. They all are described by the product of localised wave packets in 3D (times the quantum states of their constituents). The majority of physicists view quantum theory as a great success. They all say that the wave function collapsing at measurements explains in an excellent way all that we see around. They accept the postulate that our experience supervenes on this wave function. It is the Collapse with its action at a distance and randomness that goes against the spirit of physics. What I tried to explain here is that we do not need Collapse. The wave function, instead of collapsing, splits into macroscopically different world wave functions. Every one of these world wave functions explains well the experience of life in 3D. Acknowledgments I thank Michael Ridley for useful discussions. This work has been supported in part by the Israel Science Foundation Grant No. 2064/19.

References Aharonov, Y., & Vaidman, L. (1990). Properties of a quantum system during the time interval between two measurements. Physical Review A, 41, 11–20. Albert, D. Z. (1996). Elementary quantum metaphysics. In S. G. J. T. Cushing, & A. Fine (Eds.), Bohmian mechanics and quantum theory: An appraisal (pp. 277–284). Springer.

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Albert, D. Z. (2013). Wave function realism. In: A. Ney, & D. Z. Albert (Eds.), The wave function: Essays on the metaphysics of quantum mechanics (pp. 52–57). Oxford: Oxford University Press. Albrecht, A., Phillips, D. (2014). Origin of probabilities and their application to the multiverse. Physical Review D, 90(12), 123,514. Allori, V. (2017). Primitive ontology and the classical world. In Quantum structural studies: Classical emergence from the quantum level (pp. 175–199). World Scientific. Allori, V., Goldstein, S., Tumulka, R., Zanghì, N. (2011). Many worlds and Schrödinger’s first quantum theory. British Journal for the Philosophy of Science, 62(1), 1–27. Allori, V., Goldstein, S., Tumulka, R., Zanghì, N. (2014). Predictions and primitive ontology in quantum foundations: a study of examples. British Journal for the Philosophy of Science, 65(2), 323–352. Bacciagaluppi, G. (2020). The role of decoherence in quantum mechanics. In: E. N. Zalta (Ed.), The stanford encyclopedia of philosophy, fall, 2020 edn. Metaphysics Research Lab, Stanford University. Bell, J. S. (1995). The theory of local beables. In Quantum mechanics, high energy physics and accelerators. Selected papers of John S. Bell (with commentary). Singapore: World Scientific, (1976). Dziewior, J., Knips, L., Farfurnik, D., Senkalla, K., Benshalom, N., Efroni, J., Meinecke, J., BarAd, S., Weinfurter, H., & Vaidman, L. (2019). Universality of local weak interactions and its application for interferometric alignment. Proceedings of the National Academy of Sciences, 116(8), 2881–2890. Earman, J. (1986). A primer on determinism, Reidel, Boston. Everett III, H. (1957) “Relative state” formulation of quantum mechanics. Reviews of Modern Physics, 29, 454–462. Ghirardi, G. C., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34, 470–491. Maudlin, T. (2010). Can the world be only wavefunction? In: A. K. S. Saunders J. Barrett, & D. Wallace (Eds.), Many worlds? Everett, quantum theory, & reality (pp. 121–143). Oxford University Press. McQueen, K. J., & Vaidman, L. (2020). How the many worlds interpretation brings common sense to paradoxical quantum experiments. In Scientific challenges to common sense philosophy (pp. 40–60). Routledge . Ney, A. (2021). The world in the wave function: A metaphysics for quantum physics. Oxford University Press. Ney, A., & Albert, D. Z. (2013). The wave function: Essays on the metaphysics of quantum mechanics. Oxford University Press. Page, D. N. (2003) Mindless sensationalism: A quantum framework for consciousness. In Q. Smith, A. Jokic (Eds.), Consciousness: New philosophical perspectives (pp. 468–506). OUP Oxford. Pearle, P. (1976). Reduction of the state vector by a nonlinear schrödinger equation. Physical Review D, 13(4), 857. Wallace, D. (2012). The emergent multiverse: Quantum theory according to the everett interpretation. Oxford University Press. Wallace, D. (2020). Against wavefunction realism. In: S. Dasgupta, R. Dotan, & B. Weslake (Eds.), Current controversies in philosophy of science (pp. 63–74). Abingdon: Routledge. Wigner, E. P. (1995). Remarks on the mind-body question. In Philosophical reflections and syntheses (pp. 247–260). Springer. WorldSplitter (2012). Tel Aviv University. http://qol.tau.ac.il. Vaidman, L. (1998). On schizophrenic experiences of the neutron or why we should believe in the many-worlds interpretation of quantum theory. International Studies in the Philosophy of Science, 12, 245–261. Vaidman, L. (2002). Many-worlds interpretation of quantum mechanics. In: E. N. Zalta (Ed.), The stanford encyclopedia of philosophy. Metaphysics Research Lab. Stanford University.

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Vaidman, L. (2014). Quantum theory and determinism. Quantum Studies: Mathematics and Foundations, 1(1–2), 5–38. Vaidman, L. (2016). All is ψ. Journal of Physics: Conference Series, 701, 012,020. Vaidman, L. (2019). Ontology of the wave function and the many-worlds interpretation. In O. Lombardi, S. Fortin, C. López, & F. Holik (Eds.), Quantum worlds: Perspectives on the ontology of quantum mechanics. Cambridge University Press. Vaidman, L. (2020). Derivations of the born rule. In M. Hemmo, & O. Shenker (Eds.), Quantum, probability, logic: The work and influence of itamar pitowsky (pp. 567–584). Springer Nature. Von Neumann, J. (2018). Mathematical foundations of quantum mechanics. Princeton University Press.

Chapter 15

Reality as a Vector in Hilbert Space Sean M. Carroll

Abstract I defend the extremist position that the fundamental ontology of the world consists of a vector in Hilbert space evolving according to the Schrödinger equation. The laws of physics are determined solely by the energy eigenspectrum of the Hamiltonian. The structure of our observed world, including space and fields living within it, should arise as a higher-level emergent description. I sketch how this might come about, although much work remains to be done. Invited contribution to the volume Quantum Mechanics and Fundamentality: Naturalizing Quantum Theory Between Scientific Realism and Ontological Indeterminacy; Valia Allori (ed.).

[I]t would be naive in the extreme to be a ‘Hilbert-space-vector realist’: to reify Hilbert space, and take it as analogous to physical space. – Wallace (2017) In Hilbert space, nobody can hear you scream. – Aharonov and Rohrlich (2005)

Almost a century after the 1927 Solvay Conference, the question of the ultimate ontology of quantum mechanics remains unsettled. Essentially all formulations of quantum theory rely on the use of a wave function or state vector (or mathematically equivalent structures). But researchers do not agree on whether the state vector is a complete and exact representation of reality, whether it represents part of reality but needs to be augmented by additional variables to be complete, or whether it is an epistemic tool rather than a representation of reality at all. And they further do not agree on whether the state vector should be thought of as purely an element of some abstract Hilbert space, or whether there is some fundamental ontological status to a particular representation of that vector in terms of something more directly physical, such as configuration space or particles or fields in honest threedimensional “space.”

S. M. Carroll () Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, USA Santa Fe Institute, Santa Fe, NM, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_15

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Here I want to argue for the plausibility of an extreme position among these alternatives, that the fundamental ontology of the world is completely and exactly represented by a vector in an abstract Hilbert space, evolving in time according to unitary Schrödinger dynamics. Everything else, from particles and fields to space itself, is rightly thought of as emergent from that austere set of ingredients. In particular, there is no preferred set of observables or decomposition of Hilbert space. This approach has been called “Mad-Dog Everettianism” (Carroll & Singh, 2019) although “Hilbert Space Fundamentalism” would be equally accurate. Let’s see how one might end up seduced by an ideology that is so radically different from our direct experience of the world. When we are first taught quantum mechanics, we are shown how to construct quantum theories by taking classical models and quantizing them. Imagine we have a classical precursor theory defined on some phase space, expressed mathematically as a symplectic manifold , with evolution determined by some Hamiltonian function H : → R. We choose a “polarization” on phase space, which amounts to coordinatizing it in terms of canonical coordinates Q (defining “configuration space”) and corresponding canonical momenta P , where each symbol might stand for multiple dimensions. This is a fairly general setup; for N point particles moving in D-dimensional Euclidean space, configuration space is isomorphic to RDN , but we could also consider field theory, for which the coordinates are simply the values of the fields throughout space, and configuration space is infinite-dimensional. One way to construct a corresponding quantum theory is to introduce complexvalued wave functions of the coordinates alone, (Q) ∈ C.Wave functions must be normalizable, in the sense that they are square-integrable,  ∗  dQ < ∞, where  ∗ is the complex conjugate of . Momenta are now represented by linear operators ˆ Pˆ ] = Pˆ , whose form can be derived from the canonical commutation relations [Q, ˆ i h¯ (where the operator Q is simply multiplication by Q). This lets us promote the ˆ Pˆ ) (up to potential operatorclassical Hamiltonian to a self-adjoint operator Hˆ (Q, ordering ambiguities). We then posit that the wave function evolves according to the Schrödinger equation, ∂ Hˆ  = i h¯ . ∂t

(15.1)

This form of the Schrödinger equation is perfectly general, and applies to relativistic theories as well as non-relativistic ones, as long as one uses an appropriate Hamiltonian. (There may be equivalent representations using what would, ex post facto, be thought of as Lorentz-transformed Hamiltonians; if so, that should be emergent from the dynamics specified by any representative Hamiltonian.) This procedure gives us the beginnings of a quantum theory. For Everettians, it gives us the complete theory; a unitarily-evolving quantum state describes the entire ontology at a fundamental level. Other approaches require additional dynamical rules, physical structures, or some combination thereof. Here our interest is in seeing how far we can get from a minimal starting point, so the Everettian approach is appropriate. (For more on structure in the Everett interpretation, see Saunders, 2021; Wallace, 2003.)

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But how precisely should we think about this ontology? It might seem to include at least the original configuration space coordinatized by {Q}, the set of wave functions (Q), and the Hamiltonian Hˆ . This has led Albert (1996) to suggest “wave function realism”—the idea that it is the wave function (Q) that describes reality, and that the wave function is defined on configuration space, so that configuration space (3N dimensional for N particles in 3 dimensions, infinitedimensional for a field theory) is where the wave function “really lives” (see also Ney, 2013; North, 2013). This point of view has been criticized by Wallace (2017), who points out that configuration space is simply one possible way of representing the wave function; it could also be represented in momentum space (as a function of P ), or an infinite number of other choices. (Wallace also has other criticisms, and other authors criticize the idea for other reasons Allori 2013; Myrvold 2015; Ney has recently responded Ney 2021.) The ambiguity between defining wave functions in position versus momentum space is an example of a broader issue. “Quantization” does not give us a oneto-one map from classical theories to quantum ones, or even a well-defined map at all. A single classical precursor may correspond to multiple quantum theories, due to operator-ordering ambiguities. Moreover, distinct classical models may have identical quantizations, as in the dualities of quantum field theory. This makes it difficult to uniquely pinpoint what a quantum theory is supposed to be a theory “of.” In the duality between the massive Thirring model and sine-Gordon theory, a single quantum theory can arise from classical precursors describing either fundamental bosons or fundamental fermions (Coleman, 1975). In the AdS/CFT correspondence, a nongravitational field theory in D spacetime dimensions is dual to a quantum gravity theory in D + 1 dimensions (Maldacena, 1999). So from a quantum theory alone, it might be impossible to say what kinds of fields the model describes, or even the number of dimensions they live in. But the complicated relationship between classical and quantum theories is a problem for physicists, not for physics. Nature simply is quantum from the start, and the classical world arises as an emergent approximation in some appropriate limit. If our interest is in fundamental ontology, rather than focusing on quantum theories derived by quantizing classical precursors, it would make sense to consider the inverse problem: given a quantum theory, what kind of classical limits might arise within it? So let us think carefully about what it means to be given a quantum theory (at least from an Everettian perspective). Wave functions, as von Neumann noted long ago, can be added and scaled by complex numbers, and have a natural inner product  defined by (, ) =  ∗ . They therefore describe a complex, normed vector space, called Hilbert space, and the fact that the Schrödinger equation is linear means that it respects this structure. In vector-space language, the choice between expressing the wave function in configuration space or momentum space is merely a change of basis in Hilbert space, which presumably has no physical importance whatsoever. The physical quantum state—as distinguished from its representation in some particular basis—is simply a vector in Hilbert space, sometimes called the “state vector,” and written in Dirac notation as |.

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It would therefore seem natural, if our goal is to take a nature’s-eye view of things and specify the correct quantum theory of the world in its own right, to define that theory as a set of state vectors | in a Hilbert space H, evolving under the Schrödinger equation via a specified Hamiltonian Hˆ . The problem is that this is very little structure indeed. Hilbert space itself is featureless; a particular choice of Hilbert space is completely specified by its dimension d = dim H. A vector in Hilbert space contains no direct specification of what the physical content of such a state is supposed to be; there is no mention of space, configuration space, particles, fields, or any such familiar notions. Presumably all of that is going to have to somehow emerge from the dynamics, which seems like a tall order. One might imagine that we can somehow read off the physical structure being described from the explicit form of the Hamiltonian. For example, if we were handed h¯ 2 ∂ 2 1 Hˆ = − + mω2 xˆ 2 , 2 2m ∂x 2

(15.2)

we would quickly surmise that we had a simple harmonic oscillator on our hands. But that’s only because we have conveniently been given the Hamiltonian in a particularly useful basis, in this case the position basis |x. That is not part of the specification of the theory itself. Hilbert space does not come equipped with a preferred basis; we should be able to deduce that the position basis (or some other one) is useful to our analysis of the system, rather than assuming we have been given it from the start. Given that the only data that comes along with Hilbert space is its dimension d, the only other thing we have to work with is the Hamiltonian considered as an abstract operator. That does pick out one particular basis: that of the energy eigenstates, vectors satisfying Hˆ |n = En |n. (For simplicity we assume nondegenerate energy eigenvalues, so that the states {|n} define a unique basis.) There is no information contained in the specification of the energy eigenstates themselves; they are just a collection of orthonormal vectors. The information about the Hamiltonian is contained entirely in its set of eigenvalues {En }, called the “spectrum” of the Hamiltonian. A specification of a quantum theory consists entirely, therefore, of this list of real numbers, the energy eigenspectrum. There is an important caveat to this statement. If Hilbert space is non-separable (infinite non-countable dimension), there can be unitarily inequivalent representations of the canonical commutation relations (Haag, 1955). It is therefore necessary to give additional information to define the theory; typically this would amount to specifying an algebra of observables. We have good reason, however, to expect that the Hilbert space for the real world is finite-dimensional, at least if we restrict our attention to our observable universe or any other finite region of space (Banks, 2001; Bao et al., 2017; Bekenstein, 1981; Bousso, 1999; Jacobson, 2012). The reason is gravity. In the presence of gravity, to make a long story short, the highest-entropy configuration we can construct in a spherical region R of radius r is a black hole, and black holes have a finite entropy

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SBH = A/4G = πr 2 /G, where A is the area of the event horizon and G is Newton’s constant (and we set h¯ = c = 1). If we decompose Hilbert space into a tensor product of a factor describing that region and one describing the rest of the world, H = HR ⊗ HE ,

(15.3)

it follows that we have an upper bound on the dimensionality of HR , given by dim HR ≤ exp (eπr

2 /G

).

(15.4)

If R represents our observable universe with current values of the cosmological 123 parameters, this works out to approximately dim HR ≤ ee , which is large but still 1 smaller than infinity. We don’t know whether the dimension of the full Hilbert space H is finite or infinite, other than the indirect consideration that infinite time evolution in a finite-dimensional Hilbert space would lead to a proliferation of Boltzmann Brains (Carroll, 2008; Dyson et al., 2002). But for our current purposes, it suffices that operations confined to our observable universe effectively act on a finitedimensional part of Hilbert space. In that case, the set of in-principle observables is simply all Hermitian operators on HR . What can be observed in practice will depend on how we split further split Hilbert space into systems and observers, but the theory is completely specified by the Hilbert space and Hamiltonian, or in other words, by the discrete list of numbers {En } in the energy spectrum. By which we mean the spectrum of the Hamiltonian acting on HR itself, assuming that interactions with degrees of freedom in HE can be neglected for practical purposes. Henceforth we will speak as if the finite-dimensional factor HR is the relevant part of Hilbert space, and our world can be described by a unitarily-evolving state within it. Technically it is more likely to be a mixed state described by a density operator, but that can always be purified by adding a finite-dimensional auxiliary factor to Hilbert space, so we won’t worry about such details. The challenge facing such an approach should be clear. The world of our experience doesn’t seem like a vector in Hilbert space, evolving according to a list of energy eigenvalues. It seems like there is space, and objects located in space, and those objects interact with each other, and so forth. How in the world is all of that supposed to come from a description as abstract and featureless as a vector evolving through Hilbert space? To make matters seemingly worse, the actual evolution is pretty trivial; in the energy eigenbasis, an exact solution to the Schrödinger equation

1

There are a number of nuances here. In gauge theories, we cannot precisely decompose Hilbert space into factors representing regions of space. And we are somewhat cheating by invoking “regions of space” at all, although this will be a sensible notion on individual semiclassical branches of the universal quantum state. For elaboration see (Bao et al., 2017).

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from an initial state |(0) =



n ψn |n

|(t) =

is ψn e−iEn t |n .

(15.5)

n

Each component simply evolves via an energy-dependent phase factor. It seems like a long way from the complicated nonlinear dynamics of our world. This challenge accounts for the epigraphs at the beginning of this paper. Everettians tend to think that the right strategy for understanding the fundamental nature of reality is not necessarily to start with what the world seems like and to construct an ontology that hews as closely as possible to that. Rather, we should start with some proposed ontology and ask what it would seem like to observers (if any such exist) described by it. Clearly, “observers” are not represented directly by a vector in Hilbert space, nor is the world that they observe. What we can instead ask is whether there could be a higher-level description, emergent from our ontology, that can successfully account for our world. Such a description is not forced on us at the God’s-eye (or Laplace’s-Demon’s eye) view of the universe. It would always be possible to say that reality is a vector in Hilbert space, evolving through time, and stop at that. But that’s not the only thing we’re allowed to say. The search for emergent levels is precisely the search for higher-level, non-fundamental descriptions that approximately capture some of the relevant dynamics, perhaps on the basis of incomplete information about the fundamental state. The question is whether we can recover the patterns and phenomena of our experience (space, objects, interactions) from the behavior of our fundamental ontology. To get our bearings, consider the classic case of N massive particles moving in three-dimensional space under the rules of classical Newtonian gravity. The state of the system is specified by one point in a 6N-dimensional phase space. Yet there is an overwhelming temptation to say that the system “really lives” in threedimensional space, not the 6N-dimensional phase space. Can we account for where that temptation comes from without postulating any a priori metaphysical essence to three-dimensional space? There are two features of the description as N particles that make it seem more natural than that featuring a single point in phase space, even though they are mathematically equivalent. The first is that the internal dynamics of the system are more easily interpreted in the N-particle language. For example, it is immediately clear that two particles will strongly affect each other when they are nearby and the others are relatively far away. This kind of partial and approximate understanding of the dynamics is transparent in the N-particle description, and obscured in the pointin-phase-space description. The second is that the system looks like N particles. That is, in the real-world analogues of this toy model, when we observe the system by interacting with it as a separate physical system ourselves, what we immediately see are N particles. There can be multiple equivalent ways of describing the internal dynamics of a system, but the one we think of as “natural” or describing what “really exists” is often predicated on how that system interacts with the outside world.

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Similar considerations apply to familiar examples of emergence, such as treating a box of many atoms as a fluid. In this case the two descriptions are not equivalent— the emergent fluid description is an approximation obtained by coarse-graining—but the same principles apply. The internal dynamics of the particles in the box are more easily apprehended in the fluid description (fewer variables and equations are required, given some short-distance coarse-graining scale), and we can measure the fluid properties directly (using thermometers and barometers and so on), while the states of each individual atom are inaccessible to us. Emergent structures that accurately describe the dynamics of part of a system using only information accessible within the emergent description itself have every right to be thought of as “real,” even if they are not “fundamental” (Dennett, 1991; Wallace, 2012). These two considerations (internal dynamics and what we see) work in tandem: we want the information we gather by observing a system to be sufficient for us to predict its subsequent behavior. The ontology that seems most natural for us to ascribe to a physical subsystem thus depends on how that subsystem interacts with the rest of the world (Zanardi et al., 2004). For our program of Hilbertspace fundamentalism, this suggests that we should look for emergent descriptions by considering ways to factorize H into a tensor product of factors representing different subsystems, and ask how those subsystems interact with each other. (For certain situations we might also consider direct-sum structures Kabernik et al. 2020.) We are therefore interested in quantum mereology: how to decompose the whole of Hilbert space into parts such that individual subsystems have simple internal dynamics, and those dynamics are readily observed via interactions with other subsystems, given nothing but the spectrum of the Hamiltonian (Carroll & Singh 2019; for related work see Brun & Hartle 1999; Hartle 2011; Ney 2020a; Tegmark 2015). This seems ambitious, but turns out to be surprisingly tractable. Consider the most basic thing we might want to describe by such a factorization, the distinction between a “system” representing a macroscopic object exhibiting quasi-classical behavior and an “environment” describing degrees of freedom that passively monitor the system and lead to decoherence. This corresponds to expressing Hilbert space as the tensor product H = HS ⊗ HE .

(15.6)

Fixing the dimensions of HS and HE , different factorizations are related by unitary transformations that mix the two factors together. For any specific factorization, we automatically get a decomposition of the Hamiltonian into a self-Hamiltonian for the system, another self-Hamiltonian for the environment, and an interaction term: Hˆ = Hˆ S ⊗ IE + IS ⊗ Hˆ E + Hˆ int.

(15.7)

We can now ask, within the set of all possible factorizations, which ones lead to system dynamics that can be described by approximate classical behavior in appropriate circumstances?

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In this decomposition, internal dynamics of the system are governed by Hˆ S and how it is observed by the environment is governed by Hˆ int. (We assume that dim HE  dim HS and that the environment’s internal dynamics are largely irrelevant.) To recover classical behavior when the system is macroscopic, we want localized wave packets in HS to stay relatively localized and follow classical equations of motion under Hˆ S , but also for unentangled states in HS ⊗ HE to remain relatively unentangled; the environment is supposed to passively monitor the system, not rapidly reach maximal entanglement with it. It was argued by Carroll and Singh (2021) that generic Hamiltonians, as defined by their spectra, have neither of these features in any factorization; the Hamiltonian of the real world is apparently non-generic, to nobody’s surprise.2 When the Hamiltonian and the system/environment split allow for classical behavior, the density operator for the system rapidly diagonalizes in the dynamically-preferred pointer basis {|φn }, such that the corresponding pointer states are robust under environmental monitoring (Zurek, 1981). The pointer basis  ˆS = |φn  φn |. defines a pointer observable in the system’s Hilbert space, Q Pointer states are the ones that appear classical, and a standard pointer observable is the position in space of a system; a macroscopically coherent cat is described by a pointer state, but a superposition of cats in different physical configurations is not (Joos & Zeh, 1985; Riedel et al., 2012; Zurek, 1993). In the Quantum Measurement Limit, where Hˆ int dominates over Hˆ S , the pointer observable satisfies Zurek’s commutativity criterion (Zurek, 1981), [Hˆ int, Qˆ S ⊗ IE ] ≈ 0,

(15.8)

reflecting the fact that the system in a pointer state does not continually entangle with the environment. These ideas about pointer states and decoherence are usually discussed within the context of a known factorization, but they also suggest a criterion for determining the factorization that allows for classical behavior (Carroll & Singh, 2021). Given H and Hˆ , we search through all possible factorizations, and for each one we define a candidate pointer observable as the one that comes closest to achieving (15.8). This candidate pointer observable embodies the idea that external observers “see” certain features of the system, and that those features evolve classically. We can then calculate, from an initially localized and unentangled system state, the rate of delocalization and entanglement growth. The correct factorization is the one that minimizes both. Simple numerical examples verify that this criterion picks out what we usually think of as the standard system/environment split. Again, the sought-after behavior is non-generic; it doesn’t occur for random Hamiltonians, nor for random factorizations with any given Hamiltonian.

2

If you are surprised, consider for example that the unitary dynamics in real-world physics seem to be local in space to a high degree of precision. There are infinitely many more ways for a Hamiltonian to be non-local than for it to be local.

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The data contained within the spectrum of Hˆ therefore seems to provide the requisite information to pinpoint an appropriate emergent classical description, when one exists. Thus reassured, let us be somewhat more ambitious, and train our sights on the ontology that seems to accurately describe our low-energy world (Carroll, 2021): an effective quantum field theory that includes both gravity and the Standard Model of particle physics, defined in four-dimensional spacetime. We can again take guidance from our experience of the real world. In quantum field theory (modulo previously-mentioned nuances), we can think about degrees of freedom as being associated with regions of space. If we partition space into regions indexed by α, there is a corresponding decomposition of Hilbert space into a tensor product of local factors, H=



Hα .

(15.9)

α

This lets us expand the Hamiltonian as a sum of self-Hamiltonians for each region, plus interactions between pairs of regions, triplets of regions, and so on: Hˆ =

a

ha Oˆ a(self) +

ab

hab Oˆ ab

(2−pt)

+

habc Oˆ abc

(3−pt)

+ ··· ,

(15.10)

abc

where the h··· are numerical parameters. A necessary requirement for the emergence of a structure recognizable as “space,” with local dynamics therein, is that the series (15.10) does not continue indefinitely when the factorization (15.9) corresponds to local regions. Degrees of freedom only interact with a small number of nearest neighbors, not with regions arbitrarily far away, which means that we only need interactions between a small number of Hilbert-space factors to capture the entire Hamiltonian. Happily, this requirement is also essentially sufficient. As shown by Cotler et al. (2019), generic Hamiltonians admit no local factorization at all, and when such a factorization exists, it is unique up to irrelevant internal transformations (and some technicalities that we won’t go into here). Therefore, the spectrum of the Hamiltonian is enough to pick out the correct notion of an emergent spatial structure when one exists. A natural next step would be to define an emergent dynamical metric on spacetime, as in general relativity. Here results are much less definitive, but initial auguries are promising. (For alternative perspectives see Carlip 2014; Giddings 2015; Hu 2009; Huggett & Wüthrich 2013; Ney 2020b; Nielsen & Kleppe 2013; Raasakka 2017.) In the local decomposition of Hilbert space (15.9), given an overall state | we can calculate the reduced density matrix ρˆα for each factor or collection of factors, and the corresponding von Neumann entropy Sα = −Tr ρˆ log ρ. ˆ The amount of entanglement between two factors can be measured by the quantum mutual information, I (α : β) = Sα + Sβ − Sαβ .

(15.11)

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In the vacuum state of a quantum field theory, we know that the entanglement between two regions decreases monotonically (exponentially in theories with massive fields, as a power law in conformal theories) with the distance between them. We can therefore imagine turning this around, and defining a distance metric depending inversely on the mutual information.3 (In non-vacuum states, collections of particles may have arbitrary entanglement regardless of their distance; however, in realistic circumstances there are far more unexcited quantum-field modes than excited ones, so this fact makes essentially no quantitative difference.) If there exists an emergent best-fit smooth geometry, that can be uniquely determined using techniques such as classical multidimensional scaling. Thus, both the dimensionality and geometry of space can be defined using the entanglement information contained in a state |, at least if that state is near to the vacuum (Cao et al., 2017). We can go further, and show that under certain optimistic assumptions (most notably, the eventual emergence of approximate Lorentz invariance), the resulting geometry obeys Einstein’s equation of general relativity in the weak-field limit (Cao & Carroll, 2018). The basic idea, following Jacobson (1995), Jacobson (2016), is to posit that perturbations in the quantum state lead to a change in the emergent area of a codimension-one surface that is proportional to the change in entanglement entropy across that surface, δA ∝ δS.

(15.12)

The area perturbation, being a geometric quantity, can then be related to the Einstein tensor, while in the long-distance limit the entropy can be related to the stress-energy tensor. Assuming emergent Lorentz symmetry, the resulting dynamical equation for a perturbation of the vacuum (representing flat spacetime) is δGμν ∝ δTμν ,

(15.13)

just as Einstein leads us to expect. This is a provocative result, but one that shouldn’t be over-interpreted. The accomplishment is not that we recover specifically Einstein’s equation; there aren’t that many other locally Lorentz-invariant equations one could imagine for a dynamical spacetime metric. What matters is that the necessary information required for such a description to emerge can be found in an abstract quantum state evolving according to a Hamiltonian specified purely by its spectrum, without any additional ontological assistance. Moreover, while we can use entanglement

3

The resulting metric is defined on the same spacetime structure in which entanglement exists. This is in contrast to the emergence of spacetime from entanglement in the AdS/CFT context (Faulkner et al., 2014; Maldacena & Susskind, 2013; Swingle, 2012; van Raamsdonk, 2010), where entanglement is on the boundary and geometry is in the bulk. There is no inconsistency, as the procedure described here applies to weak-field situations far from any horizons, while the AdS/CFT construction extends over a cosmological spacetime where large-scale curvature is a central part of the description.

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to define an emergent metric, we haven’t shown that it is the emergent metric, the one whose geodesics define the motion of localized test particles. We can imagine how this might happen; as just one piece of the puzzle, the Lieb-Robinson bound in quantum information theory (Lieb & Robinson, 1972) provides a natural mechanism whereby light-cone structures can emerge from quantum information theory (Hamma et al., 2009). What strictly emerges from entanglement in this story is the metric on space, not on spacetime; a reconstruction of the latter depends on a procedure for stitching together spatial surfaces evolving over time. Since our entire analysis is based on the eigenvalues of the Hamiltonian appearing in the Schrödinger equation Hˆ | = i∂t |, we need to imagine that time itself is fundamental, rather than emergent. The appearance of an explicit time parameter does not imply that this parameter is in some sense preferred. Any Lorentz-invariant theory can be written in Hamiltonian form by choosing a frame; such a form will not be manifestly Lorentz invariant, but the symmetry is still there, so there is no obstacle to the emergent theory being approximately Lorentz- and diffeomorphism-invariant. It would be natural if Lorentz invariance is indeed only approximate, since we are working in a finite-dimensional factor of Hilbert space, and there are no nontrivial representations of non-compact symmetry groups on finite-dimensional vector spaces. This raises the intriguing possibility of potential experimental signatures of these ideas, as Lorentz invariance can be tested to high precision (Liberati, 2013). One might also worry about compatibility with the Wheeler-DeWitt equation of quantum gravity, which takes the form Hˆ | = 0. In this case the eigenvalues of Hˆ are seemingly irrelevant, since the world is fully described by a zero-energy eigenstate (cf. Albrecht & Iglesias 2008). But there is also no fundamental time evolution; that is the well-known “problem of time” (Anderson, 2010). A standard solution is to imagine that time is emergent, which amounts to writing the full Hamiltonian as d Hˆ = Hˆ eff − i , dτ

(15.14)

where τ is the emergent time parameter. In that case everything we have said thus far still goes through, only using the eigenvalues of the effective Hamiltonian Hˆ eff . In addition to spacetime, we still have to show how local quantum fields can emerge in the same sense as the spacetime metric. Less explicit progress has been made in reconstructing approximate quantum field theories from the spectrum of the Hamiltonian, but it’s not unreasonable to hope that this task is more straightforward than reconstructing spacetime itself. One promising route is via “string net condensates,” which have been argued to lead naturally to emergent gauge bosons and fermions (Levin & Wen, 2005). Nothing in this perspective implies that we should think of spacetime or quantum fields as illusory. They are emergent, but none the less real for that. As mentioned, we may not be forced to invoke these concepts within our most fundamental picture, but the fact that they play a role in an emergent description is highly non-trivial. (Most Hamiltonians admit no local decomposition, most factorizations admit no

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classical limit, etc.) It is precisely this non-generic characteristic of the specific features of the world of our experience that makes it possible to contemplate uniquely defining them in terms of the austere ingredients of the deeper theory. They should therefore be thought of as equally real as tables and chairs. This has been an overly concise discussion of an ambitious research program (and one that may ultimately fail). But the lesson for fundamental ontology is hopefully clear. Thinking of the world as represented by simply a vector in Hilbert space, evolving unitarily according to the Schrödinger equation governed by a Hamiltonian specified only by its energy eigenvalues, seems at first hopelessly far away from the warm, welcoming, richly-structured ontology we are used to thinking about in physics. But recognizing that the latter is plausibly a higher-level emergent description, and contemplating the possibility that the more fundamental vocabulary is the one straightforwardly suggested by our simplest construal of the rules of quantum theory, leads to a reconstruction program that appears remarkably plausible. By taking the prospect of emergence seriously, and acknowledging that our fondness for attributing metaphysical fundamentality to the spatial arena is more a matter of convenience and convention than one of principle, it is possible to see how the basic ingredients of the world might be boiled down to a list of energy eigenvalues and the components of a vector in Hilbert space. If it did succeed, this project would represent a triumph of unification and simplification, and is worth taking seriously for that reason alone. Acknowledgments It is a pleasure to thank Ashmeet Singh, Charles Cao, Spiros Mikhalakis, and Ning Bao for collaboration on some of the work described here. This research is funded in part by the Walter Burke Institute for Theoretical Physics at Caltech, by DOE grant DE-SC0011632, and by the Foundational Questions Institute.

References Aharonov, Y., & Rohrlich, D. (2005). Quantum paradoxes. New York: Wiley. Albert, D. (1996). Elementary quantum metaphysics. In J. Cushing, A. Fine, & S. Goldstein (Eds.), Bohmian mechanics and quantum theory: An appraisal (pp. 277–284). Dordrecht: Kluwer. Albrecht, A., & Iglesias, A. (2008). The Clock ambiguity and the emergence of physical laws. Physical Review, D77, 063506. Allori, V. (2013). Primitive ontology and the structure of fundamental physical theories. In A. Ney, & D. Albert (Eds.), The wave function: Essays on the metaphysics of quantum mechanics (pp. 58–75). Oxford: Oxford University Press. Anderson, E. (2010). The Problem of Time in Quantum Gravity. https://arxiv.org/abs/1009.2157 Banks, T. (2001). Cosmological breaking of supersymmetry? International Journal of Modern Physics A, A16, 910–921. Bao, N., Carroll, S. M., & Singh, A. (2017). The Hilbert space of quantum gravity is locally finite-dimensional. International Journal of Modern Physics A, D26(12), 1743013. Bekenstein, J. D. (1981). A universal upper bound on the entropy to energy ratio for bounded systems. Physical Review, D23, 287. Bousso, R. (1999). A covariant entropy conjecture. Journal of High Energy Physics, 07, 004.

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Brun, T. A., & Hartle, J. B. (1999). Classical dynamics of the quantum harmonic chain. Physical Review D, 60(12), 123503. Cao, C., & Carroll, S. M. (2018). Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space. Physical Review D, 97(8), 086003. Cao, C., Carroll, S. M., & Michalakis, S. (2017). Space from Hilbert space: Recovering geometry from bulk entanglement. Physical Review D, D95(2), 024031. Carlip, S. (2014). Challenges for emergent gravity. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 46, 200–208. Carroll, S. M. (2008). What if Time Really Exists? https://arxiv.org/abs/0811.3772 Carroll, S. M. (2021). The Quantum Field Theory on Which the Everyday World Supervenes. https://arxiv.org/abs/2101.07884 Carroll, S. M., & Singh, A. (2019). Mad-dog everettianism: Quantum mechanics at its most minimal. In A. Aguirre, B. Foster, & Z. Merali (Eds.), What is fundamental? (pp. 95–104). Springer. Carroll, S. M., & Singh, A. (2021). Quantum mereology: Factorizing Hilbert space into subsystems with quasiclassical dynamics. Physical Review A, 103(2), 022213. Coleman, S. (1975). Quantum sine-gordon equation as the massive thirring model. Physical Review D, 11, 2088–2097. Cotler, J. S., Penington, G. R., & Ranard, D. H. (2019). Locality from the Spectrum. Communications in Mathematical Physics, 368(3), 1267–1296. Dennett, D. C. (1991). Real patterns. The Journal of Philosophy, 88(1), 27–51. Dyson, L., Kleban, M., & Susskind, L. (2002). Disturbing implications of a cosmological constant. Journal of High Energy Physics, 10, 011. Faulkner, T., Guica, M., Hartman, T., Myers, R. C., & Van Raamsdonk, M. (2014). Gravitation from entanglement in holographic CFTs. Journal of High Energy Physics, 03, 051. Giddings, S. B. (2015). Hilbert space structure in quantum gravity: An algebraic perspective. Journal of High Energy Physics, 12, 099. Haag, R. (1955). On quantum field theories. Matematisk-fysiske Meddelelser, 29(12), 1–37. Hamma, A., Markopoulou, F., Premont-Schwarz, I., & Severini, S. (2009). Lieb-Robinson bounds and the speed of light from topological order. Physical Review Letters, 102, 017204. Hartle, J. B. (2011). The quasiclassical realms of this quantum universe. Foundations of Physics, 41, 982–1006. Hu, B.-L. (2009). Emergent/quantum gravity: Macro/micro structures of spacetime. In Journal of Physics: Conference Series (vol. 174, p. 012015). IOP Publishing. Huggett, N., & Wüthrich, C. (2013). Emergent spacetime and empirical (in)coherence. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 44(3), 276–285. Jacobson, T. (1995). Thermodynamics of spacetime: The einstein equation of state. Physical Review Letters, 75, 1260–1263. Jacobson, T. (2012). Gravitation and vacuum entanglement entropy. International Journal of Modern Physics D, D21, 1242006. Jacobson, T. (2016). Entanglement equilibrium and the Einstein equation. Physical Review Letters, 116(20), 201101. Joos, E., & Zeh, H. D. (1985). The emergence of classical properties through interaction with the environment. Zeitschrift für Physik B Condensed Matter, 59, 223–243. Kabernik, O., Pollack, J., & Singh, A. (2020). Quantum state reduction: Generalized bipartitions from algebras of observables. Physical Review A, 101(3), 032303. Levin, M. A., & Wen, X.-G. (2005). String net condensation: A physical mechanism for topological phases. Physical Review B, 71, 045110. Liberati, S. (2013). Tests of Lorentz invariance: A 2013 update. Classical and Quantum Gravity, 30, 133001. Lieb, E. H., & Robinson, D. W. (1972). The finite group velocity of quantum spin systems. Communications in Mathematical Physics, 28, 251–257.

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Chapter 16

Platonic Quantum Theory Jacob A. Barandes

“One’s ideas must be as broad as Nature if they are to interpret Nature.” —Sherlock Holmes, in Arthur Conan Doyle’s A Study in Scarlet, 1887

Abstract In this essay, I describe a novel interpretation of quantum theory, called the Platonic interpretation or Platonic quantum theory. This new interpretation is based on a sharp notion of ontology and on the Gelfand-Naimark-Segal (GNS) construction, in contrast to traditional approaches that start instead by assuming the textbook formalism of Hilbert spaces and wave functions. I show that the Platonic interpretation naturally emerges from the consideration of random variables that represent underlying physical properties but that lack a common sample space, in the sense that they do not share an overall probability distribution despite having individual probability distributions.

16.1 Introduction 16.1.1 The Measurement Problem Quantum theory is the most empirically successful scientific framework in human history. It is therefore remarkable that a century after the theory’s founding, there is still no consensus over its proper physical interpretation. One can trace this ongoing controversy directly to the Dirac-von Neumann axioms (Dirac, 1930; von Neumann, 1932c), which underlie quantum theory’s textbook formulation (Sakurai, 1993;

J. A. Barandes () Jefferson Physical Laboratory, Harvard University, Cambridge, MA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_16

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Shankar, 1994) and contain a famous puzzle known as the measurement problem, which is worth reviewing briefly. According to the Dirac-von Neumann axioms, a quantum system ordinarily evolves in time according to the Schrödinger equation or, more generally, according to an appropriate open-system master equation (Breuer and Petruccione 2002)— except when the system is being ‘measured,’ in which case it instead undergoes ‘collapse’ to a definite outcome. Neither the Schrödinger equation nor master equations generically single out definite measurement outcomes, whereas collapse, by stipulation, does. However, the Dirac-von Neumann axioms do not provide a rigorous definition of what physically constitutes a measurement, apart from positing that measurements are something that ‘observers’ do. Hence, textbook quantum theory is fundamentally ambiguous about when, on the one hand, to apply the Schrödinger equation or a master equation, and when, on the other hand, to apply collapse. This ambiguity is essentially the measurement problem. One might hope that the subset of the Dirac-von Neumann axioms that specifically concern measurements and collapse could be replaced by an adequately sophisticated master equation that, by itself, could somehow produce definite outcomes. Early efforts in this direction focused on decoherence (Bohm 1951; Zeh 1970; Joos & Zeh, 1985; Zurek 1994), which refers to the tendency of environmental effects to cause a quantum system’s density operator to evolve rapidly into a form that mathematically resembles a classical probability distribution. Unfortunately, this hope is obstructed by the fact that decoherence simply does not single out definite outcomes. Moreover, the textbook version of collapse is manifestly nonlocal, apparently capable of reaching out instantaneously across vast stretches of space to affect faraway systems, whereas the no-signaling/no-communication theorem (Ghirardi et al., 1980; Jordan, 1983) guarantees that physically realistic master equations are always fundamentally local.

16.1.2 A New Approach A physical theory should ideally be transparent about its ontology, epistemology, and nomology, which refer respectively to what the theory says exists in reality, how the theory characterizes the limitations of our knowledge and probabilistic predictions, and what laws capture the behavior of the theory’s ingredients. Historically, the challenge of reconciling these three basic theoretical ingredients with the measurement problem in particular, as well as with the larger issue of making sense of state vectors and Hilbert spaces more generally, has pushed physicists and philosophers in a number of different interpretational directions. Some of these interpretational directions, like the de Broglie-Bohm approach, augment the Hilbert spaces and state vectors of quantum theory with a supplementary set of ontological variables. Other directions, like the Everett ‘many worlds’ approach, deny the need for definite outcomes altogether. A case could be made that the ultimate source of all these difficulties is a paradigm that assumes a fundamental status for quantum theory’s mathematical

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ingredients—state vectors, wave functions, density operators, self-adjoint linear operators, inner products, and so forth—which I will collectively call the Hilbertspace picture. This paradigm is not sacrosanct and merits a critical re-evaluation. With a nod to Magritte:

In this essay, which summarizes a more comprehensive companion treatment (Barandes, forthcoming), I will show that by beginning at an entirely different starting place, one can derive quantum theory from a sharp ontology consisting of physical systems with basic properties that exhibit a nontrivial probability structure. This approach sequesters the ontological content of quantum theory from its epistemological and nomological content, and suggests a new interpretation of the theory that I call the Platonic interpretation or Platonic quantum theory, for reasons to be explained in the concluding section of this essay. The Platonic interpretation asserts that a quantum system has • an ontology—a real-world existence—consisting of physical states of being that are characterized by ontic properties with quantifiable underlying values, • an epistemology—a state of knowability—consisting of interdependent probability distributions that capture to the degree to which those underlying values are manifest to the external world, • and a nomology—a set of laws—consisting of a subtle but ultimately codifiable collection of rules for the foregoing ingredients. According to the Platonic interpretation, the ontic properties of a quantum system do not generically share a meaningfully correct, overarching joint probability distribution. In Platonic quantum theory, such joint probability distributions only exist under specific conditions. These conditions include the requirement that there should be negligible direct entanglement between the ontic properties in question, as encoded in the information describing the interdependencies between their individual probability distributions. As we will see, direct entanglement between relevant ontic properties can approximately disappear at the conclusion of a measurement process after decoherence by the surrounding environment, leading to the emergence of a joint probability distribution that links together the relevant ontic properties of the system being measured and the appropriate ontic properties of the measuring device, in a manner consistent with the correct empirical results. From the standpoint of the Platonic interpretation, the mathematical abstractions of Hilbert spaces and wave functions are analogous to the gauge potentials of electromagnetism—they show up only as convenient tools for calculations, and

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should not be taken to refer directly to physical aspects of reality. As a consequence, we will see that the Platonic interpretation dissolves a number of long-standing questions in quantum foundations, and also provides new ways of understanding fermionic fields. In Sect. 16.2, I will formulate classical probability theory in a manner that will streamline my later derivation of quantum theory, which I will cover in Sect. 16.3. I will then describe the measurement process in Sect. 16.4. I will conclude in Sect. 16.5 with a summary of Platonic quantum theory and comparisons with other prominent interpretations.

16.2 The Classical Case 16.2.1 Ontic States In classical physics, one can imagine that a given physical system has a set of properties that ground the system’s ontology, where the specific values of these ontic properties at any one moment collectively define the system’s ontic state. Some of these ontic properties may have values that vary from one ontic state to another, whereas other ontic properties may have fixed values across all the ontic states of the given system. Ontic properties with varying values characterize the nature and scope of the system’s possible ontic states. By contrast, one could regard ontic properties that are fixed as intrinsic features that define the kind or species of the system in question. One could also (or instead) allow such fixed ontic properties to be dispositional in the sense of playing a role in determining the system’s behavior. Note that there is no need to assert the existence of a universally applicable set of ontic properties—different kinds of classical systems can have different sets of ontic properties. For a mechanical system consisting of non-relativistic particles, these ontic properties might include their varying positions and momenta, as well as their fixed masses and charges. For a system of fields, these ontic properties would more naturally consist of local field intensities and their rates of change, setting aside ambiguities that can arise for gauge theories.1

16.2.2 Epistemic States In practical cases, a system’s specific underlying ontic state is knowable only with imperfect precision. One therefore typically assigns the system an epistemology, 1

If it turns out that a specific kind of system happens to be the only fundamental kind of system in the world, then one could potentially argue that only the ontic properties of that kind of system would fundamentally exist in reality.

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or an epistemic state, consisting of an overall probability distribution made up of probabilities p(x), p(x  ), p(x  ), . . .

(16.1)

that correspond to each of the system’s mutually exclusive ontic states, which are labeled as x, x  , x  , . . . . These labels collectively define a set  called the system’s sample space:2  ≡ {x, x  , x  , . . . }.

(16.2)

The probabilities (16.1) satisfy the usual rules of being non-negative, p(x) ≥ 0,

(16.3)

and having unit normalization,

p(x) = 1.

(16.4)

x∈

16.2.3 Random Variables To each of the system’s ontic properties, one can associate a symbol A,

(16.5)

called a random variable, that stands for a set of possible (not necessarily distinct) real values A(x), A(x  ), A(x  ), . . .

(16.6)

that depend on the system’s ontic state, where, again, the labels x, x  , x  , . . . denote the system’s different possible ontic states. For instance, one can associate a random variable Px0

2

(16.7)

Just for purposes of mathematical simplicity, I will assume that our sample spaces are always discrete sets—perhaps after some suitable coarse-graining—so that one can assign their individual elements finite, nonzero probabilities.

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to the true-or-false statement that the system’s ontic state is the one labeled by x0 , with ‘true’ represented by 1 and ‘false’ represented by 0. Then Px0 (x) is just the Kronecker delta:  1 for x = x0 , Px0 (x) = δx0 x ≡ (16.8) 0 for x = x0 . One can define any function f (A, B, C, . . . ) of our system’s random variables, such as algebraic combinations like A + B and AB, to be a random variable in its own right according to f (A, B, C, . . . )(x) ≡ f (A(x), B(x), C(x) . . . ),

(16.9)

wherever the right-hand side is well-defined (and thereby excluding, say, situations involving a division by zero). In mathematical language, our collection of random variables therefore becomes an algebra. In particular, because 12 = 1 and 02 = 0, the random variable Px0 introduced in (16.7) becomes a projector, meaning that it satisfies the idempotence condition Px20 = Px0 .

(16.10)

One also has the mutual exclusivity condition Px Px  = δxx  Px ,

(16.11)

as well as the completeness relation

Px = I,

(16.12)

x∈

where I is the identity random variable whose value is always the number 1. Moreover, for any random variable A, one can easily check that A satisfies the eigenvalue equation APx = A(x)Px

(16.13)

and has the spectral decomposition A=

A(x)Px .

(16.14)

x∈

(Despite the appearances of eigenvalue equations and spectral decompositions, remember that we are still working in the classical case here.)

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Given subsets E and F of the sample space, E, F ⊂ ,

(16.15)

one can introduce corresponding generalized projectors PE and PF according to PE ≡

Px ,

PF ≡

x∈E

Px .

(16.16)

x∈F

As one can check, the product of these generalized projectors yields the projector onto the set-theoretic intersection E ∩ F , PE PF = PE ∩ F .

(16.17)

16.2.4 C∗ -Algebras Whatever one’s views on the physical meaning of the complex numbers, it is undeniable that they are useful for capturing many patterns that arise throughout physics. If one wishes to introduce complex numbers into the present context, then one can define an involution operation ∗ on our algebra of random variables by saying that for a given complex-valued random variable Z ≡ A + iB,

(16.18)

where A and B are real-valued random variables, one can define Z ∗ as Z ∗ ≡ A − iB.

(16.19)

Equivalently, in terms of underlying values, (Z ∗ )(x) ≡ Z(x),

(16.20)

where the right-hand side denotes ordinary complex conjugation of complex numbers. It will also be helpful to define a non-negative norm Z ≥ 0 for a given random variable Z to be the supremum or least upper bound of the set of absolute values of all the possible underlying values of Z. That is, Z ≡ sup {|Z(x)|} ≡ sup{|Z(x)|, |Z(x )|, |Z(x  )|, . . . }.

(16.21)

x∈

One can use this norm to define notions of limits and continuity on our algebra of random variables. It is then convenient to extend our algebra by including all

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the idealized limit points of Cauchy sequences of its random variables, so that the algebra thereby becomes closed in the language of topology. Collectively, our set of random variables together with their algebraic structure (16.9), the ∗ involution (16.20), the norm (16.21), and the inclusion of all limit points define a C∗ -algebra A (Segal 1947). (‘C’ stands for ‘closed’ and ‘C∗ ’ is pronounced like ‘sea star.’)

16.2.5 State Maps One can then subsume all of classical probability theory, including our system’s epistemic state, into this C∗ -algebra A together with the notion of a state map ω defined to give the statistically weighted average or expectation value of any random variable A,

p(x)A(x). (16.22) ω(A) ≡ x∈

For example, one can express the individual probability p(x) of the ontic state labeled by x in terms of the state map ω and the projector Px introduced in (16.7) as ω(Px ) = p(x).

(16.23)

More generally, one can express the probability p(E) of x belonging to a subset E ⊂  of the sample space in terms of the generalized projector PE defined in (16.16) according to ω(PE ) =

p(x) = p(E).

(16.24)

x∈E

One can define the joint probability of x belonging both to E ⊂  and to F ⊂  as ω(PE PF ) = ω(PE ∩ F ) = p(E and F ).

(16.25)

One can also define the standard deviation A of a random variable A as the rootmean-square of A, A ≡



ω((A − ω(A)I )2 ) =



ω(A2 ) − (ω(A))2 ,

(16.26)

and one can define the covariance cov(A, B) of a pair of random variables A, B as cov(A, B) ≡ ω(A − ω(A)I )(B − ω(B)I ) = ω(AB) − ω(A)ω(B).

(16.27)

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The variance var(A) of A is defined to be the covariance of A with itself, var(A) ≡ cov(A, A) = ω((A − ω(A)I )2 ) = ω(A2 ) − (ω(A))2 ,

(16.28)

meaning that the standard deviation of A is the square root of the variance of A, A =



var(A).

(16.29)

Finally, one can re-express our C∗ -norm (16.21) as  Z = sup { ω(Z ∗ Z)}, ω∈S

(16.30)

where the supremum is now taken over the set S of all the system’s state maps ω.

16.2.6 The Classical Measurement Process One can neatly describe an idealized version of the classical measurement process in terms of the framework laid out above. If the epistemology of a classical ‘subject’ system is initially encoded as a nontrivial state map ω, and a measuring device identifies the subject system’s specific ontic state x, then that measuring device is now capable of determining the true underlying values A(x), B(x), C(x), . . . of all the subject system’s random variables. Hence, the measuring device is now able to employ a trivial state map ωx for the subject system that assigns unit probability to all those specific underlying values. The transition from ω to ωx is not a physical process, but merely represents the measuring device carrying out an epistemically subjective update of its information about the subject system. Indeed, a second measuring device that remains in the dark as to the measurement outcome would retain the original state map ω for the subject system. State maps are therefore effectively perspectival—that is, they depend on one’s perspective.

16.3 Quantum Theory 16.3.1 Ontic Random Variables The classical framework described above includes some assumptions that one could imagine dropping. To generalize, consider again a physical system with ontic properties represented by random variables A, B, C, . . . , but now suppose that we discard the assumption of a common sample space , meaning simply that we no longer require that there is a valid joint probability distribution for all the system’s random variables as a whole.

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From this point of view, giving up the requirement of a common sample space is therefore a matter of epistemology, not a matter of ontology. That is, the absence of a common sample space is a statement about the degree to which our system’s intrinsic features are knowable or exposed to the external world without further intervention, as encoded in the language of probabilities and in information characterizing relationships between probabilities. The lack of a common sample is not a statement that the system has no ontic states or that the system’s ontic states do not consist of ontic properties. Those ontic properties are still assumed to exist and to be represented by random variables, which I will now call ontic random variables to emphasize their connection with ontic properties. Even without a common sample space, each such ontic random variable A still has its own set of possible underlying values A(a), A(a ), A(a  ), . . .

(16.31)

labeled by a, a  , a  , . . . . These labels naturally define an individual sample space A belonging to A, A ≡ {a, a , a  , . . . },

(16.32)

and A still has an associated set of probabilities, pA (a), pA (a  ), pA (a  ), . . . .

(16.33)

16.3.2 Incommensurable Probability Distributions However, given another ontic random variable B with its own individual sample space B ≡ {b, b , b  , . . . } and probability distribution pB (b), it could be the case that pA (a) and pB (b) are incommensurable probability distributions. By incommensurable, I mean that there may not exist a physically accurate joint probability distribution p(a, b) that reliably describes the current underlying values of A and B and that yields the probability distributions pA (a) and pB (B) from the   respective marginalization rules b p(a, b) and a p(a, b). For example, the existence of a meaningfully correct joint probability distribution p(a, b) could be obstructed by detailed interdependencies between the probability distributions pA (a) and pB (b). These interdependencies could take the form of information specifying that certain physical effects that alter the probability distribution pA (a)—say, due to interactions with other systems—can alter the probability distribution pB (b), and vice versa. For a pair of ontic random variables A and B with incommensurable probability distributions, the algebraic structure of the classical case breaks down, because without a joint probability distribution p(a, b), algebraic combinations like A + B and AB lack well-defined probability distributions of their own and are therefore no

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longer random variables themselves. That is, A + B and AB are now merely formal expressions and do not have an obvious meaning at this point. Crucially, however, our system still has ontic states partially labeled by ordered pairs of the form (a, b). In any such ontic state, the underlying values A(a) and B(b) of the corresponding ontic properties are simultaneously well-defined. It is important not to confuse these underlying values A(a) and B(b) with the random variables A and B that abstractly represent them. In particular, in an ontic state partially labeled by (a, b), the underlying values A(a) and B(b) are quantities that can be added together, A(a) + B(b), or multiplied, A(a)B(b), and the results are still well-defined quantities. The assumed incommensurability of pA (a) and pB (b) merely implies that one cannot attach valid probabilities to quantities like A(a) + B(b) and A(a)B(b), and that these quantities cannot be known with certainty by any other systems that abide by the laws of physics. So even though the quantities A(a) + B(b) and A(a)B(b) exist, they are simply not represented by the formal algebraic combinations A + B or AB of the random variables A and B. Again, due to the assumed incommensurability of the individual probability distributions for A and B, formal algebraic combinations like A + B and AB are not valid random variables themselves.

16.3.3 Quantum State Maps I will call our system a quantum system if it nonetheless admits a set S of state maps ω with certain simple properties that are borrowed from the classical case, to be described below.3 In this scenario, we will see that we can go back to employing much of probability theory once again—in fact, we will eventually end up with the usual rules of quantum theory. In other words, we will see that quantum theory can be understood as a framework for managing ontic random variables that have incommensurable probabilities distributions. As to the defining properties of a quantum state map ω, I will require that it be a map ω : A0 → C from the formal algebra A0 generated by our quantum system’s ontic random variables A, B, C, . . . to the complex numbers,4 with the following desiderata taken directly from the classical case. • Normalization: ω(I ) = 1.

3

(16.34)

I will address the question of non-uniqueness of S in this essay’s concluding section. I will likewise address the status of the complex numbers in quantum theory in the concluding section.

4

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• Linearity: for complex numbers c1 , c2 , we have ω(c1 A + c2 B) = c1 ω(A) + c2 ω(B).

(16.35)

• Positivity: for Z ≡ A + iB, we have ω(Z ∗ Z) ≥ 0.

(16.36)

 Z ≡ sup { ω(Z ∗ Z)} ≥ 0

(16.37)

• C∗ -Norm: The norm defined by

ω∈S

satisfies the requirements of a C∗ -norm, with the state maps ω continuous under this C∗ -norm.5 These desiderata further imply another important identity. • Complex Conjugation:6 ω(Z1 Z2 ) = ω(Z2∗ Z1∗ ).

(16.38)

So far these properties are all purely mathematical. In order to complete the definition of a quantum state map ω, it is necessary to assign at least some of its output values a physical meaning. That physical meaning will consist of the statement that for the specific case of an ontic random variable A, the quantity ω(A) is to be understood as the expectation value of A according to that particular state map ω: ω(A) = expectation value of A according to ω.

(16.39)

Because the underlying values A(a), A(a ), . . . of any ontic random variable A are always real, A is naturally taken to be invariant under the ∗ operation, A∗ = A,

(16.40)

so the complex-conjugation identity (16.38) immediately tells us that the expectation value of A according to ω is guaranteed to be real as well: ω(A) ∈ R.

(16.41)

A C∗ -norm satisfies cZ = |c| Z (for c a complex number), Z1 + Z2 ≤ Z1 + Z2 , and Z1 Z2 ≤ Z1 Z2 , along with the C∗ -condition Z ∗ Z = Z 2 . 6 Be careful to note the different orderings of Z and Z on the two sides. One can prove this 1 2 identity using the previously stated desiderata by considering the inequality ω((Z1 + cZ2 )∗ (Z1 + cZ2 )) ≥ 0 in the two special cases c = 1 and c = i. 5

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By contrast, generic elements Z of the formal algebra A0 generated by our system’s ontic random variables are not typically ontic random variable themselves and therefore do not represent underlying ontic properties. Acting with our quantum state map ω on such a generic element Z will typically produce an output value ω(Z) that is complex-valued and does not admit a clear physical interpretation. Given a choice for our set of ontic random variables A, B, C, . . . , and given the resulting formal algebra A0 that they generate, choosing a comprehensive set S of state maps determines a specific algebraic structure on A0 . For instance, given specific elements A, B, and C of our formal algebra A0 , if an equation of the general form ω(· · · AB · · · ) = ω(· · · C · · · )

(16.42)

holds for all state maps ω in S, then we have the algebraic relation AB = C.

(16.43)

As an important example, the complex-conjugation identity (16.38) satisfied by our state maps automatically implies that the ∗ operation is product reversing: (Z1 Z2 )∗ = Z2∗ Z1∗ .

(16.44)

16.3.4 Noncommutative C∗ -Algebras As suggested by the previous identity, the resulting algebraic structure will generically be noncommutative, in the sense that some pairs A and B of our ontic random variables may fail to commute: AB = BA.

(16.45)

Hence, the algebra generated by our ontic random variables, now with the specific algebraic structure determined by our set S of state maps ω, becomes a noncommutative C∗ -algebra. We will use the symbol A in place of A0 to refer to this algebra. Notice that if two particular ontic random variables A and B indeed fail to commute under multiplication, AB = BA, then the product-reversing property of the ∗ operation entails that (AB)∗ = BA = AB, meaning that AB fails to satisfy the property (16.40) required of an ontic random variable. This observation serves as a valuable reminder that arbitrary algebraic combinations of ontic random variables will not generically be ontic random variables themselves, and will therefore play a less physically transparent role in determining the fundamental structure of our quantum system.

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Given an ontic random variable A that has possible underlying values (16.31), A(a), A(a ), A(a  ), . . . , with labels a, a  , a  , . . . constituting an individual sample space A for A, one can introduce true-or-false projectors PA,a , PA,a  , PA,a  , . . .

(16.46)

that satisfy the same kind of mutual exclusivity condition as in (16.11), PA,a PA,a  = δaa  PA,a ,

(16.47)

and the same kind of completeness relation as in (16.12),

PA,a = I.

(16.48)

a∈A

One also has analogues of the eigenvalue equation (16.13), APA,a = A(a)PA,a ,

(16.49)

and of the spectral decomposition (16.14), A=

A(a)PA,a .

(16.50)

a∈A

In particular, one can express the probabilities (16.33) for A directly in terms of the quantum state map ω and the projectors (16.46) as ω(PA,a ) = pA (a).

(16.51)

However, the counterpart to our classical expression (16.25) for joint probabilities generically gives complex numbers: ω(PA,a PB,b ) = ω(PB,b PA,a ) = ω(PA,a PB,b ).

(16.52)

If one were to try to make these quantities real and non-negative by, say, taking their absolute values, then they would no longer satisfy the correct marginalization rules. Attempting instead to replace the product PA,a PB,b with a combination like (1/2)(PA,a PB,b + PB,b PA,a ) to eliminate ordering ambiguities would not be guaranteed to give a non-negative result. Hence, if (16.52) is taken seriously, then it means that the joint probability of A(a) and B(b) simply does not exist, as a fact of the matter. Again, in the approach taken in this essay, the nonexistence of such joint probability distributions is not

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regarded as an issue of ontology, but instead as an issue of epistemology—of what is knowable or manifest to the external world about the given system without further intervention.

16.3.5 The Gelfand-Naimark-Segal (GNS) Construction So far this probabilistic framework might seem interesting but not obviously connected with quantum theory. However, there exists a remarkable theorem, called the Gelfand-Naimark-Segal (GNS) construction (Gelfand & Naimark, 1943; Segal 1947), that tells us that if we have a C∗ -algebra A and a state map ω, then • there automatically exists a corresponding Hilbert space Hω , • the involution operation ∗ on elements of the C∗ -algebra becomes the adjoint operation † on linear operators defined on that Hilbert space Hω , • all the ontic random variables A, B, C, . . . and their ∗-invariant algebraic combinations are represented as self-adjoint linear operators Aˆ ω , Bˆ ω , Cˆ ω , . . . on Hω , • and there exists a distinguished unit-normalized vector |ω  in Hω that implements the state map ω according to ω(A) = ω |Aˆ ω |ω .

(16.53)

This last equation thereby serves as a ‘dictionary’ that translates between the C∗ -algebra formulation and the Hilbert-space formulation. Furthermore, the distinguished vector |ω  is cyclic, which means that essentially7 every other vector in the Hilbert space Hω can be obtained from |ω  by acting with a particular operator representing an element of our C∗ -algebra, leading to a vector-operator correspondence (or state-operator correspondence) that plays an especially important role in the context of conformal field theories. To those familiar with textbook quantum theory, the dependence of the Hilbert space, linear operators, and vectors on the state map ω, as indicated by the appearance of ω as a subscript, may seem strange. I will explain later how the formulation of quantum theory here matches onto the theory’s textbook version.8

7

Technically speaking, the set of vectors obtained by acting with operators on the cyclic vector |ω  is a dense subset of the Hilbert space Hω . 8 When applied to the commutative C∗ -algebra appropriate to the case of a classical system, the GNS construction yields the Koopman-von Neumann formulation (Koopman 1931; von Neumann 1932a,b), a Hilbert-space picture for classical physics that has many applications in subjects like dynamical-systems theory. This classical Hilbert-space picture is not typically taken seriously as a physical statement about reality, and the Platonic interpretation essentially holds that one should apply the same skepticism toward the Hilbert-space picture in the quantum case as well.

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As an important special case of the GNS dictionary (16.53), the formula (16.51) for the probability pA (a) of the underlying value A(a) is given by pA (a) = ω |Pˆω,A,a |ω .

(16.54)

In the simplest situation in which Pˆω,A,a is a rank-one projector, there exists a vector |ψω,A,a  in Hω for which Pˆω,A,a is the outer product Pˆω,A,a = |ψω,A,a ψω,A,a |,

(16.55)

and (16.54) reduces to the famous Born rule: pA (a) = |ψω,A,a |ω |2 .

(16.56)

Furthermore, if a specific real-valued ontic random variable A has a trivial probability distribution according to the given state map ω, in the sense that pA (a) = 1 for a single value of the label a, then ω(A) = A(a) and the standard deviation A vanishes: A = 0. Hence,  †   ω | Aˆ ω − (A(a))Iˆ Aˆ ω − (A(a))Iˆ |ω  = ω((A − ω(A)I )2 ) = (A)2 = 0. It follows immediately from the positive definiteness of the Hilbert-space inner product that Aˆ ω |ω  = A(a)|ω .

(16.57)

This result is the familiar eigenvector-eigenvalue link of quantum theory.

16.3.6 Pure State Maps and Mixed State Maps In both the classical and quantum cases, one can construct new state maps out of old state maps by forming convex linear combinations. That is, given two state maps ω1 and ω2 and a real number λ in the interval 0 < λ < 1, one can define a new state map ω3 satisfying all the necessary desiderata (16.34)–(16.37) according to ω3 ≡ λω1 + (1 − λ)ω2 ,

(16.58)

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where for any element Z of our system’s C∗ -algebra, ω3 (Z) ≡ λω1 (Z) + (1 − λ)ω2 (Z).

(16.59)

Following standard terminology, one calls a state map ω pure if it cannot be expressed nontrivially as a convex linear combination of other state maps, and mixed otherwise. The GNS construction implies (see, for instance, Barandes (forthcoming)) that pure state maps produce Hilbert spaces that are irreducible representations of the given C∗ -algebra, whereas mixed state maps lead to reducible representations. In what follows, I will retain the symbol ω for pure state maps, and I will use ρ for the more general case of a state map that might be mixed. In the classical case, every state map ρ satisfies (16.22), ρ(A) =

p(x)A(x),

(16.60)

x∈

where p(x) is a probability distribution over the classical system’s fundamental sample space  = {x, x  , x  , . . . }. For any x in this fundamental sample space, one can introduce a trivial pure state map ωx that merely outputs the corresponding values of the system’s ontic random variables, ωx (A) ≡ A(x),

(16.61)

and in terms of which one can express the mixed state map (16.60) as the convex linear combination

p(x)ωx . (16.62) ρ= x∈

Thus, all pure classical state maps are trivial, and every mixed classical state map has a unique decomposition (16.62) in terms of pure state maps. If desired, one can therefore interpret a classical mixed state map as merely encoding subjective epistemic uncertainty over the system’s true ontic state—that is, over the underlying values of all the system’s ontic random variables. The story is trickier in the quantum case. By assumption, the ontic random variables of a quantum system may have incommensurable probability distributions. As a consequence, there are no state maps that assign the system a trivial overall probability distribution, as even a trivial overall probability distribution would be a disallowed joint probability distribution. (This absence of trivial overall probability distributions for quantum systems was first rigorously proved by von Neumann in 1932 (von Neumann, 1932c, 2018).) Moreover, the presence of incommensurable probability distributions implies that the quantum system’s state map contains information about their detailed interdependencies, as well as information about the degree to which potentially valid joint probability distributions fail to exist.

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Hence, even pure state maps are nontrivial in the quantum case, meaning that the line between pure and mixed state maps is blurrier than in the classical case. Mixed quantum state maps are also subtler than their classical counterparts, as noted by Schrödinger in 1936 (Schrödinger, 1936). The more complicated nature of mixed quantum state maps is due in large part to the fact that they lack unique decompositions into pure state maps, in contrast with the classical case. Mixed quantum state maps fall into two conceptually distinct categories (d’Espagnat, 1976). In the first category, one can imagine that a given quantum system would be empirically best described by a particular pure state map, but, solely because of subjective epistemic uncertainty, one is only able to say that the system’s state map is one of several possibilities ω1 , ω2 , ω3 , . . . with respective subjective probabilities p1 , p2 , p3 , . . . . In that case, one would naturally employ a mixed state map defined explicitly by ρ ≡ p1 ω1 + p2 ω2 + p3 ω3 + · · · ,

(16.63)

and this mixed state map would be said to describe a proper mixture. In the classical case, all mixed state maps describe proper mixtures, as we saw in (16.62). However, it may happen for a quantum system that the empirically most correct state map is fundamentally a mixed state map ρ, in the sense that no pure state map does as good a job at capturing the system’s state of affairs or predicting its behavior. Because mixed quantum state maps do not have unique decompositions into pure state maps, there is, in particular, no canonical decomposition of such a state map ρ as a convex linear combination of pure state maps. Consequently, one cannot obviously interpret the coefficients of any such decomposition of ρ as subjective epistemic probabilities. In this case, one is therefore left with an understanding of ρ only in its most general, abstract sense as an irreducible whole that encodes the expectation values of its host system’s ontic random variables. The quantum system’s state map ρ is then said to correspond to an improper mixture. Given a mixed quantum state map ρ, and given a (not necessarily unique) decomposition of that state map as a convex linear combination of pure state maps ω1 , ω2 , ω3 , . . . with respective coefficients p1 , p2 , p3 , . . . , ρ=

pi ωi ,

(16.64)

i

one can express the system’s corresponding distinguished vector |ρ  from the GNS dictionary (16.53) as a direct sum of the distinguished vectors |ω1 , |ω2 , |ω3 , . . . that respectively correspond to the pure state maps ω1 , ω2 , ω3 , . . . : |ρ  =

√ pi |ωi .

(16.65)

i

The direct sum appearing here is taken over the system’s universal Hilbert space Huniv , which is defined to be the direct sum of the GNS Hilbert spaces Hω

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corresponding to all the pure state maps ω in the set S of all the system’s state maps: 

Huniv ≡

Hω .

(16.66)

all pure ω∈S

The formula (16.65) ensures compatibility with the GNS dictionary (16.53): ρ(A) = ρ |Aˆ ρ |ρ  =

pi ωi |Aˆ ωi |ωi .

(16.67)

i

Alternatively, one can introduce the density operator (or density matrix) ρˆ ≡



pi |ωi ωi |.

(16.68)

i

A simple calculation then shows that one can express the right-hand side of the GNS dictionary (16.53) as the trace of the product of the density operator ρˆ with the appropriate linear operator Aˆ ρ : ρ(A) = Tr[ρˆ Aˆ ρ ].

(16.69)

16.3.7 Quantum Dynamics Before considering dynamics—especially the kind of dynamics that conserves information—it will be useful to discuss entropy and information briefly. With a notion of mixed state maps in hand, and momentarily considering the classical case, one can define the Gibbs or Shannon entropy (Gibbs, 1902; Shannon, 1948) of a classical system with state map ρ in accordance with the usual formula σρ ≡ −

pi log pi .

(16.70)

i

For the quantum case, the non-uniqueness of decompositions of mixed state maps means that a quantum system does not have a unique Shannon entropy. The system’s von Neumann entropy can be defined as the minimal Shannon entropy from among all decompositions of the system’s state map ρ as convex linear combinations of pure state maps (Hughston, 1993; Jaynes, 1957). This definition turns out to be expressible in terms of the system’s density operator ρˆ from (16.68) as ˆ σρ = −Tr[ρˆ log ρ].

(16.71)

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Turning now to dynamics, I will take information-conserving time evolution for a given quantum system to mean a map parameterized by a time parameter t and acting on the system’s C∗ -algebra that leaves the overall structure of the C∗ -algebra and all its expectation values intact, so that, in particular, the system’s entropy (16.71) remains unchanged. It then follows from a standard theorem (again, see Barandes (forthcoming)) that information-conserving time evolution can be implemented as a parameterized unitary transformation on the system’s universal Hilbert space (16.66) through a sequence of distinguished vectors |(t) ≡ |ωt  ∈ Hωt starting from an initial distinguished vector |(0) ≡ |ω0  ∈ Hω0 : |(t) = Uˆ (t)|(0).

(16.72)

Here Uˆ (t) is a unitary operator called the system’s time-evolution operator.

16.3.8 Textbook Quantum Theory We are now ready to see how the foregoing formulation of quantum theory matches onto its textbook version. Given a pure state map ω and its associated irreducible GNS Hilbert space Hω , notice that every vector |ψ in Hω defines a new map ωψ according to the ‘reverse dictionary’ ωψ (A) ≡ ψ|Aˆ ω |ψ,

(16.73)

where ωψ automatically satisfies all the required desiderata (16.34)–(16.37) of a state map. The set SHω of all such state maps that can be defined in this way from the vectors of a single Hilbert space Hω is called the folium of that Hilbert space. The Stone-von Neumann theorem (Stone, 1930; von Neumann, 1931) effectively guarantees that for a quantum-mechanical system consisting of finitely many nonrelativistic particles, the folium SHω of any one irreducible GNS Hilbert space Hω is sufficient to encompass the entire set S of the system’s possible state maps, so all the other GNS Hilbert spaces are redundant and one can pretend that H ≡ Hω is the system’s ‘one true’ Hilbert space.9 One can then drop the subscript ω from the notation completely, and one ends up with non-relativistic textbook quantum mechanics as we know it. Beyond the case of non-relativistic quantum systems in particular, Fell’s theorem (Fell, 1960) ensures that any GNS Hilbert space H that provides a faithful representation of a given quantum system’s C∗ -algebra has the resources necessary to make empirical predictions that are accurate to any finite level of precision.10

9

Technically, one must restrict to the case of GNS representations that are regular. precisely, Fell’s theorem states that if ρ is any state map for a C∗ -algebra A, and if A1 , . . . , An are any finite collection of n > 0 elements in that C∗ -algebra A, then given any faithful

10 More

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In light of Fell’s theorem, any such Hilbert space H would therefore appear to be sufficient according to an interpretation of quantum theory that, like the Platonic interpretation, regards Hilbert-space pictures not as ontological in nature, but merely as effective, instrumentalist contrivances for calculating predictions. One could simply choose whichever such faithful representation and associated Hilbert-space picture were most convenient for the application at hand. Suppose now that one is given an empirically adequate GNS representation of a specific quantum system, along with the unitary operator Uˆ (t) that implements smooth time evolution in that representation. Invoking Stone’s theorem (Stone, 1930), one can consider infinitesimal time durations t → t + dt and define the system’s Hamiltonian operator Hˆ (t) according to Uˆ (t + dt) − Uˆ (t) Hˆ (t) ≡ i h¯ , dt

(16.74)

where the reduced Planck constant h¯ is introduced here to provide the Hamiltonian operator with units of energy. The famous Schrödinger equation then follows from a few simple rearrangements: i h¯

∂|(t) = Hˆ (t)|(t). ∂t

(16.75)

16.3.9 Hilbert-Space Ingredients as Gauge Variables Having finally derived the familiar Hilbert-space picture, it is worth pointing out yet one more reason to be suspicious about taking it too seriously as an aspect of fundamental reality. Notice that we can rewrite the Schrödinger equation (16.75) in the suggestive form Dˆ t |(t) = 0,

(16.76)

representation of A with a Hilbert space H and a corresponding set of operators Aˆ 1 , . . . , Aˆ n on H, and given any positive real number  > 0, there exists a density operator ρˆ on the Hilbert space H such that |ρ(Ai ) − Tr[ρˆ Aˆ i ]| <  for all i = 1, . . . , n. See Clifton and Halvorson (2001) for a review. Given that n can be any positive integer (say, the thousandth busy beaver number raised to the power of a googolplex), and  can be any positive real number (say, 1/n for that same choice of n), it is difficult to imagine how any faithful representation of A could be empirically insufficient without A itself being empirically insufficient. In particular, no finite amount of experimental data could ever distinguish between two faithful representations of A, even if one of those representations contained certain limiting operators, called parochial observables, that did not exactly exist in the other representation, or did not exactly exist even at the level of the original C∗ -algebra A.

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with the gauge-covariant derivative operator Dˆ t defined by i ∂ Dˆ t ≡ Iˆ + Hˆ (t) ∂t h¯

(16.77)

and with the Hamiltonian Hˆ (t) playing the role of a corresponding gauge connection. As noted in Brown (1999), and as made manifest by this alternative form of the Schrödinger equation, the Hilbert-space picture features a class of non-abelian gauge transformations defined by arbitrary, local-in-time, unitary operators Vˆ (t) according to11 |(t) → |  (t) = Vˆ (t)|(t), ˆ Vˆ † (t), ˆ → Aˆ  (t) = Vˆ (t)A(t) A(t)

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ˆ † (t) ⎪ ∂ V ⎪  † , ⎭ Hˆ (t) → Hˆ (t) = Vˆ (t)Hˆ (t)Vˆ (t) − i h¯ Vˆ (t) ∂t

(16.78)

and which, in particular, leave the Schrödinger equation (16.76) unchanged in form: Dˆ t |(t) = 0 → Dˆ t |  (t) = 0.

(16.79)

The existence of this class of gauge transformations suggests that state vectors, linear operators, and Hamiltonians are mere gauge variables—akin to the gauge potentials of electromagnetism or of a Yang-Mills field theory—and so one should perhaps be skeptical about attaching too much physical meaning to the ingredients of the Hilbert-space picture. What is invariant under the gauge transformations (16.78) is the set of expectation values of all observable elements of the given quantum system’s C∗ -algebra A—which makes sense, because it is precisely those expectation values (which include all Born-rule probabilities) that make up the fundamental empirical content of a quantum system.

16.4 The Measurement Process 16.4.1 Pre-Measurement In an archetypal quantum measurement, one considers a subject system with an observable feature A, which we will take to be an ontic random variable. One 11 One

can show that the Schrödinger, Heisenberg, and interaction pictures for describing a quantum system’s time evolution correspond to specific choices of Vˆ (t). These three pictures are particularly useful for solving problems and for making empirical predictions in practice. However, the set of possible choices of Vˆ (t) vastly exceeds these three cases. For a review of non-abelian gauge transformations in the context of quantum field theory, see, for example, Weinberg (1996).

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assumes that the subject system is prepared in a pure state map ωA represented by a state vector | describing a superposition over the possible underlying values A(a) of A:

| = ca |a. (16.80) a

One then brings in a measuring device that is initially prepared in a pure state map represented by an eigenvector |d0  of its pointer variable D. Through unitary time evolution, the measuring device becomes entangled with the subject system as the measuring device measures the ontic random variable A,

| ⊗ |d0  → ca |a ⊗ |da , (16.81) a

where each vector |da  is an eigenvector of the pointer variable D representing a definite measurement result. At this step, known as a pre-measurement (Zurek, 1994), the state map ωAD of the overall system consisting of the subject system and the measuring device together is pure, and exhibits direct entanglement between the subject system’s observable A and the measuring device’s pointer variable D. This direct entanglement represents subtle, non-classical interdependencies between the respective probability distributions for A and D, with empirical consequences like interference, so the overall system’s state map ωAD does not resemble a classical proper mixture. As such—and as explained in the introductory section of this essay—the Platonic interpretation holds that the overall system’s state map ωAD does not yet describe a well-defined joint probability distribution for A and D.

16.4.2 Decoherence by the Environment However, when the environment, initially in a pure state map represented by an eigenvector |e0  of its own pointer variable E, couples to the measuring device, 

 ca |a ⊗ |da  ⊗ |e0 

a

→

ca |a ⊗ |da  ⊗ |ea ,

(16.82)

a

the resulting decoherence causes the state map of the original two systems to  that now formally resembles the separable final become a mixed state map ωAD state map following a classical measurement. As a result, there is no longer direct  entanglement between A and D, and the overall state map ωAD of the subject system and the measuring device resembles a classical proper mixture.

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At this point—again as noted in the introductory section of this essay—  the Platonic interpretation posits that the overall state map ωAD of the subject system and the measuring device now represents a well-defined joint probability distribution describing the appropriate statistical correlation between the subject system’s observable A and the measuring device’s pointer variable D. It is also precisely at this post-measurement, physical decoherence step that the famous nonlocality of quantum theory seeps into the story, with the underlying values of A and D synchronizing as needed even if the subject system and measuring device happen to be far apart in space when the environmental decoherence process (16.82) occurs.

16.4.3 Wave-Function Collapse To make future predictions about the subject system, the measuring device would then naturally switch to using an updated pure state map in its description of the subject system, replacing the initial pure state map ωA represented by | with a new state map ωa represented by the eigenvector |a corresponding to the measurement result that was actually obtained. Due to this epistemic conditionalization, the measuring device therefore sees the effective appearance of wave-function collapse:12 | → |a.

(16.83)

Just as in the classical case, however, a second measuring device that remains completely sealed off from the subject system, the first measuring device, and their local environment would continue using the classical-looking but uncollapsed  overall state map ωAD for the subject system and the measuring device. That is, just like in the classical case, state maps in the quantum case are essentially perspectival. So whereas decoherence (16.82) is a physical process that can actually entail changes in the underlying values of ontic random variables (even nonlocally), wavefunction collapse (16.83) is subjectively epistemic in nature, and merely reflects the different information available to the first and second measuring devices for making further predictions. The first measuring device’s inside view has the advantage of knowing the actual measurement outcome, whereas the second measuring device’s outside view has access to a more correct overall state map that makes it possible, in principle, to carry out manipulations like the complete erasure of the measurement process and the resulting restoration of the entire universe back to its prior configuration

12 The same sort of post-measurement conditionalization occurs explicitly in the de Broglie-Bohm interpretation, and also occurs implicitly in the Everett interpretation on a branch-by-branch basis.

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before the measurement process took place—although such manipulations would be thermodynamically impractical for all but the most microscopic of systems. As far as the inside view is concerned, the subject system then evolves from its new initial state vector |a according to whatever quantum dynamics is most empirically correct—whether that is the information-conserving Schrödinger equation (16.75) if there are no further interactions with other systems, or an opensystem master equation such as the Lindblad equation (Lindblad 1976; Breuer & Petruccione, 2002) in the case of additional external interventions. (In particular, there is no need for ad hoc guiding equations or pilot waves of the kind that are part of the de Broglie-Bohm formulation of quantum theory.) Small discrepancies between the inside view’s predictions and the outside view’s predictions are inevitable and unavoidable, but those predictions would more neatly mesh together in the sense of a suitable ensemble average if one were to consider sufficiently macroscopic measuring devices and a sufficiently isolated subject system. If the subject system and both the inside and outside systems are macroscopic, and a measurement process causes all appropriately coarse-grained ontic random variables to become sharply peaked around their expectation values, then the Ehrenfest theorem (see Sakurai (1993) for a review) would ensure that those expectation values evolve according to classical equations of motion to an excellent approximation, in agreement with our everyday experience of the world.

16.5 Conclusion 16.5.1 The Platonic Interpretation The end result of the preceding discussion is a new interpretation of quantum theory in which every quantum system has an ontology consisting of ontic properties—say, positions and momenta for a mechanical system of non-relativistic particles, or local intensities and rates of change for a field theory—together with an epistemology consisting of the varying degrees to which those ontic properties make their underlying values known to the external world, as encoded in individual probability distributions and information characterizing their relationships. These ontic and epistemic ingredients follow a complicated but ultimately summarizable set of rules, which provide quantum theory with its nomology—its laws. In particular, the individual probability distributions for a quantum system’s ontic properties may be incommensurable, in the sense that they cannot necessarily be joined together into a physically accurate, overarching probability distribution on a common sample space. Merely at the level of mathematical description, one can encode a quantum system’s ontic properties and their individual probability distributions as a set of ontic random variables, perhaps with some convenient degree of coarse-graining. One can then collectively codify those individual probability distributions, their detailed interdependencies, and their rules in terms of a set S of state maps that

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collectively determines a specific C∗ -algebraic structure on that set of ontic random variables, leading to a specific C∗ -algebra A for the quantum system in question. Note that this specific C∗ -algebraic structure—including any attendant commutation relations—therefore captures epistemic-nomic data, and is not a statement about the system’s ontology. This mathematical reformulation in terms of a set S of state maps and a C∗ -algebra A simplifies the task of identifying the circumstances under which a quantum system’s ontic properties share a joint probability distribution that accurately describes their current underlying values. Specifically, such a joint probability distribution exists when the ontic random variables representing those ontic properties commute and, moreover, when there is approximately no direct entanglement between their respective systems. In particular, this no-direct-entanglement condition is a novel ingredient of the Platonic interpretation. As explained further in Barandes (forthcoming), this condition threads the needle represented by the many no-go theorems that have cropped up over the past century, as these no-go theorems typically rely on the assumed existence of valid joint probability distributions even in the presence of direct entanglement between the relevant quantum systems. Given a C∗ -algebra A and a set S of state maps for a given quantum system, the GNS construction then yields a Hilbert-space picture for the system automatically. None of the mathematical abstractions of the Hilbert-space picture, however, are to be regarded as physically meaningful, despite providing a powerful set of tools for calculations. Metaphorically, the GNS construction therefore plays a role akin to the fire in the cave of Plato’s famous allegory, projecting shadows onto the interior cave wall that one can analogize to the ingredients of the Hilbert-space picture—hence the name “Platonic” for this interpretation of quantum theory. At a basic conceptual level, the Platonic interpretation implies that quantum theory is not really about wave functions or Hilbert spaces. Moreover, within the Platonic interpretation, one does not regard ontic random variables or their underlying values as ‘hidden variables’ that are added to quantum theory, but regards the ontic properties that they represent, their associated probabilities, and the set of interdependencies and rules that they exhibit as the true stuff of reality. From this point of view, the ingredients of the Hilbert-space picture are the true hidden variables—indeed, no one ever directly sees wave functions or linear operators.

16.5.2 Questions of Uniqueness According to Platonic quantum theory, the choice S for the set of state maps ω plays a central role in codifying the nomology for a given quantum system—again, S determines a specific algebraic structure on the system’s ontic random variables and thus, via the GNS construction, generates a corresponding Hilbert-space picture for the system. It is therefore natural to wonder whether one needs to be concerned over questions of the uniqueness of S for a specific quantum system.

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However, because S is not a matter of ontology, but merely a matter of how one chooses to represent a given quantum system’s nomology mathematically, uniqueness is not an issue. Just as the ‘ontological’ pattern of whole numbers 1, 4, 9, 16, 25, 36, 49, . . . can be codified by different-looking ‘nomological’ rules—say, the rule that we consider consecutive square integers, or the alternative rule that we start with 1 and then repeatedly add consecutive odd numbers—the ontological patterns inherent in fundamental physics can be codified by differentlooking nomologies. Moreover, specific choices for mathematically representing a nomology may be more useful in certain circumstances than other such choices. For instance, for our ‘ontological’ number sequence 1, 4, 9, 16, 25, 36, 49, . . . , computing the square of a large integer may be easier than adding up many consecutive odd numbers. As a more relevant example, quantum theory can, in principle, be formulated entirely in terms of real numbers rather than complex numbers without compromising the empirical success of the theory, provided that one is willing to work with a much more complicated mathematical representation of the nomology that includes superficially new kinds of non-locality and various√ad hoc features engineered to mimic the role played by the imaginary unit i ≡ −1 (Myrheim, 1999; Renou, 2021; Stueckelberg, 1959, 1960).

16.5.3 Comparison with Other Approaches Unlike the Everett ‘many worlds’ approach, Platonic quantum theory does not entail reifying the Hilbert-space picture, nor does it contain ‘multiple worlds’ or any ontic ingredients that ‘branch.’ Consequently, the Platonic interpretation does not need to make ontological sense of emergent branches of wave functions in many- or infinitedimensional Hilbert spaces, dissolves problems of identifying ontologically correct ways to decompose wave functions into branches, and sidesteps the difficulty of making sense of probability in a world in which all possible outcomes actually occur. In contrast to the de Broglie-Bohm approach, Platonic quantum theory does not need to invoke guiding equations or pilot waves, nor does it derive its ontic properties from the input arguments of coordinate-space wave functions or wave functionals.13 As a result, the Platonic interpretation does not encounter conceptual difficulties in handling systems like fermionic fields whose local field operators anticommute and that therefore do not have coordinate-space wave functionals in 13 For example, according to the de Broglie-Bohm approach, the ontic properties of a threedimensional particle with coordinate-space wave function (x, y, z) are the input arguments x, y, z, which are just the particle’s Cartesian position coordinates, whereas the ontic properties of a Klein-Gordon field with coordinate-space wave functional [φ] are the input arguments φ(x, y, z), which are just the Klein-Gordon field’s local values. Notice that the particle’s position operators ˆ x, ˆ y, ˆ zˆ mutually commute, as do the local Klein-Gordon field operators φ(x, y, z).

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any ordinary sense. In practice, fermionic field operators have expectation values that are ordinary complex numbers, and the Platonic interpretation therefore permits us to choose the ontic random variables for a fermionic field to be the field’s intensities and rates of change, just like for bosonic fields. The Platonic interpretation also fundamentally differs from other prominent interpretative approaches to quantum theory. Unlike the Copenhagen interpretation, the Platonic interpretation does not invoke a Heisenberg cut that places microscopic systems on an ontologically different footing from macroscopic systems. Unlike spontaneous-collapse approaches, the Platonic interpretation does not alter the dynamical laws of standard quantum theory. Finally, unlike instrumentalist approaches like QBism, the Platonic interpretation posits a specific ontology for every given quantum system. Acknowledgments I would like to thank David Albert, Jeremy Butterfield, Eddy Keming Chen, Juliusz Doboszewski, Johannes Fankhauser, Benjamin Feintzeig, Samuel Fletcher, Peter Galison, Ned Hall, Richard Healey, Mina Himwich, Carl Hoefer, David Kagan, Martin Lesourd, Logan McCarty, Tushar Menon, Ana Raclariu, Simon Saunders, Chip Sebens, Jeremy Steager, Chris Timpson, and David Wallace. I am especially grateful to Valia Allori for organizing the collection in which this essay appears, as well as to two anonymous referees for their insightful comments and questions.

References Barandes, J. A. (forthcoming). The platonic interpretation of quantum theory. Bohm, D. J. (1951). Quantum theory. Mineola: Courier Dover Publications. With Bohm’s prescient discovery of decoherence in Chaps. 22–23. Breuer, H. P., & Petruccione, F. (2002). The theory of open quantum systems. Oxford: Oxford University Press. Brown, H. (1999). Aspects of objectivity in quantum mechanics. In From physics to philosophy (pp. 45–70). Cambridge: Cambridge University Press. https://philpapers.org/rec/BROAOO-2 Clifton, R., & Halvorson, H. (2001). Are rindler quanta real? Inequivalent particle concepts in quantum field theory. The British Journal for the Philosophy of Science, 52(3):417–470. https:// www.jstor.org/stable/3541925 d’Espagnat, B. (1976). Conceptual foundations of quantum mechanics. New York : W.A. Benjamin. Dirac, P. A. M. (1930). The principles of quantum mechanics, 1st ed. Oxford: Oxford University Press. Fell, J. M. G. (1960). The dual spaces of C*-algebras. Transactions of the American Mathematical Society, 94(3):365–403. https://doi.org/10.2307/1993431 Gelfand, I. M., & Naimark, M. A. (1943). On the imbedding of normed rings into the ring of operators on a Hilbert space. Matematicheskii Sbornik, 12(54)(2):197–217. http://mi.mathnet. ru/msb6155 Ghirardi, G. C., Rimini A., & Weber, T. (1980). A general argument against superluminal transmission through the quantum mechanical measurement process. Lettere al Nuovo Cimento, 27(10):293–298. https://doi.org/10.1007/BF02817189 Gibbs, J. W. (1902). Elementary principles in statistical mechanics (1st ed.). New York: Scribner’s Sons.

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Hughston, L. P., Jozsa, R., & Wootters, W. K. (1993). A complete classification of quantum ensembles having a given density matrix. Physics Letters A, 183(1):14–18. https://doi.org/ 10.1016/0375-9601(93)90880-9 Jaynes, E. T. (1957). Information theory and statistical mechanics. II. Physical Review, 108(2):171–190. https://doi.org/10.1103/PhysRev.108.171 Joos, E., & Zeh, H.-D. (1985). The emergence of classical properties through interaction with the environment. Zeitschrift für Physik B Condensed Matter, 59(2):223–243. https://doi.org/10. 1007/BF01725541 Jordan, T. F. (1983). Quantum correlations do not transmit signals. Physics Letters A, 94(6):264. https://doi.org/10.1016/0375-9601(83)90713-2 Koopman, B. O. (1931). Hamiltonian systems and transformations in Hilbert space. Proceedings of the National Academy of Sciences, 17(5):315–318. https://doi.org/10.1073/pnas.17.5.315 Lindblad, G. (1976). On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48(2):119–130. https://doi.org/10.1007/BF01608499 Myrheim, J. (1999). Quantum mechanics on a real Hilbert space. . https://arxiv.org/abs/quant-ph/ 9905037 arXiv:quant-ph/9905037 Renou, M.-O., Trillo, D., Weilenmann, M., Thinh, L. P., Tavakoli, A., Gisin, N., Acin, A., & Navascues, M. (2021). Quantum physics needs complex numbers. . https://arxiv.org/abs/ 2101.10873, arXiv:2101.10873 Sakurai, J. J. (1993). Modern quantum mechanics (revised edition). Boston: Addison Wesley. Schrödinger, E. (1936). Probability relations between separated systems. Mathematical Proceedings of the Cambridge Philosophical Society, 32(3):446–452. https://doi.org/10.1017/ S0305004100019137 Segal, I. E. (1947). Irreducible representations of operator algebras. Bulletin of the American Mathematical Society, 53, 73–88. https://doi.org/10.1090/S0002-9904-1947-08742-5 Shankar, R. (1994). Principles of quantum mechanics (2nd ed.). New York: Plenum Press. Shannon, C. E. (1948). A mathematical theory of communication. The Bell System Technical Journal, 27(3):379–423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x Stone, M. H. (1930). Linear transformations in Hilbert space. Proceedings of the National Academy of Sciences, 16(2):172–175. https://doi.org/10.1073/pnas.16.2.172 Stueckelberg, E. C. G. (1959). Field quantisation and time reversal in real Hilbert space. Helvetica Physica Acta, 32(4):254–256. https://www.e-periodica.ch/digbib/view?pid=hpa-001:1959:32:: 145#260 Stueckelberg, E. C. G. (1960). Quantum theory in real Hilbert space. Helvetica Physica Acta, 33(4):727–752. https://www.e-periodica.ch/digbib/view?pid=hpa-001:1960:33::715#735 von Neumann, J. (1931). Die Eindeutigkeit der Schrödingerschen Operatoren. Mathematische Annalen, 104(1):570–578. https://doi.org/10.2307/1968535 von Neumann, J. (1932a). Zur Operatorenmethode In Der Klassischen Mechanik. Annals of Mathematics, 33(3):587–642. https://doi.org/10.2307/1968537 von Neumann, J. (1932b). Zusatze Zur Arbeit ‘Zur Operatorenmethode. . .’. Annals of Mathematics, 33(4), 789–791. https://doi.org/10.2307/1968225 von Neumann, J. (1932c). Mathematische Grundlagen der Quantenmechanik. Berlin: Springer. von Neumann, J. (2018). Mathematical foundations of quantum mechanics: New edition. Princeton: Princeton University Press. With the English translation by Robert T. Beyer, and edited by Nicholas A. Wheeler. Weinberg, S. (1996). The quantum theory of fields (Vol. 2). Cambridge: Cambridge University Press. Zeh, H.-D. (1970). On the interpretation of measurement in quantum theory. Foundations of Physics, 1(1):69–76. https://doi.org/10.1007/BF00708656 Zurek, W. H. (1994). Decoherence and the existential interpretation of quantum theory, or ‘No Information Without Representation’. In P. Grassberger & J.-P. Nadal (Eds.), From statistical physics to statistical inference and back (pp. 341–350). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-94-011-1068-6_23

Chapter 17

cat alive and cat dead Are not Cats! Ontology and Statistics in “Realist” Versions of Quantum Mechanics Jean Bricmont

Abstract It is often claimed that there are three “realist” versions of quantum mechanics: the de Broglie-Bohm theory or Bohmian mechanics, the spontaneous collapse theories and the many worlds interpretation. We will explain why the two latter proposals suffer from serious defects coming from their ontology (or lack thereof) and that the many worlds interpretation is unable to account for the statistics encoded in the Born rule. The de Broglie-Bohm theory, on the other hand, has no problem of ontology and accounts naturally for the Born rule.

17.1 Introduction: The Need for an Ontology I wish I could use another word than “ontology”, which sounds too “philosophical” to the ears of many physicists, but that is the word whose usage is generally accepted. However, there is no reason to be scared by this word; it simply refers to what exists in the world, atoms, stars, electrons, fairies, gods (if the latter exist) etc. Of course, to speak of an ontology, we need some theory or some idea of what the world consists of. In the classical worldview, the ontology consists of particles in motion and (electromagnetic) waves plus a certain structure of space-time. But what is the ontology postulated by quantum mechanics? That is far less clear. Of course, quantum physicists speak of atoms, radiation, elementary particles etc., as if they existed in reality, namely as being part of the ontology. But they also insist, at least most of them, that the complete description of those objects is given by their wave function (or quantum state).

J. Bricmont () IRMP, Université catholique de Louvain, Louvain-la-Neuve, Belgium e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_17

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But what does that mean? The standard meaning given to the wave function is that it allows us to compute the probabilities of various “results of measurements” performed in laboratories. So, if I tell you that an electron “out there” has such and such wave function, all it means is that if I brought this electron in a laboratory and performed experiments on it, I would get such and such result with certain probabilities. But if I ask what are the properties of the electron “out there”, the only honest answer, within ordinary quantum mechanics, is that it has none:1 no position, no velocity, no energy, no spin value (in an arbitrary direction). But then, what does it mean to say that the electron exists “out there”? It possesses intrinsic properties common to all electrons, like its mass and electric charge, but no extrinsic properties, like position, velocity etc., and it is not clear what it means to say that objects without such extrinsic properties “exist”. A possible reaction could be to restrict our ontology to measuring devices and that is sometimes what one might think that “neo-Bohrians” believe. But that can’t be right: everybody knows that measuring devices are made of more elementary particles and, if they don’t exist, how come that the measuring device does? Another reaction is to think, for one particle at least, that the wave function represent some spread out density of matter or of electric charge (and that is how Schrödinger thought of it at first, see Allori et al. (2011)) but that cannot be right either, partly because the free evolution of the wave function makes that supposed density more and more spread out, but mostly because, for more than one particle, the wave function is defined on R3N , where N is the number of particles and not on the ordinary space R3 (that is a major difference with respect to electromagnetic waves and it prevents us to think of the wave function as some kind of “field” in space-time). Finally one may think in terms of a naïve statistical interpretation (and that is probably what is in the back of the minds of most of the “no worry about quantum mechanics” physicists): particles do have properties such as position, velocity, spin etc., but we cannot know or control them-we have only access to their wave function; the latter gives the statistical distribution of the value of those quantities (through the Born rule) over sets of particles having the same wave function. And, when we perform a measurement of a property of a given particle, we learn what that value is for that particle. To spell out the naïve statistical interpretation, it means that, for each individual system, there is a map v that assigns a value to each observable A and the wave function simply assigns a probability distribution over such maps, in agreement with Born’s rule. Those values are what one calls “hidden variables”, namely variables that characterize the state of an individual quantum system beyond its wave function. It is curious that this expression is often considered by physicists not to be used in polite company, yet is the most straightforward way to give an unproblematic meaning to the wave function.

1

Unless the wave function happens to be an eigenstate of some operator.

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And, if that worked, there would indeed be no reason to worry about the meaning of quantum mechanics and we would have a decent ontology, but somewhat “hidden” from us. In fact, we will see in Sect. 17.4 that this does work, but in a subtly modified form. However, as stated here, it cannot work, because of wellestablished theorems due to Bell (1966) and to Kochen and Specker (1967), but that unfortunately are not widely known among physicists: those theorems show that, if we assign values to various properties that are simultaneously measurable according to ordinary quantum mechanics (and thus, if one such property possesses a given value, so should the others),2 then one can deduce a contradiction (see Mermin (1993) for pedagogical proofs and Daumer et al. (1996), or Bricmont et al. (2019), for a discussion of the meaning of those results). In other words, what we call the naïve statistical interpretation (individual quantum systems do have definite properties but we are only able to know their statistical distributions) does not work. Since none of the easy ways to assign a meaning to the wave function in the world “out there” works, what one could call the meaning problem or the ontology problem (what exists in the world according to quantum mechanics?) remains. This problem is quite different from the most popular problem in the foundations of quantum mechanics, the “measurement problem”, which is illustrated by Schrödinger’s cat. As this example is well known, let us simply recall that, at the end of an experiment where one measures the property of a particle that can take two values, the measuring device, or the cat if we couple the device to the cat through a poison capsule, is (leaving aside normalization factors): cat alive + cat dead .

(17.1)

And that, argued Schrödinger, cannot be a complete description of the cat, which is obviously either alive or dead but not both! The way out of this problem from the point of view of ordinary quantum mechanics is to introduce the collapse postulate: when one looks at the cat, one sees whether she is alive or dead and, depending on what one sees, one reduces the wave function of the cat (and of the particle that was measured and is thus coupled to the state of the cat) to either cat alive or cat dead . 2

Technically, it means that the operators associated to those properties or “observables” commute., More precisely, the Bell-Kochen-Specker result implies that, if H is a Hilbert space of dimension at least four, and if A is the set of self-adjoint operators on H, there does not exist a map v : A → R such that: (1) ∀O ∈ A, v(O) is an eigenvalue of O.

(2) ∀O, O  ∈ A with [O, O  ] = OO  − O  O = 0, v(OO  ) = v(O)v(O  ). Obviously, there cannot be a statistical distribution of maps that do not exist.

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Since this is a deus ex machina from the point of view of the linear Schrödinger evolution, justifying it is often viewed as the main problem in foundations of quantum mechanics. What we are suggesting here is that there is a deeper problem: neither cat alive nor cat dead are cats: they are functions defined on a high dimensional space R3N (putting aside the fact that there are different kinds of particles and that they may also have a spin) while cats are located in R3 . And it is not clear what it means to say that cat alive or cat dead are descriptions of cats, let alone “complete descriptions” of them.3 Like all wave functions they allow us to predict results of measurements done on the objects that they “describe”, but nothing else. What most people do is to mentally identify cats and wave functions of cats. But that is exactly what we argue is illegitimate. It is curious that even critics of the Copenhagen interpretation like Einstein and Schrödinger did not, to my knowledge, challenge that identification. However, de Broglie did say in his report to the 1927 Solvay Conference: “it seems a little paradoxical to construct a configuration space with the coordinates of points that do not exist”. He also remarked that, if “the propagation of a wave in space has a clear physical meaning, it is not the same as the propagation of a wave in the abstract configuration space” (Bacciagaluppi & Valentini, 2009, p. 346). Just as in the well-known Magritte painting “this is not a pipe”, a wave function defined on a set of points that would constitute a cat, if those points corresponded to particle positions, is simply not a cat if there are no particles. Let us now consider the main proposals to solve the measurement problem, while keeping in mind the problem of meaning (of the wave function) or the “ontology problem”.

17.2

Statistics and Ontology in the Many-Worlds Interpretation

The many-worlds interpretation was introduced in 1957 by Hugh Everett III. We will first discuss what we call the “naïve many-worlds interpretation”. Then, we will give a mathematical formulation of it, due to Allori et al. (2011), and finally discuss a more sophisticated version of the many-worlds interpretation which is favored by some of its defenders.

17.2.1 The Naïve Many-Worlds Interpretation The naïve view is that, when the proverbial cat (or any other macroscopic device) finds itself in a superposed state, such as (17.1), then, instead of undergoing a 3

This idea is emphasized in Tumulka (2018) Sect. 5.1.

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collapse by fiat as in ordinary quantum mechanics, both terms simply continue to exist. But how can that be possible? We always see the cat alive or dead but not both! The short answer is that they both exist, but in different “worlds”. Hence, still thinking naively, whenever an experiment leads to a macroscopic superposition, the world splits into two or more worlds, depending on the number of distinct macroscopic states produced by that experiment, one for each possible result. But why do I always perceive only one of the results? It is simple: I, meaning my body, my brain (and thus also my consciousness) gets entangled with the states of the cat, so there are two or more copies of me also, one seeing the dead cat in one world, another seeing the live cat in another world. And that, of course, is also true for everything else: every molecule in the entire world gets to be copied twice (maybe not instantaneously, but that is another question). In his original paper (Everett, 1957), Everett stressed that “all elements of a superposition4 are ‘actual’, none any more ‘real’ than the rest.” Everett felt obliged to write this because “some correspondents” had written to him saying that, since we experience only one element of a superposition, we have only to assume the existence of that unique element. This shows that some early readers of Everett were already baffled by the radical nature of the “many-worlds” proposal.5 Putting aside the issue of ontology (to be discussed in Sect. 17.2.2), as well as the weirdness of the multiplication of “worlds”, one should ask whether the many worlds scheme is coherent. Consider first the Born rule. Suppose that the probabilities of having the cat alive or dead, as a result of an experiment, are ( 12 , 12 ). And suppose that I decide to repeat the same experiment successively many times, with different particles (and cats) but all having the same initial wave function. After one experiment, there are two worlds, one with a dead cat and a copy of me seeing a dead cat and one with a live cat and a copy of me seeing a live cat. Since both copies of me are in the same state of mind as I was before the first experiment (after all, both copies are just copies of me!), each of them repeats that experiment. Then, we have four worlds, one with two consecutive dead cats, one with two consecutive live cats and two with one dead cat and one live cat. “I” (by that I mean each copy of me in each of those four worlds) repeat the experiment again: we have now eight worlds, with consecutive cats as follows (d=dead, l=live): ddd, ddl, dld, dll, ldd, ldl, lld, lll. So there is one “history” of worlds with three dead cats, one history of worlds with three live cats, three histories of worlds with one live cats and two dead ones, three histories of worlds with one dead cat and two live ones. Now, continue repeating that experiment: for every possible sequence of outcomes, there will be some of my “descendants” (by that I refer to the copies of me, that exist in all the future worlds) that will see it. There will be a sequence of worlds in which the cats are always alive and another sequence where they are always dead.

4

Superposition refers to sums such (17.1) (Note of J.B.). See Freire (2015) for an historical perspective on Everett (and other “dissidents” with respect to the quantum orthodoxy) and Barrett (2018) for a discussion of various interpretations of Everett’s ideas, many of which depart from the original ones.

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There are also many sequences of worlds where the cats are alive one quarter of the time and dead three quarters of the time, and that is true for any other statistics different from ( 12 , 12 ). So that we can be certain that many of our descendants will not observe Born’s rule in their worlds. Of course, this is also true for many of our “cousins”, i.e., descendants, like us, of some of our ancestors, if many identical experiments have been made in the past in each of those worlds. But one could argue, on the basis of the law of large numbers, that, at least in the vast majority of worlds, the Born rule will be obeyed, since in the vast majority of worlds the frequencies of dead and live cats will be close to ( 12 , 12 ). But what happens if, instead of being ( 12 , 12 ), the probabilities predicted by quantum mechanics are, say, ( 34 , 14 )? We will still have two worlds coming out of each experiment, because these experiments have two possible outcomes. So, the structure of the multiplication of worlds is exactly as when the predicted probabilities were ( 12 , 12 ). But now, if one applies the law of large numbers as above, one arrives at the conclusion that, in the vast majority of worlds, the quantum predictions will not be observed, since our descendants will still see the cats alive in approximately 12 of the worlds, and the cats dead also in approximately 12 of the worlds, instead of the ( 34 , 14 ) frequencies predicted by the Born rule. This is prima facie a serious problem for the many-worlds interpretation. There have been many proposals to solve this problem and it would be too long and too technical to discuss all of them here. Some authors have argued that one should count the worlds differently, by weighting them with the coefficients that appear in the Born rule (e.g. Everett in Everett (1957), but also Graham in DeWitt and Graham (1973)). However, this is circular: the goal is to explain where the Born rule comes from in a deterministic theory, not to impose it from the start. Hence, I find these proposal unsatisfactory, and the problem remains.6 Another “solution” is to give to low probability worlds (according to Born’s rule) a lower degree of existence or of reality (see Vaidman (1998) or Damour (2006, p. 155), quoted in Maudlin (2012)), but it is unclear what it means to live in a low reality world since we cannot compare that life with one in a world with a high degree of existence: indeed, different worlds don’t interact with each others. And this “solution” is quite contrary to Everett’s original idea that “all elements of a superposition are ‘actual’, none any more ‘real’ than the rest” (Everett, 1957). Moreover, by assigning the low probability worlds (according to Born’s rule) a lower degree of existence, even though they exist just as much as the high probability worlds, the theory becomes borderline incoherent: if something exists, it has the property of “being existent” simpliciter.

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I develop these objections in Bricmont (2016) (Sect. 6.1). See also Norsen (2017) (Chap. 10) and Maudlin (2019) (Chap. 6).

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Another attempt to solve this problem, advocated by David Deutsch and David Wallace is to justify the Born rule by appealing to betting strategies by a rational agent (see, e.g., Deutsch (1999), Wallace (2003, 2007) and Maudlin (2014) for a critique). But what have betting strategies to do with physics?7 What we call here the naïve view is rather vague and is not explicit about its ontology. A paper by Allori et al. (2011) addresses this issue.

17.2.2 A Precise Many-Worlds Interpretation Allori et al. (2011) associate a continuous matter density with the wave function of a system of N particles. For each x ∈ R3 , and t ∈ R, one defines m(x, t) =

N

i=1

 mi

R3N

δ(x − xi )|(x1, . . . , xN , t)|2 dx1 . . . dxN ,

(17.2)

where (x1 , . . . , xN , t) is the usual wave function of the system at time t. This equation makes a connection between the wave function defined on the highdimensional configuration space and an object, the matter density, existing in our familiar space R3 . Here, the ontology does not consist only of the wave function, but also of the matter density. In our three-dimensional world, there is just a continuous density of mass: no structure, no atoms, no molecules, no brain cells, etc., just an amount of “stuff”, with high density in some places and low density elsewhere. Let us note that, if the God of the physicists was trying not to be malicious8 and if the many-worlds version of Allori et al. (2011) is true, then He failed badly: indeed, it means that we were wrong all along when we “discovered” atoms, nuclei, electrons, etc., and that we are lying to schoolchildren when we tell them that matter is mostly void with a few pieces of matter (the atoms) here and there. Indeed, in the theory of Allori et al. (2011), matter is continuous after all, with higher and lower density in some places, and we have simply been fooled by this version of the manyworlds interpretation of quantum mechanics into thinking that it is not. Of course, we might be forced to accept this ontology if there were independent reasons for doing so, like a greater explanatory power of that theory, but we do not see how this goal is reached. Moreover, this many-worlds theory does not solve the problem of the probabilities discussed in Sect. 17.2.1. Coming back to our example with two possible outcomes, one having probability 34 and the other 14 , the density of matter will be different in the world where ones “sees” the outcome having probability 34 and in the one having probability 14 . But what difference does it make? In which way does

7 8

For more on all this, see e.g. Maudlin (2019) (Chap. 6). The famous quote “The Lord God is subtle, but malicious he is not” is due to Einstein.

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having a smaller or larger matter density affect my states of mind?9 And if it does not, we are back to the problem that, if one repeats many times the experiment whose outcomes have probabilities ( 34 , 14 ), most of my descendants (some of course having a small matter density) will see massive violations of the Born rule. In any case, some defenders of the many-worlds interpretation dismiss both this approach and the naïve one and prefer to speak of a pure wave function ontology, something to be discussed now.

17.2.3 The Pure Wave Function Ontology In this ontology, the world is made of a universal wave function and nothing else.10 There are, in actual fact, no cats, pointers, brains, etc., situated in ordinary threedimensional space, but only a mathematical object, a complex function defined on an abstract space, containing all the possible configurations of particles and fields in the universe. In the language of Bell (1984), there are no “local beables” in that theory, i.e., things that exist (beables) and that are localized in R3 . But it is important to realize that the ontology of the “pure wave function” manyworlds interpretation is just a function defined on a space of “configurations” (of particles and fields), but without any particles or fields being part of the ontology. Indeed, if we put particles and fields, existing in ordinary three-dimensional space, into our ontology, then we get back to the “naïve” picture of worlds constantly multiplying themselves. But it is very difficult to see how to make sense of this pure wave function ontology, in particular how to relate it to our familiar experience of everyday objects situated in three-dimensional space. If we think of the live and dead cat, in this ontology, it just means that the wave function has a high value on configurations corresponding to the dead cat and also on configurations corresponding to the live cat, but without there being particles constituting the cat (either alive or dead).11

9

This problem is similar to the one posed by postulating a lower degree of existence for low probability worlds, see Sect. 17.2.1. 10 For a more detailed discussion, see the section on what is called GRW0 in Allori et al. (2008), and also Maudlin (2012) and Norsen (2014). 11 In Norsen (2014), the author explains why the pure wave function ontology is solipsism “for all practical purposes” because the world of the pure wave function ontology has a status similar to the idea that we are just brains in a vat, with our brains manipulated from the outside in just such a way as to make our illusory conscious experiences what they are. For more discussion of this ontology, see the collection of essays edited by Ney and Albert (2013).

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17.3 Ontologies for the Spontaneous Collapse Theories In Ghirardi et al. (1986) Ghirardi, Rimini, and Weber (GRW) introduced a modified version of quantum mechanics in which wave functions spontaneously collapse.12 To be precise, in their model, the wave function evolves according to the Schrödinger equation most of the time, but there is a set of spacetime points (yi , ti ) chosen at random, such that the wave function (x1 , . . . , xN , t) for a system of N particles is multiplied at the chosen times ti by a Gaussian function in the variable xk (k chosen uniformly among 1, . . . , N), centered in space at the chosen space points yi . The probability distribution of these random points is determined by the wave function of the system under consideration at the times when they occur, and is given by the familiar ||2 distribution. This ensures that the predictions of the GRW theory will (almost) coincide with the usual ones. The above-mentioned multiplication factors localize the wave function in space, and for a system of many particles in a superposed state, effectively collapse the quantum state onto one of the terms. Now, the trick is to choose the parameters of the theory so that spontaneous collapses are rare enough for a single or for a few particles in order to ensure that they do not lead to detectable deviations from the quantum predictions, but are frequent enough to ensure that a system composed of a large number of particles, say N = 1023 , will not stay in a superposed quantum state such as (17.1) for more than a split second; imagine a system whose wave function is a product of many wave functions for the individual particles composing the system. If even one of those wave functions collapses to 0 at some location, the product wave function of the system will also collapse to 0, since a product of factors, one of which equals 0, vanishes.13 The GRW theory is not the same as ordinary quantum mechanics, since the spontaneous collapse theory leads to predictions that differ from the usual ones, even for systems made of a small number of particles. But the parameters of the theory are simply adjusted so as to avoid being refuted by present experiments, which is not exactly an appealing move. Moreover, as for the many-worlds interpretation, there is also the problem of making sense of a pure quantum state ontology (even when the latter collapses, since the collapsed quantum state is still just a function defined on a high-dimensional space), which is called the “bare” GRW theory in Allori et al. (2008). Two solutions have been proposed to give a meaning to the GRW theory beyond the pure wave function ontology: the matter density ontology (Ghirardi et al., 1995) often denoted GRWm, and the flash ontology (Bell, 1987) denoted GRWf. In the first solution, one associates the continuous matter density defined in (17.2) with the wave function of a system of N particles. So the ontology here is exactly

12 For

reviews and further discussions of those theories, see Ghirardi (2011), Ghirardi et al. (1995), Allori et al. (2008), Goldstein et al. (2012), Norsen (2017) (Chap. 9), Maudlin (2019) (Chap. 4). 13 In actual fact, in GRW theories, wave functions do not collapses exactly to 0, but to a very small value. This raises additional problems but we will not discuss them.

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the same as in the version of many worlds due to Allori et al. (2011), but with a different time evolution of the quantum state: whereas it was the pure Schrödinger evolution in the many-worlds case, it is that evolution interrupted at some random times by random collapses in the GRWm theory. This has the advantage that there is, in practice, only one macroscopic world. In the flash ontology, one has a world made only of spacetime points at the center of the Gaussian multipliers of the wave function that “collapse” it. No particles, no fields, nothing at all, except a “galaxy” of spacetime points, called “flashes”. But again, if either the GRWm or the GRWf theories are true, then the God of the physicists is quite malicious. In the GRWm case, this is true for the same reason as for the version of many worlds described in Allori et al. (2011). On the other hand, if the flash ontology was true, then we have been fooled into thinking that there exists something most of the time (like atoms): if we take the visible universe since the Big Bang, it has contained only finitely many flashes. Since the flashes are all there is in that ontology, this means that, most of the time, the universe is just empty. Of course, as we said in Sect. 17.2.3, one might be forced to accept this ontology if the theory had greater explanatory power or greater empirical adequacy. But, and this is the most important point, empirical adequacy of any GRW theory would mean that ordinary quantum mechanics is empirically wrong, since the predictions of both theories differ, at least in principle (and that difference is not visible in practice because the parameters of GRW theories are chosen so as to avoid possible refutations). So, if one found that a GRW prediction is right where it differs from ordinary quantum mechanics, it would be a major revolution in physics and we might be forced to worry about those weird ontologies. But this hasn’t happened yet and I would suggest not to worry about what to do after the revolution before that revolution has occurred.

17.4 Ontologies for the de Broglie-Bohm Theory In the de Broglie-Bohm theory (Bohm,  1952a,b),the complete state of a system with N variables at time t is specified by (t), X(t) , where (t) is the usual quantum   state, (t) = (x1 , . . . , xN , t) and X(t) = X1 (t), . . . , XN (t) ∈ R3N are the actual positions of the particles.14

14 Our

presentation of the de Broglie-Bohm theory follows the one of Bell (2004) and of Dürr et al. (2013) rather than the one of Bohm (1952a,b). This approach is actually close to the original one of de Broglie, see Bacciagaluppi and Valentini (2009). Many expositions of the de Broglie–Bohm theory are available, see, e.g., Albert (1994) or Tumulka (2004) for elementary introductions and Bacciagaluppi and Valentini (2009), Bohm and Hiley (1993), Bricmont (2016), Dürr and Teufel (2009), Dürr et al. (2013), Goldstein (2013), Towler (2009) for more advanced ones. There are also pedagogical videos made by students in Munich, available at: https://cast.itunes.uni-muenchen.de/vod/playlists/URqb5J7RBr.html.

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The theory assumes that the particles have positions at all times, and therefore trajectories, independently of whether one measures them or not. The evolution of the state (, X) is given by two laws: 1.  obeys the usual Schrödinger equation at all times. The quantum state never collapses. 2. The evolution of the positions of the particles is determined by the quantum state at time t. One writes, for spinless particles, (x1 , . . . , xN , t) = R(x1 , . . . , xN , t) exp iS(x1 , . . . , xN , t), and the dynamics is simply   d 1 Xk (t) = ∇Xk S X1 (t), . . . , XN (t), t , dt mk

(17.3)

∀k = 1, . . . , N and mk being the mass of the particle with label k. So the ontology consists here simply of particles in motion, as in classical physics, but guided by a wave that itself is defined on R3N and not on R3 , as would be the case for electromagnetic waves. The theory reproduces all the usual quantum predictions, accounts for their statistical aspects through natural assumptions on initial conditions and explains the absence of “hidden variables” other than positions (see the references in footnote 14 for more explanations). The way it accomplishes that latter feat is by showing that what are usually called “measurements” in ordinary quantum mechanics are in reality interactions between the system and the measuring device;15 thus, the latter does not measure any property of the particle (except for position measurements) and the fact that the map v mentioned in Sect. 17.1 does not exist has no consequence for the de Broglie-Bohm theory (except to make that theory more natural). The de Broglie-Bohm theory is thus a statistical theory, but unlike the naïve one mentioned in Sect. 17.1, it is consistent and is not refuted by the theorems of Bell (1966) and Kochen and Specker (1967). For extensions of that theory to field ontologies, see e.g. Struyve (2010) and references therein. We may also add that, as stressed in Valentini (2012), the de Broglie version of the theory, which was introduced before 1927, was not trying to solve the measurement problem, because that “problem” arose only within the post-1927 “orthodox” version of quantum mechanics.

15 For example, one could “measure” the spin of a particle in a given direction, with a Stern-Gerlach apparatus, starting with exactly the same initial wave function and initial position for the particle, but with two different orientations of the gradient of the magnetic field in the apparatus, and obtain two opposite results; thus no “spin property” of the particle has been measured; see Daumer et al. (1996), Bricmont et al. (2019) for more details.

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17.5 Comparison of Ontologies To get a feeling for how odd the continuous matter and the flash ontologies are, compared to the particle ontology of the de Broglie–Bohm theory, consider what happens, in those ontologies, with the thought experiment of Einstein’s boxes16 (discussed e.g. in Norsen (2005)): imagine a box containing just one particle that is cut in two parts; one part is sent to New York, the other to Tokyo. The wave function of the particle is a superposition of a wave function located in one box + a wave function located in the other box. In ordinary quantum mechanics, when a person in New York opens its half-box and sees the particle, the wave function collapses on the part of the wave function located in that half-box and, if he doesn’t see it, the wave function collapses on the part located in the other half-box.17 In the de Broglie–Bohm theory, nothing surprising happens: the particle is in one of the half-boxes all along and is found where it is. But in the GRWm theory, there is one-half of the matter density of a single particle in each half-box. But when one opens one of the half-boxes, the evolution of the wave function of the particle becomes coupled with a macroscopic measuring device and many collapses occur quite rapidly, so that the matter density suddenly jumps from being one-half of the matter density of a single particle in each half-box to being the full matter density of a particle in one half-box and nothing in the other. Of course, this happens randomly, so that the matter density may jump from the half-box that one opens to the other half-box. In any case, there is a nonlocal transfer of matter in the GRWm theory, while there is no such thing in the de Broglie–Bohm theory, and not even anything nonlocal when one deals with only one particle. In the GRWf theory, there is simply nothing in either half-box, just a wave function traveling so to speak with the half-boxes. When one opens one of the halfboxes and the wave function of the particle becomes coupled with a macroscopic device, there is suddenly a “galaxy of flashes” appearing (randomly) in one of the two half-boxes, which we interpret as meaning that the particle is in that half-box. Again, the theory is more nonlocal, in a sense, than de Broglie–Bohm, since, by opening one half-box, one may very well trigger a galaxy of flashes in the other half-box, on which nothing is done.18

16 See

Maudlin (2011) (Chap. 10) for a similar discussion of the GRWm and GRWf ontologies, as opposed to the de Broglie–Bohm ontology. 17 Some people might argue that cutting the box in two already collapses the wave function. Since we discuss here gedanken experiments, it is not easy to determine what happens. But here is a modification of that experiment that avoids this objection: imagine that the box is inside a long cylinder, stretching from New York to Tokyo if you wish, and that we remove the two sides of the box perpendicular to the long side of the cylinder. Then the wave function will spread itself in the cylinder and the particle can be detected later on one side or the other of the cylinder. 18 For other criticisms of the GRWf theory, in particular in what sense is it really Lorentz invariant, see Esfeld and Gisin (2014).

17 cat alive and cat dead Are not Cats! Ontology and Statistics in. . .

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In any case, this example should suffice to convince any reasonable person of the superiority in terms of naturalness of the de Broglie–Bohm theory over its alternatives (many-worlds à la Allori et al. (2011) or GRWm or GRWf). Acknowledgments I thank two referees for useful comments, Sheldon Goldstein and Tim Maudlin for many discussions on Bohmian mechanics and the many-worlds theories, and specially Valia Allori for illuminating exchanges on the topics discussed here.

References Albert, D. (1994, May). Bohm’s alternative to quantum mechanics. Scientific American, 270, 32– 39. Allori, V., Goldstein, S., Tumulka, R., & Zanghì, N. (2008). On the common structure of Bohmian mechanics and the Ghirardi–Rimini–Weber theory. British Journal for the Philosophy of Science, 59, 353–389. Allori, V., Goldstein, S., Tumulka, R., & Zanghì, N. (2011). Many-worlds and Schrödinger’s first quantum theory. British Journal for the Philosophy of Science, 62, 1–27. Bacciagaluppi, G., & Valentini, A. (2009). Quantum Mechanics at the Crossroads. Reconsidering the 1927 Solvay Conference. Cambridge: Cambridge University Press. Barrett, J. (2018). Everett’s relative-state formulation of quantum mechanics. Stanford Encyclopedia of Philosophy. Bell, J. S. (1966). On the problem of hidden variables in quantum mechanics. Reviews of Modern Physics, 38, 447–452. Reprinted as Chap. 1 in Bell (2004). Bell, J. S. (1984). Beables for quantum field theory. CERN-TH 4035/84. Reprinted as Chap. 19 in Bell (2004). Bell, J. S. (1987). Are there quantum jumps? In C. W. Kilmister (Ed.), Schrödinger. Centenary celebration of a polymath (pp. 41–52). Cambridge: Cambridge University Press. Reprinted as Chap. 22 in Bell (2004). Bell, J. S. (2004). Speakable and unspeakable in quantum mechanics. Collected papers on quantum philosophy (2nd ed.). With an introduction by Alain Aspect. Cambridge: Cambridge University Press (1st ed., 1987). Bohm, D. (1952a). A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Physical Review, 85(2), 166–179. Bohm, D. (1952b). A suggested interpretation of the quantum theory in terms of “hidden” variables. II. Physical Review, 85(2), 180–193. Bohm, D., & Hiley, B. J. (1993). The undivided universe. London: Routledge. Bricmont, J. (2016). Making sense of quantum mechanics. Berlin: Springer. Bricmont, J., Goldstein, S., & Hemmick, D. (2019). Schrödinger’s paradox and proofs of nonlocality using only perfect correlation. Journal of Statistical Physics, 180, 74–91. Damour, T. (2006). Once upon Einstein. Wellesley, MA: A.K. Peters. Daumer, M., Dürr, D., Goldstein, S., & Zanghì, N. (1996). Naive realism about operators. Erkenntnis, 45, 379–397. Deutsch, D. (1999). Quantum theory of probability and decisions. Proceedings of the Royal Society of London A, 455, 3129–3137. DeWitt, B. S, & Graham, R. (Eds.) (1973). The many-worlds interpretation of quantum mechanics. Princeton: Princeton University Press. Dürr, D., Goldstein, S., & Zanghì, N. (2013). Quantum physics without quantum philosophy. Berlin: Springer. Dürr, D., & Teufel, S. (2009). Bohmian mechanics. The physics and mathematics of quantum theory. Berlin: Springer.

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Esfeld, M., & Gisin, N. (2014). The GRW flash theory: a relativistic quantum ontology of matter in space-time? Philosophy of Science, 81, 248–264. Everett, H. (1957). ‘Relative state’ formulation of quantum mechanics. Reviews of Modern Physics, 29, 454–462 (1957). Reprinted in DeWitt and Graham (1973) (pp. 141–149). Freire, O. J. (2015). The quantum dissidents. Rebuilding the foundations of quantum mechanics (1950-1990). Berlin: Springer.. Ghirardi, G. C., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34, 470–491. Ghirardi, G. C., Grassi, R., & Benatti, F. (1995). Describing the macroscopic world: closing the circle within the dynamical reduction program. Foundations of Physics, 25, 5–38. Ghirardi, G. C. (2011). Collapse theories. In E. N. Zalta (Ed.), The Stanford Encyclopedia of philosophy (Winter 2011 ed.). http://plato.stanford.edu/entries/qm-collapse/ Goldstein, S. (2013). Bohmian mechanics. In E. N. Zalta (Ed.), The Stanford Encyclopedia of philosophy (Spring 2013 ed.). Available on plato.stanford.edu/archives/spr2013/entries/qm-bohm/ Goldstein, S., Tumulka, R., & Zanghì, N. (2012). The quantum formalism and the GRW formalism. Journal of Statistical Physics, 149, 142–201. Kochen, S., & Specker, E. P. (1967). The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics, 17, 59–87. Maudlin, T. (2011). Quantum non-locality and relativity: Metaphysical intimations of modern physics (3rd ed.). Oxford: Wiley-Blackwell. Maudlin, T. (2012). Can the world be only wavefunction? In S. Saunders et al. (Eds.), Many Worlds?: Everett, quantum theory, and reality (Chap. 4). Oxford University Press. Maudlin, T. (2014). Critical study. D. Wallace, The emergent multiverse: Quantum Theory according to the Everett interpretation, Oxford University Press, 2012. Noûs, 48, 794–808. Maudlin, T. (2019). Philosophy of physics. Quantum theory. Princeton: Princeton University Press. Mermin, N. D. (1993). Hidden variables and the two theorems of John Bell. Reviews of Modern Physics, 65, 803–815. Ney, A., & Albert, D. (Eds.) (2013). The wave function: Essays on the metaphysics of quantum mechanics. New York: Oxford University Press. Norsen, T. (2005). Einstein’s boxes. American Journal of Physics, 73, 164–176. Norsen, T. (2014). Quantum solipsism and nonlocality. International Journal of Quantum Foundations (electronic journal). Available at www.ijqf.org/archives/1548 Norsen, T. (2017). Foundations of quantum mechanics: An exploration of the physical meaning of quantum theory. Cham: Springer International Publishing. Struyve, W. (2010). Pilot-wave theory and quantum fields. Reports on Progress in Physics, 73, 106001. Towler, M. (2009). De Broglie–Bohm pilot-wave theory and the foundations of quantum mechanics. Lectures, available at www.tcm.phy.cam.ac.uk/~mdt26/ Tumulka, R. (2004). Understanding Bohmian mechanics — A dialogue. American Journal of Physics, 72, 1220–1226. Tumulka, R. (2018). Paradoxes and primitive ontology in collapse theories of quantum mechanics. In S. Gao (Ed.), Collapse of the wave function (pp. 134–153). Cambridge: Cambridge University Press. Vaidman, L. (1998). On schizophrenic experiences of the neutron or why we should believe in the many-worlds interpretation of quantum theory. International Studies in the Philosophy of Science, 12, 245–261. Valentini, A. (2012). de Broglie–Bohm pilot-wave theory: Many worlds in denial? In S. Saunders et al. (Eds.), Many Worlds?: Everett, quantum theory, and reality (pp. 476–509). Oxford University Press. Wallace, D. (2003). Everett and structure. Studies in History and Philosophy of Modern Physics, 34, 87–105. Wallace, D. (2007). Quantum probability from subjective likelihood: Improving on Deutsch’s proof of the probability rule. Studies in History and Philosophy of Modern Physics, 38, 311–332.

Chapter 18

Cosmic Hylomorphism vs Bohmian Dispositionalism Implications of the ‘No-Successor Problem’ William M. R. Simpson and John M. Pemberton

Abstract The primitive ontology approach to Bohmian mechanics seeks to account for quantum phenomena in terms of particles that follow continuous trajectories and a law of nature that describes their temporal development. This approach to explaining quantum phenomena is compatible with a dispositional account of the wave function. In this paper, however, we argue that dispositional models which posit powers that are stimulated according to the instantaneous configuration of the particles and manifest an instantaneous velocity profile are subject to a ‘nosuccessor dilemma’: either time must be discrete rather than continuous, or the powers fail to determine the particles’ trajectories. What is needed is for the cosmos to have a power to manifest a teleological process that determines the particles’ trajectories, where this process is metaphysically prior to their instantaneous configurations. Keywords Primitive ontology approach · Quantum mechanics · Causal powers · Causal process · Teleology · Hylomorphism · Aristotelianism

18.1 Introduction One of the challenges confronting scientific realists is the problem of specifying the ontological status of the wave function in quantum mechanics. In primitive ontology approaches to quantum mechanics, the wave function enters the account through the

W. M. R. Simpson () Wolfson College, University of Cambridge, Faculty of Philosophy, Faculty of Divinity, Cambridge, UK e-mail: [email protected] J. M. Pemberton Centre for the Philosophy of the Natural and Social Sciences, London School of Economics, London, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_18

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nomological or dispositional role it plays in the dynamics of a primitive distribution of matter in three-dimensional space (or four-dimensional spacetime) (Allori et al., 2008; Esfeld, 2014; Esfeld et al., 2013). Maudlin has argued that the choice of quantum dynamics for the wave function comes down to two options (Maudlin, 1995): either we should adopt something like the modified Schrödinger dynamics proposed by Ghirardi, Rimini and Weber (Carlo et al., 1986), in which the wave function undergoes spontaneous collapse, or something like the pilot wave theory of de Broglie and Bohm (Bohm, 1951, 1952; de Broglie, 1928), which includes an additional equation of motion for a particle configuration whilst preserving Schrödinger’s original dynamics. Our focus in this paper will be on the second of these two alternatives, and will rely upon the first-order interpretation of the pilot wave theory advanced by Dürr, Goldstein and Zanghí, in which the particles obey an equation of motion that specifies their instantaneous velocities (rather than their accelerations) (Dürr et al., 1992). The velocities of the particles depend on both the positions of the particles and the universal wave function. For those who adopt a primitive ontology approach to quantum mechanics and reject primitivism about laws, there are two ways of spelling out the role of the wave function in the temporal development of the particle configuration: namely, by appealing to a Humean account of laws, in which laws summarise regularities in the spatiotemporal distribution of the particles, or to some form of dispositionalism, in which the change in the distribution of the particles is underwritten by powers (Esfeld et al., 2017). In the ‘Super-Humean’ ontology recently proposed by Esfeld, for example, the particles are nothing over and above the distance relations in which they stand, and their spatiotemporal distribution is entirely contingent (Esfeld and Deckert, 2017). By contrast, in the theory of ‘Cosmic Hylomorphism’ recently proposed by Simpson, the particle configuration comprises a substance with an intrinsic power to choreograph the trajectories of the particles, and the particles are caught up within a process in which their causal powers are continually changing (Simpson, 2021). In this paper, we seek to put forward an objection to a rival dispositional model proposed by Suarez, which attributes permanent ‘Bohmian dispositions’ to the individual particles (Suárez, 2015), where these powers are stimulated by the instantaneous configuration of the particles and manifest an instantaneous velocity profile. We shall argue that this alternative form of dispositionalism falls foul of a ‘no-successor dilemma’: either time must be regarded as discrete, or the powers fail to determine the particles’ trajectories. In order for powers in a Bohmian cosmos to underwrite change, we claim, the Bohmian cosmos as a whole must have a power to manifest a teleological process in which the particles are directed to follow certain trajectories,1 where this process is metaphysically prior to any instantaneous configurations of the particles. Cosmic Hylomorphism is therefore to be preferred to Bohmian dispositionalism.

1

A telological process is one directed toward some telos, ie. end.

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The discussion is structured as follows. In Sect. 18.2, we motivate and lay out a dispositionalist version of the primitive ontology for Bohmian mechanics based on Suarez’s proposal, in which the particles are endowed with intrinsic powers to change their velocities. In Sect. 18.3, we argue that this proposal is susceptible to the no-successor dilemma. Bohmian dispositionalism thus shares a fundamental problem with a number of powers ontologies extant in the literature. In Sect. 18.4, we suggest that dispositionalists should embrace a more thoroughgoing Aristotelianism, in which powers manifest processes which are metaphysically unified and fundamentally directed towards their telos.

18.2 Bohmian Dispositions 18.2.1 Bohmian Primitive Ontology According to the primitive ontology approach to quantum mechanics, the world is made of a distribution of matter, and the wave function plays a nomological or dispositional role in its temporal development. In the primitive ontology proposed by Goldstein et al. (2005a,b) for the pilot wave theory of de Broglie and Bohm (Bohm, 1951, 1952; de Broglie, 1928), the matter consists of primitive particles.2 In the contemporary interpretation of the pilot wave theory championed by Dürr, Goldstein and Zanghí under the name of ‘Bohmian mechanics’ Dürr et al. (1992), the particles obey an equation of motion which is first-order with respect to time that determines their instantaneous velocities (rather than accelerations). According to this interpretation, the particles do not have any intrinsic physical properties like mass or charge, but each particle is assigned a position Qi , and the equation of motion depends upon both the positions of all the particles and a universal wave function ψ.3 The Schrödinger equation and the guiding equation together comprise the non-classical dynamics of a particle configuration that exists independently of our observations, and the wave function induces a velocity field ψ vi that choreographs the trajectories of all the particles. The Bohmian theory, for all practical purposes, is empirically equivalent to standard quantum mechanics. Although the particles have determinate positions, we cannot know where all of the particles are, and must confine ourselves to

2

For recent textbook studies, see Bricmont (2016); Dürr and Teufel (2009); Dürr et al. (2012); Holland (1993). 3 The particles are attributed gravitational mass, but the COW experiment suggests mass delocalises over the wave function and need not be regarded as an intrinsic property (Brown, 1996).

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probabilistic predictions of observable outcomes which satisfy the Born rule.4 Nonetheless, the physical state of the world at time t is completely specified by both the wave function and the positions of the particles: (ψ, {Q1 , . . . , QN })t . It has recently been argued that sophisticated quantum field theories can also be interpreted in terms of an ontology of persisting particles that move on continuous trajectories following a deterministic law (Deckert et al., 2019). A primitive ontology approach to Bohmian mechanics enjoys a number of advantages over other ways of interpreting this theory. Unlike multifield realism (Chen, 2017; Forrest, 1988; Hubert and Romano, 2018), it avoids the dualism of postulating a multitude of fields which exist in ordinary physical space that are somehow controlled by a wave function which exists in configuration space.5 Multifield realism leads to an interaction problem, in which the causal mechanism between physical space and configuration space is mysterious (Belot, 2012; Solé, 2013). Unlike configuration space realism (Albert, 1996; Ney, 2013; North, 2013), it avoids exchanging ordinary three or four-dimensional space as the theatre of reality for the abstract 3N-dimensional space of the wave function. Configuration space realism leads to a problem of perception, in which the arena of perceptual experience must somehow be shown to ‘emerge’ from something very unlike the ordinary physical space in which scientists conduct experiments (Belot, 2012; Monton, 2002; Solé, 2013). There is more than one way, however, of specifying the role of the wave function in the temporal development of the primitive ontology in physical space.

18.2.2 Instantaneous powers Suarez argues skilfully for a dispositional interpretation of the role of the wave function in which the wave function represents dispositional properties of the particle configuration (Suárez, 2015). After all, the wave function itself is (in principle) subject to evolution through time, in accordance with Schrödinger’s dynamics, yet we normally conceive laws as determining the temporal development of objects within their domain, rather than being themselves subject to temporal evolution. If this ‘law’ is grounded in the powers of the particles, however, then the time-dependence of the wave function can be explained. In Suarez’s account, when it is applied to the first-order interpretation of Bohmian mechanics,6 the particles are endowed with their own ‘Bohmian dispositions’ to 4 Agreement with the Born Rule is secured if the initial configuration of the particles at t may be supposed to be randomly distributed with a probability distribution ρ(t = 0) = |ψt=0 |2 . It then follows as a consequence of the Schrödinger equation and the Bohmian law of motion that this relationship will hold at time t > 0 for the distribution ρ(t) = |ψt |2 . 5 For further discussion of the multifield option, see (Suárez, 2015, pp. 3211–3212). 6 We recognise that Suarez’ theory need not be applied globally to all particles nor is it restricted to the first-order interpretation of the Bohmian equation of motion. These qualifications are imposed within the primitive ontology approach to Bohmian mechanics.

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change their velocities (Suárez, 2015). Each of the particles exercises a power to change its instantaneous velocity, which depends upon the instantaneous spatial configuration of all of the particles (however far apart they may be). The spatial configuration of the particles at time t, which may be explicated in terms of the distance relations between the particles, serves as a stimulus condition for the velocity profile v ψ (t), which is specified by the Bohmian equation of motion for the particle configuration. The nature of these causal powers is closely tied to the first-order interpretation of this guiding equation, which determines for every ψ possible particle position Qi a velocity vector vi that depends on the wave function ψ. According to Suarez, the velocity field specified by this equation ‘describes the dispositions of every particle configuration in any real or imaginary system of particles’ (Suárez, 2015, p.3216). Since it seems rather extravagant to think of each of these particles as having an infinite number of numerically distinct powers, we may think of each particle as instantiating a multi-track power instead, where each ‘track’ of a power corresponds to a different instantaneous velocity profile and is stimulated by a different instantaneous spatial configuration. The powers of the particle do not change in time: they simply yield different instantaneous manifestations according to how the particles are configured. Esfeld et. al. have also developed a dispositional model, in which the configuration as a whole instantiates a power (Esfeld et al., 2013). Likewise, in this alternative model, we can think of the power that is instantiated by the particle configuration as being a multi-track power. There is no need in either of these models to include another physical entity, in addition to the particle configuration, in order to activate these powers. Rather, for the Bohmian dispositionalist, the manifestation of the power corresponding to the appropriate track is spontaneously actualised whenever the particles happen to reside in the correct positions.

18.3 The No-Successor Problem We wish to call into question the claim that Bohmian dispositions, whether they belong to the particles individually or to the particle configuration as a whole, can explain change through time in a world that is governed by the laws of Bohmian mechanics. Our objection to Bohmian dispositionalism is based on an argument recently advanced by Pemberton against the concept of ‘state-state powers’ (Pemberton, 2021), which adapts Bertrand Russell’s sceptical objections to the concept of causation (Russell, 1913). Russell conceived the law of causality as a determination relation between a cause ‘event’, which occurs at a particular instant in time, and an effect event, which also occurs at a particular instant. The central insight behind his sceptical attack is that, just as there are no successors in the set of real numbers, an instantaneous effect cannot occur just after an instantaneous cause without there being a time gap

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between them, assuming time to be continuous rather than discrete. Yet postulating a temporal gap between a cause and its effect, or insisting upon the simultaneity of the cause event and the effect event, turns out to be problematic in a number of ways. So it shall prove, we claim, for the Bohmian dispositionalist who invokes Bohmian dispositions to explain how different configurations of the particles cause the velocity profile of the particles to change, when Russell’s argument is recast in terms of the ‘activation’ (cause) and ‘manifestation’ (effect) states of these intrinsic dispositions.7 Yet the solution is not to give up on causality, we shall argue, but to drop Russell’s atomised conception of change.

18.3.1 State-State Powers Many contemporary accounts of powers assume that once a power is in some suitable activation state, a manifestation state will occur. When the power is in circumstances sufficient for its manifestation, it will bring about this manifestation by some kind of natural necessity. We shall categorise these kinds of powers as ‘state-state powers’. Suarez, for instance, appears to be committed to such a conception of powers. In his model, an activation state in a Bohmian disposition results from an instantaneous spatial arrangement of the particles {Q1 , . . . , QN }, whilst a manifestation state corresponds to an instantaneous velocity profile v ψ ({Q1 (t), . . . , QN (t)}) (Suárez, 2015). Likewise, Esfeld et al. seem committed to such a conception of powers, even though there is just one Bohmian disposition in their model which is instantiated by the whole particle configuration (Esfeld et al., 2013). Let us imagine a Bohmian world in which the powers of the particle configuration are in a collective activation state A at time t  , and in which they yield a manifestation state M at time t, which is the new velocity profile of the particle configuration. We shall assume A and M to be distinct states which obtain instantaneously during the development of the particle configuration. We make this assumption because each instantaneous configuration of the particles is supposed to stimulate a numerically distinct Bohmian disposition (or a numerically distinct track of a multi-track disposition) to manifest a different velocity profile. If these Bohmian dispositions are not to interfere with one another, it is reasonable to assume that only one velocity profile may be manifested at a given moment, which is the velocity profile that is specified by the differential equation of motion for the particle configuration. The question is: does M occur at the same moment of time as A, at t = t  , or does M occur just after A, at some later moment of time, t > t  ?

7

See Mumford and Anjum (2011) for a contemporary account of causation in terms of powers or dispositions.

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On the one hand, suppose that M occurs at the same instant of time as A such that no time elapses between the stimulation of the Bohmian disposition and the manifestation of a new velocity profile of the particles. In that case, it appears that the powers of the particles do not underwrite change in the configuration of the particles. Let us return to considering this option in a moment. On the other hand, suppose that M occurs after A. In that case, the particles would not change their velocities at the same moment that the appropriate stimulus condition is satisfied, but at some subsequent moment, t > t  . But just how much time should elapse between A and M? As a first attempt at tackling this question, we might try putting a temporal gap between A and M of finite duration. In that case, the particles would remain at certain velocities for finite periods of time, before manifesting their powers and changing their velocity profiles. However, this time lapse is inconsistent with the theory of Bohmian mechanics, which does not represent the particles as following zig-zag trajectories through space, but represents their velocities as continually changing and their paths as smooth and continuous. More troubling, however, is the fact that we can offer no intelligible account of such a delay. As Russell complained: it ‘seems strange – too strange to be accepted, in spite of the bare logical possibility—that the cause, after existing placidly for some time, should suddenly explode into the effect’ (Russell, 1913, p.5). What could be the reason for such a delay between the cause and the effect? It is rather difficult to say. Just how much time is required before the cause can produce the effect? Any answer would seem to be arbitrary. It may seem tempting, as a second attempt, to try and mitigate the problem by making the temporal delay between A and M infinitesimal, adopting a Leibnizian notion of infinitesimal quantities in order to reconcile the sequential conception of causation embodied in Bohmian dispositionalism with the differential equation of motion for the particle configuration specified by Bohmian mechanics. Yet modern analysis has dispensed with infinitesimals, as Huemer and Kovitz observe (Huemer and Kovitz, 2003): the ‘delta and epsilon’ proofs developed by Cauchy and Weierstrass demonstrate that a rate-of-change quantity such as the velocity profile of the Bohmian particle configuration can be understood in terms of relationships between finite quantities.8 Moreover, this second attempt merely reiterates the question: why this infinitesimal delay? As a third attempt, we might allow that activation and manifestation states should be temporally contiguous, but conceive this cause-and-effect pairing as obtaining on semi-open intervals of time: the activation-cause taking place during the interval [T − δ, T ); the manifestation-effect taking place during [T , T + δ  ). In that way, they appear temporally distinct, without any time elapsing between them. Besides being ad hoc, however, this solution is subject to another of Russell’s objections.

8

Recent work on infinitesimals in non-standard mathematics should not be read as challenging the contemporary mathematical orthodoxy which rejects the use of infinitesimals within standard mathematical methods, especially those applicable to the physical world.

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Suppose the activation state obtains during [T − δ, T ). Let us consider the first half of this period: [T − δ, T − δ/2). Since this period is not temporally contiguous with the manifestation state, it cannot give rise to the manifestation state. In that case, it must be the activation state during the second of this period that gives rise to the manifestation state: [T − δ/2, T ). By repeating this analysis, however, we can deduce that the causally salient part of the activation state obtains during the interval, Lim[T − δ, T ) as δ → 0, which is simply the empty set. In other words, the causally salient part of the activation state never obtains at any time. In a fourth attempt, we might allow activation and manifestation states to be temporally contiguous by treating time as discrete. This is a coherent solution to the problem, but it comes at a theoretical cost: in Bohmian mechanics, time is treated as continuous. To adopt a discrete conception of time would be to concede that any version of Bohmian mechanics which resembles the one under consideration, including Bohmian treatments of quantum field theories which are interpreted in terms of particles that move on continuous trajectories following a deterministic law, are merely effective theories which fail to refer to any fundamental physical constituents. Perhaps we should return, then, to the possibility of A and M being simultaneous, allowing time to flow independently of the activation and manifestation of powers by admitting change as a primitive (like the Super-Humean), or relinquishing the idea that time flows by embracing four-dimensionalism. A defender of Suarez’s account might reply that Bohmian dispositions are not susceptible to any Russelliantype objections to their causal efficacy: after all, the positions of the particles sets the synchronic velocities of the particles via the velocity field, the velocities of the particles in turn sets the particles’ trajectories over time, and hence their positions over time. What is the difficulty? Granted, in Bohmian mechanics the position of the particles at time t  sets their synchronic velocity profile at t  . And if time were discrete, then the velocity of a particle at t  would determine its position at the next contiguous moment. But if time is continuous, then for any point P on the worldline of a particle that one may pick after t  , there is another point that lies in between at which its velocity will change, so its trajectory to P is not determined at t  . To determine subsequent positions of the Bohmian particles we must establish the velocity of the particles over an interval of time (t  , t  + δ), where δ > 0. But the velocity at each point in this interval is not yet available: in the primitive ontology approach to Bohmian mechanics, the Bohmian equation of motion is first-order in time, specifying velocities rather than accelerations, and the Bohmian dispositions just determine the instantaneous velocity profile of the particle configuration. Only where we have the acceleration of the particle over (t  , t  + δ) as a continuous function of time (at least, the Lebesgue measure of the subset of points of discontinuity is zero), may we determine the trajectories. We thus arrive at the no-successor dilemma for state-state powers such as Bohmian dispositions: either (i) the manifestation state occurs at the same time as the activation state, but the powers do not determine the particles’ trajectories, or (ii) the manifestation state is temporally contiguous with the activation state,

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but Bohmian mechanics mistakenly describes time as continuous. Neither of these options is compatible with a realist approach to the theory of Bohmian mechanics that explains the role of the wave function in terms of the particles (or the configuration) having Bohmian dispositions.

18.3.2 State-Process Powers Suppose we seize the first horn of the dilemma, but reconceive the manifestation of a power as a process that exists through time, in an attempt to deflect the accusation that the powers fail to underwrite continuous trajectories (Pemberton, 2021). The activation state of the power and its manifestation state could share a single point in time, yet the manifestation that is brought about by the power might extend beyond that point continuously through time. Such a view would rule out both Suarez’s and Esfeld’s conception of Bohmian dispositions, which manifest instantaneous velocity profiles. Is there a revised conception of a manifestation process which can avoid the no-successor dilemma? As a first attempt, we might think of a manifestation process as a ‘causal line’ consisting of a dense infinity of similar states or events that lack any necessary connections between them. Such a set of events might be picked out by its association with the transmission of a conserved quantity. However, it is difficult to see how a power can be thought of as manifesting a process that consists merely of a series of disconnected states or events, if the activation state of the power cannot be said to necessitate any of the members of this series. Even if the power might be said to necessitate the first member of this series, it does not necessitate anything that follows, so we are confronted once again with the no-successor dilemma, which can be applied to the activation state of the power and the first member of its manifestation process. Secondly, we might conceive of a manifestation process as a dense series of temporally distinct states in which each state is necessitated by one and the same causal power. In that case, however, all but one of the states comprising this process would fail to be temporally contiguous to the activation state of the power. Such a move merely amplifies the no-successor problem by introducing a temporal delay between the activation state of the power and the elements of which its manifestation process is comprised. How can we explain the delay between the activation state of the power and each of the manifestation states? Thirdly, we might conceive of a manifestation process as a series of states in which each state necessarily manifests the subsequent member, until some final state in the series is achieved. The power may be said to manifest the process by necessitating the first element of the process, which then necessitates the second, and so on. However, the no-successor dilemma can just as readily be applied to a state within the manifestation process and the subsequent state that it is supposed to necessitate. If all the elements comprising the process are simultaneous, then this

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manifestation process does not exist through time, and the power which manifests this process does not underwrite change. Fourthly, we might consider the manifestation process to be a unified whole— that is, a single manifestation that arises from the activation state. Such a process could not be interrupted, for otherwise the manifestation might fail to occur, even though the activation state has occurred, yielding (supposedly) a manifestation. But how could it be that such a process through time should take place without any possible interruption? This would seem to require some additional, implausible story.

18.4 Cosmic Hylomorphism 18.4.1 Powers with Aristotelian-Timing Another possibility remains: suppose we retain the idea of a power having a manifestation process which exists through time, but reject the idea of a power having an instantaneous activation state. Pemberton suggests that the best candidates for avoiding the no-successor problem are ‘powers with Aristotelian-timing’, which are powers that are active during the period of their manifesting (Pemberton, 2021). It is supposed that such powers have bearers and that the manifesting of these powers is the changing through time of the bearers, which must persist through the period of this change. Such a conception of powers demands a thoroughgoing rejection of the socalled At-At theory of change presupposed by Russell in his infamous attack on causation (Russell, 1903, pp. 469–473). In the At-At account, change amounts to the instantiation of different properties at different times, rather than something becoming different (Cleland, 1990). A process of change, from this point of view, is grounded in the sum of its temporal parts.9 A process of manifestation that is brought about by powers with Aristotelian-timing, however, is the changing through time of the bearers of these powers. The temporal parts of a process, from this point of view, are grounded in a process which exists through time. For Aristotle, a process of change is a continuous thing, and a continuous thing is more unified than the contemporary notion of a ‘mereological sum’. The spatial or temporal parts of a continuous thing are potential parts, rather than actual parts, which derive their identity from the physical whole of which they are part, rather than possessing their identities independently of the whole. They are abstractions from something which persists through space and time, rather than having a separate existence of their own.10

9

The At-At conception of change was christened by Geach as ‘mere Cambridge change’. [Aristotle, Physics, Book III, v227a10-14] translated in Aristotle (1996).

10 See

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This Aristotelian account of powers, unlike Suarez’s or Esfeld’s conception of powers, is able to avoid the no-successor dilemma. Since the activation of the powers which bring about the manifestation process obtains during the same temporal interval in which this process obtains, the activation and manifestation of the powers are simultaneous. Nonetheless, the powers still underwrite change, since the manifestation which they bring about by natural necessity is a process which exists through time. We find such an account of powers and processes to be implemented in Cosmic Hylomorphism, which provides an alternative metaphysical model for Bohmian mechanics (Simpson, 2021).

18.4.2 Cosmic Form According to Aristotle, as he has often been understood, substances should be analysed in terms of both their ‘matter’ and ‘form’, where form is said to determine the matter of a substance and explain the unity of the substance as a whole. Cosmic Hylomorphism combines the primitive ontology approach to quantum mechanics with a contemporary adaptation of Aristotle’s doctrine of hylomorphism (Koons, 2014), reconceiving the primitive ontology of Bohmian mechanics such that the particles depend for their natures and identities upon a ‘Cosmic Form’ (Simpson, 2021). In this metaphysical model, the particles have causal powers to change their velocities in conformity with the Bohmian equation of motion, but these powers derive their identities from the Cosmic Form and they are subject to change through time.11 The Cosmic Form and the particles thus compose a single substance, which is identified according to the Cosmic Form. The empowered particles are integral parts of this Cosmic Substance, lacking any physical identities apart from the cosmic whole of which they are parts. We may think of the Cosmic Form as conferring a monadic nature upon the Cosmic Substance which evolves through time toward its telos—like a seedling growing into a tree – and of the particles as expressing this nature in having their causal powers evolve over time.12 The Cosmic Substance is thus a metaphysically unified whole which exercises a power to choreograph the trajectories of its particles through time, whilst the wave function of quantum mechanics represents this intrinsic power.13 Likewise, the process of change involving these particles is a metaphysically unified whole, which derives its identity from the Cosmic Substance,

11 According to Simpson, the powers of the particles are metaphysically grounded in the Cosmic Form, which unites itself to the particles by grounding their causal powers (Simpson, 2021). 12 See [Aristotle, Physics, Book II] translated in Aristotle (1996). See Waterlow (1982) on Aristotelian natures. 13 Different accounts of the goal of this cosmic process might be given by panpsychists (Bruntrup & Jaskolla, 2016) and by those seeking to revive a neo-Platonic conception of a ‘world soul’ (Dumsday, 2019).

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since the empowered particles are integral parts of this substance. It derives its unity from the Cosmic Form, since the Cosmic Form is directed toward a single telos. Cosmic Hylomorphism is preferable to Bohmian dispositionalism on a number of grounds (Simpson, 2021). For instance, a God who prefers the evolution of the world to express the elegant mathematical form of the Schrödinger equation seems more likely to choose a Cosmic Form to implement this choice, rather than the elaborate conspiracy between the particles that is involved in Bohmian dispositionalism. Moreover, the existence of a Cosmic Form explains the diachronic and trans-world sameness of the cosmic power that grounds the Bohmian equation of motion. We cannot explore all of these advantages here. Our concern is merely to highlight how Cosmic Hylomorphism avoids the no-successor problem, unlike Bohmian dispositionalism. This is evident in its conception of the cosmic power and of the cosmic process that involves all of the particles. In the first place, the cosmic power in this metaphysical model exhibits the requisite property of Aristotelian-timing, since its activation and manifestation exist simultaneously during a period of time in which the particles are subject to change. We are to think of the bearer of this cosmic power as the Cosmic Substance, which is an entity that persists in time throughout this process of change. In the second place, the process of change is the requisite kind of Aristotelian process, since it is a teleological process which is unified by the telos of the Cosmic Substance. We should think of any instantaneous configuration of the particles as potential parts of the cosmic process which must be abstracted from the cosmic process as a whole, rather than as actual parts of this process which have their identity independently of the whole, since the powers of the particles which bring about change in their spatial configuration are grounded at any time in the Cosmic Form of the substance.

References Albert, D. Z. (1996). Elementary quantum metaphysics. In Bohmian mechanics and quantum theory: An appraisal (pp. 277–284). Dordrecht: Springer. Allori, V., Goldstein, S., Tumulka, R., & Zanghì, N. (2008). On the common structure of Bohmian mechanics and the Ghirardi-Rimini-Weber theory: Dedicated to GianCarlo Ghirardi on the occasion of his 70th birthday. The British Journal for the Philosophy of Science, 59(3):353– 389. Aristotle (1996). Oxford world’s classics: Aristotle: Physics. Oxford World’s Classics: Aristotle: Physics. Oxford: Oxford University Press. Belot, G. (2012). Quantum states for primitive ontologists. Foundations of Physics, 2, 67–83. Bohm, D. (1951). Quantum theory. Englewood Cliffs: Prentice-Hall. Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of “Hidden” variables. I. Physical Review, 85(2):166–179. Bricmont, J. (2016). Making sense of quantum mechanics (Springer International Publishing, Cham, 2016). Brown, H. R. (1996). Bovine metaphysics: Remarks on the significance of the gravitational phase effect in quantum mechanics. In Perspectives on quantum reality (pp. 183–193). Dordrecht: Springer.

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G. Bruntrup & L. Jaskolla (eds.) (2016). Panpsychism: Contemporary perspectives. Oxford: Oxford University Press. Carlo Ghirardi, G., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34(2):470–491. Chen, E. K. (2017). Our fundamental physical space: an essay on the metaphysics of the wave function. The Journal of Philosophy, 114(7):333–365. Cleland, C. E. (1990). The difference between real change and “Mere” Cambridge change. Philosophical Studies, 60, 1–25. de Broglie, L. (1928). La nouvelle dynamique des quanta [The new dynamics of quanta] Electrons et photons. Rapports et discussions du cinquième Conseil de physique tenu à Bruxelles du 24 au 29 octobre 1927 sous les auspices de l’Institut international de physique Solvay. Paris: Gauthier-Villars (pp. 105–132). English translation. In G. Bacciagaluppi & A. Valentini, (eds.), Quantum theory at the crossroads: Reconsidering the 1927 Solvay conference (pp. 341–371). Cambridge: Cambridge University Press. Deckert, D.-A., Esfeld, M., & Oldofredi, A. (2019). A persistent particle ontology for quantum field theory in terms of the Dirac sea. The British Journal for the Philosophy of Science, 70(3):747–770. Dumsday, T. (2019). Breathing new life into the world-soul? Revisiting an old doctrine through the lens of current debates on special divine action. Modern Theology, 35(2):301–322. Dürr, D., & Teufel, S. (2009). Bohmian mechanics. New York: Springer. Dürr, D., Goldstein, S., & Zanghì, N. (1992). Quantum equilibrium and the origin of absolute uncertainty. Journal of Statistical Physics, 67, 843–907. Dürr, D., Goldstein, S., & Zanghì, N. (2012). Quantum physics without quantum philosophy. New York: Springer. Esfeld, M. (2014). The primitive ontology of quantum physics. Studies in History and Philosophy of Modern Physics, 47(C):99–106. Esfeld, M., & Deckert, D.-A. (2017). A minimalist ontology of the natural world. London: Routledge. Esfeld, M., Hubert, M., Lazarovici, D., & Durr, D. (2013). The ontology of Bohmian mechanics. The British Journal for the Philosophy of Science, 65(4):773–796. Esfeld, M., Lazarovici, D., Lam, V., & Hubert, M. (2017). The physics and metaphysics of primitive stuff. The British Journal for the Philosophy of Science, 68, 133–161. Forrest, P. (1988). Quantum metaphysics. New York: Blackwell. Goldstein, S., Taylor, J., Tumulka, R., & Zanghì, N. (2005a). Are all particles identical? Journal of Physics A: Mathematical and General, 38(7):1567–1576. Goldstein, S., Taylor, J., Tumulka, R., & Zanghı, N. (2005b). Are all particles real? Studies in History and Philosophy of Modern Physics, 36(1):103–112. Holland, P. R. (1993). The quantum theory of motion. An account of the de Broglie–Bohm causal interpretation of quantum mechanics. Cambridge: Cambridge University Press. Hubert, M., & Romano, D. (2018). The wave-function as a multi-field. European Journal for Philosophy of Science, 8(3):521–537. Huemer, M., & Kovitz, B. (2003). Causation as simultaneous and continuous. Philosophical Quarterly, 53(213):556–565. Koons, R. (2014). Staunch vs. faint-hearted hylomorphism: Toward an Aristotelian account of composition. Res Philosophica, 91(2):151–177. Maudlin, T. (1995). Three measurement problems. Topoi, 14(1):7–15. Monton, B. (2002). Wave function ontology. Synthese, 130(2):265–277. Mumford, S., & Anjum, R. L. (2011). Getting causes from powers. Oxford: Oxford University Press. Ney, A. (2013). Ontological reduction and the wave function ontology. In A. Ney & D. Z. Albert (eds.), The wave function. Oxford: Oxford University Press. North, J. (2013). The structure of a quantum world. In A. Ney & D. Z. Albert (eds.), The wave function. Oxford: Oxford University Press.

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Pemberton, J. (2021). Powers - The no-successor problem. Journal of the American Philosophical Association, 7(2):213–230. Russell, B. (1903). The principles of mathematics. Cambridge: Cambridge University Press. Russell, B. (1913). On the notion of cause. Proceedings of the Aristotelian Society, 13, 1–26. Simpson, W. M. R. (2021). Cosmic hylomorphism. European Journal for Philosophy of Science, 11(28). Solé, A. (2013). Bohmian mechanics without wave function ontology. Studies in History and Philosophy of Modern Physics, 44(4):365–378. Suárez, M. (2015). Bohmian dispositions. Synthese, 192(10):3203–3228. Waterlow, S. (1982). Nature, change, and agency in Aristotle’s physics: A philosophical study. Oxford: Oxford University Press.

Chapter 19

The Governing Conception of the Wavefunction Nina Emery

Abstract I distinguish between two different ways in which the wavefunction might play a role in explaining the behavior of quantum systems and argue that a satisfactory account of quantum ontology will make it possible for the wavefunction to explain the behavior of quantum systems in both of these way. I then show how this constraint has the potential to impact two quite different accounts of quantum ontology.

The question of what the wavefunction represents is the central question of quantum ontology.1 Just as one could not understand classical mechanics if one knew that f =m

d 2x dt 2

was one of the dynamical laws but did not know what f, m and x represent, one cannot understand quantum theory if one knows only that Schrödinger’s equation i

d ψ = Hˆ ψ dt

is one of the dynamical laws but does not know what ψ—the wavefuction— represents.

1

I take it that this claim is compatible with a primitive ontology approach as exemplified in Allori et al. 2008 and Allori 2013. As I understand these views, the question of what the wavefunction represents is still central, but it turns out that, given what the wavefunction represents, there must be more to quantum ontology than just the wavefunction.

N. Emery () Mount Holyoke College, South Hadley, MA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_19

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This paper articulates one route by which we might approach the question of what the wavefunction represents. This route starts by focusing on the explanatory role of the wavefunction. I first distinguish between two different aspects of this explanatory role, and then argue that it is important that we respect not just one, but both of these aspects. The result is what I call the governing conception of the wavefunction. The governing conception of the wavefunction isn’t itself an answer to the question of what the wavefunction represents, but it places significant and heretofore under-appreciated constraints on the possible ways by which we might answer this question. I don’t expect that everyone will endorse the governing conception of the wavefunction, but for those who don’t, the discussion below should help clarify the potential costs of their view. The challenge will then be to articulate why those costs are worth paying. One quick clarification before we begin. Readers who are familiar with debates about the metaphysics of laws will have some sense of the direction in which I am headed, just from the title of the paper. But it’s worth emphasizing up front that one can endorse the governing conception of the wavefunction even if one does not endorse the governing conception of laws.2 (Indeed my reading of the literature suggests that several prominent philosophers do just this.) I won’t say anything here about whether that combination of positions is all things considered the best combination. Nor will I say anything about whether one can endorse the specific argument that I give below for the governing conception of the wavefunction without endorsing a similar argument for the governing conception of laws. These are good questions, but they are questions for another time.

19.1 What the Wavefunction Must Do Here’s a claim that should not be at all controversial: The wavefunction plays a key role in explaining the behavior of quantum systems. Although this claim should not be controversial, it will be important. So it is worth going through a few examples.

19.1.1 Three Examples First, consider the well-known double-slit experiment that is used to illustrate the fact that quantum particles sometimes exhibit wave-like behavior. In this experiment, we fire a stream of electrons at a wall that has two small slits in it, near the center. Some of the electrons pass through the slits and hit a detection screen on the far side of the wall. If one of the slits but not the other is open, we see an unsurprising result: there are a lot of hits on the detection screen near the center,

2

See Beebee 2000 and Maudlin 2007 for discussion of the governing conception of laws.

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and fewer as one gets farther toward the top or the bottom. But if both slits are open we see something quite different. Some areas near the center of the screen register a lot of hits, but some areas—areas that registered many hits when only one slit was open—suddenly register few or no hits at all. (More specifically we see what physicists call an interference pattern.) This result is surprising. Any complete account of the behavior of electrons will need to explain it. Any complete account of the behavior of electrons, in other words, will need to give a satisfying answer to the following question: DS

Why is it that when we send lots of particles through a double slit, there will be points near the center of the detection screen that register few or no hits?

The answer to this question has two parts. First, there is a claim about the wavefunction of the particles when they reach the detection screen. In this experiment, when the particle reach the detection screen, the wavefunction for each particle has an amplitude close to zero at a number of points near the center of the detection screen. Call these points the central low points. Second, there is a claim about the relationship between the amplitude of wavefunction of a particle at a certain point and the probability of finding that particle in that location. According to Born’s rule, if the amplitude of the wavefunction is α at a point, then the probability of finding the particle at that point is |α|2 . Combined with the initial claim that we made about the amplitude of the wavefunction at the central low points, Born’s Rule entails that the probability of finding each particle at one of the central low points is close to zero. This is why, even when we send lots of particles through a double slit, there will be points near the center of the detection screen that register few or no hits. Here is the second example. We can think of the nucleus of a radium-226 atom as containing a number of alpha particles (each consisting of two protons and two neutrons). The alpha particles themselves do not have enough energy to overcome the forces that keep them bound in the nucleus. But if you observe enough radium226 atoms, some of them will spontaneously emit an alpha particle. This is an illustration of the phenomena of quantum tunneling. To use a metaphor common in physics texts, we can model the forces keeping the alpha particles in the nucleus as the walls of a well—a “potential well”—that contains the particles. The particles don’t have enough energy to get up over the walls of the well, but if you wait long enough you will occasionally see them “tunnel” through those walls. This result is surprising. Any complete account of the behavior of alpha particles will need to explain it. Any complete account of the behavior of alpha particles, in other words, will need to give a satisfying answer to the following question: R

Why is it that, if we observe enough radium-226 atoms, we will see some of them will spontaneously emit an alpha particle?

The answer to this question again begins with a claim about the wavefunction of the particles in a radium-226 atom. In particular the wavefunction of each alpha particle has a non-zero amplitude outside the nucleus that contains it. Combined with Born’s Rule, this entails that the probability of each of those particles being

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found outside the nucleus is greater than zero. This is why, if we observe enough radium-226 particles, some of them will spontaneously emit an alpha particle. One final example. It is possible to prepare pairs of silver atoms in a special way that puts them into what physicists call the singlet state. When two silver atoms are in the singlet state it doesn’t matter how far apart they are or how careful we are to keep them from sending signals to one another—if we measure the spin of both particles in a particular direction, the measurements will always have opposite outcomes. This is an example of quantum entanglement. This result is surprising. Any complete account of the behavior of silver atoms will need to explain it. Any complete account of the behavior of silver atoms, in other words, will need to give a satisfying answer to the following question: S

Why is it that, regardless of how far apart they are, two particles in the singlet state are always measured to have opposite spin?

Once again, the answer to this question begins with a claim about the wavefunction of the particles. But in this case, the particles, taken individually do not have a wavefunction. There is only the wavefunction for the system as a whole. This wavefunction takes as inputs states of the system as a whole—for instance, particle 1 having spin up in the z-direction while particle 2 has spin down in the z-direction, or particles 1 and 2 both having spin up in the z-direction. The wavefunction of a system that is in the singlet state is such that, according to Born’s rule, the probability of the two particles having opposite spin in a particular direction is 1, while the probability of the two particles having the same spin is 0. This is why, regardless of how far apart they are, two particles in the singlet state are always measured to have opposite spin.

19.1.2 Two Types of Why Questions In all three examples above we had a phenomena that needed to be explained and we did so by appealing to the wavefunction. More carefully, in each case, the question we needed answered was a question about the relative frequency with which the outcome of some experiment was observed. And in each case, the answer involved pointing out that the wavefunction of the system in question had a certain form. In conjunction with Born’s rule, this fact about the wavefunction then entailed that the probability of the outcome in question matched the observed relative frequency. What these examples show, therefore, is that the wavefunction plays a key role in explaining the behavior of quantum systems. This much I think is uncontroversial. But the reason it is uncontroversial is that we haven’t said anything at all about what it means to explain the behavior of a quantum system. Consider again the three requests for explanation that we saw above. One way of interpreting these questions is as questions about why we should expect the relevant behaviors:

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DSE

RE SE

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Why should we expect that, when you send lots of particles through a double slit, there will be points near the center of the detection screen that register few or no hits? Why should we expect that, if we observe enough radium-226 particles, we will see some of them will spontaneously emit an alpha particle? Why should we expect that, regardless of how far apart they are, two particles in the singlet state are always measured to have opposite spin?

Call this kind of why question a why-should-we-expect question. The examples discussed above demonstrate that everyone should be on board with the idea that the wavefunction explains the behavior of quantum systems in the sense that the wavefunction plays a key role in answering why-should-we-expect questions. This just follows from the fact that Born’s rule is the standard rule for predicting the behavior of quantum systems, and the fact that the wavefunction plays a central role in Born’s rule. Crucially, however, when we asked our original why-questions with respect to each of the examples above, we might have meant something different. We might instead have been asking about the reason why the behavior in question happened: DSR

RR SR

What is the reason why, when we send lots of particles through a double slit, there are points near the center of the detection screen that register few or no hits? What is the reason why, if we observe enough radium-226 particles, we see some of them spontaneously emit an alpha particle? What is the reason why, regardless of how far apart they are, two particles in the singlet state are always measured to have opposite spin?

Let’s call these kinds of questions reason-why questions.3 I think it would be a mistake to try to argue that one—and only one—of the two kinds of why-questions just described is the right kind of question to be interested in when we are looking for an explanation of some phenomena. Both have a plausible claim on playing such a role. But I also think that it would be a mistake not to clearly distinguish between which of these two kinds of questions you are after when you are looking for an explanation. This is because it is often the case that we can have a good answer to a why-should-we-expect question without having a good answer to the corresponding reason-why question.4 Consider, for instance, a case where we have reliable testimony. If someone who has just arrived from the 3 Bradford Skow (2016) has recently made extensive use of the terminology of ‘reasons why’. Although there are obvious similarities between my view and his (e.g. causes are paradigm examples of reasons why), there are also important differences (e.g. I take reasons why to provide explanations, whereas Skow does not). 4 We can also have a good answer to a reason-why question without having a good answer to the corresponding why-should-we-expect question. Consider, for instance, cases where the explanans confers only low probability on the explanandum. In Scriven’s well known example, someone’s having syphilis might be the reason why they got paresis, even though their having syphilis is not a good answer to the question “why should we expect them to get paresis?” because having

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relevant direction tells us that the highway is closed, we might have a good answer to the question of why we should expect the highway to be closed without having any answer at all as to the reason why the highway is closed. Or consider a case in which you have inductive support for something happening. The fact that every sample of salt that you have examined in your chemistry lab has been soluble, for instance, might be a perfectly good answer to the question of why we should expect the next sample of salt to soluble as well, while telling us nothing whatsoever about the reason why the sample is soluble. In general, all you need to establish a good answer to a why-should-we-expect question is a good epistemic rule. But as the examples above show, not all good epistemic rules for figuring out what is the case involve identifying the reason why it is the case. The fact that we can have a good answer to why-should-we-expect questions without having a good answer to the corresponding reason-why questions means that it is quite easy to end up talking past one another when we start talking about the explanatory role of an entity like the wavefunction. Someone who thinks that the wavefunction only needs to answer why-should-we-expect questions about the behavior of quantum systems may be satisfied with a particular account of quantum ontology while someone who thinks that the wavefunction needs to answer reasonwhy questions finds the very same account lacking. (We will see a concrete example of this in Sect. 19.2.1.) For this reason alone, it’s worth distinguishing these two types of why questions and being more clear about which kind of question one thinks the wavefunction is supposed to answer. So what exactly does it take to answer reason-why questions? This is a difficult question, and much of what can be said in response to it will be highly controversial. But here is a relatively neutral starting point: paradigm examples of good answers to the question “What is the reason why X” involve identifying either (i) the cause of X or (ii) the grounds of X. What we want to know, for instance, when we want to know the reason why the highway is closed is what has caused the closure. And what we want to know, when we want to know the reason why sodium is soluble, is what it is about the nature of sodium that results in it being soluble.5 This suggests that good answers to reason why questions in general involve identifying dependence relations. Paradigm examples of dependence relations are causation and grounding, but insofar as there are other kinds of dependence relations besides causation and grounding, those dependence relations, too, would underwrite

syphilis, though a necessary condition for getting paresis, still only gives someone a 25% chance of developing the latter condition (Scriven 1959). 5 As an anonymous referee pointed out to me, we sometimes answer a why-should-we-expect question by pointing to the cause or the ground of the explanandum. For instance, if I asked, “Why should we expect the highway to be closed?” it would be reasonable to answer by saying, “Because there is a snowstorm and the plows aren’t running.” I take this point to be compatible with everything I have said here. In some cases (not all, cf. the previous footnote) it is possible to answer why-should-we-expect questions by identifying a dependence relation. But in general answering why-should-we-expect questions doesn’t require identifying such a relation. What is distinctive about reason why questions is that they do require identifying such a relation.

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good answers to reason why questions. In principle, at least, reason-why questions could also involve identifying something that stands in a novel dependence relation R to X.6

19.1.3 Why Reason Why Questions Must Be Answered Returning to the observation that we started with at the beginning of this section, we can now see that two different things might be meant by the claim that the wavefunction plays a key role in explaining the behavior of quantum systems. WE WR

The wavefunction plays a key role in answering why-should-we-expect questions about the behavior of quantum systems. The wavefunction plays a key role in answering reason-why questions about the behavior of quantum systems.

As I said above, everyone should agree with WE . WE just follows from the fact that Born’s Rule is the standard rule for predicting the behavior of quantum systems, and the fact that the wavefunction plays a central role in Born’s Rule. My view, however, is that we should not only accept WE . We should accept both WE and WR . This is what it means to adopt the governing conception of the wavefunction. Why think that in addition to playing a central role in answering why-should-weexpect questions about the behavior of quantum system, the wavefunction also plays a central role in answering reason-why questions about the behavior of quantum systems? First and foremost notice that if we can’t answer reason-why questions like DSR , RR , or SR by appealing to the wavefunction, then we can’t answer them at all. To give up on WR is to either admit that there is no reason why quantum systems behave the way they do, or to admit that even if there is such a reason, we cannot identify that reason using our best science. To give up on WR , therefore, is a significant cost. In my experience, most philosophers and physicists alike recognize this. But many of them still have the following concern: What if any account of quantum ontology on which WR comes out true is an account where the wavefunction represents an entity that is strange or novel or otherwise the kind of the thing that we would prefer not to have in our metaphysics? In that kind of case, the costs of giving up on WR might be worth paying. 6

Some philosophers are rightly cautious regarding the notion of grounding. But note that perhaps the most prominent way of rejecting the notion of grounding, due to Jessica Wilson (2014), is to argue that although there are many distinct non-causal dependence relations, there isn’t any single coherent notion of grounding that groups them together. Those who are attracted to Wilson’s approach should simply include all of the relevant non-causal dependence relations as possible ways of answering reason-why questions. Those who instead think that there just is no such thing as non-causal dependence at all, should feel free to ignore future references to grounding as a kind of dependence relation (though note that this will make it harder to make sense of several of the candidate views regarding quantum ontology).

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In response to this kind of concern, I think it is highly instructive to consider some historical cases in which scientists have found themselves in a similar situation.7 Consider, for instance, Pauli’s introduction of a massless, chargeless, previously undetected particle—the neutrino—in the 1930s to explain the apparent loss of energy and momentum in beta decay. The neutrino was a strange and novel kind of particle. No one wanted to admit the existence of such an entity, and Pauli himself called the neutrino a “desperate remedy”.8 But as strange as it was, it had to be admitted. For there had to be some reason why energy and momentum appeared to be lost during beta decay. Or consider Faraday’s introduction of the electromagnetic field in the 1850s and the further development of that idea by Maxwell and Thomson. None of these physicists was quite sure what the electromagnetic field was, but they were quite certain that it existed.9 Why? Because there had to be some reason why various electromagnetic phenomena happened in the way that they did. Or, finally, to take a more contemporary example, consider the introduction of dark energy in cosmology following the observation of the accelerating rate of expansion of the universe in the 1990s. Even today, although there is widespread consensus that dark energy exists, there is little consensus as to what it is. Dark energy is, first and foremost, whatever explains the accelerating rate of expansion of the universe.10 All of these cases are nuanced, and deserve a more detailed discussion than I have time for here. But on a relatively straightforward understanding, they all have a common structure. In each of these cases, physicists observed an unexpected pattern in the data. And in each case they were, however reluctantly, willing to introduce a type of entity that was highly strange or novel (or both) in order to explain that phenomena. Indeed in each case, the kind of entity that was introduced was the kind of thing that provided a good answer to not just the question of why we should expect the pattern in question to occur; it also answered reason-why questions about that pattern. The fact that beta decay results in the production of a neutrino is the reason why there appears to be energy and momentum lost during beta decay. The fact that the magnetic field has a certain form is the reason why the iron filings arrange themselves in a certain pattern. And the fact that there is a certain amount

7

I go through these cases in more detail in Emery Forthcoming. A detailed discussion of this case can be found in Brown 1978. 9 Suggestions ranged from the field being instantiated by an ether of contiguous, unobservable particles, which transmitted the electromagnetic forces, to it being a collection of lines of force that existed independently, to it being a fluid filled with vortex tubes. See Faraday 1852, Maxwell 1861, Hesse 1962 (especially chapter 8) and Harman 1982. 10 See Carroll 2007, lectures 14–17. Note that I am using the term ‘dark energy’ in an expansive sense that encompasses the notion of vacuum energy. This usage is in keeping both with early discussions of dark energy (e.g. Turner 2001) and recent summary discussions (e.g. Carroll 2007), but is not universal. If we reserve the term ‘dark energy’ for those hypotheses that would provide a dynamical explanation, it is no longer true that there is a consensus regarding the existence of dark energy. 8

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of dark energy in the universe is the reason why the rate of expansion of the universe is accelerating. What these kinds of cases suggest, then, is that when we have a robust pattern in the data, we need to identify some reason why that pattern occurs, even if doing so comes with costs in terms of the kind of entities that we need to introduce into our metaphysics. What these kinds of cases suggest, in other words, is that even if it commits us to an account of quantum ontology that involves highly strange or novel entities, we ought to find some way to accept both WR and WE . Indeed if we take the dark energy example as a guide, then even if all other analyses fail, we should accept the governing conception of the wavefunction.11 Obviously that is not an ideal situation—ideally we would be able to say something more about what the wavefunction is or subsume it under a category of entity with which we already somewhat familiar. But what the examples above show is that we do not need any guarantee of the ideal situation being actual before accepting that the wavefunction answers the relevant reason why questions. The way in which a theory is explanatorily impoverished if it fails to answer reason-why questions is the kind of consideration that trumps virtually any metaphysical scruples we might antecedently have.

19.1.4 The Governing Conception of the Wavefunction Here’s where we are so far. I have distinguished between why-we-should-expectthat questions and reason-why questions and argued that the wavefunction must represent the kind of thing that is the reason why quantum systems behave the way they do. I will call this view the governing conception of the wavefunction. The Governing Conception of the Wavefunction. The wave function represents something that is the reason why quantum systems behave the way they do.12 Given what I said at the end of Sect. 19.1.2 about what it takes to provide good answers to reason-why questions, the governing conception of the wavefunction can be further spelled out as follows: the wavefunction either represents something that causes quantum systems to behave the way they do; or the wavefunction represents something that grounds the behavior of quantum systems; or the wavefunction

11 Perhaps, for instance, all we can say is that the wavefunction is a sui generis entity that answers the relevant reason-why questions. See Maudlin 2013. 12 In general—and certainly in the examples in section 1.1—I take it that the wave function will also answer why-should-we-expect questions about the behavior of quantum systems. Note that we may want to leave open the possibility that the wavefunction plays a role in answering reason-why questions about the behavior of quantum systems even when that behavior has a low probability of occurring. If that is correct then the wavefunction may not always answer why-should-we-expect question about the behavior of quantum systems.

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represents something that is in some other way the reason why quantum systems behave the way they do. Notice that the governing conception isn’t itself an account of quantum ontology. It does not tell us what the wavefunction represents. Instead it is a constraint on such accounts. In part 2 of the paper I will say more about how this constraint impacts a couple of candidate theories of quantum ontology. Before going on to discuss what the wavefunction could represent, however, it is worth saying a bit more about two ways in which one might resist the argument just given for the governing conception of the wavefunction. The first way of resisting the argument is to insist that the only genuine explanatory demands are demands for answers to why-we-should-expect questions. On this view, reason-why questions are either unimportant or non-sensical. The first thing to say in response to this option is just that it is surprising. It seems as though we can sensibly distinguish between questions about why we should expect some phenomena and questions about the reason why that phenomena occurred and that the latter are important. But perhaps more concretely, anyone who takes this route owes us some kind of story about what was going on in the historical cases described in Sect. 19.1.3. Why do we ever feel pressured to introduce surprising new entities to answer reason-why questions about patterns in the data, if reason-why questions aren’t important? The second way to resist the argument in Sect. 19.1.3 is to try to identify some middle ground between reason-why questions and why-should-we-expect questions, and then to argue that all we need from an account of quantum ontology is something that plays a role in explanation in this “middle ground” sense. On this view, it isn’t enough just to identify some epistemic rule that will allow you to predict the phenomena in question. You need to do something more; but that something more falls short of identifying the reason why the phenomena occurred. Of course, before we can really evaluate this way of resisting the argument we need to know more about this “middle ground” sense of explanation. But it is worth emphasizing that any account along these lines will also need to reckon with the historical cases described above.13 In what sense did the sorts of entities introduced in those cases satisfy the need for the relevant kind of explanation? Until we have an answer to this question we should focus on accounts of quantum ontology on which the wavefunction answers reason-why questions about the behavior of quantum systems in addition to why-should-we-expect questions.

13 A common thought along these lines involves some sort of appeal to unification. It isn’t enough to show that the explanandum is to be expected—you have to show that it is to be expected in a way that unifies it with other phenomena. But what notion of unification is relevant here? Think again about the historical cases. In what way did introducing the neutrino or the electromagnetic field or dark energy unify the phenomena to be explained with other phenomena? I’m not claiming that this question can’t be answered. But given how novel the electromagnetic field and dark energy were (in the latter case, still are) a satisfying answer will require quite a bit of philosophical work.

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19.2 What the Wavefunction Could Represent Let’s turn now to the question of what the wavefunction could represent. The central idea is that the argument above—which is an argument about what the wavefunction must do—constrains the possible answers to the question of what the wavefunction could represent in interesting ways. Due to space, however, I will have to focus on two specific points. The first, I think, is relatively obvious, but deserves to be stated more clearly in the literature. The second, I think is more surprising.

19.2.1 The Governing Conception and Epistemic Accounts of the Wavefunction Here is the first point. Given the governing conception of the wavefunction, the wavefunction cannot merely represent the degrees of belief that a particular observer (or group of observers) should have in various possible outcomes of an experiment. Why? Because the degrees of belief that an agent should have just are not the kinds of things that can answer reasons-why questions about the behavior of quantum system. Consider, for instance, the following accounts of the wavefunction: QBism. The wavefunction represents the degrees of belief that an observer should have in the outcomes of various measurements given that the observer started with coherent initial degrees of belief and updated consistently using Bayes Theorem.14 Pragmatism. The wavefunction represents the degrees of belief that observers should have in the outcomes of various measurements given the kinds of creatures that we are and the way in which we are epistemically situated in the world.15

Neither of these accounts would allow the wavefunction to provide answers to reason-why questions about quantum phenomena unless one thinks that the reason why quantum phenomena occur is, in part, some fact about us, the agents investigating those phenomena. Let us call facts about an individual’s initial credence distribution, the way in which they update those credences, the kinds of creatures we are and the way we are epistemically situated in the world epistemic facts. If we accept both the governing conception of the wavefunction and either QBism or pragmatism, we will be committed to epistemic facts being part of the reason why quantum systems behave the way they do. This is not an inconsistent position, but it is a position that the vast majority of us would, I assume, like to avoid. The reason why quantum systems behave the way they do has nothing whatsoever to do with the details of our epistemic situation as individuals, or as human agents, investigating the world.

14 See 15 See

Caves et al. 2002 and Fuchs et al. 2014. Healey 2012 and 2017.

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Of course, if one is willing to give up the governing conception of the wavefunction—if one thinks that the wavefunction merely plays a role in answering why-should-we-expect questions about the wavefunction—either a QBist account or a pragmatic account would appear to be perfectly well equipped to meet the explanatory demands that arise from observing the behavior of quantum systems. The details of our epistemic situation as individuals and as human agents investigating the world are of course quite relevant to why we should expect quantum systems to behave in various ways. This is worth emphasizing. If you don’t care about reason-why questions, there is no explanatory pressure to go beyond the kind of epistemic view captured by QBism or pragmatic accounts. So although the point that if one accepts the governing conception of the wavefunction then one should not endorse QBism or pragmatic accounts is straightforward, this is only true because we have been clear as to what kind of explanatory demand is involved in this way of understanding the wavefunction. Insofar as one just says, for instance, we should not adopt QBism because QBism doesn’t respect the explanatory role of the wavefunction, it is quite easy to end up in a rather confusing dialectic. In my experience, this happens often in conversation, and it has also played out explicitly in the literature. Consider, for instance, the worry voiced in Timpson 2008 and the reply found in Fuchs and Schack 2015. Timpson complains that QBism has “troubles with explanation” because “we are not interested in agents’ expectation that [a certain quantum system will behave a certain way]; we are interested in why it in fact does so” (2008, p. 600). Timpson, in other words, wants any interpretation of the quantum formalism to be able to answer reason-why questions about the behavior of quantum systems. Fuchs and Schack respond that the “explanation offered by quantum theory have a similar character to explanations offered by probability theory” and that “probability theory explains the agent’s expectations” (2015, 7–8). In other words, their response is that a QBist interpretation does a perfectly adequate job in answering why-should-weexpect questions about the behavior of quantum systems. The disagreement here is not really over what QBism can do, it is over what kind of explanation is required from a scientific theory of the sort that QBism purports to be.

19.2.2 The Governing Conception of the Wavefunction and Configuration Space Realism Let’s turn now to a second way in which the governing conception of the wavefunction constrains possible accounts of what the wavefunction represents. One currently popular account of the wavefunction is what I will call configuration space realism.16 According to this view, the wavefunction represents a field in an

16 This

view also goes by the name wavefunction realism.

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extremely high-dimensional space.17 I will argue that the governing conception of the wavefunction makes it quite difficult to be a configuration space realist. One of the motivations for configuration space realism is that it is supposed to be the most straightforward way of interpreting the quantum formalism. This motivation has two distinct components. The first is the idea that the wavefunction must represent a physical object. Philosophers of physics have pointed, for instance, to an analogy with classical mechanics similar to the one made at the beginning of this paper. Just as Newton’s second law describes how the properties of certain physical objects—particles—change over time, we should think of Schrodinger’s equation as describing the properties of a physical object—whatever the wavefunction represents—changing over time.18 The second component of the motivation is the idea that if the wavefunction represents a physical object, then it must represent a field. This thought turns on the fact that the wavefunction is a function—it takes inputs from a specified domain and outputs a value. In other mathematical formalisms—for instance in the formalism for classical electromagnetism—functions are associated with fields.19 At this point, however, the configuration space realist faces a complication. Think back to the example involving quantum entanglement at the end of Sect. 19.1.1. What that example showed is that the wavefunction is not defined over a space where each point corresponds to the possible properties of individual particles. Instead it is defined over a space where each point in the space corresponds to a complete specification of all of the degrees of freedom of the system as a whole. This means that if the wavefunction represents a field, it does not represent as field in ordinary 3-D space. Instead it represents a field in an extremely high dimensional space where each dimension corresponds to one degree of freedom for the system.20 The wavefunction of the universe as a whole, therefore, will represent a field in a space that has something like 3 × 1080 dimensions.21 This space is often called configuration space. So configuration space realism is the view that the wavefunction represents a field in configuration space. But of course configuration space is not the space of our ordinary experience. Nor is it the space in which we do physics. The kinds

17 Advocates

of configuration space realism include Albert 1996, Loewer 1996, Ney 2012 and 2013 and North 2013. I have argued against this view on quite different grounds in Emery 2017. The papers in Albert and Ney 2013 provide a helpful introduction to the topic. 18 See Albert 1996, 277; Ney 2012, 532, Lewis 2004, 714. 19 See Albert 1996, 278. 20 See the appendix of Ney 2012 for a detailed discussion of this point. 21 Philosophers of physics get this number by assuming that all degrees of freedom can be captured by thinking about the location of a particle in 3D space. (For instance, the way that we measure spin is by sending the particles through a magnetic field that separates the particles into two groups. We then interpret one group as the particles that have spin up in the relevant direction and one group as the particles that have spin down in that direction.) A rough estimate is that there are 1080 particles in the universe. So if the above assumption is correct then we can capture all of the degrees of freedom by using a space of 3 × 1080 dimensions.

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of experiments described in Sect. 19.1.1—the kinds of experiments that led to the development of the quantum formalism—are experiments in 3D space involving 3D entities. The configuration space realist therefore faces a challenge: they need to explain how the wavefunction-field in configuration space is related to the 3D entities like electrons and silver atoms and magnets and detection screens in 3D space. Following Callender (2015), let’s call this the lost in space problem.22 Configuration space realists are well aware of the lost in space problem. In response they strive to come up with an account of how the wavefunction might “enact” 3D entities or how to “find” the 3D world in the wavefunction.23 But once one accepts the governing conception of the wavefunction, it becomes more clear what a satisfactory response to this problem would need to involve. If we adopt the governing conception of the wavefunction and configuration space realism, then we must think that the wavefunction-field in configuration space is the reason why 3D entities in 3D space behave the way they do. In other words, an advocate of the governing conception of laws who wants to be a configuration space realist will be explicitly committed to there being dependence relations between the highdimensional space in which the wavefunction field exists, and the 3D space in which our physics labs and experiments are located. In order to resolve the lost in space problem, the configuration space realist will therefore need to make sense of these inter-spatial dependence relations.24 That is to say, the configuration space realist will need to make sense of a dependence relation in which the relata exist in distinct physical spaces. Now, inter-spatial dependence relations aren’t always strange of novel. Consider a 3D cube and the 2D square that makes up one side of that cube. There are straightforward inter-spatial dependence relations between these two entities: the 2D square is a part of the 3D cube. But in the case of configuration space realism the issue is more complicated. As everyone in the debate agrees, no three of the dimensions within the high-dimensional space correspond to our ordinary 3D space. There is no sense in which 3D space is a part of the high-dimensional space. Instead

22 As

I read it, the lost in space problem is the same problem that Ney (2017 and 2021) calls ‘the macro-object problem’. 23 The “enacting” terminology is found in Albert 2015. Ney 2017 uses the terminology of “finding” the world in the wavefunction. 24 Once the lost in space problem for configuration space realism is laid out so explicitly, configuration space realists may want to retreat to a somewhat different version of their view. According to this alternative version, which I call configuration space monism, the wavefunction field in the high-dimensional space is all that there is. The 3D entities of our everyday experience (and our physics labs) are just an illusion. Configuration space monism neatly eliminates the lost in space problem as stated above, since there no longer is a 3D space or 3D entities. But it does so by giving up on the governing conception of the wavefunction. The wavefunction no longer answers reason-why questions about the behavior of quantum systems. (It might, of course, still answer reason-why questions about aspects of our experience that seem like they are quantum systems.)

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each dimension of the high-dimensional space corresponds to one degree of freedom in the system that exists in the 3D space.25 The configuration space realist who wants to make sense of the governing conception of the wavefunction, therefore, must either make sense of inter-spatial causation, inter-spatial grounding, or some kind of novel inter-spatial dependence relation.26 I suspect that most philosophers will think that positing these kinds of interspatial dependence relations is costly—all else being equal it would be better to have a single space in which things depend on one another than to have two genuine physical spaces such that what happens in one of those spaces is the reason why the entities in the other space behave the way they do. So the question becomes whether the costs associated with these kinds of inter-spatial dependence relations are worth paying. A full answer to this question would require an in depth examination of the alternatives to configuration space realism, and I don’t have space to go in to that sort of examination here. But let me say something briefly about why I think configuration space realism faces a challenge. As I noted above, one motivation for configuration space realism is that it is a straightforward interpretation of the quantum formalism. But now that the lost in space problem is clearly on the table, we should ask straightforward in what sense? Yes, the configuration space realist has faithfully followed the standard ways of interpreting the formalism of classical theories, but by doing so they have introduced a kind of dependence relation that is wholly foreign to classical physics. Why think that it is really so important to be straightforward in the very specific way that the configuration space realist is straightforward, when their theory is not at all straightforward in other ways? A second, often mentioned motivation for configuration space realism is that it preserves locality at the fundamental level.27 I don’t have space to go into a full explanation of locality (and the related concept of separability) here. Suffice it to say that any interpretation of the quantum formalism that involves only 3D space and 3D entities will involve a non-local dynamics in the following sense: it will admit that what happens at one point in 3D space depends on what happens at other points in 3D space without there being any kind of signal or causal influence that travels through space to connect those two points. Configuration space realism avoids this kind of non-locality in the space in which the wavefunction is defined. Consider

25 Ney

2012 section 3 includes a helpful discussion of this point and why this also makes the high-dimensionality of configuration space different than the high-dimensionality of other physical theories, like string theory. 26 As I understand it, the account found in Ney 2021 (section 7.4), according to which threedimensional entities are part of the wavefunction involves a inter-spatial dependence relation— specifically inter-spatial constitution. According to Ney, three-dimensional objects are part of the high-dimensional wavefunction. Ney acknowledges that this stretches the ordinary notion of ‘part’ according to which a part and a whole occupy the same physical space, but points out that we also think that abstract objects have parts, even though they don’t exist in physical space at all. 27 See the discussion in Ney 2021, section 3.

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a case in which there are two particles that are separated in 3D space but that are nonetheless entangled with respect to their spin states. In configuration space, this system doesn’t involve multiple particles at all. There is only the wavefunctionfield, which has different amplitudes at different points, each of which represent a complete specification of the properties of the system as a whole in 3D space.28 Philosophers of physics have rarely been explicit about why exactly they think that preserving locality at the fundamental level is important.29 Of course, having a non-local dynamics makes doing science significantly harder—we can’t rule out possible sources of influence on some phenomenon just by examining what is nearby and testing for causal signals entering the relevant region.30 But surely the configuration space realist can’t care about this reason for prioritizing locality. After all, the space in which we do science is 3D space, and the configuration space realist is still committed to the dynamics being non-local in that space. They have only eliminated non-locality in the high-dimensional, fundamental space, and we don’t do science in the high-dimensional, fundamental space. Another reason why one might want to avoid non-locality is that a non-local dynamics is novel or strange. In short, if one is at all conservative in one’s metaphysics then one should want to avoid non-locality.31 But notice that once we have accepted the governing conception of laws, the configuration space realist is also committed to something quite novel and strange. The configuration space realist is positing multiple physical spaces, connected by genuine dependence relations– that is far from a conservative account of what the world is like! All in all, the configuration space realist who also accepts the governing conception of the wavefunction seems to be on fairly shaky footing. On the one hand, the main motivations for their theory seems to be premised on the idea that it is relatively straightforward and conservative. On the other hand, in order to make sense of the fact that the wavefunction-field answers reason-why questions about the behavior of quantum systems, they need to explicitly accept a metaphysics that is deeply surprising. This doesn’t mean configuration space realism is a non-starter. But in order to defend their view against rival accounts of quantum ontology, the configuration space realist (insofar as they endorse the governing conception of the wavefunction, at least) is going to need to get deep in the weeds with respect to precisely what kind of straightforwardness and precisely what kind of conservatism is important when interpreting physical theories, and why. These kinds of arguments are rarely decisive.

28 In the Bohmian version of configuration space realism, the high-dimensional space will include the wavefunction field and a single “uber particle”. I will set this complication aside. 29 Ney 2021, section 3, contains the first real in-depth treatment of this question. 30 This idea can be traced back to Einstein 1948. 31 As I read her, this is the position that Ney 2021 ultimately ends up endorsing.

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19.3 Conclusion When making claims about the explanatory role of the wavefunction it is important to distinguish between the claim that the wavefunction plays a role in answering why-should-we-expect questions about the behavior of quantum systems, and the claim that the wavefunction plays a role in answering reason-why questions about that behavior as well. If we only accept the former, then the explanatory role of the wavefunction places few, if any significant constraints on what the wavefunction can be. One of the key claims of this paper, however, has been to argue that we should not only accept the former claim–we should accept the latter claim as well. The view that the wavefunction also plays a role in answering reason-why questions about the behavior of quantum systems is what I call ‘the governing conception of the wavefunction’. Insofar as we accept the governing conception of laws, it will constrain the possible accounts of quantum ontology that we might give. First, and most obviously, we should reject QBism and pragmatic accounts of the wavefunction. Second, it will be quite challenging to be a configuration space realist. And of course these are only the first steps in a much more detailed analysis that would involve the discussion of alternative accounts of quantum ontology besides the ones mentioned here and the ways in which the governing conception of the wavefunction impacts those accounts.32 That is work yet to be done. But in closing let me also note that insofar as we accept the governing account of the wavefunction, our understanding of what the wavefunction could be will only be as good as our understanding of various possible dependence relations. It may be, then, that those who are attracted to this view should not focus exclusively on the existing literature on quantum ontology, but also make sure they are immersed in discussions of causation, grounding, governance, and other kinds of dependence relations that tend to take place in metaphysics texts. A better understanding of these relations has the potential to reveal both novel complications for existing accounts of quantum ontology, and to inspire alternative accounts that have been overlooked or misunderstood in the current literature.33

32 One might wonder whether the governing conception of the wavefunction is compatible with the view that the wavefunction represents a law or the view that the wavefunction represents the dispositional properties of the particles in the quantum system. The answer is that it will depend on whether one can make sense of the idea that laws or dispositional properties either cause the behavior of quantum systems, ground the behavior of quantum systems, or stand in some other dependence relation to that behavior. One way to put this is to say that combining the governing conception of the wavefunction with a nomological or dispositional account of the wavefunction will require making sense of the idea that laws or dispositional properties are able to govern. 33 Many thanks to the audience at the Foundations of Physics Workshop at Harvard University, especially Jacob Barandes and Barry Loewer, and to an audience at the University of Toronto. Thanks also to two anonymous referees for this volume.

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Ney, A., & Albert, D. Z. (Eds.). (2013). The wave function: Essays in the metaphysics of quantum mechanics. Oxford University Press. Scriven, M. (1959). Explanation and prediction in evolutionary theory. Science, 30, 477–482. Skow, B. (2016). Reasons why. Oxford University Press. Timpson, C. (2008). Quantum Bayesianism: A study. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 39(3), 579–609. Turner, M. (2001). Dark matter and dark energy in the universe. Particle Physics and the Universe, 2001, 210–220. Wilson, J. (2014). No work for a theory of grounding. Inquiry, 57, 1–45.

Chapter 20

Representation and the Quantum State Richard Healey

Abstract Alternative views of quantum states are often expressed using the language of representation. It is important to distinguish three questions here: What is a quantum state? How may a quantum state be represented? What, if anything, does a quantum state represent? I defend answers to these questions against alternatives. In brief, a quantum state is an objective relational property of a physical system that describes neither its intrinsic physical properties nor anyone’s epistemic state. A quantum state is representational (in my preferred sense of that term) and many quantum states are real. Since its primary role is to assign Born probabilities to certain physical events involving the system, a quantum state may be represented in quantum theory by any mathematical object that facilitates this role. If it represents anything, a quantum state represents the objective probabilities it yields in this way.

20.1 Introduction Alternative views of quantum states are often expressed using the language of representation. For example, after titling his 2014 review article “Is the quantum state real?”, Leifer went on to say “The question of just what type of thing the quantum state, or wavefunction, represents, has been with us since the beginnings of quantum theory.” It is important to distinguish three questions here: What is a quantum state? How may a quantum state be represented? What, if anything, does a quantum state represent? I shall defend answers to these questions against alternatives. In brief, a quantum state is an objective relational property of a physical system that describes neither its intrinsic physical properties nor anyone’s epistemic state. Since its primary role is to assign Born probabilities to certain physical events involving the system, a quantum state may be represented in quantum theory by any mathematical object that can play this role. If it represents anything, a quantum state represents the objective probabilities it yields in this way.

R. Healey () University of Arizona, Tucson, AZ, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality , Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_20

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The paper continues like this. Section 20.2 surveys a variety of equivalent ways a quantum state may be represented in order to serve its function in quantum theory. Section 20.3 argues that a quantum state is not a physical entity, while Sect. 20.4 argues that it is not a physical magnitude. In Sect. 20.5 I argue that a system’s quantum state does not represent its intrinsic physical properties or those of threedimensional space. While few, if any, of these arguments are original or decisive, I take them to motivate consideration of an alternative that can help improve our understanding of quantum theory. So in Sect. 20.6 I develop a relational account of a quantum state as what I call an extrinsically physical property of a system, contrasting this with Rovelli’s relational quantum mechanics and QBism. In Sect. 20.7 I ponder whether this makes a quantum state representational: in my sense it is representational and many quantum states are real. Section 20.8 says what quantum states may represent and why this makes them modal properties. In Sect. 20.9 I show why quantum states give a naturalist a reason to reject Representationalism.

20.2 How Quantum States May Be Represented Schrödinger (1936) used the word ‘representative’ to refer to a wave function, but a quantum state may be represented by many different mathematical objects. A quantum state may be represented by a wave function in the Heisenberg, Schrödinger or interaction picture. This wave function may be in position, momentum or energy representation. In a Hilbert space of square-integrable functions it may be represented by an equivalence class of functions that differ only on a set of measure zero. It may be represented by a vector or a ray of vectors in an abstract Hilbert space: a more general class of quantum states then includes mixed states representable not by vectors but by density operators on the space. In algebraic quantum theory a quantum state is represented by a normed, positive, linear functional from a C* algebra into the complex numbers. There are states on sub-algebras of the von Neumann algebra of bounded linear operators on a Hilbert space that cannot even be represented by density operators. What all these different representations of a quantum state have in common is their role in yielding probabilities when (the appropriate form of) the Born rule is applied to them. The negative conclusion of this section is that it is both a category mistake and an unwarranted restriction on the scope of inquiry to identify the question “What does the quantum state represent?” with the question “What does the wave function represent?” But implicit in the very diversity of mathematical representations of quantum states is a positive suggestion about what a quantum state is and what, if anything, it represents. Generating probabilities is a key role of a quantum state, no matter how that state is represented. This suggests a preliminary answer to the nature of quantum states: a quantum state is whatever it takes to generate the Born probabilities it represents. I think this is basically the right answer, though more needs to be said to clarify and justify it.

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20.3 Quantum States Are Not Physical Entities One objection arises immediately. If all a quantum state does is to yield a bunch of probabilities then that state is not real: but we should accept the reality of a central posit of a spectacularly successful theory like quantum theory. What lies behind this objection is a conception of what it is for a central posit of a physical theory to be real. Oxygen is posited as a physical entity by a successful theory of combustion, and to say that it is real is to say that the physical world contains such an entity. On this conception, quantum states would be real if and only if the physical world contains such entities as quantum states. But (the objector continues) probabilities are not physical entities,1 and if all a quantum state does is to generate probabilities then it is not real either. I’ll defend the reality of quantum states in Sect. 20.7 and the objective probabilities they yield in Sect. 20.8. But first, in deciding whether they are real, quantum states should not be thought of as (potential) physical entities. A typical physical entity such as a speck of dust or a sample of oxygen bears physical properties and relations and has physical parts, all of which may change with time. It interacts causally with other physical entities. Typical physical events are also entities with physical properties, relations and parts: They have spatiotemporal locations and they interact causally with other physical entities. Atypical physical entities may lack some of these characteristics. Democritus’s atoms had no parts and their intrinsic properties did not change, though their changing spatial relations were supposed to underlie all other physical change; classical physics contemplated point events with no spatiotemporal extent and fields located everywhere in spacetime. By contrast, a quantum state has no physical properties, relations or parts: it has no spatiotemporal location and does not interact causally. These claims may be disputed. The ground state of a hydrogen atom is often depicted as an electron cloud of varying density surrounding a central proton. But this does not locate the state itself, but merely represents the position-probability distribution (or probability amplitude—a real number in this case) for the electron in accordance with the Born rule. Some views privilege a particular representation of a quantum state. Ney (2020) advocates wave function realism, according to which the wave function of the universe represents a field in a high-dimensional configuration space. For a wave function realist, this field is a physical entity occupying the whole of this highdimensional analogue of the ordinary three-dimensional space alleged to emerge from it dynamically. The universal wave function is supposed to act on matter (represented by a point in configuration space), confirming its physicality. But wave function realism does not square well with how physicists typically use quantum

1 When the subjectivist statistician Bruno De Finetti (1968) famously wrote PROBABILITY DOES NOT EXIST he meant to deny that the physical world contains any such thing, just as it contains no phlogiston, fairies or witches.

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states in all forms of quantum theory, not to represent the entire universe all at once but to predict and explain the behavior of selected physical subsystems (such as the proton, helium atom, or ammonia molecule; superfluid helium-3, a Heisenberg ferromagnet, or an entangled photon pair) by assigning them quantum states and using these to generate probabilities concerning them and their fourdimensional spacetime environment. There is nothing in these applications to warrant privileging a configuration-space wave function as representative of the quantum states assigned, or to show these states are physical entities.

20.4 Quantum States Are Not Physical Magnitudes Bell (2004) introduced the term ‘beable’ to refer to whatever a theory takes to be physically real—as what may be “described in ‘classical terms’, because they are there.” As examples he mentioned the settings of switches and knobs and currents needed to prepare an unstable nucleus. These are not physical entities; each is a physical magnitude. The famous EPR paper (1935) used the phrase ‘element of physical reality’ to refer to such quantities in arguing that there are some that quantum mechanics fails to describe. Perhaps quantum states are themselves such elements of reality—real physical magnitudes, not real physical entities? Wallace and Timpson’s (2010) spacetime state realism suggests that a density operator is a magnitude representing a spacetime region’s physical properties the way a real number T represents a room’s temperature properties. Anticipating an incredulous stare, Wallace and Timpson say (op. cit., p. 710): There need be no reason to blanche at an ontology merely because the basic properties are represented by such objects: we know of no rule of segregation which states that only those mathematical items to which one is introduced sufficiently early on in the schoolroom get to count as possible representatives of physical quantities, for example!

If general relativity represents a quantity in a region of spacetime by an order-two tensor field, why should quantum theory not represent the physical properties of a region of spacetime by a density operator on a Hilbert space? There is a good answer. Each component of the stress-energy tensor field at a spacetime point gives the flux at that point of a 4-momentum component across a 3-hypersurface. Each of the 16 components of this tensor in a coordinate system has a real-numbered value on a common cardinal scale to satisfy the tensor’s transformation properties. The components of any tensorial magnitude (scalar, vector, or higher order) have values on some cardinal scale. But a density operator on a Hilbert space is none of these, and its matrix elements in a basis for that space are not the values of any magnitude on any cardinal scale. A quantum state is not a physically real magnitude.

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20.5 A Quantum State Does Not Represent Any (Intrinsic) Physical Properties The instantaneous physical state of a system of n classical particles may be represented by a point in a 6n-dimensional phase space that determines the value of every other dynamical variable of the system. Associated with each measurable region of phase space is a physical property that the system possesses just in case the phase space point representing its state lies in that region. Some have thought a system’s quantum state plays an analogous role in quantum theory with phase space replaced by Hilbert space, each subspace N of which is associated with a physical property the system definitely has if the density operator W representing its state projects onto N but otherwise has only with probability given by TrWPN , where PN is the projection operator uniquely corresponding to N. But the analogy between phase space and Hilbert space as bearers of a probability measure breaks down as a consequence of results due to Gleason (1957) and others. Gleason proved that the only measures on (closed) subspaces of a Hilbert space of dimension greater than 2 are those generated by some density operator. No such measure is dispersion-free, in that it assigns either 0 or 1 to each subspace in a way that can be interpreted as saying which properties corresponding to subspaces are possessed and which are not. These negative results show that a quantum state does not represent a system’s physical properties the way a phase space point represents the dynamical properties of a classical system. Attempts to evade this conclusion by adopting a kind of quantum logic governing reasoning about statements assigning physical properties to a quantum system have the character of a degenerating interpretative research program. But the framework of ontological models (see Harrigan & Spekkens, 2010; Leifer, 2014) explores the possibility that a quantum state yields a probability distribution over some space of random variables, perhaps of a wholly novel kind, the points of which may be parametrized by a variable λ whose value more completely describes the system assigned that quantum state. In that framework, Pusey, Barrett and Rudolph (PBR) (2012) proved a result they took to show both that suitably prepared quantum states are physically real and that they are properties of the systems so prepared. The basic assumption of the ontological models framework is that after the quantum state of a system has been prepared, the value of the variable λε  provides a complete specification of the physical properties of that system. But a preparation procedure for a state represented by vector ψ may yield one of many distinct real states λ, with a probability distribution μψ (λ) on . So distinct quantum states ψ 1 , ψ 2 result in different distributions μ1 (λ), μ2 (λ). These might or might not overlap: if they did, then distinct quantum states would be compatible with the same underlying real state λ. The parameter λ is also supposed to specify a fine-grained probability measure over all possible outcomes of measurements of magnitudes (“observables”) to which a quantum state assigns a probability via the Born rule. Indeed, that Born

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probability is assumed to arise by integrating this fine-grained measure weighted by the distribution μψ (λ) over the space . For PBR, the variable λ represents the real state of a system assigned a quantum state represented by ψ following some preparation procedure. They consider a state represented by λ to be physically real because: 1. it determines all actual physical properties of the system, and 2. for each observable, it determines a probability distribution over the possible outcomes of a measurement of that observable. Assuming that systems that are prepared independently have independent physical states, PBR proved that the quantum state assigned to each system is uniquely determined by the value of λ that provides a complete specification of the actual physical properties of that system. Because it is determined by the physically real state λ, they conclude that this quantum state is also a physically real property of the system. But the proof establishes its conclusion only if a system has a state meeting both conditions (1) and (2). The ontological models framework simply assumes that it does. But a quantum state meeting condition (2) may be real even though it does not meet condition (1): and (pace PBS) a system may have a real physical state meeting condition (1) but not condition (2). The PBR theorem does not show that a quantum state is a physical property of the system to which it is assigned. Some views of quantum theory assign a descriptive function to a privileged universal quantum state. For non-relativistic quantum mechanics this would be the state of all n particles in the universe (for some finite n). Monton (2006) takes this universal quantum state to be pure, and to determine a holistic physical property of the universe through the eigenstate → eigenvalue link: the universe’s particles have property P at time t if its state vector |ψ(t) > is an eigenstate of a projection operator uniquely corresponding to P with eigenvalue 1. If this description were complete, then no particle would ever have a precise position, since a well-defined configuration-space wave-function has no corresponding projection operator. But some take Bohmian mechanics to offer a clear version of non-relativistic quantum mechanics that also includes a precise trajectory for every particle in the universe determined by a universal wave-function ψ(x1 , x2 , ..., xn , t) in accordance with a law sometimes called the Bohmian guidance equation. This universal wave-function can then be understood either to represent a holistic physical property of the n particles at t, or to represent an n-place physical relation among the points of space they then occupy. The latter option has been called the multi-field conception of the wave function (by Belot, 2012), as opposed to viewing it as a field on configuration space. The former option has been proposed as a way of fleshing out a so-called nomological conception of the wave function—as something required to state the law determining particle trajectories. The idea here is to regard

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this holistic property of the particles as a disposition grounding that law (see Esfeld et al., 2014).2 The main problem I see with either of these options is that it applies only in the artificially restricted context of a particular take on non-relativistic quantum mechanics that assumes the existence of a universal wave function. But quantum theory is applied much more widely outside of that context, where the quantum state of a system is not represented by a wave function and it is not assumed that there is any universal wave function. Quantum states play their role in all applications without that assumption.

20.6 A Quantum State Is an Extrinsically Physical Property of a System A property may be either intrinsic or extrinsic to its bearer. Mass is an intrinsic property of an electron (at least according to classical physics)—a property it has in and of itself, without regard to the existence or properties of anything else: Being lighter than a proton is an extrinsic property because it involves the electron’s relation to the proton. Here this is a physical relation to something physical: I’ll call such an extrinsic property extrinsically physical. Being married is an extrinsic property but it is not extrinsically physical since it also involves a person’s social or legal relations to another. These change with the death of a distant spouse but this is no instantaneous action at a distance since it involves no immediate change in the person’s intrinsic properties. Having a particular position or velocity is not an intrinsic property of a classical particle insofar as it implicitly depends on its relation to a reference frame. To highlight the contrast with a system’s quantum state I shall ignore this complication and treat these and all other classical dynamical properties determined by a system’s phase space point as intrinsic properties. But a quantum state is an extrinsically physical, not an intrinsic, property of a physical system. QBists and others regard a quantum state as a state of knowledge or opinion of an agent and take this to imply that it is not a property of the system to which the agent assigns it. The system does have the extrinsic property of being assigned that state by that agent. But this is not an extrinsically physical property because it depends on an intentional relation to an agent, not a physical relation to a physical object or situation. A quantum state is ontic in Leifer’s (2014) sense, and not a state of knowledge or opinion of some agent (though an agent may come to know this state).

2 Suarez (2015) has proposed an “intermediate” view with a dispositional velocity field defined at each point of space that also faces the following problem.

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...an ontic state refers to something that objectively exists in the world, independently of any observer or agent. In other words, ontic states are the things that would still exist if all intelligent beings were suddenly wiped out from the universe. op. cit. p. 69

Because a quantum state is an extrinsically physical property, a system in given circumstances may have more than one quantum state, each relative to a different physical relatum. But a system has a quantum state only relative to something physical. Everything in the previous paragraph accords with Rovelli’s (1996) relational quantum mechanics. For Rovelli, the physical item relative to which a system has a quantum state is a distinct physical system. He maintains that just as “the observer” to which velocities must be relativized in Galilean relativity may be any physical object (such as a table lamp), so also “the observer” to which the state of a physical system must be relativized in quantum mechanics may be any physical system (such as an electron). In his view Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world. (1996, p. 1650)

But it is difficult if not impossible to reconcile this view with the objectivity of physical description in quantum theory. By examining how quantum states are used in applications of quantum theory one can arrive at a better relational view. Quantum theory is applied not in order to describe a physical system but in order to assign probabilities to a range of statements about it. Such a probability assignment is useful for an agent in a particular physical situation whose physical conditions make it impossible to determine which statement in that range is true. What makes a quantum state an extrinsically physical property of a system is not its relation to another physical system but to such a physical situation. A system has a quantum state relative to a physical situation because of features of the physical environment that constrain the information accessible to any agent that may (or may not) happen to be in that situation. That is why a system’s quantum state is ontic and not epistemic: to say the system has that state is not to say what any observer knows, believes or suspects of it. A localized agent has direct epistemic access only to the contents of their momentary past light-cone: but this may permit indirect epistemic access to events outside it. Quantum theory mediates this access because a quantum state relative to the agent’s momentary spacetime location yields probabilities concerning such possible events. Bohm’s version of the EPR Gedankenexperiment provides a memorable illustration of such relativity. A pair of spin ½ particles has a physically extrinsic property represented by the spin-singlet state vector. The z-spin component of particle R is measured with spinup outcome. Relative to points to the (absolute) future of that R-measurement event, the quantum spin state of the L particle is represented by the vector |↓ > z , a spindown z-spin eigenstate: but relative to other points, L’s spin state is represented by a reduced density operator—the two dimensional identity operator I.

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|↓ > z and I each represent the state of particle L (relative to different spacetime locations), and the change from I to |↓ > z is not a physical event, caused by the measurement on particle R. It is a change in relativization from one spacetime region to another to accommodate the change in physical situation of any agent Alice whose world-line entered the future of the z-spin measurement on particle R. As her physical situation changed, so would the quantum state that such a (merely hypothetical) Alice should assign to particle L to yield the relevant Born probabilities for her new situation. The spin-up outcome on R is the physical reason why any such Alice should update her quantum state for L after gaining access to new information about what happened to R. The spacetime location of an actual or merely hypothetical agent may present a less fundamental physical barrier to informational access. In the Gedankenexperiment of Wigner’s friend, Wigner has no informational access to the contents of his isolated friend’s laboratory and so cannot observe the outcome of her quantum measurement even when this does lie in his past light-cone. The Schrödinger picture state vector of his friend’s laboratory evolves unitarily relative to Wigner’s physical situation so after her measurement the system she measured has a mixed quantum state relative to this external physical situation. But this is consistent with that system’s having a pure quantum state relative to the physical situation of his friend after she has measured it.

20.7 Is a Quantum State Representational? The term ‘representational’ has been used in recent discussions to classify opposed views of the nature of quantum states.3 The pragmatist view of quantum states I have taken elsewhere (2012a, 2017a, b) and assumed here has sometimes been classified as non-representational. But usage has not been consistent. To clear the air I shall distinguish several possible senses of the term while explaining an important sense in which a quantum state is representational in this view. A quantum state could count as representational simply by being represented by a wave function or other mathematical object. Since representation is an intentional concept, a quantum state would then count as representational whether or not it exists. A quantum state is representational in this minimal sense whether or not it is real, and so are the philosophers’ stone, caloric and the properties of being dephlogisticated or at absolute rest. One could choose to say that a quantum state is non-representational if a quantum state is neither a physical entity nor magnitude and does not represent a system’s intrinsic physical properties. Since Sects. 20.3, 20.4 and 20.5 ruled out these views of a quantum state one would then call a quantum state non-representational.

3

See, for example, Krissmer (2018), Bub (2019), Wallace (2020) and several papers in French and Saatsi (2020), Glick et al. (2020), Hemmo and Shenker (2020).

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An alternative use of ‘representational’ ties it to truth-aptness. Emotivists claimed that an ethical statement such as “Stealing money is wrong” expresses an emotion rather than stating a fact and so cannot be evaluated for truth. They maintained that despite its subject-predicate form, this statement is not truth-apt and the term ‘wrong’ is non-representational. Similarly, someone could deny that a statement ascribing quantum state ψ to a system is truth-apt and so the predicate ‘being in quantum state ψ’ counts as non-representational, on the grounds that this statement does not state a fact but expresses an epistemic attitude. But while the primary function of a statement ascribing a quantum state to a system is not to describe or represent that system’s intrinsic physical properties, that statement does have a truth-value (relative to an appropriate physical situation), since a quantum state is an extrinsically physical property of a system. In this sense, a quantum state is representational. Indeed, a statement ascribing a quantum state to a system is useful only to the extent that statement is true (true enough, that is: see Elgin, 2017) and so the state is real. It is because there are many such statements ascribing a quantum state to a system that I consider those quantum states to be real. Wallace (French & Saatsi, p. 87) has offered yet another understanding of what it is for a quantum state to be representational. He considers the classic Schrödinger-cat state α |live cat> + β |dead cat> which unitary quantum theory can straightforwardly produce. If the quantum state can be understood representationally— that is, if distinct quantum states correspond to distinct objective ways a physical system can be—and if the theory is unsupplemented by hidden variables, then it looks as if such a state must somehow represent a cat that is simultaneously alive and dead.

Since a quantum state is an extrinsically physical property of a system, that system may have distinct quantum states, each relative to a different physical situation. There is nothing subjective about its having these ontic states, and in that sense each represents a distinctive objective way for that system to be. But a cat’s death certainly involves a change in its intrinsic physical properties. The passage suggests that two ways a physical system can be count as objectively distinct only if these involve incompatible intrinsic physical properties of that system. Entanglement-swapping features quantum states that count as non-representational on this narrower understanding of objective distinctness (see Healey, 2017b). In entanglement swapping a particular pure entangled state is assigned to a system after a measurement is performed on a second, distant system: which state is assigned depends on the outcome of that measurement. Assuming it is a local event, in the absence of retro- or spacelike causation the distant measurement does not alter any intrinsic property of the first system. So it has the same intrinsic physical properties no matter which of two or more distinct pure quantum states it is assigned. In that sense, these distinct quantum states do not correspond to distinct objective ways that physical system can be. That is why these states count as non-representational in this narrow sense of objective distinctness. But there are other cases in which a quantum state is assigned to a system following a procedure that is said to prepare or put the system into that state, suggesting that being in this quantum state involves having associated intrinsic

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physical properties. A Stern-Gerlach (SG) experiment is a paradigm of such a procedure. Spin ½ atoms in a beam may be detected by one of two detectors placed on the z-axis symmetrically above and below the incoming beam after passage through the magnet’s inhomogeneous magnetic field. If the upper detector is replaced by some experimental equipment, any atom that is subsequently detected in this experiment is said to have been prepared in a quantum spin eigenstate |↑ > z through its local interaction with the magnetic field. Naively, the incoming beam has been split by the magnet into an upper “wavepacket” of positive z-spin atoms and a lower “wave-packet” of negative z-spin atoms. But as Wigner (1963) pointed out, unitary evolution of the quantum state vector during passage through the magnet results in an entangled superposition of the translational and spin quantum states of the atoms. Wessels (1997) called passage through a z-oriented SG magnet a mere pseudo-preparation of z-spin eigenstates in upper and lower beams on the grounds that the (reduced) quantum spin state of emerging atoms was therefore a mixture rather than a pure state. She noted that something similar is true of most if not all actual laboratory preparation procedures. But these are real state preparations, involving a local interaction warranting assignment of a superposed state followed by a conceptual selection of one component with a view to a possible later local measurement-type interaction involving the target system. The selected state does not represent an intrinsic property of this system (like positive z-spin). A different selection would not have represented a different intrinsic property: it would merely have selected the state relative to a different possible subsequent local measurement-type interaction. State preparation does not prepare a system’s intrinsic physical properties.4

20.8 What Quantum States May Represent, and Why This Makes Them Modal A quantum state is an extrinsically physical property of a physical system. It is representational in the sense that a statement ascribing this property to a system is truth-apt and may be true; when it is true, that state is objectively real, like a speck of dust. But a speck of dust does not represent anything: does a quantum state? In my (2017b) I said that if a quantum state represents anything, it is the objective probabilistic relations between its backing conditions and its advice conditions. Backing conditions describe physical situations and processes on which the state supervenes: advice conditions are magnitude claims of the form Mε  to which the

4

The state of an individual system can sometimes be prepared with no selection step by controlling its interactions with its environment, as in laser cooling or just letting the system relax when coupled to a vacuum (Fröhlich & Schubnel, 2016). While it is tempting to conclude this works by modifying its intrinsic properties, certification of the procedure by quantum tomography on many similarly prepared systems does not establish this conclusion.

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Born rule assigns probabilities when legitimately applied. This requires the system to be involved in an appropriate interaction—one modeled by rapid and robust decoherence of a quantum state.5 Once one understands the function of a quantum state it matters little whether one chooses to say that it represents the objective probabilistic relations between its backing conditions and its advice conditions. Many true statements supervene on these backing conditions. What distinguishes a statement assigning a system’s quantum state is that it implies objective probabilistic relations between its backing conditions and its advice conditions via the Born rule. Since probability is a modal concept—a quantified possibility—if a system has a quantum state this means that this statement has modal content because it implies many modal statements. What makes Born probabilities objective is not that they are determined by all local matters of particular fact but that they offer authoritative advice to any user of quantum theory on how to set their credences (coherent degrees of belief) in magnitude claims in a physical situation that blocks any more direct epistemic access to the truth of those claims. The advice carries this authority insofar as adjusting one’s credences to accord with Born probabilities is on the whole the most reliable way of forming expectations in a situation of uncertainty. Objective probabilities arise outside of quantum theory, in statistical physics and so-called games of chance. The function of probability is to guide belief (and hence action) in a situation of uncertainty: different situations require different objective probabilities, in applications of quantum theory and elsewhere. A probability could not adequately serve this function if its value were determined by any actual frequency because following its guidance may not yield the expected results, even in the long term. A quantum state has modal content because its function is to yield modal objective probabilities. This content is doubly modal because these probabilities concern sets of possibilities, where each set pertains to a different hypothetical decoherence context, at most one of which is actual.

20.9 Representationalism and the Quantum State Price (2011, 2013) has argued in favor of a distinctive view he calls subject naturalism by contrast with object naturalism. The object naturalist holds that ultimately all there is is the world of science and that all genuine knowledge is scientific knowledge. The subject naturalist instead maintains that philosophy needs to begin with what science tells us about ourselves—that we are natural creatures, and that philosophy must proceed by acknowledging this fact. An object naturalist has a proto-theory about language involving the assumption of: Representationalism: The function of statements is to represent ‘worldly’ states of affairs and...true statements succeed in doing so. (Price, 2013, p. 24)

5

For further details see my (2012b).

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This leads to problems in view of the striking mismatch between the rich world of ordinary discourse and the sparse world apparently described by science. For there are many apparently true statements that don’t seem to line up neatly with facts of the kind uncovered by natural science. These include not only normative statements, in ethics and elsewhere, but also statements about probability, possibility and causation—even when these occur within science itself. Statements attributing quantum states to physical systems provide a striking illustration within fundamental science of the superiority of subject naturalism over an object naturalism burdened with Representationalism. For one who accepts quantum theory, the fundamental facts about the world are stated by magnitude claims. Truths about quantum states are not magnitude claims, although they supervene on them. The function of a statement attributing a quantum state is not to represent a ‘worldly’ state of affairs, even though many such statements are true. Representationalism fails for these statements. The function of a quantum state is to offer good advice to any suitably-placed agent on how to set credences concerning magnitude claims whose truth-values they are not in a position to determine more directly. This function is exercised through application to the quantum state of the Born rule and adjustment of credences to match the probabilities it yields. Science tells us that we (and potentially other kinds of agents) are spatiotemporally localized natural creatures whose physical situation limits epistemic access to many physical states of affairs. Such an agent is able reliably to improve its epistemic state with respect to a physical system by applying the Born rule to the appropriate quantum state for one in the agent’s physical situation. For each such situation this yields a plethora of probability distributions, each pertaining to a possible circumstance in which the system may find itself. By adjusting credences to match these probabilities the agent is better prepared to face the unknown.

References Bell, J. S. (2004). Speakable and unspeakable in quantum mechanics (Rev. ed.). Cambridge University Press. Belot, G. (2012). Quantum states for primitive ontologists. European Journal for Philosophy of Science, 2, 67–83. Bub, J. (2019). What is really there in the quantum world? In A. Cordero (Ed.), Philosophers look at quantum mechanics (pp. 217–233). Springer. De Finetti, B. (1968). Probability: The subjectivistic approach. In R. Klibansky (Ed.), La Philosophie Contemporaine (pp. 45–53). La Nuova Italia. Einstein, A., Podolsky, B., & Rosen, N. (1935) (EPR) “Can quantum-mechanical description of physical reality be considered complete?” Physical Review, 47: 777–80. Elgin, C. (2017). True enough. MIT Press. Esfeld, M., Lazarovici, D., Hubert, M., & Dürr, D. (2014). The ontology of Bohmian mechanics. British Journal for the Philosophy of Science, 65, 773–796. French, S., & Saatsi, J. (2020). Scientific realism and the quantum. Oxford University Press.

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Fröhlich, J., & Schubnel, B. (2016). The preparation of states in quantum mechanics. Journal of Mathematical Physics, 57, 042101. Gleason, A. M. (1957). Measures on the closed subspaces of a Hilbert space. Journal of Mathematics and Mechanics, 17, 59–81. Glick, D., Darby, G., & Marmodoro, A. (2020). The foundation of reality. Oxford University Press. Harrigan, N., & Spekkens, R. W. (2010). Einstein, incompleteness, and the epistemic view of quantum states. Foundations of Physics, 40, 125–157. Healey, R. (2012a). “Quantum theory: A pragmatist approach”, British. Journal for the Philosophy of Science, 63, 729–771. Healey, R. (2012b). Quantum decoherence in a pragmatist view: Dispelling Feynman’s mystery. Foundations of Physics, 42, 1534–1555. Healey, R. (2017a). The quantum revolution in philosophy. Oxford University Press. Healey, R. (2017b). Quantum states as objective informational bridges. Foundations of Physics, 47, 161–173. Hemmo, M., & Shenker, O. (2020). Quantum, probability, logic. Springer. Krissmer, R. (2018). Representation lost: The case for a relational interpretation of quantum mechanics. Entropy, 20, 975. Leifer, M. (2014). Is the quantum state real? An extended review of -ontology theorems. Quanta, 3, 67–155. Monton, B. (2006). Quantum mechanics and 3N-dimensional space. Philosophy of Science, 73, 778–789. Ney, A. (2020). The world in the wave function. Oxford University Press. Price, H. (2011). Naturalism without mirrors. Oxford University Press. Price, H. (2013). Expressivism, pragmatism and representationialism. Cambridge University Press. Pusey, M. F., Barrett, J., & Rudolph, T. (2012). On the reality of the quantum state. Nature Physics, 8, 475–478. Rovelli, C. (1996). Relational quantum mechanics. International Journal of Theoretical Physics, 35, 1637–1678. Schrödinger, E. (1936) Probability relations between separated systems. Mathematical Proceedings of the Cambridge Philosophical Society, 32, 446–452. Suarez, M. (2015). Bohmian dispositions. Synthese, 192, 3203–3228. Wallace, D. (2020). “On the plurality of quantum theories”, in French and Saatsi (2020): 78–102. Wallace, D., & Timpson, C. (2010). Quantum mechanics on spacetime I: Spacetime state realism. British Journal for the Philosophy of Science, 61, 697–727. Wessels, L. (1997). The preparation problem in quantum mechanics. In J. Earman & J. Norton (Eds.), The cosmos of science (pp. 243–273). University of Pittsburgh Press. Wigner, E. (1963). The problem of measurement. American Journal of Physics, 31, 6–15.

Part IV

Indeterminacy

Chapter 21

Quantum Mechanics Without Indeterminacy David Glick

Abstract Metaphysical indeterminacy in the context of quantum mechanics is often motivated by the eigenstate-eigenvalue link. However, the sparse view of Glick (Thought J Philos 6(3):204–213, 2017) illustrates why it has no such implications. Other links connecting quantum states and property ascriptions— such as those associated with the GRW theory—may introduce indeterminacy, but such indeterminacy may be viewed as merely representational and is susceptible to familiar treatments of vagueness. Thus, I contend that such links fail to provide a compelling motivation for quantum metaphysical indeterminacy.

21.1 Quantum Metaphysical Indeterminacy 21.1.1 Metaphysical Indeterminacy Quantum mechanics has often been associated with indeterminacy. In contrast to familiar cases of vagueness, the indeterminacy involved has seemed to some to be worldly as opposed to merely representational. There are several possible explications of this notion of worldly or metaphysical indeterminacy. First, there is the metaphysical supervaluationist approach developed by Barnes and Williams (2011).1 According to the supervaluationist, a proposition is vague just in case it admits of multiple precisifications, each of which assign it a truth value, but some

To appear in V. Allori (ed.) Quantum Mechanics and Fundamentality: Naturalizing Quantum Theory between Scientific Realism and Ontological Indeterminacy, Springer Nature. 1

See also Akiba (2004) and Barnes (2010).

D. Glick () Department of Philosophy, University of California, Davis, Davis, CA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_21

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of which disagree about its value. Such vagueness is typically attributed to our epistemic or representational limitations. According to Barnes and Williams, metaphysical indeterminacy occurs when actuality admits of multiple precisifications, each of which is represented by an ersatz possible world. On this view, reality is indeterminate just in case there are propositions about actuality that are true at some but not all candidate ersatz worlds. An alternative account of metaphysical indeterminacy is provided by the determinable-based approach of Wilson (2013, 2017). This view eschews the precisifications of the supervaluationist and instead allows for indeterminate states of affairs. The guiding idea is that a state of affairs is indeterminate just in case it involves the instantiation of a determinable property without a unique determinate.2 There are two ways this could occur: either a determinable property could have more than one determinate (“glutty” indeterminacy) or no determinates (“gappy” indeterminacy). Glutty indeterminacy involves the possessions of multiple determinates of a single determinable in a relativized or degree-theoretic fashion. Wilson (2013) gives the example of an iridescent feather that is both red (from one perspective) and blue (from another) as a case of glutty indeterminacy. Below we will see some potential examples of gappy indeterminacy in the context of quantum mechanics. In the discussion of quantum indeterminacy that follows, I’ll focus on Wilson’s determinable-based understanding of metaphysical indeterminacy. There are two reasons for this. First, there is a concern that the metaphysical supervaluationist approach cannot be applied to the case of quantum indeterminacy. Several authors have noticed that no-go results such as the Kochen-Specker theorem seem to rule out the possibility of maximal and precise ersatz possible worlds that the approach seems to require (Darby, 2010; Skow, 2010). This is an open area of debate and tangential to my primary concerns.3 Second, much of the articulation and defense of quantum indeterminacy occurs in the context of the determinable-based approach (Bokulich, 2014; Calosi & Wilson, 2019, 2021; Lewis, 2016). This chapter proceeds as follows. In the remainder of this section, I briefly introduce quantum indeterminacy in the context of so-called “orthodox” quantum mechanics and the alternative offered by Glick (2017). In Sect. 21.2, I turn to the GRW theory and the case for indeterminacy there. I argue that, while the GRW theory does introduce vagueness via the links between quantum states and properties, this indeterminacy may be viewed as representational. Finally, I conclude in Sect. 21.3 by highlighting two remaining issues: alternative interpretations and emergent indeterminacy.

2

A familiar example of the determinable-determinate relationship is red and scarlet. Scarlet is a particular way of being red, hence it is a determinate of the determinable red. Note that a property may be a determinate at one level of analysis, but a determinable at a “deeper” level—Venetian scarlet is a determinate of the scarlet determinable. 3 See Darby and Pickup (2019) for an attempt to resolve this challenge facing the metaphysical supervaluationist and Corti (2021) for criticism of their attempt.

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21.1.2 Quantum Indeterminacy Assuming the determinable-based understanding, quantum mechanics is alleged to involve metaphysical indeterminacy in so far as it describes systems that lack unique determinate values of physical quantities. For instance, consider a particle characterized by a quantum state of spin that is a superposition of up and down in the x direction, ψ = c1 |↑x  + c2 |↓x . In order to move beyond the quantum state description, we need a principle linking it to certain properties (“observables”). The best known of these is a tenet of orthodox quantum mechanics:4 Eigenstate-Eigenvalue Link (EEL): A system A has a value v of property P if the quantum state of A is in an eigenstate of the associated operator Oˆ with eigenvalue v. Applied to the particle, EEL implies that it lacks both the value “up” and “down” of x-spin. Thus, one may be inclined to regard it as instantiating the determinable of x-spin without a unique determinate of it.5 There are two ways this could go: either it lacks any determinate of x-spin (gappy) or it possesses more than one determinate (glutty). Initial applications of determinable-based metaphysical indeterminacy to quantum mechanics focused on the gappy understanding (Bokulich, 2014; Wolff, 2015), but more recently, some authors have advocated for the glutty view. For instance, the particle could be said to possess value “up” to a certain degree (given by its modulus-squared coefficient) and value “down” to a certain degree (Calosi & Wilson, 2019, 2021). However, EEL alone doesn’t imply indeterminacy of either form. It has the form of a biconditional between a quantum state description and the attribution of a specific value of an observable. The situations alleged to give rise to indeterminacy are those where the quantum state isn’t in an eigenstate of the observable under consideration. It follows that we cannot attribute a specific value of that observable. But, as Glick (2017) observes, there is a clear distinction between (determinately) lacking a property and possessing an indeterminate value of that property.6 So, EEL is compatible with a view that eschews indeterminacy: the sparse view according to which systems don’t possess properties—determinate or determinable—for which

4

It’s not entirely clear what constitutes “orthodox” quantum mechanics. For my purposes here, I assume that it involves the eigenstate-eigenvalue link, the collapse postulate, and Born’s rule. See Wallace (2019) for a criticism of this view and Gilton (2016) for a defence of the role of the eigenstate-eigenvalue link in “orthodox” or “textbook” quantum mechanics. 5 I will be challenging this inclination below. For now, a motivating idea might be that we can measure the particle’s x-spin, and when we do so, it will be found either up or down. So, the particle in question is the kind of thing that can possess a precise value of x-spin even if it doesn’t have one at the moment. This might incline one to regard it as possessing the x-spin determinable without a unique determinate of it (until it’s measured). 6 Consider, for instance, category mistakes. The number two lacks a determinate mass, but this does not imply that its mass is indeterminate because it lacks the determinable as well.

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they aren’t in an eigenstate. On the sparse view, a particle in a superposition of x-spin up and down simply lacks x-spin. While the sparse view may illustrate that the EEL is strictly compatible with metaphysical determinacy, if the sparse view is sufficiently implausible it may be alleged that any reasonable understanding of orthodox quantum mechanics will involve genuine metaphysical indeterminacy. Indeed, some allege that the sparse view has the implausible consequence that particles are not located in spacetime and that measuring a particle will cause it to pop into existence (Calosi & Wilson, 2021). In order to see why such worries are misplaced, let’s briefly reconsider location on the sparse view of orthodox quantum mechanics.

21.1.3 Quantum Location To simplify matters, consider a particle that’s confined to a region X, which is divided into two subregions A and B. We can write its position state as a superposition c1 |A + c2 |B, where |A is the quantum state associated with being in region A and likewise for |B and B. From the Born rule we know that the probability of finding the particle in region A is given by ||c1 ||2 , and ||c2 ||2 for region B. Moreover, we know that the probability of finding the particle in the region X = A ∪ B is 1. Given this, what should we say about the particle’s position? EEL implies that the system is (determinately) located in the region X, as it is in an eigenstate of the associated operator with eigenvalue 1. With respect to the regions A and B, EEL precludes attributing to the system a (determinate) location in either region. Thus, the sparse view (indeed, any version of orthodox QM) is committed to saying that the particle has a location—namely, it is located in X. But, despite being located in X, the particle isn’t located in either of the subregions that X comprises, A and B. This may require a revision of our concept of position in light of quantum mechanics. In particular, it motivates denying Precise Location. Precise Location: being located in a region X is a determinable with being located in xi ∈ X as determinates. According to Precise Location, being located in a region X admits of further specificity in terms of a proper subregion of X. Perhaps, as one moves to even greater levels of specificity, the location of a system will bottom out in regions that are exactly the same size as the object they contain.7 But Precise Location doesn’t require this. All that is required is that the property associated with being in a region X is, at the relevant level of specificity, a determinable property with its only determinates corresponding to being located in proper subregions of X.

7

Of course, it’s often unclear what the “size” of a quantum system is. So, the relevant notion of a maximally precise location might be the smallest region to which the system can in principle be confined.

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Precise Location leads to position indeterminacy when applied to the case described above. The particle will possess the maximally-unspecific position determinable with the determinate being located in X. At a deeper level of specificity, it will possess the determinable being located in X. However, it will lack a unique determinate of that determinable. Either it will possess no determinate or it will possess multiple determinates (e.g., each to a degree less than one). One reaction to this is to embrace position indeterminacy in quantum theory, another is to challenge the assumption of Precise Location. Imprecise Location: determinate.

being located in a region X can serve as an absolute

On Precise Location, being located in X is both a determinate of a more general location determinable and also a determinable with more precise location properties as determinates. Imprecise Location, by contrast, allows for regions that are intuitively larger than the physical system to serve as absolute determinates— i.e., determinates that are not themselves determinables. How does this differ from indeterminacy about position? After all, Wilson (2013) refers to states of affairs involving determinables lacking unique determinates as both “indeterminate” and “imprecise.” But absolute determinates are not bare determinables—only the latter implies that an object has an indeterminate position. On Imprecise Location, if asked where a particle confined to a region X(> xp 8 ) is located, the answer is simply “in region X.” If asked where in region X the particle is located, the proper response is not that it is indeterminate, but rather, that the question rests on a mistaken assumption, namely, Precise Location. According to Imprecise Location, there is nothing indeterminate about having a location given by a region that is intuitively larger than the object in question. This understanding of position is consistent with the standard formalism of ordinary non-relativistic quantum mechanics. If we assume that observables are associated with self-adjoint operators on Hilbert space, then there is a problem with applying EEL to a position observable that incorporates Precise Location. The corresponding operator will have a continuous spectrum rather than discrete eigenstates. In order to apply EEL, we need to consider operators that project onto the subspace corresponding to being in some finite region of space X. So, if we take this seriously as a guide to thinking about position in quantum mechanics, position should always be understood in terms of questions of the form: “Is the particle confined to region X?” Moreover, there will always be some smallest region X beyond which the answer is always “no.” If the system isn’t in an eigenstate of any operator corresponding to being located in a proper subregion xi ∈ X, then EEL precludes attributing to it a location more precise than X.9

8 Let x denote a subregion of X that would be regarded as a maximally precise location for the p particle. See previous footnote. 9 Notice that there is nothing wrong with asking whether we will find the particle in the region xi ∈ X if we were to measure its position. To answer this question we use Born’s rule, not EEL.

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Imprecise Location allows us to see why the sparse view doesn’t imply that particles aren’t located in spacetime and don’t pop into existence upon measurement. In realistic scenarios, it may be impossible to ascribe a position more specific than the entirety of space. Even after measurement, particles will never be perfectly localized for any finite period of time. This may be unsatisfying, but such is orthodox quantum mechanics. Notice that the fan of indeterminacy will be compelled to adopt a parallel position: that position is almost always indeterminate. For instance, realistic position measurements cannot be understood as removing indeterminacy given that the particle isn’t precisely localized after a measurement either (at least not for any finite period of time). The sparse view allows for a non-zero probability of finding a particle in a region in which it isn’t located, but the implausibility of this claim rests on an understanding of measurement as revealing preexisting properties, which is rejected by orthodox quantum mechanics. A more significant problem with Imprecise Location is that intuitively quantum theory provides a number of different ways in which a system can be located in X. There are any number of quantum states that are eigenstates of a projector onto the subspace associated with being located in X, for instance: the equally-weighted superposition √1 (|A + |B); non-equally-weighted superpositions c1 |A + c2 |B 2 (where c1 = c2 ); a more precise location xi within X (|xi ). Each of these imply that an ideal measurement will find the particle in X with probability 1. Thus, Imprecise Location may be unsatisfying as an explication of the location of the system as it’s unable to make important distinctions between the various states which seem to correspond to different ways the system may be located in X. A natural way to correct this is to allow for determinates associated with each such quantum state. Quantum Location: being located in the region X is a determinable with being in the quantum state ψi among its determinates, where P (being located in X|ψi ) = 1. Quantum Location allows for many different ways of being located in the region X. For instance, being in a particular superposition of |A and |B is one such way. EEL doesn’t license the attribution of any precise position observables in such a state, but so long as there is some self-adjoint operator associated with the quantum state, then it will be in an eigenstate of that operator. Thus, EEL will license the attribution of properties such as being in an equally-weighted superposition of being located in A and being located in B.10 One might complain that such a property isn’t a candidate for a physical property, but it has a clear connection to measurement outcomes (via Born’s rule) and enters into causal/nomic relations with other physical properties. If one allows specific superpositions to represent novel determinates of position, then one gains additional resources that may allow

10 The quantum state will not be in a superposition of the operator invoked in EEL, but we may wish to describe the state in terms of a distinct operator with clear physical significance—in this case, an operator with |A and |B as eigenstates.

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the sparse view to explain phenomena seen as challenging for the view, such as interference phenomena.11 The case of position generalizes. If a system is described by a quantum state that is a superposition of x-spin—ψ = c1 | ↑x  + c2 | ↓x —what is the indeterminate property? Suppose the system in question is an electron. Then we know it has a spin of h2¯ , and we can use a two-dimensional Hilbert space to characterize its spin. So, having a spin value is a determinable with a unique determinate. EEL provides no basis for attributing the determinable having an x-spin. Indeed, it is often said that (in orthodox quantum mechanics) having a spin in one direction precludes having a spin in an orthogonal one. As in the position case, it seems better to limit our ontological commitments to those properties which are determinate: either (a) only attribute x-spin when the system is in an eigenstate of x-spin or (b) allow for superpositions of x-spin to count as determinates of x-spin. On either approach, we never find determinables without unique determinates. On the first approach, a particle in a superposition of x-spin simply lacks x-spin (until measured). On the second approach, such a particle has the property of being in a particular superposition of x-spin, where such a property is a unique determinate of x-spin. Again one might wonder how Quantum Location (and its generalization) differs from indeterminacy. Perhaps saying the system is in a particular superposition of x-spin (ψ = c1 | ↑x  + c2 | ↓x ) could be understood as possessing the “up” and “down” determinates to degrees given by the squared modulus of their coefficients. There are at least two problems with this proposal. First, EEL provides no basis for this claim. The state is not an eigenstate of x-spin with multiple eigenvalues, so another principle would need to be invoked.12 Second, and more significantly, distinct quantum states lead to the same expectation values for x-spin values, for instance: the mixed state representing a system as x-spin “up” or x-spin “down” with equal probability; a system in an eigenstate of y-spin orthogonal to x; a system in an eigenstate of z-spin orthogonal to x and y. Thus, the property associated with being in a particular superposition of x-spin cannot be identified with the weighted possession of x-spin determinates.13

11 Of course, critics of the sparse view may allege that the quantum state must be given some metaphysical analysis, and it’s here that metaphysical indeterminacy arises. Quantum Location is a denial of this demand. The property possessed by a system in a particular superposition of more precise position states is just that. Superpositions are novel kinds of determinates, not to be further analyzed in terms of the observables that appear in their arguments. 12 Calosi and Wilson (2019) propose to replace EEL with DEEL, a principle that posits degrees of possession proportional to the coefficients of the arguments of a superposition. However, unlike EEL, DEEL is not part of the standard formulation of orthodox QM. Moreover, it goes beyond what is needed for orthodox QM to solve the measurement problem. See Sect. 21.3. 13 This means that Quantum Location (and its generalization) is distinct from the glutty indeterminacy view of Calosi and Wilson (2019). The former takes a particular superposition to be a novel kind of determinate of the determinable associated with the operator that defines the basis. The latter takes a particular superposition to correspond to a plurality of determinates (each corresponding to eigenstates) each possessed to a degree less than 1. The present point is that

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In sum, the argument for metaphysical indeterminacy in orthodox quantum mechanics rests on two assumptions. First, that particles possess determinables corresponding to operators of which they are not in eigenstates. Second, that these determinables have only the eigenstates of the corresponding operators as their determinates. Each of these assumptions may be challenged, leading to two versions of orthodox QM without indeterminacy. Rejecting the first assumption leads to the sparse view on which one can only attribute properties to particles in special circumstances, or when the properties in question are very general. This meager ontology may be unsatisfying, but represents a natural way of thinking about orthodox QM and doesn’t require denying that particles have locations or pop into existence. Quantum location (and its generalization) rejects the second assumption by expanding the determinates of determinables associated with operators to include superpositions of their eigenstates. This view has the benefit of attributing properties corresponding to superpositions while retaining the one-to-one correspondence of determinables and determinates.

21.2 Indeterminacy in GRW? The sparse view shows that EEL fails to establish quantum metaphysical indeterminacy. But many regard orthodox quantum mechanics (and EEL) as problematic. Other interpretations may provide a more compelling basis for quantum indeterminacy. In particular, discussion of the problem of tails in the GRW theory has given rise to modifications of EEL which may be thought to provide a basis for metaphysical indeterminacy.

21.2.1 The GRW Theory The measurement problem results from the conflict between the linear, unitary evolution of the quantum state according to Schrödinger’s equation and the apparent fact that measurements have unique determinate results. The GRW theory attempts to solve the problem by replacing deterministic linear Schrödinger dynamics with a stochastic non-linear dynamics.14 In particular, GRW dynamics involve spontaneous localization events (“hits”) centered on a random point c and occurring

the views are inequivalent because the quantum state contains more information than a weighted collection of determinates. 14 The GRW theory is due to Ghirardi et al. (1986). It’s the simplest and best known of a family of collapse interpretations. For more, see Ghirardi and Bassi (2020).

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randomly with an average frequency λ. The crucial points for our discussion are the following: • Localizations (hits) are very unlikely for individual particles, but amplify rapidly with entanglement making it extremely likely for a hit to occur immediately for macroscopic systems. • Hits have the effect of applying a Gaussian distribution of width σ centered on a random point c. • The probability of a given point being the center of localization is given by the squared modulus of the amplitude of the wavefunction in accordance with Born’s rule. Like orthodox quantum mechanics, GRW is an indeterministic theory, but it differs in precisely specifying the dynamics that lead to wavefunction collapse.15 With suitable values for the frequency of hits λ and the width of the Gaussian σ , GRW may be regarded as consistent with the current empirical evidence for quantum mechanics.

21.2.2 The Problem of Tails In orthodox quantum mechanics, EEL acts as a principle linking quantum states and observables. In GRW, however, EEL must be replaced by a more forgiving linking principle. The reason is that hits fail to localize the wavefunction precisely to a point. Indeed, while localizations cause the majority of wavefunction amplitude to be concentrated near c, the tails of the wavefunction will extend arbitrarily far away from c. Even after measurement, a particle will not be in an eigenstate of the operator associated with being located in the neighborhood of c, so EEL does not allow the attribution of a localized position to the particle. This is a problem as the GRW theory aims to solve the measurement problem by changing the dynamics of quantum mechanics so as to deliver determinate measurement outcomes.16 To solve this problem, several proposals for a revised link have been offered. In conjunction with reasonable assumptions, these principles allow the GRW theory to yield determinate outcomes for many kinds of quantum measurements. However,

15 It may be thought that indeterminism alone implies indeterminacy, and indeed, some motivate metaphysical indeterminacy by consideration of the “open future” (Barnes & Cameron, 2009). However, the metaphysical status of the future is largely independent of whether there are stochastic laws of nature. The path from indeterminism to indeterminacy is not at all straightforward and inevitable. 16 Armed with only EEL, the GRW theory is unable to secure determinate measurement outcomes because it precludes the assignment of reasonably precise positions to macroscopic systems like pointers that constitute such results. Again, the only position properties licensed by EEL are those that attribute a location in the region to which a system is strictly confined. Often this region will be no less than the entirety of space.

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they also introduce a certain degree of indeterminacy. First, consider Albert and Loewer’s fuzzy link. Fuzzy link: Particle x is in region R if and only if the proportion of the total squared amplitude of x’s wave function which is associated with points in R is greater than or equal to 1 − p (Albert & Loewer, 1996, p. 87). Generalizing somewhat, the fuzzy link allows one to attribute a property to a system when its quantum state is sufficiently close to an eigenstate of the associated operator. “Sufficiently close” is captured by the 1 − p term, which acts as a cutoff below which a determinate property cannot be ascribed. Combined with GRW’s non-linear stochastic dynamics and assumptions about the measurement process, the fuzzy link allows for determinate outcomes. A position measurement induces a hit to the wavefunction of a particle, after which it is sufficiently close to an eigenstate of the operator Pˆc with eigenvalue 1 for the fuzzy link to ascribe the determinate property of being located near c.17 This determinate position property accounts for the measurement outcome, namely a detection near c. The fuzzy link introduces indeterminacy into the ascription of observables in GRW. While EEL is unambiguous about the conditions under which observables are to be attributed, the fuzzy link requires imposing a threshold of “sufficiently close to an eigenstate.” This threshold has features characteristic of vagueness: whatever threshold is chosen will be arbitrary, subject to disagreement and contextdependence, and will admit of borderline cases. Such vagueness is not unproblematic, but is familiar from ordinary cases involving the boundaries of mountains and clouds, baldness, etc. This would suggest that the strategies for dealing with familiar vagueness will apply here as well. In particular, approaches such as supervaluationism and epistemicism could be used to avoid the implication of metaphysical indeterminacy. A supervaluationist could identify a number of precisifications of “sufficiently close,” each free from indeterminacy, but which disagree about the precise threshold. An epistemicist could claim that there is a precise threshold that lies beyond our ken. Both approaches relegate the resultant indeterminacy to the representational domain—the indeterminacy is the result of our language or epistemic limits, not the world itself. More recently, Peter Lewis has proposed another linking principle with a more direct connection to metaphysical indeterminacy, the vague link. Vague link: A system has a determinate value for a given determinable property to the extent that the squared projection of its state onto an eigenstate of the corresponding operator is close to 1, where the determinate value is the eigenvalue for that eigenstate (Lewis, 2016, p. 90). The vague link allows for properties to be attributed in degrees. For a system sufficiently close to an eigenstate, the vague link says that the system possesses 17 Pˆ is a projector onto the subspace of Hilbert space associated with being located in some region c C including c.

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the property to an extent close to 1. Crucially, it also allows that such a system possesses other properties to a minimal extent. For example, a particle localized around c would possess the property of being in the neighborhood of c to a very high degree, but would also possess the property of being outside the neighborhood of c to a very low degree. On its own, the vague link fails to serve the required function for the GRW theory: it will not deliver (unique) determinate outcomes for quantum measurements. One possible response would be to reject the demand for determinate measurement outcomes, but such a stance removes much of the appeal of dynamical collapse theories like GRW. At any rate, this isn’t the approach taken by Lewis when presenting the vague link. According to the vague link, my coffee mug almost entirely possesses the determinate property of being on top of my desk, but it also very slightly possesses the determinate property of being inside the drawer. Because the degree of possession of competing properties is so slight, for all practical purposes I can say that the coffee mug is on the desk. (Lewis, 2016, pp. 90–91)

This suggests that the vague link is to be supplemented with a further principle that allows one to ascribe observables for all practical purposes when the degree of possession is sufficiently high. This allows the GRW theorist to say that measurements have determinate outcomes (for all practical purposes). But, again it does so at the cost of introducing indeterminacy into the ascription of observables. What counts as a sufficiently high degree of possession will exhibit the same familiar characteristics of vagueness as the fuzzy link—arbitrariness, disagreement, borderline cases, context dependence, etc. Moreover, these features will be susceptible to the same representational analysis. What counts as a sufficiently high degree may be subject to further precisification or may be beyond our ken. Of course, the phrase “for all practical purposes” suggests that Lewis may be happy to grant that such indeterminacy in the ascription of observables is merely representational. So, the vague link is similar to the fuzzy link FAPP (for all practical purposes) but it differs in that, strictly speaking, it is naturally understood as positing glutty indeterminacy. The opponent of metaphysical indeterminacy must deny this aspect of the link, but there is very little cost in doing so. Quantum indeterminacy is typically thought to reside outside of measurement contexts, but on the fuzzy link indeterminacy is pervasive even after a maximally precise measurement. This is why it’s unable to solve the measurement problem in the context of the GRW theory without ascending to the FAPP level of description. Compare a view that posits degrees of possession for the property of baldness, where a very bald person possesses baldness to a degree close to 1 and a very not bald person possesses baldness to a degree close to 0. This does nothing to resolve or clarify the vagueness of our application of the predicate “bald.” We have just pushed the problem back to what degree of possession is sufficient to apply the predicate (FAPP). Similarly here, positing degrees of possession leaves unchanged the question of when we should ascribe a determinate value of some observable. The only remaining motivation for this aspect of the fuzzy link is the thought that non-eigenstates must involve

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metaphysical indeterminacy, but the considerations here are unchanged from the discussion of orthodox quantum mechanics above.18 Thus, both links allow for the attribution of determinate properties (at least FAPP) after measurement on the GRW theory, but in so doing introduce a certain degree of indeterminacy. Such indeterminacy shares many of the features of familiar cases of vagueness outside of quantum mechanics, but Lewis (2016) alleges that it’s unique in at least two respects: (1) there is no continuum of property values in which a vague boundary occurs and (2) it isn’t the result of composition.19 I will address each point in turn. First, it’s not obvious that there isn’t a continuum of properties in which the boundary occurs. Recall that according to Quantum Location, each quantum state is taken to describe a specific determinate property. In the context of a position measurement, different quantum states could be arranged in a continuum with respect to the degree of localization to which they correspond. Even if one rejects this association of quantum states with determinates, there is nevertheless an underlying continuum of quantum states (and corresponding probabilities) in which the threshold for possession of a given observable is located according to the fuzzy link. And again, where to locate this threshold within the continuum of quantum states (or probabilities) exhibits the characteristic features of familiar cases of vagueness. On the vague link, there is an additional level of properties possessed to varying degrees, which may also be ordered in a continuum. Along this continuum of degrees of possession, one must establish some threshold that allows for ascription of the relevant observable for all practical purposes. As with that of the fuzzy link, this threshold shares the characteristics of familiar cases of vagueness and is susceptible to the usual representational treatments. Second, the indeterminacy here isn’t the result of composition, but composition is not the only source of indeterminacy familiar from cases outside of quantum mechanics. The indeterminacy of a mountain, or a cloud, or some other medium-sized object may be understood in compositional terms. For instance, the indeterminacy of a mountain can be taken to concern which molecules are part of the mountain and which are not. The case of observables in GRW is not like this. As best we know, electrons aren’t composed of anything, and so any indeterminacy associated with them cannot be the result of their composition. However, there are other instances of indeterminacy involving familiar (“non-quantum”) objects that

18 Of course, the GRW theory is a more “realist” interpretation than orthodox quantum mechanics, so one might think that we are owed more of a story about non-eigenstates. However, rejecting metaphysical indeterminacy doesn’t require silence about non-eigenstates. In addition to providing probabilities for various measurement outcomes, such states can also provide a basis for the attribution of novel properties along the lines of Quantum Location as outlined in Sect. 21.1.3. 19 Lewis takes the GRW theory to involve indeterminacy that is both metaphysical and distinct from familiar non-quantum cases of indeterminacy. Here I treat the distinctness claim independently with the aim to rebut the claim that there is anything distinctively “quantum” about the indeterminacy introduced by the GRW links.

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are independent of their material composition. The indeterminacy exhibited by these cases admits of representational analyses that make no appeal to composition. Consider an ordinary doorstop. Whether a given object counts as a doorstop is a matter of it performing a certain function, namely, holding open a door. Clearly it can be vague whether a given object satisfies this functional role. For instance, one may count a rock as a doorstop, but only if it is sufficiently large or heavy to hold open the door, a matter that admits of degrees. Now, while it may be true that a rock is composed of material parts, this fact is irrelevant to the indeterminacy surrounding whether it is a doorstop or not. A representational view locates the indeterminacy in our representation of the object as a doorstop, and hence, denies that it is an instance of metaphysical indeterminacy. The representational approach is compatible with either supervaluationism or epistemicism as applied to the specification of the functional role characterizing the entity rather than its composition—the indeterminacy dissolves if the function of holding open a door is specified or known with sufficient precision. Observables in GRW may be viewed as functional entities. Each property has an associated probability of resulting in a certain measurement result. Indeed, it is this connection that gives observables (and quantum mechanics more generally) empirical significance. This suggests that the indeterminacy introduced by the GRW theory can be understood in terms of whether the system satisfies a given functional role. Consider again the case of location near c. If the probability of being found near c is sufficiently high, then it is natural to regard the particle as possessing the determinate property, being located near c. But the probabilities form a continuum corresponding to the extent to which the particle plays the functional role associated with being located near c. Both the fuzzy and vague links may be seen as attempts to ascribe observables to systems when they are almost certain to deliver the corresponding measurement result. This is simply another way of representing the nature of the functional role of observables in quantum mechanics. Thus, even if some quantum systems are not composed of smaller parts, the indeterminacy of property attribution in the GRW theory needn’t be seen as novel or metaphysical. It can be seen as an instance of the familiar vagueness concerning whether an entity satisfies a certain functional role. Such vagueness can be readily understood as representational indeterminacy.

21.3 Remaining Issues In this chapter I have argued that two approaches to quantum mechanics can be understood without invoking metaphysical indeterminacy. In the case of orthodox quantum mechanics, the sparse view demonstrates that EEL does not require metaphysical indeterminacy. In the case of the GRW theory, the problem of tails requires modifying EEL in a way that introduces some indeterminacy. However, such indeterminacy may be regarded as merely representational.

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21.3.1 Other Interpretations Several issues remain. First, there are other interpretations of quantum mechanics beyond those considered here, and nothing I’ve said rules out that they may involve metaphysical indeterminacy. Indeed, there are variants of orthodox quantum mechanics and the GRW theory that posit indeterminacy. For example, Calosi and Wilson (2021) propose replacing EEL with a degree-theoretic variant: DEEL: A quantum system S has a definite value v for an observable O to a degree √ y iff y is the absolute value of the coefficient of the associated eigenvector having eigenvalue v in the quantum state of S (Calosi & Wilson, 2019, p. 2621). However, unlike EEL, DEEL is not part of the standard formulation of orthodox quantum mechanics, nor is it needed for the orthodox solution to the measurement problem. In order to secure determinate measurement outcomes in the context of orthodox quantum mechanics, we only need EEL, not DEEL. In the context of the GRW theory, Lewis’s vague link is naturally understood as positing glutty metaphysical indeterminacy. But, as noted above, the vague link alone is unable to solve the measurement problem in the context of the GRW theory as it fails to secure determinate measurement outcomes after a hit. Thus, neither link plays a role in the solutions to the measurement problem offered by these interpretations. Of course, the availability of indeterminacy-free versions of these interpretations isn’t an argument for them over their indeterminacy-involving counterparts. Ultimately, as with many metaphysical issues in the sciences, the question of metaphysical indeterminacy is underdetermined. However, there is prima facie reason to prefer avoiding metaphysical indeterminacy in physical theories when it is possible to do so. Other things being equal, we should avoid interpreting our physical theories in a way that commits us to controversial metaphysical theses.

21.3.2 Emergent Indeterminacy A second outstanding issue concerns the possibility of emergent metaphysical indeterminacy.20 Suppose that an interpretation deploys a link from the quantum state to observables that introduces some kind of indeterminacy. For example, the GRW links discussed above allow for situations where there can be indeterminacy surrounding the location of a particle after a position measurement. Any such indeterminacy will be emergent in that it is derived from the fundamental description provided by the quantum state of the system via the link in question. But just because something is emergent doesn’t mean it’s not real, so this fact alone fails to tell against quantum metaphysical indeterminacy. Indeed, this is once again a

20 For

a defense of emergent quantum metaphysical indeterminacy, see Mariani (2021).

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matter that is underdetermined by one’s choice of interpretation qua solution to the measurement problem. However, there is more to say about the status of emergent metaphysical indeterminacy: whether one countenances it has more to do with one’s attitude toward ordinary cases of non-fundamental indeterminacy than anything specific to quantum theory. This means that quantum theory is unlikely to make much of a difference to the debate—those who find metaphysical indeterminacy in the everyday world irrespective of quantum theory can maintain their beliefs and likewise for opponents of metaphysical indeterminacy. First, consider the situation in the GRW theory. Suppose the position wavefunction of a particle has support within the region R but also outside of it. Is the particle located in R? Given that the system isn’t in an eigenstate of the associated operator (PˆR ), it will depend on the details of our link, which may (or may not) introduce indeterminacy concerning the position of the particle. Such indeterminacy can be given either a deflationary or inflationary reading. The former would say the indeterminacy concerns the attribution of the predicate is located in R and may rely on standard treatments of vagueness to resolve the indeterminacy. The latter approach would regard the system’s location as genuinely metaphysically indeterminate—even though there is no indeterminacy in the quantum state, the link introduces indeterminacy in the location properties of the particle. It’s worth noting that Barnes (2014) regards emergent metaphysical indeterminacy as impossible in principle. Barnes argues that if the fundamental level is fully determinate, and it determines the emergent level, then there is nowhere for indeterminacy to come from. However, Lewis’s vague link provides a potential counterexample. For a system like the particle considered above, its quantum state description will give rise to multiple determinates of the location determinable (each possessed to an extent less than 1).21 Alternatively, one can adopt Albert and Loewer’s fuzzy link and only attribute determinate properties. There remains residual indeterminacy in the specific conditions under which we can attribute determinate properties, but this can be regarded as an ordinary case of representational vagueness. Second, consider Wallace’s (2012) version of the Everett interpretation. On this view, the fundamental ontology is described by the universal quantum state, which evolves unitarily according to the Schrödinger equation.22 This fundamental ontology gives rise to a vast emergent ontology of many quasi-classical worlds, each

21 Barnes doesn’t share Wilson’s determinable-based account of metaphysical indeterminacy, however, one may develop an understanding of the vague link that fits with the metaphysical supervaluationist account. In the present case of a position measurement in GRW, one could posit a candidate ersatz world corresponding to each location where the wavefunction has support. Then, the position of the system will be metaphysically indeterminate in that the truth of a proposition of the form the system is located in region R will differ between candidate ersatz worlds, hence it will be indeterminate whether the system is located in R. 22 In the case of quantum field theory, Wallace advocates for spacetime state realism, which understands the universal quantum state in terms of density operators assigned to regions of spacetime. On this view, the fundamental ontology includes spacetime (regions) and properties corresponding to density operators (see Wallace & Timpson, 2010).

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populated by a full complement of macroscopic objects. As Wallace recognizes, “it is commonplace in emergence for there to be some indeterminacy” (Wallace, 2012, p. 101). The number of worlds, the objects they contain, and even human minds can exhibit indeterminacy. One reason for this is the process of environmental decoherence that Wallace’s account relies on, but equally important is the functionalist account of emergent ontology he adopts. As noted above, functionalist criteria often introduce a certain amount of indeterminacy. It follows that if one endows such criteria with metaphysical significance—i.e., regards them as existence criteria— then emergent metaphysical indeterminacy results. There is nothing incoherent about such a view, which would give rise to widespread metaphysical indeterminacy in the quantum world. There is, however, an alternative. One could take a more deflationary attitude toward the emergent multiverse as a way of representing the universal quantum state in terms that we can more readily understand. As Wallace (2002) argues, the many worlds of the Everett interpretation could be seen as analogous to global planes of simultaneity in relativity, indeterminacy in the properties of which is the result of our representational limitations, not the world itself.23 What these examples show is that emergent indeterminacy in quantum interpretations may be regarded as metaphysical or merely representational. Which version of the interpretation one prefers will depend on their general attitude toward indeterminacy in non-fundamental ontology. If one finds metaphysical indeterminacy in everyday properties like baldness and heaphood, they may find it in quantum observables as well. But, if one prefers to adopt a deflationary strategy in everyday contexts, they are free to do the same here. A link connecting the quantum state to observables may be taken to be part of our representation of the quantum world, in which case any indeterminacy such a link introduces will be representational as well. Acknowledgments Many thanks to Claudio Calosi, Sam Fletcher, Dana Goswick, Peter Lewis, Cristian Mariani, Alyssa Ney, Paul Teller, Jessica Wilson, and audiences at the Dartmouth Workshop on Quantum Indeterminacy, the University of California Davis, the Society for the Metaphysics of Science, and the California Quantum Interpretation Network.

23 For example, whether two events are simultaneous is indeterminate in special relativity. Such indeterminacy is naturally regarded as representational given that instants—global planes of simultaneity—are artifacts of our representation of spacetime. Analogously, if the Everettian multiverse is “a more useful description of an entity whose perfect description as a physical system lies (at least for the moment) beyond our ability to comprehend directly”(Wallace, 2002, p. 654), then any indeterminacy it engenders will be representational as well.

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References Akiba, K. (2004). Vagueness in the world. Noûs, 38(3), 407–429. Albert, D. Z., & Loewer, B. (1996). Tails of Schrödinger’s cat. In Perspectives on quantum reality (pp. 81–92). Springer. Barnes, E. (2010). Ontic vagueness: A guide for the perplexed 1. Noûs, 44(4), 601–627. Barnes, E. (2014). Fundamental indeterminacy. Analytic Philosophy, 55(4), 339–362. Barnes, E., & Cameron, R. (2009). The open future: Bivalence, determinism and ontology. Philosophical Studies, 146(2), 291. Barnes, E., & Williams, J. R. G. (2011). A theory of metaphysical indeterminacy. In K. Bennett & D. W. Zimmerman (Eds.), Oxford studies in metaphysics (Vol. 6, pp. 103–148). Oxford University Press. Bokulich, A. (2014). Metaphysical indeterminacy, properties, and quantum theory. Res Philosophica, 91(3), 449–475. Calosi, C., & Wilson, J. (2019). Quantum metaphysical indeterminacy. Philosophical Studies, 176(10), 2599–2627. Calosi, C., & Wilson, J. (2021). Quantum indeterminacy and the double-slit experiment. Philosophical Studies, 1, 1–27. Corti, A. (2021). Yet again, quantum indeterminacy is not worldly indecision. Synthese, 199, 5623–5643. Darby, G. (2010). Quantum mechanics and metaphysical indeterminacy. Australasian Journal of Philosophy, 88(2), 227–245. Darby, G., & Pickup, M. (2019). Modelling deep indeterminacy. Synthese, 198, 1685–1710. Ghirardi, G., & Bassi, A. (2020). Collapse theories. In E. N. Zalta (Ed.), The Stanford Encyclopedia of philosophy (Summer 2020 ed.). Metaphysics Research Lab, Stanford University. Ghirardi, G. C., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34(2), 470. Gilton, M. J. (2016). Whence the eigenstate–eigenvalue link? Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 55, 92–100. Glick, D. (2017). Against quantum indeterminacy. Thought: A Journal of Philosophy, 6(3), 204– 213. Lewis, P. J. (2016). Quantum ontology: A guide to the metaphysics of quantum mechanics. Oxford University Press. Mariani, C. (2021). Emergent quantum indeterminacy. Ratio, 34(3), 183–192. Skow, B. (2010). Deep metaphysical indeterminacy. The Philosophical Quarterly, 60(241), 851– 858. Wallace, D. (2002). Worlds in the Everett interpretation. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 33(4), 637–661. Wallace, D. (2012). The emergent multiverse: Quantum theory according to the Everett interpretation. Oxford: Oxford University Press Wallace, D. (2019). What is orthodox quantum mechanics? In A. Cordero (Ed.), Philosophers look at quantum mechanics (pp. 285–312). Cham: Springer International Publishing. Wallace, D., & Timpson, C. G. (2010). Quantum mechanics on spacetime I: Spacetime state realism. The British Journal for the Philosophy of Science, 61(4), 697–727. Wilson, J. M. (2013). A determinable-based account of metaphysical indeterminacy. Inquiry, 56(4), 359–385. Wilson, J. M. (2017). Are there indeterminate states of affairs? Yes. In E. Barnes (Ed.), Current controversies in metaphysics (Chap. 7, pp. 105–119). New York: Routledge. Wolff, J. (2015). Spin as a determinable. Topoi, 34(2), 379–386.

Chapter 22

Derivative Metaphysical Indeterminacy and Quantum Physics Alessandro Torza

Abstract This chapter argues that quantum indeterminacy can be construed as a merely derivative phenomenon. The possibility of merely derivative quantum indeterminacy undermines both a recent argument against quantum indeterminacy due to David Glick, and an argument against the possibility of merely derivative indeterminacy due to Elizabeth Barnes. Keywords Quantum logic · Classical logic · Logical space · Logical realism · Bivalence · Fundamentality · Naturalness

22.1 Introduction It is a near platitude that a sizable part of our utterances are indeterminate in truth value.1 Because the shirt I am wearing is a shade lying somewhere between green and blue, my utterance of ‘this shirt is green’ is not quite true, but not false either. And when asked where I live in my hometown, I say ‘near the historical center’ mainly because everyone knows where it is, and not because I live particularly close to the historical center, although I do not live far from it, either. In the last century, the philosophical consensus used to be that indeterminacy can only originate in the way we represent the world, never in the nonrepresentational world. Russell (1923, p. 85) wrote that “apart from representation, whether cognitive or mechanical, there can be such thing as vagueness or precision: things are what they are.” Dummett (1975, p. 314) would go as far as to claim that “the notion that

1

That near platitude has been nevertheless denied by those who recommend an epistemicist understanding of vagueness (Williamson, 2002).

A. Torza () Instituto de Investigaciones Filosóficas, UNAM, Circuito Mario de la Cueva, Ciudad Universitaria, Coyoacán, Mexico e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_22

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things might actually be vague, as well as being vaguely described, is not properly intelligible.” Likewise, according to Lewis (1986, p. 212) “the only intelligible account of vagueness locates it in our thought and language. The reason it’s vague where the outback begins is not that there’s this thing, the outback, with imprecise borders; rather there are many things, with different borders, and nobody has been fool enough to try to enforce a choice of one of them as the official referent of the word ‘outback’.” Rhetoric aside, however, little has been provided in the way of arguments against the notion of indeterminacy in the world, aka metaphysical indeterminacy (MI). Sure, Evans (1978) and Salmon (1981, p. 338) have offered a clever and terse disproofs of the possibility of vague objects. But the Evans-Salmon line of argument only shows that an object cannot have indeterminate de re identity—it does not rule out objects that are vague in other respects,2 nor does it rule out ways for reality to display indeterminacy that do not involve any vague objects. And although Williamson (2003) has provided a general argument against de re indeterminacy, his conclusion rests on the specific limitations of his model theory of choice, rather than on any substantive metaphysical considerations. My diagnosis is that what kept philosophers from buying into MI was the lack of suitable concepts that would allow them to theorize about it, rather than any specific argument.3 The general attitude has indeed changed to some degree now that a number of characterizations of MI have been put forward, as in Akiba (2004), Barnes and Williams (2011), Darby and Pickup (2021), Smith and Rosen (2004), Torza (2021), and Wilson (2013). Potential manifestations of MI include (i) the ‘fuzzy’ objects of the macroscopic world, such as clouds, mountains and persons; (ii) future contingents and the open future; and (iii) quantum indeterminacy. Putative instances of iii include (iii.a) the failure of value definiteness of quantum observables; (iii.b) the vague identity of quantum objects; and (iii.c) the count indeterminacy arising in quantum field theory. Although a good part of what I will be saying is the result of general features of my favorite way of understanding MI, the focus of this chapter will be on iii.a. The failure of value definiteness is a typically quantum-mechanical phenomenon whereby a system fails to have any determinate value of an observable at a time. Given the eigenstate-eigenvalue link, the failure of value definiteness follows from the fact that a quantum state which is a superposition of eigenstates of an observable is in general not an eigenstate of that same observable.4 For example, a particle in

2

For example, it has been argued that objects can have indeterminate coincidence (Akiba, 2000), and indeterminate distinctness (Akiba, 2014). 3 Barnes (2010) has drawn a similar moral. 4 The eigenstate-eigenvalue link, a postulate of so-called ‘orthodox’ quantum mechanics (Gilton, 2016), states that a system has property O with value λ iff the quantum state of the system is in an eigenstate of the associated operator Oˆ with eigenvalue λ. It is worth mentioning that the orthodoxy of the eigenstate-eigenvalue link has been challenged by Wallace (2019). For discussion of the eigenstate-eigenvalue link vis-à-vis quantum indeterminacy, see Calosi and Wilson (2018), Fletcher and Taylor (2021).

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a superposition of position states has no determinate value of position. Although the interpretative details may differ depending on the philosophical methodology being employed, there appears to be a growing consensus that the failure of value definiteness constitutes evidence of MI (see Bokulich (2014), Calosi and Mariani (2020, 2021), Calosi and Wilson (2018, 2021), Darby (2010), Darby and Pickup (2021), Fletcher and Taylor (2021), Mariani (2021), Skow (2010), Torza (2020, 2021)). A dissenting voice is Glick (2017), who has argued that no evidence of MI is to be found in quantum theory. His overarching argument is as follows: 1. Orthodox quantum theory provides no evidence of fundamental MI. 2. The main realist interpretations of quantum theory provide no evidence of fundamental MI. 3. Therefore, the main interpretations of quantum theory provide no evidence of fundamental MI (from 1, 2). 4. MI cannot be derivative. 5. Therefore, the main interpretations of quantum theory provide no evidence of MI (from 3, 4). A comprehensive assessment of Glick’s argument lies outside the scope of this work (see Calosi and Mariani (2020), Calosi and Wilson (2021) for criticism). I am going to focus my attention on the thesis of line 4. Glick’s belief that there is no derivative MI can be evinced from the following passage, in which he argues that the failure of value definiteness does not bring about MI, if observables are derivative entities: “If, by contrast, one took the properties to be ontologically derivative and quantum states to be fundamental, there would be little room for metaphysical indeterminacy. [. . . ] Any indeterminacy would occur at the non-fundamental level and hence may be viewed as eliminable” (p. 206, my emphasis). But as already noted elsewhere, derivative does not amount to eliminable. We can all agree that tables are derivative entities (whatever ‘derivative’ means), while retaining our belief in the existence of tables. Indeed, the unpopularity of revisionist ontological doctrines, such as mereological nihilism, is partly explained by the fact that they demand us to give up on the existence of the medium-sized dry goods of naive physics. Perhaps Glick thinks that although not everything which is derivative is eliminable, some things are, and MI is one such thing. If that is the underlying thought, however, we have not been given any arguments. Here is a different line of reasoning that could be pursued on Glick’s behalf. In lieu of 4, one could think that there is derivative MI only if there is fundamental MI—in other words: 4∗ . MI cannot be merely derivative. By substituting 4∗ for 4 in the argument, Glick can still draw his conclusion. For if we have reason to reject fundamental MI, and if lack of fundamental MI entails lack of derivative MI, we have reason to reject MI, period. Interestingly, Barnes

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(2014) has offered a defense of 4∗ . If her disproof of merely derivative MI turns out to be conclusive, Glick could piggyback on that. I make two claims: Barnes’ argument for 4∗ is invalid; and, given my preferred characterization of MI, 4∗ is false. In this paper I will defend both claims, and conclude that for all we know merely derivative MI arises in quantum physics.

22.2 Metaphysical Indeterminacy As observed in the previous section, run-of-the-mill indeterminacy originates in the way we represent reality. According to the standard account, representational indeterminacy is semantic in character: it is rooted in the meaning of particular subsentential expressions, such as predicates and names. The go-to semantic theory of indeterminacy is the supervaluationism of Fine (1975), which characterizes a term as indeterminate just in case its meaning is compatible with different precisifications, that is to say, with different assignments of extensions in actuality.5 A sentence is said to be indeterminate in truth value just in case it is true on some precisifications, and false on others. The supervaluationist picture can be generalized in a most natural way by taking precisifications to be assignments of intensions, rather than extensions. Accordingly, a term is indeterminate just in case its meaning is compatible with different functions from worlds to extensions. A sentence is said to be indeterminate in content if its meaning is compatible with multiple functions from worlds to truth values—or, equivalently, with multiple sets of worlds. Now, let us identify coarse-grained facts (or states of affairs) with sets of worlds. Say that a fact F obtains at world w if w ∈ F ; and that F obtains simpliciter if it obtains at the actual world @. If sentence p is not indeterminate in content, let [p] be the fact that p; and if it is indeterminate in content, let [p]1 , [p]2 , . . . be the facts associated with the different precisifications of the language. The following holds: FACT 1.

On the supervaluationist picture, sentence p is indeterminate in truth value just in case (i) it is indeterminate in content, and (ii) @ ∈ [p]i and @ ∈ / [p]j , for some i, j .

The left-to-right direction of FACT 1 highlights that, on the most popular semantic account, all truth-value indeterminacy is indeterminacy in content. Nevertheless, there appear to be sentences having indeterminate truth value but determinate content, as exemplified by Aristotle’s problem of the open future.6 5

A precisification must be defined for all terms at once, in order to preserve penumbral connections, cf. Fine (1975, p. 271). 6 I am saying that there ‘appear’ to be such cases because, on two prominent characterizations of MI—namely, the metaphysical supervaluationism of Barnes and Williams (2011), and the determinable-based account of Wilson (2013)—MI does not involve truth-value gaps. However, there are independent reasons for being skeptical of such approaches, since metaphysical super-

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If there is no fact of the matter now as to whether there will be a sea battle tomorrow, the sentence ‘there will be a sea battle’ is neither true nor false. Yet each term occurring in it is semantically precise, and so the sentence cannot pick out different intensions on different precisifications. This class of cases suggests that the supervaluationist characterization of truth-value indeterminacy is too restrictive, as it prejudges the possibility of indeterminacy originating in the language-independent world. From now on I will therefore be assuming a definition of truth-value indeterminacy which is neutral as to the source of the indeterminacy, to the effect that p is indeterminate in truth value just in case p is neither true nor false. Truth-value indeterminacy in this sense is entailed by, but does not entail supervaluationist truth-value indeterminacy. The above considerations suggest a negative characterization of MI as indeterminacy that cannot be eliminated by precisifying the content of our assertions (Torza (2020), cf. Barnes (2010, p. 604)): IND − . MI arises if there is a sentence p

which is indeterminate in truth value but not

in content. According to IND− , MI occurs just when there is a sentence which is neither true nor false, and yet picks out exactly one fact. However, MI can also be characterized directly, as the phenomenon arising when there is no fact of the matter about something. Fleshing out this alternative characterization will require that we say more about the structure of logical space. A logical space is a space of possibilities. In order for a class of facts to constitute a logical space, they need to be closed under a number of logical operations such as negation, conjunction etc. (cf. Rayo (2017)).7 A caveat: logical operations, as objects in logical space mapping facts to facts, should not be confused with logical operators, which are linguistic items mapping formulas to formulas. For example, the negation operator ‘not’ is a logical constant having a negation operator as its semantic value. Likewise for conjunction, disjunction etc. Accordingly, if F is the fact that grass is green, the negation of F is the fact that grass is not green.

valuationism is unable to subsume quantum indeterminacy (Darby, 2010; Skow, 2010), whereas the determinable-based theory has a hard time making sense of a number of phenomena such as the open future, indeterminate identity, and indeterminate existence (Barnes & Cameron, 2016). Moreover, it has been argued that the determinable-based account is inadequate in the way it deals with quantum indeterminacy as well (Fletcher & Taylor, 2021; Torza, 2021). For a comparison between the present approach and the determinable-based account see Lewis (2022) in this volume. 7 However, Turner (2016) has defended the idea that the relations holding between facts are quite different from the familiar logical ones, and are akin to geometrical relations. Although Turner’s view is both fascinating and compelling, discussing it would take me far afield. Suffice to say that everything I say here could be restated within Turner’s theory.

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A logical space can be represented as a structure S = S, @, TS , −S , #S . . ., such that: 1. S is a set of states (worlds). Among them is a distinguished item @, the actual state.8 2. Facts are sets of states. The universal set S is the necessary fact; the empty set is the impossible fact. 3. A fact F is said to obtain at state w (in symbols, TS (F, w)) if w ∈ P ; it is said to obtain simpliciter if TS (F, @). 4. Logical operations are operations on facts: −S F is the negation of F ; F #S G is the conjunction of F and G; etc. One caveat: the above characterization is largely independent of questions in modal metaphysics, such as whether worlds are concrete or abstract, or about the nature of facts. All I am assuming is that logical space, whatever it is, instantiates the structure defined above. Likewise, when I speak of states (worlds) as points or vectors in a structure, it is being assumed that states (worlds) play the relevant structural role, and not that they are literally points or vectors. Armed with those tools we can now state the idea that, relative to a logical space S, MI amounts to there being no fact of the matter about something (Torza, 2021): IND + .

MI arises if there is a fact F such that neither F nor −S F obtains.

Prima facie, IND− and IND+ provide quite different characterizations of MI, in that the former is semantic in character, whereas the latter defines a property of logical space without making any detour through language. As it turns out, however, given some background assumptions the two characterizations are provably equivalent: Given a logical space S and a language L interpreted on S, there is a sentence p of L which is indeterminate in truth value but not in content iff there is a fact F in S such that neither F nor −S F obtains.9

FACT 2.

8

If the logical space is the space of a dynamical system, @ should be a function of time, rather than a constant. For present purpose, this complication can be set aside. 9 Proof. If p of L is determinate in content, it picks out a unique fact [p] in S . And if it is indeterminate in truth value, neither TS ([p], @) nor TS (−S [p], @) is the case. So, [p] is a fact such that neither it nor its negation obtains. Conversely, if F is a fact in S such that neither it nor its negation obtains, let p be a sentence of L such that [p] = F . Hence, p is not indeterminate in content. Moreover, p is neither true nor false. QED. This proof hinges on two background assumptions. One is that the object language L contains no irreferential terms, such as ‘Vulcan’ or ‘God’. For otherwise there could be a sentence p∗ such as ‘Vulcan is a gas planet’ which, by not picking out any fact, is indeterminate in truth value (on some semantic accounts, at least) but not in content (since indeterminacy in content requires that it pick out multiple facts); and yet there would be no fact [p∗ ] such that neither it nor its negations obtains (Torza, 2020). The other background assumption is that every fact F must be expressible in L. Insofar as one may reject either assumption, and so the equivalence, I take IND + to be my official characterization of MI. (Also notice that IND + , unlike IND − , does not involve any

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Because of the equivalence between IND− and IND+ , we can speak of MI with no ambiguity.

22.3 Fundamentality and Derivativeness Logical spaces of different kinds correspond to different logics. For example, classical logical spaces differ from intuitionistic logical spaces in that classical negation is involutive, unlike its intuitionistic counterpart. One fact that will play a crucial role in the ensuing discussion is that the class of states of a physical system can live in logical spaces of different kinds that agree about the assignment of values to physical quantities. Consider a Hilbert space H associated to a given quantum system. The class of all states (unitary vectors) in H can be embedded in a classical logical space C, where facts are arbitrary sets of vectors, negation is set-theoretic complementation, and disjunction is set-theoretic union. The same class of states can also be embedded in a quantum logical space Q, where facts are sets of vectors closed under linear combination, negation is orthocomplementation, and disjunction is span10 (Birkhoff & von Neumann, 1936). Note that the rays in H are maximally specific facts, i.e., facts about the system’s having a value of a particular observable. Since C and Q on H contain the same rays, the two spaces will agree with respect to the obtaining of facts about the assignment of values to physical quantities (e.g., about whether the system is spin up along a particular direction). In other words, the classical and the quantum logician will only disagree about the truth value of logically complex sentences—in particular, sentences that contain either a negation or a disjunction. Although a realist attitude towards orthodox quantum theory arguably favors Q over C as representing the space of possibilities associated with a quantum system (Torza, 2021, sec. 3.2; Fletcher & Taylor, 2021), there is no conclusive strategy for picking one option over the other on the basis of empirical evidence alone. So, if we think that there is such a thing as the One True logical space of a given quantum system, the choice appears to be underdetermined by the physics.11 This is relevant to the present discussion because, if it is underdetermined whether the states of a

representational machinery, and so can provide a reductive analysis of MI.) Nevertheless, it is both interesting and illuminating that IND + can be cast in semantic terms as IND − , given suitable qualifications. 10 The orthocomplement of a fact F is the set of vectors that are orthogonal to each vector in F ; the span of facts F, G is the closure of the union of F and G under linear combination. 11 Quantum logic was famously defended as the One True logic in Putnam (1968) on the grounds that it provides a solution to the measurement problem. Maudlin (2005) has argued against Putnam, and concluded that there is no reason to replace classical with quantum logic. Although I agree with Maudlin that quantum logic is of no help in addressing the measurement problem, I reject his conclusion. Indeed, quantum logic is a consequence of accepting either the eigenstate-eigenvalue link (Fletcher & Taylor, 2021) or the EPR criterion of reality (Torza, 2021, sec. 3.2).

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quantum system live in classical or quantum logical space, the question of quantum MI will also be underdetermined. For if the states of an arbitrary quantum system define a classical logical space, MI cannot arise. Indeed, it is trivially the case that every fact F of a classical logical space C is such that either it obtains or it does not obtain (i.e., either TC (F, @) or not TC (F, @)). But because the classical negation −C is complementation, it follows that F is such that either it or its negation obtains (i.e., either TC (F, @) or TC (−C F, @)). On the other hand, if the states of a quantum system define a quantum logical space, MI can and will arise. For example, when a system composed by a single electron is in a superposition of zspin states described by the equation |ψ = √1 (| ↑z e + | ↓z e ), neither the fact [e 2 is z-spin up] nor the fact [e is z-spin down] obtains. Since the quantum negation −Q is orthocomplementation, [e is z-spin down]=−Q [e is z-spin up]. Hence, the fact [e is z-spin up] is such that neither it nor its negation obtains (i.e., neither TQ ([e is z-spin up], @) nor TQ (−Q [e is z-spin up], @)). Since the empirical evidence does not let us select a unique way of carving the logical space associated with a quantum system, we seem to be faced with underdetermination about logic. This raises a challenge to logical realism, the view that there is One True logic, and that this is so in virtue of the way the mind and language-independent world is like. The logical realist could rejoin as follows: we can countenance multiple logical spaces, without thereby surrendering to conventionalism, as long as they are ordered with respect to their relative fundamentality. In that way, a physical system will be associated with One True logical space, namely the fundamental one, as well as a number of nonfundamental logical spaces.12 In the quantum case, talk of fundamental vs derivative logical spaces can be cashed out in (at least) two ways. The first strategy involves the realism about structure articulated and defended in Sider (2011)—very roughly, the idea that there is a metaphysically primitive and privileged way of carving reality into natural properties, facts, etc. A realist about structure who regards classical logic as fundamental will take there to be a metaphysically privileged way of carving out the class of states of a quantum system, namely the classical one. This brand of realist can countenance the existence of alternative ways of carving the same class of states into a logical space, as long as those ways are nonfundamental. The alternative carvings, despite being metaphysically second-rate, can be helpful relative to specific theoretical or practical goals. Of course, fundamentality talk is no solution to logical underdetermination if we have no criteria to identify which logical space is fundamental. Fortunately, there are sensible constraints on fundamentality that the realist can appeal to in order to make progress. One such constraint is that, given a physical system, every nonfundamental

12 For

a discussion of logical realism see McSweeney (2019), Tahko (2021).

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logical space must be reducible to the fundamental one in the following sense (Torza, 2021, sec. 6, cf. Sider, 2011, ch. 7): COMPLETENESS . Let W be a class of states. If S is the fundamental logical space on W , and S  is any logical space on W , every fact in S  is equivalent to some

fact in S.

The rationale for COMPLETENESS is that the facts of a nonfundamental space should be nothing over and above the facts in the fundamental space.13 In order to avoid irrelevant complications, I am taking fact equivalence to be cointensionality (although more fine-grained notions of fact equivalence could be employed, as in Correia and Skiles (2019), Dorr (2016)). Thus, facts are equivalent just in case they obtain at the exact same states. Since we are modeling facts as sets of states, and since sets are extensional entities, fact equivalence reduces to strict identity. Now, given a set W of states of a quantum system, it is compatible with COMPLETENESS that a classical logical space C on W be fundamental and a quantum logical space Q on W be nonfundamental, but not the other way around. For recall that in Q only sets of vectors closed under linear combination count as facts, whereas in C any arbitrary set of vectors is a fact. So, the facts in Q are a proper subset of the facts in C, which entails that every fact in Q is equivalent to a fact in C, but not vice versa. This guarantees that a realist about fundamental structure can regard a system’s classical space as fundamental and its quantum space as nonfundamental, but not vice versa. I now turn to a strategy for justifying fundamentality talk in the quantum case without relying on a metaphysical notion of structure. The reason why in quantum logic facts are sets closed under linear combination is that the quantum logician identifies facts with experimental facts, which is to say, facts whose (non)obtaining can be established by experimental means.14 But experiments are procedures for determining the value of some quantum observable (position, momentum, spin, etc.). Therefore, the facts that live in a quantum logical space Q are facts about the possible values of physical observables. For the classical logician, on the other hand, any set of vectors in a Hilbert space is a fact about the relevant system. So, the facts of a classical space C need not be associated with a possible experiment, nor are they defined by reference to physical observables. With that being said, here is the sketch of how classical and quantum logic may coexist as two pictures of one and the same reality. According to quantum logic, logical space is the space of experimental facts—the facts involving physical observables. This is the familiar picture suggested by orthodox quantum theory.

13 But

see Torza (2021, sec. 6) for an argument to the effect that COMPLETENESS is too stringent, and should be replaced with a weaker condition of ‘collective completeness’. Although that weakening has important consequences in the discussion on quantum MI, I must set it aside for reasons of space. 14 This is due to the constraints that the Born rule sets on the possible outcomes of quantum experiments (Birkhoff & von Neumann, 1936; Torza, 2021, sec. 3.2).

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At the fundamental level, however, reality does not involve either experiments or observables. Fundamentally, the world is isomorphic to a vector in Hilbert space, and facts are regions of that space, i.e., sets of possible positions for the state vector. This picture is exemplified by the Hilbert Space Fundamentalism which Sean Carroll defends in this volume: Here I want to argue for the plausibility of an extreme position among these alternatives, that the fundamental ontology of the world is completely and exactly represented by a vector in an abstract Hilbert space, evolving in time according to unitary Schrödinger dynamics. Everything else, from particles and fields to space itself, is rightly thought of as emergent from that austere set of ingredients. (Carroll, 2022)

If something like Hilbert Space Fundamentalism is true, the structure of observables encoded in Q is nonfundamental, and grounded in C. On this theory, logical space is fundamentally classical and derivatively quantum. I have outlined two ways of motivating the view that C is fundamental and Q is derivative: one from first principles, and one inspired by foundational work in physics. As I am about to argue, such a view bears on our central question. Let us first define what it is for MI to arise fundamentally, derivatively, and merely derivatively: FMI.

Given a class W of states, fundamental MI arises if MI arises relative to a fundamental logical space on W ; derivative MI arises if MI arises relative to a nonfundamental logical space on W ; merely derivative MI arises if MI arises derivatively but not fundamentally.

Since MI arises in quantum logical spaces, but not in classical logical spaces, we can draw the following corollary: be the class of possible states of a given quantum system. If C on W is fundamental, and Q on W is nonfundamental, MI will arise merely derivatively.

FACT 3. Let W

My first main claim is now established: the assumption that 4∗ . There is no merely derivative MI which I have put forward on behalf of Glick, is unjustified. Consequently, the revised argument against quantum MI from Sect. 22.1 is unsound. For all we know, MI arises in quantum mechanics.

22.4 Against Barnes I showed in the previous section that MI can be merely derivative, and I did so constructively by providing an example from quantum physics. However, Barnes (2014) has offered an argument purporting to show that merely derivative MI cannot possibly arise. The goal of this section is to show that her argument is inconclusive. Although replies to Barnes have already been offered (Eva, 2018; Mariani, 2021), I

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will put forward a different line of resistance based on the observation that in general there are multiple ways of carving logical space compatibly with the evidence. Barnes’ argument purports to show that derivative MI does not arise unless fundamental MI also arises. Let us start with some definitions, relative to a logical space S: 1. A set F of facts entails a fact F if F obtains whenever each fact in F obtains. 2. A set F of facts is said to be complete if it entails every fact or its negation. Here is a sketch of Barnes’ proof strategy, reformulated for consistency with the present conventions. First, she supposes that there are a fundamental description of reality, and a derivative description of reality. She also assumes crucially (and implicitly) that both the facts picked out by statements of the fundamental description and the facts picked out by statements of the derivative description—call them fundamental and derivative facts, respectively—coexist in the same logical space. Finally, she assumes that the set of obtaining fundamental facts is complete.15 Suppose now that the fundamental is determinate, i.e., that each fundamental fact is such that either it or its negation obtains. Since the fundamental facts form a complete set, they entail each derivative fact or its negation. The fundamental being determinate, it follows that each derivative fact is such that either it or its negation obtains. Therefore, the derivative is also determinate. It can be concluded that merely derivative indeterminacy cannot arise. The point where I distance myself from Barnes is the one I have flagged, which is her implicit assumption that both fundamental and derivative facts coexist in one logical space. As the quantum case suggests, such an assumption is unjustified, since different logical spaces can be defined on the same set of states, and different spaces correspond to descriptions of reality differing by their relative fundamentality. For example, we can describe a quantum system as fundamentally isomorphic to a vector in Hilbert space, which defines a classical logical space; and derivatively as made up of observable properties such as position and momentum, which defines a quantum logical space. But once we account for the existence of multiple logical spaces ordered by relative fundamentality, Barnes’ line of reasoning becomes invalid, as can be evinced from the following reformulation of her argument (tailored to the quantum case): Consider a quantum system associated with a fundamental classical logical space C and a nonfundamental quantum logical space Q. Because MI does not arise in C , the set C of facts that obtain in C is complete. Now pick a fact F in Q. Because the facts in Q are a subset of the facts in C , F is also a fact in C . Therefore, C entails either F or its negation. It follows that either F or its negation obtains. Hence, every fact in Q is such that either it or its negation obtains. Therefore, merely derivative MI does not arise in Q.

15 This

assumption is Barnes’ version of the COMPLETENESS condition I discussed in Sect. 22.3.

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The above argument is invalid because “either F or its negation obtains” is ambiguous between the following two readings: α. Either F or −C F obtains. β. Either F or −Q F obtains. Notice that “C entails either F or its negation” should be read as “C entails either F or −C F”, since the entailment takes place in C. From that α follows. But it does not follow from α that every fact in Q is such that either it or its negation obtains, because in Q the negation of F is −Q F , and the obtaining takes place in Q. On the other hand, whereas it follows from β that every fact in Q is such that either it or its negation obtains, β does not follow from the fact that C entails either F or its negation −C F . The moral should be straightforward. It did not occur to Barnes that logical operations are relative to a logical space, and that one and the same set of states can be embedded in different logical spaces endowed with different sets of operations. In the present case, although classical negation prevents MI from arising in C, quantum negation allows for gaps in Q and, therefore, for merely derivative MI. Thus, Barnes’ argument fails because it rests on the unwarranted assumption that the fundamental and the derivative obey the same logic. One might object that Barnes and I are talking past each other, since we assume different characterizations of MI. Now, it is true that her own characterization of MI—the metaphysical supervaluationism developed for example in Barnes and Williams (2011)—neither requires nor postulates gaps in logical space, unlike the view I articulated in Sect. 22.2. But our disagreement concerning the nature of MI is irrelevant to the question as to whether her argument against merely derivative MI is valid. For although on her theory the failure of bivalence is not a necessary condition for MI, it still is a sufficient condition. Indeed, metaphysical supervaluationism says that MI arises just when there is a fact F such that it neither determinately obtains nor determinately fails to obtain. But if a fact does not obtain, it does not determinately obtain; and if it does not fail to obtain, it does not determinately fail to obtain. So, there is MI in my sense only if there is MI in Barnes’ sense (although not vice versa). If MI in my sense arises in Q, it will also arise according to Barnes.16

22.5 Conclusions I have sketched two ways of motivating the view that every quantum system is associated with a fundamental classical logical space, as well as a nonfundamental quantum logical space—one based on realism about fundamental structure, and one based on foundational work in physics. Since MI arises in quantum logical spaces

16 It is also worth mentioning that Darby (2010), Skow (2010) and Fletcher and Taylor (2021) have independently argued that metaphysical supervaluationism is unable to model quantum MI.

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but not in classical logical spaces, I have concluded that quantum physics can be interpreted as giving rise to merely derivative MI, which is to say, MI arising only at the nonfundamental level. This result has a twofold corollary: it undermines an argument against quantum MI which improves on the one offered in Glick (2017); and it provides a counterexample to an argument against merely derivative MI due to Barnes (2014).

References Akiba, K. (2000). Vagueness as a modality. The Philosophical Quarterly, 50(200), 359–370. Akiba, K. (2004). Vagueness in the world. Noûs, 38, 407–29. Akiba, K. (2014). A defense of indeterminate distinctness. Synthese, 191(15), 3557–3573. Barnes, E. (2010). Ontic vagueness: A guide for the perplexed. Noûs, 44(4), 601–627. Barnes, E. (2014). Fundamental indeterminacy. Analytic Philosophy, 55(4), 339–362. Barnes, E., & Cameron, R. (2016). Are there indeterminate states of affairs? No. In E. Barnes (Ed.), Current controversies in metaphysics (pp. 120–132). Routledge. Barnes, E., & Williams, J. R. G. (2011). A theory of metaphysical indeterminacy. In D. Zimmerman & K. Bennett (Eds.), Oxford studies in metaphysics (Vol. 6, pp. 103–48). Oxford: Oxford University Press. Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. The Annals of Mathematics, 37(4), 823–843. Bokulich, A. (2014). Metaphysical indeterminacy, properties, and quantum theory. Res Philosophica, 91(3), 449–475. Calosi, C., & Mariani, C. (2020). Quantum relational indeterminacy. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 71, 158– 169. Calosi, C., & Mariani, C. (2021). Quantum indeterminacy. Philosophy Compass. https://doi.org/ 10.1111/phc3.12731 Calosi, C., & Wilson, J. M. (2018). Quantum metaphysical indeterminacy. Philosophical Studies, 38(3), 1–29. Calosi, C., & Wilson, J. (2021). Quantum indeterminacy and the double-slit experiment. Philosophical Studies, 178(10), 3291–3317. Carroll, S. (2022). Reality as a vector in Hilbert space. In V. Allori (Ed.), Quantum mechanics and fundamentality. Naturalizing quantum theory between scientific realism and ontological indeterminacy. Synthese Library. Springer. Correia, F., & Skiles, A. (2019). Grounding, essence, and identity. Philosophy and Phenomenological Research, 98(3), 642–670. Darby, G. (2010). Quantum mechanics and metaphysical indeterminacy. Australasian Journal of Philosophy, 88(2), 227–245. Darby, G., & Pickup, M. (2021). Modelling deep indeterminacy. Synthese, 198, 1685–1710. Dorr, C. (2016). To be F is to be G. Philosophical Perspectives, 30(1), 39–134. Dummett, M. (1975). Wang’s paradox. Synthese, 30(3–4), 201–232. Eva, B. (2018). Emerging (in)determinacy. Thought: A Journal of Philosophy, 7(1), 31–39. Evans, G. (1978). Can there be vague objects? Analysis, 38(4), 208. Fine, K. (1975). Vagueness, truth, and logic. Synthese, 30(3–4), 265–300. Fletcher, S. C., & Taylor, D. E. (2021). Quantum indeterminacy and the eigenstate-eigenvalue link. Synthese, 199(3–4), 1–32. Gilton, M. J. R. (2016). Whence the eigenstate-eigenvalue link? Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 55, 92–100.

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Glick, D. (2017). Against quantum indeterminacy. Thought: A Journal of Philosophy, 6(3), 204– 213. Lewis, D. (1986). On the plurality of worlds. Wiley-Blackwell. Lewis, P. J. (2022). Explicating quantum indeterminacy. In V. Allori (Ed.), Quantum mechanics and fundamentality. Naturalizing quantum theory between scientific realism and ontological indeterminacy. Synthese Library. Springer. Mariani, C. (2021). Emergent quantum indeterminacy. Ratio, 34(3), 183–192. Maudlin, T. (2005). The tale of quantum logic. In Y. Ben-Menahem (Ed.), Hilary Putnam (pp. 156–187). Cambridge University Press. McSweeney, M. M. (2019). Logical realism and the metaphysics of logic. Philosophy Compass, 14(1), e12563. Putnam, H. (1968). Is logic empirical? In R. Cohen & M. Wartofsky (Eds.), Boston studies in the philosophy of science (Vol. 5, pp. 216–241). Dordrecht: Reidel. Rayo, A. (2017). The world is the totality of facts, not of things. Philosophical Issues, 27(1), 250– 278. Russell, B. (1923). Vagueness. Australasian Journal of Philosophy, 1(2), 84–92. Salmon, N. (1981). Reference and essence. Princeton University Press. Sider, T. (2011). Writing the book of the world. Oxford University Press. Skow, B. (2010). Deep metaphysical indeterminacy. Philosophical Quarterly, 60(241), 851–858. Smith, N. J. J., & Rosen, G. (2004). Worldly indeterminacy: A rough guide. Australasian Journal of Philosophy, 82, 185–198. Tahko, T. (2021). A survey of logical realism. Synthese, 198(5), 4775–4790. Torza, A. (2020). Quantum metaphysical indeterminacy and worldly incompleteness. Synthese, 197, 4251–4264. Torza, A. (2021). Quantum metametaphysics. Synthese, 199(3), 9809–9833. Turner, J. (2016). The facts in logical space: A tractarian ontology. Oxford University Press. Wallace, D. (2019). What is orthodox quantum mechanics? In A. Cordero (Ed.), Philosophers look at quantum mechanics. Springer. Williamson, T. (2002). Vagueness. Routledge. Williamson, T. (2003). Vagueness in reality. In M. J. Loux & D. W. Zimmerman (Eds.), The Oxford handbook of metaphysics. Oxford University Press. Wilson, J. M. (2013). A determinable-based account of metaphysical indeterminacy. Inquiry, 56(4), 359–385.

Chapter 23

Explicating Quantum Indeterminacy Peter J. Lewis

Abstract In recent years there has been a robust but inconclusive debate over the existence and nature of indeterminacy in the world as described by quantum mechanics. I suggest that the inconclusive nature of the debate stems from starting from a metaphysical theory of indeterminacy. I propose instead framing the issue as a Carnapian explication project: start with the informal notion of indeterminacy used by physicists, and consider how best to make that concept precise. I defend a precisification based on von Neumann’s interpretation of quantum states, and consider the nature of the indeterminacy that results.

23.1 Quantum Indeterminacy: The Challenges The dispute about whether there is indeterminacy in the world is long and inconclusive. At first glance, it seems like quantum mechanics ought to provide a quick, empirical resolution to the debate: prima facie, a photon in a superposition of right-polarized and left-polarized states has an indeterminate polarization. But quantum mechanics has not provided any such resolution; the controversy drags on. In this paper, I suggest some reasons for this impasse, and lay out a path forward. The typical argument that quantum mechanics exhibits genuine indeterminacy goes something like this: First, a general account of indeterminacy is constructed, and second, it is argued that quantum mechanics is an instance of that general account (e.g. Calosi & Wilson, 2019). But this strategy runs into problems at each step. At the first step, the general accounts of indeterminacy are controversial, both individually (Darby, 2010; Glick, 2017) and because they compete with each other (Wilson, 2013). And at the second, even though quantum mechanics is arguably our best case for an instance of indeterminacy in the world, it remains a deeply problematic theory. There is no consensus over what, if anything, quantum

P. J. Lewis () Dartmouth College, Hanover, NH, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_23

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mechanics tells us about the nature of the physical world (Lewis, 2016), and so any proposed argument from quantum mechanics to indeterminacy faces the objection that it rests on a mistaken interpretation of quantum mechanics (Glick, 2017). Of course, this might just be the nature of philosophical debate. But I want to suggest a strategy that might avoid some of these controversies. The strategy is grounded in a broadly Carnapian methodology. As long as we see the issue of indeterminacy as one of discovery, analogous to scientific discovery of empirical facts, the impasse over its existence seems inevitable. Both the description of what it is we are aiming to discover in the world, and the claim that in quantum mechanics we have discovered it, remain controversial. But once we realize that the issue turns in large part on constructing an appropriate conceptual structure to use, we can make progress. In his Logical Foundations of Probability, Carnap suggests that constructing an appropriate concept is a matter of explication. Explication consists of taking an imprecise “prescientific” concept and transforming it into a precisely-defined “scientific” concept (Carnap, 1950, 3). Of course, a concept can be precisified in a number of different ways; Carnap suggests that we should judge potential explications by how fruitful they are likely to be for the scientific enterprise, weighing this against the potentially competing considerations of simplicity and similarity to the prescientific conception (1950, 5). The targets of explication for Carnap in this work are the concepts of confirmation and probability: they are used extensively but without precision in the sciences, and by paying attention to the way they are used, we can see how to make them precise in a scientifically fruitful way. My contention here is that the same is true of indeterminacy. That is, by looking at quantum mechanics, we find widespread appeal to an imprecisely defined concept of indeterminacy, and we can also see how that concept can be made more precise in a way that is likely to be fruitful within physics. In the next section, I argue that indeterminacy plays an important role in quantum mechanics. In Sect. 23.3, I argue that constructing a precise version of this concept is straightforward: the structure needed to represent indeterminacy precisely is already present in the Hilbert-space formalism, and indeed a canonical interpretation of the formalism by von Neumann already provides the required precisification. In Sect. 23.4, I defend the resulting account of indeterminacy: the road to this account passes a number of conceptual choice-points leading to different interpretations of quantum states, but I argue that the canonical one is superior as an explication. In Sect. 23.5, I consider some more general questions about the nature of the concept of indeterminacy explicated here: is it of the same kind as everyday instances of indeterminacy, e.g. concerning the extent of the Australian Outback, and is it genuinely metaphysical indeterminacy?

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23.2 The Role of Indeterminacy in Quantum Mechanics Open any textbook on quantum mechanics, and you will find a passage something like this: Let us summarize our description of the z component of the angular momentum of a photon. Corresponding to this physical quantity we have an operator, i.e. a matrix, S. Photons that are in an eigenstate, | R or | L, of this operator can be assigned a definite value of the z component of angular momentum . . . Any other photon state | ψ cannot be assigned a definite value of angular momentum. (Baym, 1969, 17)

If we read “definite” as “determinate”, this says that in many states (in fact, in all states except for a few special ones), the photon’s angular momentum is indeterminate. Similar language can be found in the experimental literature. For example, Juffmann et al. (2009) produce multi-slit self-interference between C60 molecules, and then image the individual molecules making up the resulting interference pattern. They describe the imaged individual molecules as “localized”, but the molecules passing through the slits as “delocalized” (Juffmann et al., 2009, 3); again, we can read the positions of the former as determinate and the positions of the latter as indeterminate. Of course, the fact that physicists use a concept of indeterminacy doesn’t tell us what kind of concept they are using. The textbook passage is closely followed by a description of the Born rule, via which a state fixes the probabilities that the various possible values of angular momentum would be revealed by an appropriate measurement. Since probabilities are often interpreted epistemically, this is naturally read as suggesting that a photon whose state is not an eigenstate nevertheless has a perfectly determinate angular momentum property, but one that we do not know prior to measurement. In that case, the concept of indeterminacy would apply to our knowledge of the world, not to the world itself. However, an epistemic understanding of the quantum state faces well known difficulties, in the form of various no-go theorems (Bell, 1964; Kochen & Specker, 1967; Pusey et al., 2012). These theorems each show that, subject to plausible assumptions, an epistemic understanding of the quantum state is incompatible with the predictions of the theory. This does not mean that an epistemic understanding of the quantum state is absolutely precluded; the assumptions underlying each theorem can be denied (Leifer, 2014). But for present purposes I will set epistemic interpretations of the quantum state aside. I also assume that we take a form of realism for granted; that is, we don’t take the quantum state to be merely a guide to what we should believe (Healey, 2012). In other words, for present purposes I assume that the quantum state is a representation of the world rather than of our knowledge. Given that assumption, the conclusion that a concept of worldly indeterminacy plays an important role within quantum mechanics apparently follows straightforwardly. The above-mentioned photon, when its state is not | R or | L, has a non-zero probability, on measurement, of being found to be right-polarized and a non-zero probability of being found to be left-polarized. If these probabilities are not to be understood epistemically, it looks like we must say that the photon has neither

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polarization prior to measurement. If the state is not to be understood instrumentally, it looks like a non-eigenstate describes a state in which the photon has neither polarization. But since, due to quantization, right-hand and left-hand polarization are the only polarization properties the photon can (determinately) have, the natural thing to say is that the photon’s polarization is indeterminate prior to measurement. Furthermore, this kind of indeterminacy arguably plays a role in the kind of explanations quantum mechanics gives. Consider self-interference of C60 molecules again. The “delocalization” of each molecule as it passes through the slits is apparently required to explain interference; a molecule that passes determinately through one slit cannot exhibit self-interference. If the superposition is understood in terms of molecules with indeterminate locations, an explanation of interference is at least possible, insofar as the set of trajectories between which each molecule is indeterminate includes trajectories passing through both slits and converging at the screen (Calosi & Wilson, 2021). To the extent that physicists think of interference phenomena in terms of overlap of delocalized trajectories of a single molecule, it looks like they are helping themselves to the notion that molecule trajectories can be indeterminate. Indeed, part of the motivation of researchers like Juffmann et al. (2009) in exploring interference effects in larger and larger molecules is to find the extent of this indeterminacy: where does the indeterminacy characteristic of the quantum micro-world give way to the determinacy of the observable macro-world? It would be going too far to assert that quantum mechanics requires a concept of indeterminacy: as we shall see in Sect. 23.4, it is possible to eliminate the indeterminacy from quantum mechanics instead of explicating it. But for the moment, let us proceed under the assumption that indeterminacy is a concept worth retaining, and consider how it should be made precise.

23.3 The Structure of Quantum Indeterminacy In fact, such a precisification has largely been done for us, and indeed is almost as old as quantum mechanics itself. In the first systematic account of the structure of quantum mechanics, von Neumann (1932) constructs an interpretation that incorporates indeterminacy. We can think of his interpretation as an explication of quantum indeterminacy in Carnap’s (1950) sense—a way of making the concept precise in such a way as to be fruitful for the fledgling discipline of quantum mechanics. The state of a system is represented in quantum mechanics by a vector in a Hilbert space. An observable—a set of experimentally accessible, mutually exclusive properties of the system—is represented by a Hermitian operator on the Hilbert space that takes a vector as input and produces a vector as output. And, as noted in the previous section, a system (determinately) has one of the properties that make up the observable if, and only if, its state is an eigenstate of the corresponding

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operator—a vector that is mapped to a scalar multiple of itself by the operator.1 The eigenstates of a Hermitian operator are mutually orthogonal vectors. A system whose state is not one of the eigenstates of the operator does not determinately have any property from among those that make up the observable, and is, to that extent, indeterminate. But here we need to be a little more careful. In the Juffmann apparatus, a C60 molecule that is “delocalized”—that is not determinately approaching any specified slit—can still be described as determinately “encountering” the slits, since it is determinately located in the tube containing the slits and not anywhere else (Juffmann et al., 2009, 2). The molecule does not determinately have a location at sufficiently fine grain, but nevertheless if we consider a disjunction of those fine-grained locations, the molecule determinately has the disjunctive property. To understand indeterminacy, then, we need to understand its logic. This is precisely what von Neumann (1932, 130–134) provides us with. Consider an operator with n mutually orthogonal eigenstates φ1 through φn , corresponding to n mutually exclusive properties p1 through pn . Let Pi be the proposition that the system has property pi . Then Pi obtains (the system has property pi ) iff the state is φi . The disjunction Pi ∨ Pj obtains (the system has pi or pj ) iff the state lies in the subspace spanned by φi and φj —if it lies somewhere in the plane formed by arbitrary weighted sums αφi + βφj . The negation ¬Pi obtains (the system lacks pi ) iff the state is orthogonal to φi . More generally, for propositions Qi and Qj that may themselves be compound, Qi ∨ Qj obtains iff the state lies in the subspace formed by arbitrary weighted sums of the vectors in the subspaces corresponding to Qi obtaining and to Qj obtaining. The negation ¬Qi obtains iff the state lies in the orthocomplement of Qi —the subspace formed by arbitrary weighted sums of vectors orthogonal to every vector in the subspace corresponding to Qi obtaining. This logic has become known as quantum logic (Birkhoff & von Neumann, 1936). Quantum logic is non-classical: bivalence fails, as does the compositionality of disjunction. In particular, since the state of a system can be such that it is neither in Qi nor orthogonal to Qi , the system can be indeterminate with respect to the corresponding property, in the sense that it neither has it nor lacks it. Indeed (as Torza (2021) suggests2 ), we can define indeterminacy in this way: Definition 1: A system exhibits indeterminacy iff there is at least one property p such that the system neither has p nor lacks p.

The eigenstates of a Hermitian operator form a complete basis: they span the vector space, so that any vector in the space can be expressed as a linear combination of eigenstates. This means that the set of mutually exclusive properties {p1 , p2 , . . . pn } picked out by an operator is complete in a corresponding sense: the disjunction P1 1

From hereon I will generally drop the word “determinately”: to say that a system has a property (lacks a property) means that it determinately has that property (determinately lacks that property). 2 Torza (2021) defines indeterminacy in terms of facts rather than property possession. The definition here can be regarded as a special case of Torza’s for facts about the possession of a property; this is the only kind of fact considered in this paper.

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∨ P2 . . . ∨ Pn necessarily obtains.3 But it does not follow that the system has any one of the basic properties: it can be indeterminate whether the system has pi for each pi . This gives us an alternative (and equivalent4) definition of indeterminacy: Definition 2: A system exhibits indeterminacy iff it fails to have each of a non-empty complete set of mutually exclusive properties {p1 , p2 , . . . pn }, where a set of properties is complete iff the disjunction P1 ∨ P2 . . . ∨ Pn necessarily obtains.

Given this precisification of the concept of indeterminacy, we can see how the claims of physicists about the properties of quantum systems make sense. Consider C60 interference again, and consider a location observable whose eigenstates correspond to the molecule being in a particular small region of space, say a 100 nanometer cube. A C60 molecule passing through the slits is not in an eigenstate of this location observable, and hence cannot be said to determinately pass through any particular slit. Nevertheless, the state of the molecule does lie within the subspace corresponding to the large disjunction of these small regions that makes up the tube leading to the slits; hence the molecule can correctly be described as encountering the slits.5 The location of the molecule is indeterminate at one scale, but not at another.

23.4 The Conceptual Landscape The account of indeterminacy in quantum mechanics presented so far has historical precedence: it is the earliest, and has become canonical. But that doesn’t mean that it is the best account. In laying out the account, I have breezed past some conceptual choice-points. In particular, there are choices to be made concerning the association of properties with quantum states, the logic obeyed by those property ascriptions, and the definition of indeterminacy. So let us revisit these choices a little more systematically to better examine whether the original explication of indeterminacy is the one we should retain.

3

The nature of the necessity here is an interesting question. It is not a logical necessity that a disjunction of this form is true. Rather, the source of the necessity is the property structure of the world. Does this make it a metaphysical necessity? This question is briefly taken up in Sect. 23.5. 4 Suppose a system satisfies Definition 1, and let q be the property of lacking p. Then the set {p, q} is a non-empty complete set of mutually exclusive properties, and the system fails to have each of them, hence satisfying Definition 2. Conversely, suppose a system violates Definition 1: for each property pi in {p1 , p2 , . . . pn }, it either has pi or lacks pi . Since there is no vector that is orthogonal to every eigenstate φi , the system cannot lack every property in the set. Hence it must have some property in the set, in violation of Definition 2. 5 It is worth noting that the state of the molecule will not be exactly an eigenstate of being located in the tube, since it will have a very small term corresponding to tunneling through the tube. Here and in the following I assume that such small terms can be ignored, but the justification for this claim would take us too far afield; see Lewis (2016, 86).

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First, I followed the textbooks in asserting that, for a given observable, a system determinately has a property from among those making up the observable when its state is an eigenstate of the corresponding operator, and otherwise is indeterminate regarding those properties. So, for example, the z-spin operator for a spin-1/2 particle has two eigenstates, spin-up and spin-down. The particle has the z-spinup property when its state is the spin-up eigenstate, the z-spin-down property when its state is the spin-down eigenstate, and otherwise has indeterminate z-spin. Call this non-classical property ascription. This is not the only way to describe these states. One could instead say that, for a given observable, a system determinately has a property from among those making up the observable when its state is an eigenstate of the corresponding operator, and otherwise determinately lacks that property. So, for example, a spin-1/2 particle has the z-spin-up property when its state is the spin-up eigenstate, the z-spindown property when its state is the spin-down eigenstate, and otherwise lacks both the z-spin-up property and the z-spin-down property. Call this classical property ascription.6 The choice regarding property ascription is clearly connected to the choice of logic. Non-classical property ascription allows that a system can neither have nor lack a given property, which is most naturally accommodated by a three-valued logic such as quantum logic: for a non-eigenstate of z-spin, it is neither true nor false that the system has z-spin-up. Classical property ascription allows the adoption of a fully classical logic: for every property, a system either has it or lacks it, and for a non-eigenstate of z-spin, it is false that the system is z-spin-up and false that it is z-spin-down. This pair of choices is also connected to the choice of a definition of indeterminacy. The former pair is naturally associated with Torza’s (2021) definition of indeterminacy: a system exhibits indeterminacy iff there is at least one property p such that the system neither has p nor lacks p. The latter pair is more naturally associated with Wilson’s (2013) definition of indeterminacy: a system exhibits indeterminacy iff it lacks every determinate property for a given determinable property. So, for example, a system that is not in an eigenstate of z-spin lacks the z-spin-up property and lacks the z-spin-down property, and these are the only two determinate properties for the z-spin determinable.7 So we have two packages of conceptual choices: Torza’s (2021) and Wilson’s (2013). Both yield the conclusion that there is indeterminacy in quantum mechanics, via different routes. But there are other combinations of choices that do not yield indeterminacy. If we combine classical property ascription and classical logic with Torza’s definition of indeterminacy, we get a conceptual structure according to 6

Classical property ascription violates von Neumann’s association of every proposition with a projection: “The particle lacks the z-spin-up property” does not correspond to a projection. This asymmetry between having a property and lacking a property may or may not be problematic, depending on whether one thinks there is a legitimate distinction here. 7 See also Bokulich (2014) and Wolff (2015) for discussion of a determinable-based definition of indeterminacy.

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which non-eigenstates determinately lack the relevant property, so that there is no property such that a system neither has it nor lacks it, and hence no indeterminacy.8 Similarly, if we combine non-classical property ascription and quantum logic with Wilson’s definition of indeterminacy, we get a conceptual structure according to which non-eigenstates do not determinately lack the relevant property, hence there is no state that determinately lacks every property for a determinable, and hence no indeterminacy. How should we judge these competing conceptual structures? Recall Carnap’s (1950, 5) four criteria for evaluation of a potential explication: the explicated concept should be exact and fruitful, and it should also be simple and similar to the imprecise concept on which it is based. These criteria may compete with each other; it is the best overall package that should be adopted. I submit that explications that do away with indeterminacy altogether fare poorly under these criteria. Elimination of an imprecise prescientific concept is typically an option in explication projects; one can replace the imprecise concept with nothing, and let other concepts do the work. Sometimes elimination is a reasonable option. For example, it is reasonable to hold that the only biologically respectable explication of race is one that does away with the concept altogether (Spencer, 2018). But elimination is clearly a radical option, faring very poorly on the similarity criterion—the null concept is clearly very different from any substantive prescientific concept. It is only a reasonable option when retaining the concept in question would be worse. In the case of race, a precise, biologically respectable definition arguably fares poorly on the simplicity criterion—it would have to be gerrymandered and disjoint in order to bear any relation to our ordinary race concept—and also on the fruitfulness criterion—generalizations over races may fail precisely because of this disjoint structure. In the case of indeterminacy, there are no such downsides to retaining the concept. The conceptual choices that eliminate indeterminacy are no simpler than those that retain it: in both cases, you need to pair a particular rule for property ascription and its associated logic with a particular definition of indeterminacy, so simplicity is on a par.9 And the concept of indeterminacy, as noted in Sect. 23.2, has been quite fruitful in the development of quantum mechanics, appearing in standard expositions and arguably motivating a good deal of experimental work. So while skeptics like Glick (2017) are surely right that we can do without a concept of indeterminacy in describing the quantum world, this doesn’t address the question of whether we should. Glick’s “sparse view” of properties holds that a system that lacks each determinate property for a determinable also thereby lacks 8

For this reason, Torza (2020, 4261) argues that adopting a non-classical logic is essential to an account of quantum indeterminacy. But as Calosi and Wilson (2021) note, this is only so if one also adopts Torza’s preferred definition of indeterminacy; it does not follow under Wilson’s definition. 9 There may be simplicity trade-offs: a classical logic is arguably simpler than a quantum logic, but Torza’s definition of indeterminacy is arguably simpler than Wilson’s. My point is just that there is no clear winner on simplicity grounds.

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the determinable, and hence does not exhibit indeterminacy according to the letter of Wilson’s definition. This has the effect of combining classical property ascription with an account of indeterminacy according to which a system that lacks each determinate property for an observable does not constitute a case of indeterminacy. This combination of conceptual choices was considered above; there is nothing wrong with it as a conceptual structure, but since the concept of indeterminacy is a fruitful one, it is not one we should feel motivated to adopt. The remaining question, then, is how we should judge the two packages of views that endorse quantum indeterminacy: Torza’s and Wilson’s. This is a trickier issue, and perhaps less important—either will probably do fine. My endorsement of Torza’s scheme above largely follows from considerations of similarity: it sticks closer to the way physicists have understood the theory—e.g. through the work of von Neumann—and this mode of understanding has also been extremely fruitful. To be more specific: according to Wilson’s scheme, a particle that is not in a location eigenstate (at a certain degree of fine-grainedness) determinately lacks each location, and hence lacks their disjunction. Torza (2020, 4262) interprets this as meaning that the particle fails to have a position—that it is nowhere in space. Calosi and Wilson (2021) correct this interpretation: being located in space, they say, is a matter of having a determinable property, which cannot be reduced to a disjunction of determinates. But the disjunctive account of coarse-grained location and its associated non-classical logic are part and parcel of the way physicists think about quantum systems: they are encapsulated in von Neumann’s interpretation, which is, after all, canonical. One could replace this familiar conceptual structure with an unfamiliar structure of determinables and determinates extracted from the metaphysics literature, but it takes quite a lot of philosophical work to spell out this alternative, and such a revision seems ultimately unnecessary. It is worth noting, though, that a Carnapian approach embraces conceptual pluralism. I have defended a particular approach to indeterminacy based on fruitfulness for physics and similarity to concepts already used by physicists. But there may be other goals, and other constituencies, relative to which Wilson’s approach is preferable. And even in physics, it may prove useful to employ classical property ascription and classical logic on occasion. For example, when explaining how von Neumann’s interpretation of quantum state works, it may be useful to stress that when a particle is in a superposition of being located at A and being located at B, it is not located at A, and it is not located at B, so there is a sense in which “The particle is either at A or at B” is false. Two further views of indeterminacy are worth commenting on. First, I have been concentrating on Wilson’s “gappy” account of indeterminacy, according to which a system exhibits indeterminacy iff it lacks each determinate property for some determinable. But she also constructs a “glutty” account of indeterminacy (2013, 367), and promotes it as an account of quantum indeterminacy (Calosi & Wilson, 2021). According to the “glutty” account, a system exhibits indeterminacy if it has (to some degree) more than one determinate property for some determinable. It is natural to combine this with classical logic and an account of property possession according to which a system possesses a property (to some degree) unless its state

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is orthogonal to the relevant eigenstate. Again, we get a serviceable account of indeterminacy. This account has the advantage that it allows us to take at face value physicists’ claims that e.g. a particle can be “in two places at once”, whereas if we adopt Torza’s account have to interpret this as loose talk that should be eliminated from precise science. But accepting a “glutty” account adds a further complication in addition to an appeal to the determinate/determinable distinction: it needs an account of relativized or partial property possession, and this is neither simple nor familiar. Second, there is another prominent account of indeterminacy in literature, namely the precisificationist approach of Barnes and Williams (2011). According to this account, a system exhibits indeterminacy iff there are two or more precisificational possibilities—complete, classical property ascriptions—each of which does not misrepresent reality. I don’t think this account fits well with the kind of indeterminacy we find in the quantum world. As Darby (2010) and Skow (2010) argue, the precificational possibilities violate quantum no-go theorems: a system that satisfies a complete, classical property ascription could not reproduce quantum mechanical empirical predictions. If the precificational possibilities don’t reproduce well-confirmed quantum mechanical predictions, it looks like they do misrepresent reality: a quantum system couldn’t be represented by any of the precisificational possibilities and still behave like a quantum system. While a precisificationist account may apply well to cases of vagueness, it doesn’t apply so well to quantum indeterminacy, where the indeterminacy does not (obviously) result from any kind of vagueness. The relationship between quantum indeterminacy and indeterminacy due to vagueness is taken up in the following section.

23.5 Remaining Issues I have argued for a particular account of quantum indeterminacy over others. In particular, I have argued that there is no reason to reject the canonical precisification of quantum indeterminacy based on the work of von Neumann in favor of an account based on concepts drawn from metaphysics. But there are further important issues that I can only touch on briefly here and will have to remain for future work. The first concerns the relationship between quantum indeterminacy and common or garden indeterminacy. Is the indeterminacy one finds in quantum mechanics of the same kind as the indeterminacy one finds in the everyday world, such as the extent of the Outback or of my lawn? If not, this might lead to a kind of skepticism: although we have used the word “indeterminacy” to describe the concept we are explicating, if it is not the same as everyday indeterminacy, perhaps we are not explicating indeterminacy at all, but rather some sui generis quantum concept that we might better call superposition. There are clearly important differences between quantum indeterminacy and ordinary indeterminacy. As just noted, ordinary indeterminacy involves vagueness, whereas there is no obvious vagueness in the quantum case: under either classical or

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non-classical property ascription, the line between having a property and lacking it, or between having it and being indeterminate, is perfectly precise. Furthermore, ordinary indeterminacy generally disappears when one moves to a more fundamental description: while the extent of my lawn may be indeterminate, the exact proportion of grass to other plants at any location is not indeterminate, or at least, not in the same way. In the quantum case, on the other hand, it is not clear whether any more fundamental description is available. Admittedly, various underlying ontologies for the quantum world have been posited, for example by Bohmian, spontaneous collapse, and many-worlds theories of quantum mechanics (Lewis, 2016, 179). Glick (2017) argues against quantum indeterminacy in part based on the claim that these underlying ontologies are always fully determinate; indeed, one might take it as a desideratum that the properties of the underlying ontology should be determinate. But all of these posited underlying accounts face problems (Lewis, 2016, 70), and even setting these problems aside, it is not obvious that reducibility to a determinate underlying ontology undermines claims to indeterminacy at the higher level (Calosi & Mariani, 2021). In fact, if quantum indeterminacy is reducible to a determinate underlying ontology, the quantum case parallels the ordinary case quite closely. But even if there are differences between quantum indeterminacy and ordinary indeterminacy, there are also significant similarities, in particular regarding property ascription. Consider a region in the vague border between my lawn and the surrounding native plants. It seems natural to say that it neither determinately has the “lawn” property nor determinately lacks it, in line with the definition of indeterminacy advocated in Sect. 23.3 for quantum systems. That is, despite the differences between quantum indeterminacy and ordinary indeterminacy, the same concept of indeterminacy seems to be applicable. There is clearly a lot more work to be done here, but it looks initially plausible, at least, that quantum indeterminacy and ordinary indeterminacy can be conceived under the same general conceptual structure. Given that quantum indeterminacy really deserves the name, a further question concerns how we should understand the resulting indeterminacy. A common question about indeterminacy is whether it is metaphysical or merely semantic (Barnes & Williams, 2011, 104; Wilson, 2013, 360; Torza, 2021). In Carnapian vein, my inclination is to duck the question. Consider the question of whether there are human races. In part, this is an empirical question: it depends on facts about the morphology and phylogeny of certain creatures in the world. In part, it is a terminological question: when do we count two humans as being of the same race? Once we answer the empirical and terminological questions, we determine whether there are human races in the world. If there is a further question concerning whether what we are doing counts as metaphysics, that question needs to be spelled out. Similarly for the question of whether there is indeterminacy. In the quantum case, this depends on both empirical facts concerning the behavior of the physical world, and terminological choices concerning what kinds of property structure we count as exhibiting indeterminacy. Once we answer these questions, we determine whether there is indeterminacy in the world. Again, if there is a further question concerning

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whether what we are doing counts as metaphysics, that question needs to be spelled out.10 Clearly, though, such a stance raises deep metaontological questions, for example concerning naturalness (Torza, 2021). I have not attempted to address such questions here. Acknowledgments I would like to thank Jeff Barrett, Claudio Calosi, David Glick, Chris Hitchcock, Mahmoud Jalloh, Christian Mariani, Chip Sebens, Amie Thomasson, Alessandro Torza, Jessica Wilson, and two anonymous referees for very helpful comments on earlier drafts of this paper.

References Barnes, E., & Williams, J. R. G. (2011). A theory of metaphysical indeterminacy. In K. Bennett & D. Zimmerman (Eds.), Oxford studies in metaphysics (Vol. 6, pp. 103–148). Oxford University Press. Baym, G. (1969). Lectures on quantum mechanics. Addison-Wesley. Bell, J. S. (1964). On the Einstein-Podolsky-Rosen paradox. Physics, 1, 195–200. Birkhoff, G., & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–843. Bokulich, A. (2014). Metaphysical indeterminacy, properties, and quantum theory. Res Philosophica, 91, 449–475. Calosi, C., & Mariani, C. (2021). Quantum indeterminacy. Philosophy Compass, 16, e12731. Calosi, C., & Wilson, J. (2019). Quantum metaphysical indeterminacy. Philosophical Studies, 176, 2599–2627. Calosi, C., & Wilson, J. (2021). Quantum indeterminacy and the double-slit experiment. Philosophical Studies, 178, 3291–3317. Carnap, R. (1950). Logical foundations of probability. University of Chicago Press. Darby, G. (2010). Quantum mechanics and metaphysical indeterminacy. Australasian Journal of Philosophy, 88, 227–245. Glick, D. (2017). Against quantum indeterminacy. Thought, 6, 204–213. Healey, R. (2012). Quantum theory: A pragmatist approach. British Journal for the Philosophy of Science, 63, 729–771. Juffmann, T., Truppe, S., Geyer, P., Major, A. G., Deachapunya, S., Ulbricht, H., & Arndt, M. (2009). Wave and particle in molecular interference lithography. Physical Review Letters, 103, 263601. Kochen, S., & Specker, E. P. (1967). The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics, 17, 59–87. Leifer, M. S. (2014). Is the quantum state real? An extended review of ψ-ontology theorems. Quanta, 3, 67–155. Lewis, P. J. (2016). Quantum ontology. Oxford University Press. Pusey, M. F., Barrett, J., & Rudolph, T. (2012). On the reality of the quantum state. Nature Physics, 8, 475–478. Skow, B. (2010). Deep metaphysical indeterminacy. The Philosophical Quarterly, 60, 851–858. Spencer, Q. (2018). Racial realism I: Are biological races real? Philosophy Compass, 13, e12468.

10 One might insist that metaphysical indeterminacy is indeterminacy at the fundamental level. Then the question of whether quantum indeterminacy is metaphysical would reduce to the question of whether it is fundamental. As just mentioned, this is not a settled issue.

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Torza, A. (2020). Quantum metaphysical indeterminacy and worldly incompleteness. Synthese, 197, 4251–4264. Torza, A. (2021). Quantum metametaphysics. Synthese, 199, 9809–9833. Wilson, J. M. (2013). A determinable-based account of metaphysical indeterminacy. Inquiry, 56, 359–385. Von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik. Springer. Wolff, J. (2015). Spin as a determinable. Topoi, 34, 379–386.

Chapter 24

Defending the Situations-Based Approach to Deep Worldly Indeterminacy George Darby and Martin Pickup

Abstract This paper concerns metaphysical indeterminacy and, in particular, the issue of whether quantum mechanics gives motivation for thinking the world contains it. In a previous paper (Darby G, Pickup M. Synthese 198:1685–1710, 2021), we have offered one way to think about metaphysical indeterminacy which we take to avoid some issues arising from certain features of quantum mechanics (such as the Kochen-Specker theorem). This approach has recently been criticised by Corti (Synthese, forthcoming), and we take this opportunity to respond. Our paper will therefore reply to Corti’s argument, but we also take it as a case study in ‘naturalistic metaphysics’ and hence to contribute to a more general discussion of the relationship between philosophy of science and analytic metaphysics.

24.1 Introduction The question whether quantum mechanics involves metaphysical indeterminacy has received much recent attention.1 This attention is focused both on the issue of whether quantum mechanics can be a motivation for positing worldly unsettledness of the type captured by theories of metaphysical indeterminacy, as well as on the issue of how a theory of metaphysical indeterminacy could capture the supposed unsettledness allegedly arising in quantum mechanics. Quantum mechanics is a particular motivation for positing metaphysical indeterminacy on a certain sort of naturalistic metaphysical approach. According

1

For examples illustrating the development of the debate, see French and Krause (2003), Chibeni (2004), Calosi and Wilson (2019), Torza (forthcoming).

G. Darby () Oxford Brookes University, Oxford, UK e-mail: [email protected] M. Pickup University of Birmingham, Birmingham, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_24

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to this view, metaphysics should be read off physics, and (quantum) physics tells us that the world itself is indeterminate. This is highly contentious, and can be doubted for a number of different reasons. (We are each sympathetic to some of these reasons.) Nevertheless, even granting the moves necessary to get such a view going, internal problems arise. A prominent way of thinking about indeterminacy in the metaphysics literature ends up being incompatible with a natural way of getting indeterminacy from quantum mechanics. As one of us has argued (Darby, 2010) this is (roughly) because on that way of thinking reality is supposed to be indeterminate between maximally specific ways for things to be, whereas the Kochen-Specker theorem shows that there is no maximally specific way for things to be. So: You can interpret QM as involving genuine metaphysical indeterminacy if you really must, but will then require a different account of its nature. In a later paper (Darby & Pickup, 2021) we have suggested that one way of providing that account which makes use of situation semantics – a tool put to various uses in analytic metaphysics. Briefly, the idea is that when reality is indeterminate between ways for things to be, these ways for things to be are fully precise but not maximal. Situations, as parts of possible worlds, can naturally model this approach. When reality is unsettled about whether something is the case, that thing is the case in some but not all of the (partial) ways things could be. Corti (forthcoming) has recently responded to this model, arguing that it does not after all capture metaphysical indeterminacy as found in quantum mechanics. The reason for this turns out to revolve around which propositions are true or false in the relevant situations. We had in mind propositions such as “The particle is spin-up in the x-direction”, “The particle is spin-down in the y direction”, etc. The argument of Corti (forthcoming), on the other hand, revolves around propositions such as “The system is in state psi-”. This question, of what propositions and situations metaphysicians can legitimately use in setting up an account of metaphysical indeterminacy, is connected to prior questions of whether physics drives metaphysics or vice versa. Our focus in the paper is primarily on the first-order question of how to set up an account of metaphysical indeterminacy using situation semantics that does justice to quantum phenomena, but we also take it to illuminate some of those debates in metametaphysics.

24.2 The Darby/Pickup Account One family of approaches to metaphysical indeterminacy are known as meta-level accounts.2 A meta-level account of metaphysical indeterminacy sees the unset-

2

Jessica Wilson is responsible for this terminology. See, e.g., her (2017).

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tledness of the world as arising from unsettledness between distinct, determinate candidates for the way the world is. A very influential meta-level account is given by Barnes and Williams in their (2011). On this view, metaphysical indeterminacy consists in there being a number of different ersatz possible worlds which do not determinately fail to represent reality. So if some proposition is metaphysically indeterminate, it is true in (at least) one possible world which is a candidate for actuality, and false in (at least) one possible world which is a candidate for actuality. To speak somewhat poetically, the metaphysical indeterminacy of the proposition consists in the world being undecided about whether it is represented accurately by a possible world in which the proposition is true or by a possible world in which it is not. The BW account, however, suffers difficulties when applied to the very case that seems the most naturalistically plausible example of genuinely worldly indeterminacy: quantum mechanics. As has been shown independently by Darby (2010) and Skow (2010), quantum mechanics gives rise to a distinctive deep indeterminacy. This deep indeterminacy arises because of constraints like the Kochen-Specker theorem, which dictates that certain groups of propositions just cannot all be assigned determinate truth-values together. This means that an ersatz possible world which assigned truth-values to all such propositions would determinately fail to represent reality, and the BW model is thus inadequate for these cases. In a previous paper, we have offered a fix for this problem. The core idea is that situations, rather than possible worlds, should be used to model metaphysical indeterminacy. For the sake of brevity, not much will be lost by considering situations here as simply parts of possible worlds.3 This solves the problem because the situations which are candidates for actuality, and which the world is unsettled between, need not be complete. In other words, they can give truth-values to some but not all propositions (unlike possible worlds). According to our account of metaphysical indeterminacy, then, a proposition is metaphysically indeterminate when it is true in some situation which is a candidate for representing actuality, and false in some other such situation.

24.3 Corti’s Objection In a recent paper, Corti (forthcoming) offers a criticism of our account. He argues that our account fails, and that this highlights a broader point about the inadequacy of meta-level accounts in treating quantum indeterminacy as worldly indecision. In this section, we will outline what we take to be the core objection. Corti’s argument is that the model we present assigns incorrect truth-values to propositions. In particular, Corti claims that we are committed to taking a

3

For more detail about situations, see Kratzer (2020) and Barwise and Perry (1983) as a starting point.

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determinate feature of reality as indeterminate, which falsifies our approach. To see why this is so, we’ll briefly restate the argument. This restatement is not entirely innocent: we are adapting the properties and propositions Corti uses into our own preferred terms. This is philosophically significant, as will be discussed later in the paper. But for the sake of showing how the objection is supposed to undermine our view, we will present it this way. Suppose a quantum system is prepared so that it is x-spin-up. Then it is in a superposition for z-spin. This entails (again, assuming that one goes down the route of interpreting QM as involving genuine worldly indeterminacy in the first place) indeterminacy about the system’s z-spin. In the situations-based way of thinking about this, this is captured by asserting that there are two distinct situations s1 and s2 where the following propositions are true: s1 : the system is z-spin-up s2 : the system is z-spin-down

Both situations are candidates for representing actuality because neither of them determinately misrepresents it. This is what superposition consists in, on the model we explore. (NB: to say that something is z-spin-up is to say that it has a certain property – and this is not, or at least not obviously, the same as saying that its state vector is |+x >) There is another situation to mention, s4 (we follow Corti’s numbering here).4 In s4 the very same system is x-spin-up. s4 : the system is x-spin-up

So far, we are happy to accept that these situations are all candidates to represent actuality, and that these are the propositions true in them. Note that there may or may not be other propositions true in these situations, depending on exactly which situations we are choosing for s1 , s2 and s4 . But let’s assume for now that these are the minimal situations in which these propositions are true. The issue arises, according to Corti, because our view commits us to accepting problematic additional situations as candidates for representing actuality. One in particular is the following: s5 : the system is in a superposition of x-spin-up and x-spin-down5

This is supposed to be a problem because it is determinately the case that the system is x-spin-up, and so determinately not in a superposition with respect to x-spin.

4

We are leaving out s3 , which Corti takes to be a situation in which the propositions true in s1 and s2 are both true. We wouldn’t accept that such a situation is a candidate for actuality: it is contradictory. (In fact, there is no such situation, candidate for actuality or otherwise.) We take it that s3 is supposed to combine s1 and s2 in some way. We were careful to be explicit that in any such situation, neither proposition is true (or false): s3 would be a situation in which each proposition is indeterminate. 5 Corti actually describes two situation (s and s ), with different x-spin superpositions. The 5 6 general criticism can be stated without this detail.

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According to Corti, we are obliged to admit s5 as a candidate for representing actuality, and (again, according to him) s5 entails the falsehood that the system’s x-spin is metaphysically indeterminate. Why is it that we are forced to accept s5 as a candidate for representing actuality? Corti’s answer is that the proposition true in s1 is importantly related to the proposition true in s5 . In particular, the proposition true in s1 only differs from the proposition true in s5 by there being a ‘mathematical object’ in one which is replaced by an ‘equivalent’ ‘mathematical object’. Employing a principle he terms Equivalent Candidates for representing Actuality (ECA), this entails that the situation in which the latter is true is also a candidate for representing actuality.

24.4 Reply to Corti Although there are a number of points where we disagree with Corti’s paper, for the sake of simplicity we’ll restrict our comments to this central argument. The core move of the argument is that by asserting that s1 is a candidate for representing actuality, we are thereby committed to also accepting the problematic s5 as a candidate for representing actuality. We agree that this would be a problem, but deny that there is any such commitment. To begin with, it is worth underlining that our account of metaphysical indeterminacy is not that there is indeterminacy in the world whenever a proposition is neither true nor false in a situation which is a candidate for representing actuality. This is far too broad. Even if reality were fully determinate, portions of that reality (i.e. situations) would fail to settle the truth-value of propositions about other parts of the world. Rather, metaphysical indeterminacy arises when there is a conflict between situations which are both candidates for representing actuality. With this in mind, let’s look at s5 in a bit more detail. The proposition which is true in s5 states that the system is in an x-spin superposition. Given our model, and given that superpositions are being interpreted as indeterminacy, this would have to mean that there are a pair of distinct situations which are both candidates for representing actuality such that the system is x-spin-up in one and x-spin-down in the other. The property of being superposed is therefore a meta-level property (in keeping with this meta-level account of metaphysical indeterminacy), which a system has in a situation in virtue of the properties of that system in certain other situations. There is certainly a situation which is a candidate for actuality in which the system is x-spin-up, namely s4 . But we do not accept that there is any corresponding situation which is a candidate for actuality in which the system is x-spin-down. So, there is no situation which is a candidate for actuality in which the system is in a superposition of spin up and x-spin-down. But Corti thinks we must accept that there is such a situation (namely s5 ). This is because (i) there is a candidate situation (s1 ) where the system has the property of being z-spin-up and (ii) the property of being z-spin-up is connected to the

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property of being in an x-spin superposition in such a way that the instantiation of the former in a situation which is a candidate for reality ensures that the latter is also instantiated in a situation which is a candidate for reality. The second step here is encoded in Corti’s ECA principle. We will shortly discuss this principle. But before we do, it is worth highlighting that situation theory is specifically designed to allow fine-graining that undermines the motivation for ECA. The properties of being indeterminate for x-spin and being z-spin-up are clearly distinct properties. The first is a meta-level property, while the second obtains (uninformative as this is) just when the system is z-spin-up in that situation. So, a (possible) situation in which the system is superposed in x-spin and the situation in which the system is z-spin-up look importantly different. Why, then, does Corti think that there is an intimate connection between the candidacy for actuality of these different situations? The answer to this revolves around Corti’s ECA principle: Consider a situation s1 that is a candidate for representing actuality and verifies only a proposition p1 which contains a mathematical object o1 . Any other situation s2 that differs from s1 only in that it makes true a proposition p2 which is obtained by simply replacing o1 with o2 , where the latter is a mathematical object equivalent to the former (i.e. o1 = o2 ), is also a candidate for representing actuality. (p. 11)

This principle is connected to Corti’s version of metaphysical naturalism. It is justified as follows: Such a principle seems to be intuitively reasonable. Let us see how it works by presenting a toy example. Suppose it is metaphysically indeterminate how many oranges there are in the fridge; assume further that there might just be either three or four. According to Darby and Pickup’s view, there is a possible situation in which there are three oranges, and a possible situation in which there are four, but neither describes correctly nor misrepresents the actual world (and therefore the propositions ‘there are three oranges in the fridge’ and ‘there are four oranges in the fridge’ are indeterminate, being true and false in at least one situation). The principle (ECA) simply guarantees that if the possible situation in which there are three oranges is a candidate for representing actuality, then also the situations that verify respectively only the propositions ‘there are two plus one oranges in the fridge’ or ‘there are four minus one oranges in the fridge’, and so on, are candidates for representing actuality. (ibid.)

A number of points are worth noting about this principle and its application: First of all, this has nothing in particular to do with indeterminacy, but is more about the workings of situations: The orange example shows that the (distinct?) situations verifying the (distinct?) propositions that there are three oranges, that there are 2 + 1 oranges, and that there are 4 − 1 oranges, etc, will all be candidates for representing reality if there are 3 oranges. Second, it is not entirely obvious that the proposition that the proposition that there are three oranges in the fridge and the proposition that there are 2 + 1 oranges in the fridge are really distinct propositions. (Are they both distinct from the proposition that there are 1 + 2 oranges in the fridge?) If propositions are sets of worlds, then it would seem not, for example.

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Third, it is also not entirely obvious, and depends on the details of the metaphysics of situations, that the situation verifying the proposition that there are three oranges in the fridge and the situation verifying the proposition that there are 2 + 1 oranges in the fridge are really distinct situations. If situations are individuated by the propositions true in them, then this is parasitic on the previous paragraph. If, by contrast, they are individuated by the entities they contain and the properties they instantiate, then plausibly exactly the same entities and properties are in each situation (even if the propositions are distinct). Either way, it is a substantive and controversial claim that there can be distinct situations differing only in whether they verify ‘there are 3 oranges in the fridge’ or ‘there are 2 + 1 oranges in the fridge’. On the other hand, one could no doubt construe things in such a way that the required distinctions can be made – perhaps we are dealing with one proposition concerning the oranges and the number 3, and another proposition concerning the oranges and the numbers 1 and 2. Then perhaps there would be a non-trivial sense in which there are two situations that are both candidates for representing actuality. Of course one could also do that without involving the oranges at all: The situation that verifies the proposition that 1 + 2 = 3 and the situation that verifies the proposition that 4 – 1 = 3 would also both be candidates for representing actuality. In the terms used in the definition of ECA, the first situation differs from the second “only in that it makes true a proposition [the proposition that 4 – 1 = 3] which is obtained by simply replacing [1 + 2] with [4 − 1], where the latter is a mathematical object equivalent to the former”. Again, the exact meaning of ECA depends on what it is for a proposition to contain a mathematical object, which in turn depends on what propositions are, and what mathematical objects are, and on what is meant by “equivalent”. One might mean that the terms are equivalent, i.e. co-referential, but presumably not, because here Corti is talking about objects, not terms. The mathematical objects might be identical (as is suggested by “o1 = o2 ”), or isomorphic; or one sequence of operators might be equivalent to another sequence of operators by having the same effect. Or, of course, the mathematical objects themselves might be representationally equivalent, perhaps because they represent the same physical state, for example, but that depends on the details of the interpretation. Our point in labouring this is that principles like ECA, and the “naturalistic” approach to metaphysics that underlies them, mask a number of crucial assumptions. In essence, Corti’s argument goes like this: Consider a particle in an x-spin eigenstate. Then we would think of it (assuming, as usual, that quantum mechanics is being interpreted as involving worldly indeterminacy) like this: (1) It is indeterminate whether the particle is z-spin-up or z-spin-down.

And, on the situations way of thinking, that entails: (2) the situation of the particle’s being z-spin-up is a candidate for actuality.

And presumably we should also accept:

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(3) The particle is not indeterminate for x-spin.

So, again on our way of thinking: (4) The situation of the particle’s being indeterminate for x-spin is not a candidate for actuality.

But suppose we also had: (5) To be z-spin-up is to be indeterminate for x-spin.

Then (2) and (5) would entail: (6) The situation of the particle’s being indeterminate for x-spin is a candidate for actuality.

Which contradicts (4). But that assumption (5) just seems to be false – why should what it is to be z-spin-up be the same as what it is to be indeterminate for x-spin? This gap is supposed to be closed by the ECA, but of course the ECA can’t apply to any of the statements above, because none of them describes, at least obviously, a situation that verifies a proposition that “contains a mathematical object”. You can get mathematical objects into the picture like this (still using the example of a particle with determinate x-spin): (1 ) It is indeterminate whether the particle is in state |+z> or |-z>. So (2 ) the situation of the particle’s being in state |+z> is a candidate for actuality

And then argue that, since the particle is supposed to be in an x-spin eigenstate, and since |+z> is not an x-spin eigenstate, there is a candidate for actuality that determinately misrepresents it, which would be a bad result. But this time, as far as we can see, (1 ) is straightforwardly false. It is not indeterminate whether the particle √ is in state |+z> or |–z> but rather determinate that it is in the state 1/ 2 |+z> + |–z>, which is straightforwardly neither |+z> nor |–z>. What is indeterminate is whether the particle is z-spin-up. Equating being z-spin-up with being in state |+z> is to make some deep assumptions about the connection between mathematical formalism representing the states of quantum systems and the properties of those quantum systems. So, the point is that the argument only gets going if couched in terms of properties rather than state vectors – but then the ECA, which revolves around “mathematical objects” like state vectors, just doesn’t apply.

24.5 Conclusion Have we established that quantum indeterminacy is worldly indecision? Of course not – that would involve adopting a realist position in the philosophy of science, and advocating for a particular interpretation of quantum mechanics (i.e. solution to the measurement problem, e.g. some collapse interpretation), and putting a particular

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metaphysical spin on it so that the indeterminacy involved is “worldly”, and then spelling out the metaphysical theory about how that worldly indecision is to be understood. The challenges and obstacles for that approach may be insurmountable – in particular, it may be that Calosi and Wilson (2019) have shown that metaphysical supervaluationist approaches are poorly motivated for a wide range of interpretations. Partly for this reason, and partly because we are at least somewhat drawn to the idea that there is no such thing as worldly indeterminacy at all, we took the supposed indeterminacy in QM, and the fact that it appears prima facie to be “deep”, in Skow’s terms, as simply a suggestive motivation for something that it ought to be possible to account for. We do think that the machinery offered by situation theory – already used in various areas of metaphysics – offers a way of doing so that, unlike “standard” metalevel accounts of metaphysical indeterminacy, does justice to the general features of quantum mechanics that motivate the idea of worldly indeterminacy in the first place. For the same reason, we think it worth noting that those general features of the formalism don’t translate into straightforward difficulties for the account in the way that Corti argues. More broadly, what we take this to show is that care is needed, especially in this domain, to disentangle internal criticisms of metaphysical models from external criticisms arising from naturalistic metametaphysical assumptions. Our situations-based account of quantum metaphysical indeterminacy is explicitly provisional on the controversial moves needed to get the game started. But closing up the gap in which the theory would sit requires controversial assumptions of its own.

References Barnes, E., & Williams, R. (2011). A theory of metaphysical indeterminacy. Oxford Studies in Metaphysics, 6, 103–148. Barwise, J., & Perry, J. (1983). Situations and attitudes. The MIT Press. Calosi, C., & Wilson, J. (2019). Quantum metaphysical indeterminacy. Philosophical Studies, 176, 2599–2627. Chibeni, S. (2004). Ontic vagueness in microphysics. Sorites, 15, 29–41. Corti, A. (forthcoming). Yet again, quantum indeterminacy is not worldly indecision. Synthese. Darby, G. (2010). Quantum mechanics and metaphysical indeterminacy. Australasian Journal of Philosophy, 88(2), 227–245. Darby, G., & Pickup, M. (2021). Modelling deep indeterminacy. Synthese, 198, 1685–1710. French, S., & Krause, D. (2003). Quantum vagueness. Erkenntnis, 59, 97–124. Kratzer, A. (2020). Situations in natural language semantics. In E. N. Zalta (Ed.). The Stanford Encyclopedia of Philosophy (Fall 2020 Edition). https://plato.stanford.edu/archives/fall2020/ entries/situations-semantics/ Skow, B. (2010). Deep metaphysical indeterminacy. Philosophical Quarterly, 60(241), 851–858. Torza, A. (forthcoming). Quantum Metametaphysics. Synthese. Wilson, J. (2017). Are there indeterminate states of affairs? Yes. In E. Barnes (Ed.), Current controversies in metaphysics (pp. 105–125). Taylor & Francis.

Chapter 25

Metaphysical Indeterminacy in the Multiverse Claudio Calosi and Jessica M. Wilson

Abstract One might suppose that Everettian quantum mechanics (EQM) is inhospitable to indeterminacy (MI), given that, as A. Wilson (The nature of contingency: Quantum physics as modal realism. Oxford University Press, Oxford, 2020) puts it, “the central idea of EQM is to replace indeterminacy with multiplicity” (77). But as Wilson goes on to suggest, the popular decoherence-based understanding of EQM (DEQM) appears to admit of indeterminacy in both world number and world nature, where the latter indeterminacy–our focus here–is plausibly metaphysical. After a brief presentation of DEQM, we bolster the case for there being MI in world nature in DEQM. The remainder of the paper is devoted to a comparative assessment of the two main approaches to MI for purposes of accommodating this MI–namely, a metaphysical supervaluationist approach (as per Barnes and Williams (A theory of metaphysical indeterminacy. In: Bennett K, Zimmerman DW (eds) Oxford studies in metaphysics, vol 6. Oxford University Press, Oxford, pp 103–148, 2011)) and a determinable-based approach (as per Wilson (Inquiry 56:359–385, 2013) and Calosi and Wilson (Philosophical Studies 176:2599–2627, 2018; Philosophical Studies 178:3291–3317, 2021)). We briefly describe each approach, then offer five arguments in favour of a determinable-based approach to world nature MI in DEQM.

25.1 Introduction One might suppose that Everettian quantum mechanics (EQM) is inhospitable to indeterminacy (MI), given that, as A. Wilson (2020) puts it, “the central idea of EQM is to replace indeterminacy with multiplicity” (77). But as Wilson goes on

C. Calosi University of Geneva, Geneva, Switzerland e-mail: [email protected] J. M. Wilson () University of Toronto, Toronto, ON, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_25

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to suggest, the popular decoherence-based understanding of EQM (henceforth: DEQM) appears to admit of indeterminacy in both world number and world nature, where the latter indeterminacy—our focus here—is plausibly metaphysical. After a brief presentation of DEQM (Sect. 25.2), we bolster the case for there being MI in world nature in DEQM (Sect. 25.3). The remainder of the paper is devoted to a comparative assessment of the two main approaches to MI for purposes of accommodating this MI—namely, a metaphysical supervaluationist approach (as per Barnes and Williams 2011) and a determinable-based approach (as per Wilson 2013 and Calosi and Wilson 2018 and 2021). We briefly describe each approach (Sect. 25.4), then offer five arguments in favour of a determinable-based approach to world nature MI in DEQM (Sect. 25.5).

25.2 Decoherence-Based EQM (DEQM) We start with a brief overview of DEQM. Here and in the next section we periodically excerpt from Wilson’s (2020) presentation, for continuity with the discussion of indeterminacy in DEQM later in this paper. Consider a simple superposition state such as that at issue in the case of Schrödinger’s cat: |ψ = c1 |Live Cat + c2 |Dead Cat

(25.1)

On the face of it, such a superposition state represents a system as being in a single indefinite or indeterminate state. But what does this come to, exactly? On a common understanding (see Wallace 2008, 40; Wilson 2020, 77), the crucial insight at the core of the EQM approach is that a superposition state such as (25.1) may be taken to represent a multiplicity of systems, each in a familiar definite or determinate state, rather than a single system in an unfamiliar indeterminate state. As Wilson puts it, “the central idea of EQM is to replace indeterminacy with multiplicity” (77). To be sure, the supposition that the multiplicity at issue involves multiple systems represents a development of Everett’s own take on his (1957) theory (the ‘Relative State Interpretation’), as involving multiple states of a single system. Everett’s take was driven largely by concern with the measurement problem—in brief, the question of how to bridge the gap between the world as Schrödinger’s equation (deterministically but indeterminately, via superpositions) expresses it as being, and the world as we (indeterministically but determinately, via components of superpositions output from individual measurements) experience it as being. Rather than bridge this gap in ad hoc fashion via a supposed ‘collapse’ of the wave function upon measurement, Everett suggested that measurements (e.g., opening the box) result in entanglements generating relative states, so that, e.g., a single cat is dead relative to one substate (or class of substates) and alive relative to another substate (or class of substates).1 1

Our own view is that Everett’s relative-state interpretation has a lot going for it—and moreover, is appropriately seen as accommodating metaphysical indeterminacy of a specifically ‘glutty’ variety, along lines of the multiple relativized determination discussed in Wilson 2013 and Calosi and

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The central idea was developed in influential fashion in Dewitt 1968 and 1970, with the multiplicity at issue involving multiple individual systems, and indeed multiple worlds. There are many variations on the so-called ‘Many Worlds’ interpretation of quantum mechanics (see Saunders et al. 2010), aiming to address stated concerns with, e.g., the reintroduction of something like collapse in talk of ‘splitting’ of worlds upon measurement, or with the seeming need to stipulate a preferred basis (reflecting that different bases for characterizing the universal quantum state generate different worlds), or with the introduction of profligate fundamental ontology.2 Among these variants, the approach to EQM found most promising of late has been that developed by Saunders (1993, 1994) and Wallace (2008, 2012), along with other ‘Oxford Everettians’, according to which the multiplicity of worlds is understood in terms of the branching structure induced by decoherence. As Wilson (2020) puts it: The most significant step towards a plausible version of EQM came when, in the early 1990s, progress in technical work on decoherence was applied to the preferred basis problem in EQM by Saunders (1993, 1994, 1995). Decoherence theory can be used to model the quantum-mechanical interactions between a system and its environment [. . . ]. The essence of decoherence is that a broad range of quantum systems evolve in such a way as to suppress to a negligible level the interference terms representing interactions between components of the state of the system corresponding to distinct macroscopic properties. (80)

DEQM’s popularity reflects its neat handling of each of the aforementioned concerns. First, the approach provides a principled and plausible answer to the measurement problem.3 Decoherence suppresses interference—not entirely, but to a degree sufficient to prise apart the components of a given superposition state— in a way consonant with ordinary experience, without requiring, e.g., conscious observers to make a collapse-inducing ‘measurement’.4 Second, decoherence phenomena fix the preferred basis, in an approximate but non-stipulative way which is fine for all practical purposes (for short: FAPP): Although decoherence suppresses interference between macroscopic superpositions, it does not eliminate this interference altogether. The idea behind decoherence-based EQM is that a preferred basis is approximately picked out by decoherence, to a degree of approximation easily high enough to explain the fact that superpositions of macroscopic states are unobserved and effectively unobservable. (Wilson 2020, 80)

Wilson 2018 and 2021. Development of this suggested reading, and its connection to Rovelli’s ‘Relational Quantum Mechanics’ (as in, e.g., Rovelli 1996), must await another occasion. 2 As on Deutsch’s (1985) proposal, according to which infinitely many worlds correspond to each individual physically possible history. 3 The measurement problem can be phrased as the joint inconsistency of the following three statements: (i) the quantum state provides a complete description of a quantum system; (ii) the quantum state always evolves according to the Schrödinger equation, and (iii) measurement results are unique. Everettian Quantum Mechanics in general solves the problem by rejecting (iii). Correspondingly, it is worth noting that decoherence per se does not provide a solution to the measurement problem. Thanks to a referee here. 4 The worlds generated by decomposition are sometimes described as ‘semi-classical’, in being (and notwithstanding that decoherence does not entirely eliminate interference between components of macroscopic superpositions) compatible with the world as we ordinarily experience it.

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Third, in DEQM the generation of ‘Everett worlds’ requires no new fundamental ontology. At the fundamental level there is just one highly structured object: the universal quantum state (the ‘universe’). Decoherence produces dynamically robust patterns in the universal state; these patterns represent a multiplicity of different worlds and objects within those worlds—the ‘multiverse’.5 Correspondingly, Everettian worlds are derivative entities, ‘grounded’ (to speak schematically) in the fundamental quantum state. DEQM is naturally embedded in decoherent histories interpretations of quantum theory. These interpretations use what is sometimes called the Heisenberg picture, where operators representing measurable quantities change over time, while the quantum state remains constant. This is in contrast with the intertranslatable Schrödinger’s picture, where operators remain constant while the quantum state evolves. As applied to DEQM, observables corresponding to the whole state of a world at each time t are represented by a projection operator. Different possible observable possibilities are represented by orthogonal projections Pˆi summing up to unity, as per (25.2) and (25.3) below: Pˆi Pˆj = δij Pˆi

Pˆi = 1

(25.2) (25.3)

i

Equation (25.2) says that any two distinct projection operators are orthogonal, whereas Eq. (25.3) says that such projection operators sum up to unity. These equations are intended to encode that any two observational possibilities describing the state of the whole world are mutually exclusive (25.2), and the collection of such possibilities is—so to speak—exhaustive (25.3). A partition of projection operators meeting the conditions in (25.2) and (25.3) provides a ‘coarse graining’ of the universal state; a coarse graining in turn generates histories Hi —time-ordered sequences of time-dependent projection operators: Hi = Pˆ in (tn )Pˆ in−1 (tn−1 ) . . . Pˆ i0 (t0 )

(25.4)

A history is a sequence of operators describing the whole state of the world, with one operator describing the entire observable state of the world for each time. Correspondingly, a history provides a maximal description of the world at each time. According to DEQM, individual such histories are potentially suited to represent individual worlds, and sets of such histories are potentially suited to represent complete multiverses. For this promise to be fulfilled, however, the histories must be sufficiently causally isolated—sufficiently well-decohered—that they can provide a basis for accommodating our experience of macroscopic phenomena as

5

Here and elsewhere we use ‘multiverse’ to refer to the multiplicity of Everettian worlds, following Wallace (2012) and Wilson (2020).

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comparatively determinate. Effectively, what is required here is that the histories be well-decohered enough to be dynamically independent (i.e., such that independent probabilities can be assigned to those histories). This requirement is usually cashed out in terms of the ‘medium decoherence condition’ (see Gell-Mann and Hartle 1993), where ρ is a density operator for the initial state of the universe, and Tr is the trace: Tr(Hi ρHj† ) ≈ Tr(Hi ρHi† )δij

(25.5)

On DEQM, histories meeting condition (25.5) are taken to represent a multiplicity of Everettian worlds—the (or a) multiverse.6 Decoherence is here thought of as involving the suppression of quantum interference as a result either of internal interaction within a system, or of external interaction with the environment, and in cases of decoherence the component terms in the superposition behave semiclassically (see note 4) in that we observe no interaction between them. It is in this sense that decoherence produces a multiplicity of comparatively independent, causally isolated systems and worlds—worlds which are independent and causally isolated ‘FAPP’—a process referred to as branching. As Wilson (2020) describes it: Branching occurs whenever decoherence becomes sufficient to render different histories effectively causally isolated, for example when a dust particle becomes entangled with a radiation bath environment so that the components of the particle’s state corresponding to superposition of macroscopic properties become negligible compared to the components corresponding to reasonably precise macroscopic properties. Branching may be thought of as a transition from a particle not yet correlated with its environment and with a relatively indeterminate location, to multiple particles correlated with their environments, each with a relatively determinate location. (84)

Situating DEQM in the decoherent histories formalism also provides a basis for capturing the aforementioned approximate nature of decoherence, since different coarse-grainings may satisfy the conditions. Decoherence results in fewer candidate coarse-grainings and associated bases—again, sufficient to capture failures to experience macro-superpositions—but does not narrow these down to one.

25.3 Metaphysical Indeterminacy in World Nature We are now in position to see how, notwithstanding the usual gloss on EQM as replacing indeterminacy with multiplicity, there remains room for indeterminacy on DEQM. Indeed, Wilson (2020) maintains that two sources of indeterminacy remain on this view: First, in contemporary decoherence-based EQM the space of Everett worlds is indeterminate with respect to the number of worlds it includes. Different coarse-grainings may

6

See note 5.

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each give rise to decoherent history spaces satisfying the decoherence conditions, and nothing in the theory picks out one over the other as the uniquely correct space of Everett worlds. Second, Everett worlds are indeterminate in nature: a world for example may fail to determine which of the two slits an electron travels through, if the electron wavefunction does not decohere in the process. (172)

Each form of lingering indeterminacy can be seen as reflecting that, as above, decoherence as a mechanism for suppression of interference “does not eliminate this interference altogether”. That decoherence is only approximate entails that different coarse-grainings generate different and differently numbered multiverses of Everett worlds. No coarse-graining is (meta)physically privileged. As Wallace (2012) puts is, there is no “natural grain”. Wilson takes the absence of privileged grain to indicate that there is indeterminacy in world number. Here we are primarily interested in indeterminacy in world nature, so at this point leave aside indeterminacy in world number. That the suppression of interference in decoherence is approximate also means that branching results in worlds with relatively determinate as opposed to absolutely determinate values of the observables at issue (e.g., position)—hence, there is indeterminacy in world nature. As Wilson observes, the two indeterminacies are deeply linked: The more fine-grained our partition of a consistent history space, the more histories there are and the more determinate each history is—up to the point at which the decoherence condition is not satisfied. It is a vague matter where this point is located. However coarsely or finely we grain a decoherent history space, events within individual Everett worlds will exhibit some (indeterminate) degree of indeterminacy in their properties. It is determinate that there is qualitative indeterminacy in the worlds, but it is indeterminate exactly how much indeterminacy there is. (180–1)

Importantly, this lingering indeterminacy is insuperable: “there is no way of coarse-graining the history space as to make the actual Everett world fully determinate” (181). How should these forms of indeterminacy in DEQM be handled? Wilson (2020) says, somewhat confusingly, that “both indeterminacies ought to be regarded as epistemic or semantic in origin; however, indeterminacy of world nature [. . . ] may usefully be understood as a novel example of emergent ontic indeterminacy” (173). Key here is that both forms of indeterminacy are associated with the ‘semi-arbitrary’ status of coarse-grainings (or the choice of a specific such coarse-graining), where the arbitrariness at issue is taken (by Wilson) to reflect semantic or (less plausibly) epistemic underdetermination. As he sees it, indeterminacy in world number is just a matter of semi-arbitrary coarse-graining; however, indeterminacy in world nature has an additional metaphysical component: The indeterminacy of the actual world is representational, in the sense that it depends on a semi-arbitrary choice of coarse-graining; but it is also worldly, in the sense that a complete description of the actual world fails to eliminate this indeterminacy. (182)

So Wilson is friendly to the idea that indeterminacy in world nature in DEQM is properly metaphysical; and in considering how such MI should be treated,

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he suggests that either a metaphysical supervaluationist (or ‘precisificationist’)7 account of the sort endorsed by Barnes and Williams (2011) or a determinablebased account of the sort endorsed by Wilson (2013) might do the trick—though he registers that a supervaluationist approach faces difficulties of the sort highlighted by Darby (2010), Skow (2010), and others, and that a determinable-based approach has several advantages, and “seems a more natural fit” with DEQM in various respects. We are also friendly to taking indeterminacy in world nature in DEQM to be properly metaphysical, and also see a determinable-based approach as more naturally accommodating such MI—more naturally than supervaluationism, in particular—and in the remainder of the paper will develop these lines of thought. We think there is more to say about the status and proper treatment of indeterminacy in world number in DEQM, but due to considerations of space leave this for another occasion.8 We start by filling in the case, first, for there being indeterminacy in world nature, and second, for this indeterminacy being reasonably taken to be ‘worldly’ or metaphysical. We offer three arguments for there being indeterminacy in world nature.9 The first argument pertains to the fact that even in cases of decoherence of the sort giving rise to branching and associated Everett worlds, there may remain undecohered states. As above, the central idea of EQM is to “replace indeterminacy with multiplicity”, such that superposition states such as (25.1) can be taken to represent not single systems with unfamiliar indeterminate properties—that is, as involving indeterminacy—but rather multiple systems, each with familiar

7

The notion of precisification here is modeled on that associated with a supervaluationist theory of vagueness (see, e.g., Fine 1975), on which a precisification is a complete and maximal set of sentences, each having a determinate truth value. 8 In lieu of a fuller treatment: we are inclined to think that indeterminacy in world number is also aptly treated as properly metaphysical indeterminacy, and moreover is best treated in determinable-based terms. Wilson (2020) suggests that indeterminacy in world number can be given a semantic supervaluationist treatment in line with classical logic and semantics, along lines of the non-standard supervaluationist approach proposed by McGee and McLaughlin (1995). However, first, it would be more systematic to treat indeterminacy in world number and in world nature similarly; but it is not plausible to treat the lingering indeterminacy in world nature as semantic. And second, a semantic supervaluationist treatment of indeterminacy in world number along lines of the ‘preferred precisification’ approach of McGee and McLaughlin is subject to the same difficulties that Wilson highlights with an epistemic approach to indeterminacy—namely, that it presupposes, implausibly, that there is a single preferred precisification/basis/decoherent history space/multiverse. 9 Quantum indeterminacy is commonly motivated by attention to the Eigenstate-Eigenvalue link (EEL), according to which a quantum system has a definite value v for an observable O iff it is in an eigenstate of O having eigenvalue v. However, some proponents of DEQM (e.g., Wallace, 2019) are skeptical about the link, and Wilson (2020) never mentions it in his discussion. So in what follows we offer arguments that do not rely on EEL (or any variants thereof). See Calosi and Mariani 2021 for discussion of how EEL has been taken to motivate quantum indeterminacy, and see Calosi and Wilson 2018 for an argument that, given EEL, the phenomenon of incompatible observables entails residual indeterminacy in Everett worlds.

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determinate properties. And as above, on DEQM, multiplicity is only a matter of decoherence—it is not a matter of conscious observers, new fundamental ontology, or anything else. It follows that in the absence of decoherence as applying to a superposition state such as (25.1), the state cannot be given a multiplicity reading, but must rather be given an indeterminacy reading.10 But as Wilson (2020) observes, decoherence sufficient unto branching and the associated generation of Everett worlds is compatible with any given Everett world containing superposition states that are not decohered, as when “the world fails to determine which of the two slits an electron travels through, if the electron wavefunction does not decohere in the process” (172). So DEQM is compatible with there being Everett worlds containing superposition states which cannot be given a multiplicity reading, but which must rather be given an indeterminacy reading. And in practice this will often (always?) be the case. The second argument—a variation on the first, from a different direction— pertains to the connection between component interference and state indeterminacy. As Wallace (2012, 61) notes, the state |ψL = |Live Cat

(25.6)

instantiates a structure that represents a live cat, and the state |ψD = |Dead Cat

(25.7)

instantiates a structure that represents a dead cat. Again, according to DEQM, there is a reading of state (25.1) (the superposition of states 25.6 and 25.7) on which it represents not one cat in an indeterminate state, but two cats—one in state (25.6), the other in state (25.7). Yet as Wallace goes on to observe: In general, even in a theory with linear equations, like electromagnetism or quantum theory, adding together two states with certain structures might cause those structures to overlap and cancel out, so that the structure of the resultant state cannot just be read off from the structures of the components. Indeed, in both electromagnetism and quantum theory, the technical term for this “cancelling out” is the same: interference (62).

Here Wallace connects multiplicity to the absence of canceling out of the structures associated with the components—in other words, to the absence of interference. More precisely, on DEQM it is required not that there be no interference, but rather that any interference be negligible, as per the medium decoherence condition. When interference between components of a superposition state is

10 The

line of thought here presupposes a disjunctive premise to the effect that superposition states must be interpreted either as indeterminacy states (involving a single system having an indeterminate property) or as multiplicity states (involving multiple systems having determinate properties). In general, this disjunctive premise might not available, for there might be other readings of superposition states. However, here we are concerned not with all available interpretations of superposition states, but just with what interpretations are available given DEQM as standardly motivated; and here the disjunctive premise is in place.

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negligible, then the multiplicity reading is available. It remains, however, that branching is compatible with there being some superposition states for which the interference is not negligible. For any such states, the multiplicity reading of states like (25.1) is not available, and the states must rather be given an indeterminacy reading (see note 10). And again, in practice this will often (always?) be the case. Hence it is, for example, that the double-slit experiments have the empirical interference results that they do. The third argument pertains to the fact that even in cases of decohered states, the decoherence is approximate, rendering the values of the associated observables comparatively or relatively determinate, not absolutely determinate. For purposes of generating an Everett world, all that is required is that decoherence renders certain states determinate FAPP—determinate enough, in particular, to accommodate ordinary experience in response to the measurement problem. Even so, decoherence does not eliminate all interference between components of a given superposition, and hence the values of the associated observables in the state components are rendered only comparatively or relatively determinate—that is, to some small extent indeterminate. Hence there is lingering indeterminacy even in cases of decohered superposition states.11 The upshot of the previous arguments is that there is indeterminacy in world nature in DEQM, associated with both cohered and decohered states. Is this indeterminacy moreover metaphysical in nature? Wilson thinks so: “indeterminacy in world nature may be thought of as a naturalistic form of metaphysical indeterminacy” (182).12 We agree, but it is worth saying a bit more by way of substantiating the claim. We might start with the presumed core insight of EQM. Here the choice presented is as between a reading of superposition states as involving ‘unfamiliar indeterminate properties’ and one involving multiplicity along with determinate properties; but indeterminacy in properties is metaphysical indeterminacy. And as we have just argued (and as Wilson agrees), the EQM strategy as cashed via DEQM does not, after all, result in elimination of the ‘unfamiliar indeterminate properties’; at best, it renders some of them more determinate. That may be good enough to resolve the measurement problem that was Everett’s main focus, but there remains some portion of the seeming MI originally at issue—namely, that associated with there being indeterminate quantum properties associated with not-completelydecohered superposition states. It is also worth noting that standard motivations for or presuppositions required for treating a given case of indeterminacy as semantic or epistemic are not in place for MI in world nature. To start, semantic or epistemic treatments are most 11 That there is indeterminacy even in cases of decohered states would appear to undercut the claim that (with exceptions for cases where micro-superpositions are magnified to a larger scale) “determinacy will tend to be associated with macroscopic states of affairs, with extensive indeterminacy restricted to microscopic states of affairs” (Wilson 2020, 181). On the contrary, decohered macroscopic states will still be a locus of indeterminacy. 12 See also Lewis 2016.

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often directed at cases of vagueness involving borderline cases and associated Sorites-susceptibility;13 but indeterminacy in world nature does not obviously involve borderline cases. Moreover, semantic accounts also typically proceed on the assumption that relevant expressions in the relevant language(s) are vague; but the mathematical language of the decoherent histories formalism is not vague. To be sure, Wilson thinks that indeterminacy in world number can be handled in semantic or epistemic terms, on grounds that coarse-grainings’ being ‘semiarbitrary’ provides a basis for treating indeterminacy in world number as reflecting a kind of ambiguity or ignorance in which representation correctly describes the actual world. But as he observes, these strategies don’t carry over to the case of indeterminacy in world nature, for having resolved any representational indeterminacy, indeterminacy in world nature will remain. Finally, an epistemic approach to indeterminacy in world nature presupposes that each Everett world is maximally precise, such that any indeterminacy reflects just our ignorance about which precise way a given Everett world is. But in DEQM there is no way to make Everettian worlds maximally precise; for (as noted above) at a certain point, the medium decoherence condition (25.5) will fail to be satisfied, and the precise worlds in question will fail to be members of the set of decoherent histories. We will return to this line of thought in Sect. 25.5.1. There thus appears to be good reason to take there to be indeterminacy in world nature in DEQM, and to take this indeterminacy to be properly metaphysical. At least this seems reasonable contingent on there being an account of MI up to the task of making sense of MI on DEQM—as we will argue there is, below. Supposing that we (and Wilson) are correct that there is MI in world nature in DEQM, this is of significance to the metaphysics of MI, not just as a naturalistically motivated general case study, but also because the indeterminacy at issue in DEQM is distinctively derivative, on the assumption (which we are here granting) that the fundamental ontology of the theory—given solely by the universal quantum state— is maximally determinate.14 Some have argued that any metaphysical indeterminacy there might be must be fundamental (Barnes 2014) or that any derivative metaphysical indeterminacy there might be is eliminable (Glick 2017). As we discuss further below, Barnes’s argument applies only to a certain approach to metaphysical indeterminacy, different from the approach we endorse; and in our (2021) we argue that Glick’s argument doesn’t go through. In any case, the case of derivative MI in DEQM stands as a challenge to both arguments.

13 See

Calosi and Mariani 2021. grant that the sole fundamental entity in DEQM—the universal wavefunction/quantum state—is maximally determinate in that its properties are maximally determined (as, e.g., the property of having such and such amplitude and phase at such and such point). The further metaphysical picture behind DEQM is not perfectly clear. To start, there are several incompatible ways to interpret the universal wavefunction. Moreover, there remains controversy over how to recover derivative entities from the universal wavefunction. Just to mention two options: Wallace (2012) takes derivative entities to be patterns in the universal wavefunction, whereas Ney (2021) takes them to be parts.

14 We

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25.4 Two Approaches to MI Given that DEQM involves MI in world nature, the question remains of how to account for this indeterminacy. In this and the following section, we offer an answer to this question. We start with brief summaries of the two main approaches to MI: first, metaphysical supervaluationism; second, a determinable-based approach.

25.4.1 Metaphysical Supervaluationism A metaphysical supervaluationist account of MI takes a ‘meta-level’ approach to MI, according to which MI involves its being indeterminate which state of affairs, of some range of determinate/precise states of affairs, obtains. As Barnes (2010) expresses the general idea: It’s perfectly determinate that everything is precise, but [. . .] it’s indeterminate which precise way things are. (622)

Somewhat more specifically, Barnes and Williams (2011) say: When p is metaphysically indeterminate, there are two possible (exhaustive, exclusive) states of affairs—the state of affairs that p and the state of affairs that not-p—and it is simply unsettled which in fact obtains (113–14).

Here the sense of a ‘possible’ state of affairs—where states of affairs may be local or global (i.e., entire worlds)—is one restricted to ‘admissible’ possibilities: possibilities that are compatible with what is actually the case, in that they do not determinately misrepresent reality.15 This leads to the following characterization: Metaphysical Supervaluationism: It is metaphysically indeterminate whether P iff there are two possibly admissible, exhaustive and exclusive states of affairs (SOAs): the SOA that p and the SOA that ¬p, and it is indeterminate which of these SOAs obtains.

On the face of it, such a view might seem well-suited to accommodating the sort of quantum metaphysical indeterminacy (QMI) that is our concern here: [There is] a suggestive parallel between the terms in the superposition and the idea [. . . ] of precisifications. One of the terms in the superposition [. . . ] is a term where the cat is alive, the other is not; that is reminiscent of multiple ways of drawing the extension of ‘alive’, on some of which ‘the cat is alive’ comes out true, on some, false. (Darby 2010, 235)

15 Otherwise it would be settled that such an (incompatible) state of affairs (possibility) does not obtain.

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Crucially, the precisifications that are identified with superposition terms are maximal—or complete—and classical, hence indeterminacy-free: Importantly, given our picture of indeterminacy, all the worlds in the space of precisifications are themselves maximal and classical (Barnes and Williams 2011, 116).

25.4.2 Determinable-Based MI A determinable-based approach to MI was initially proposed in Wilson 2013, and defended in Wilson 2016; it has been applied to the case of quantum metaphysical indeterminacy in Bokulich 2014, Calosi and Wilson 2018 and 2021, and elsewhere. A determinable-based account takes an ‘object-level’ approach to MI, according to which indeterminacy is located in indeterminate states of affairs themselves, and where what it is for a state of affairs to be indeterminate is more specifically cashed in terms of a certain pattern of instantiation of determinable and determinate features: Determinable-based MI: What it is for an SOA to be MI in a given respect R at a time t is for the SOA to constitutively involve an object (more generally, entity) O such that (i) O has a determinable property P at t, and (ii) for some level L of determination of P, O does not have a unique level-L determinate of P at t (Wilson 2013: 366).

There are two ways in which an object (system) can have a determinable but no unique determinate of that determinable (at a level L, etc.; henceforth we suppress this qualification): either there is no candidate determinate (‘gappy MI’), or there are too many candidate determinates, preventing attribution of a unique such determinate (‘glutty MI’). There are, moreover, two variations on the ‘glutty’ theme, whereby: 1. multiple determinates are instantiated, albeit in relativized fashion; or 2. multiple determinates are instantiated, each to degree less than one. As discussed in Wilson 2013 and 2016, a determinable-based approach to MI differs from a supervaluationist approach in various important ways, including that a determinable-based account reduces MI to a pattern of instantiation of determinable and determinate properties, and so (unlike a supervaluationist account) does not take MI to be primitive (a point to which we return below); a determinable-based account does not introduce propositional indeterminacy, and so (unlike a supervaluationist account) does not require introducing an indeterminacy operator into one’s semantics or logic; and a determinable-based account is more generally thoroughly compatible with classical logic and semantics, and so (unlike a supervaluationist account) requires no revision in these classical theories.

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25.5 Supervaluationist vs. Determinable-Based Treatments of MI in World Nature in DEQM We now offer five arguments aimed at establishing that a determinable-based approach has a clear comparative advantage over a metaphysical supervaluationist approach, so far as accommodating MI in DEQM is concerned.

25.5.1 The Argument from Imprecise Histories Histories cannot be maximally precise: after a certain point, they fail to meet the medium decoherence condition (5). Roughly (see Gell-Mann and Hartle 1990 for technical details), if the medium decoherence condition fails to be met, the histories (could) interfere. And if they interfere, it is impossible to assign them independent probabilities—as is required by the formalism, if histories are to represent somewhat semi-classical worlds. As Gell-Mann and Hartle (1990) note: [C]ompletely fine-grained histories [. . . ] cannot be assigned probabilities; only suitable coarse-grained histories can. (433)

Completely fine-grained histories are those histories in which every value of every projection operator is specified. It follows that it is not possible to assign a precise value to every projection operator, if a history is going to qualify as a decoherent history. Decoherent histories represent Everett worlds. So, Everett worlds cannot be maximally precise.16 Now, a determinable-based approach to MI can take the failure of decoherent histories—Everett worlds—to be maximally precise at face value, as representing (for a given system) the system’s having a given determinable property—say, having a certain life status, in the case of Schrödinger’s cat, or having traveled between the emittor and the detector, in the case of the double-slit experiment—without the system’s having a unique determinate of the determinable. Not so for metaphysical supervaluationism. An application of this approach would most naturally be seen as identifying precisifications with decoherent histories. But precisifications are supposed to be classical: maximally precise and indeterminacy-free. Since decoherent histories are not maximally precise/determinacy-free, supervaluationist precisifications cannot be identified with decoherent histories. Equivalently: decoherent histories do not qualify as admissible precisifications. One might try to identify precisifications with suitable fine-graining of decoherent histories, as Wilson (2020) suggests:

16 Gell-Mann and Hartle (1990) consider different

ways of coarse-graining completely fine-grained histories; one approach proceeds by specifying ranges of values rather than precise values to associated observables (434).

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Can we nonetheless find suitable candidates for [ontic precisificiations] within EQM? One prospect is that they might be identified with quantum consistent histories [. . . ]. In order to play the role of ontic precisifications, the consistent history space in question would need to be maximally fine-grained. The decoherence conditions fail for these fine-grained consistent histories, so they are not dynamically decoupled from one another and quantum modal realists ought not to regard them as representing genuine alternative possibilities. Still, these consistent histories may be apt to play a different role in the metaphysics of quantum modal realism: the role of ontic precisifications in a Barnes–Williams-style model of metaphysical indeterminacy. (182)

But this strategy won’t work—and not just because the the decoherence conditions fail for such fine-grained histories, rendering them unsuitable for being genuine possibilities by lights of Wilson’s quantum modal realism. The more general problem is that the failure of the decoherence conditions means that there is no reason to expect that interference effects will be negligible. And as discussed previously, with interference comes indeterminacy—contra the supervaluationist supposition that precisifications are indeterminacy-free. Nor does it make sense to simply stipulate that (to some extent indeterminate) Everett worlds have multiple classical precisifications; for (in addition to such precisifications’ not being admissible, on the usual understanding of admissibility as requiring compatibility with the actual world) this would undercut the core contention of DEQM, according to which the multiplicity of Everett worlds is generated by decoherence alone. The upshot is that a determinable-based approach can, while a metaphysical supervaluationist approach cannot, accommodate MI in DEQM.

25.5.2 The Argument from Interference In Sect. 25.2, we argued that MI in world nature in DEQM is strictly related to interference—such MI is present on DEQM when and only when there are residual interference effects.17 As we argue in our (2021), a determinable-based approach to MI can accommodate, and indeed provides the basis for, an intelligible explanation of quantum interference. There, and here, we use the case of quantum self-interference in the

17 Recall the argument in Sect. 25.2. There we observed that DEQM can resort to the multiplicity reading of superposition states—the reading that eliminates indeterminacy—iff interference effects are negligible. In what follows we are going to discuss an experimental setting in which such effects are not negligible—namely, the double-slit experiment. In such cases, we contend, we are left with the indeterminacy reading.

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double-slit experiment as our case-in-point.18 Simplifying a bit, we can ascribe to each particle traveling from the source to the screen detector in the double-slit experiment the following superposition state: |ψ = c1 |A + c2 |B

(25.8)

Here |A represents the state of the particle’s traveling from emitter to detector through slit A but not slit B, and |B represents the state of the particle’s traveling from emitter to detector through slit B but not slit A. On a determinable-based account, the MI associated with double-slit indeterminacy is understood as follows: [T]he associated QMI reflects that, on any given pass of the experiment, the emitted particle has the determinable property having traversed the region between source and detector (which property is itself a determinate of position or of being spatiotemporally located), but does not have a unique determinate of that determinable, due to too many of the determinates of the determinable, associated in particular with the states |A and |B, being instantiated, in glutty fashion (Calosi and Wilson 2021).

As above, glutty MI can be cashed out in at least two ways: one in which the relevant object (system) has different determinates relative to different perspectives, and one in which its has the different determinates to a degree less than 1. On the relativization variant of glutty MI as applied to the case at hand, while superposition prevents attributing a unique trajectory to the particle, there remains a sense in which the particle can, in relativized fashion, consistently travel through both slits at a time. The claim is then that these relativized instantiations can interact, consonant with self-interference. On the degree-theoretic variant of glutty MI the particle has both determinates associated with states |A and |B, to degrees |c1 |2 and |c2 |2 respectively. The claim is then that these degreed instantiations can interact, consonant with self interference. (Again, see our 2021 for further details about these implementations of glutty MI.) Either way, interference is the result of the relevant particle’s having—either in relativized fashion, or to a degree less than one—each of the causally efficacious determinate properties associated with |A and |B. By way of contrast, as we argue in our (2021), a metaphysical supervaluationist account of MI does not have the resources to explain the existence of the interference patterns characteristic of the double-slit experiment: On [metaphysical supervaluationism], a superposition is a state whose precisifications are given by the terms of the superposition. Superposition QMI is then taken to reflect its being indeterminate which term (or associated property) of the superposition obtains. How does such an approach fare as an account of the double-slit experiment? Not well. For the supervaluationist, indeterminacy is unsettledness about which one of a range of maximally precise states of affairs obtains. On this view, it is determinate that only one such state of affairs obtains, notwithstanding that it is indeterminate which one obtains. Hence in the case of the double-slit experiment, the supervaluationist takes the superposition QMI at issue to reflect its being indeterminate which one of the states |A or |B obtains. On this

18 The discussion to follow is abbreviated, for considerations of space. See Calosi and Wilson (2021) for further details.

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account, there is no question of there being any sense in which both states obtain; again, it is determinate that only one of the states obtains. But if only one of the states obtains, then there’s no physical basis for the interference characteristic of the double-slit pattern.

More generally, and for the same reason, metaphysical supervaluationism cannot accommodate quantum interference as associated with superpositions. But MI on DEQM precisely consists in the presence of interference. Hence metaphysical supervaluationism cannot accommodate MI on DEQM.19

25.5.3 The Argument from Nonfundamental MI As previously noted, the MI in DEQM is derivative: it attaches to nonfundamental rather than fundamental ontology. A determinable-based approach to MI can accommodate derivative MI, since this approach is compatible with MI’s being either fundamental or derivative. As above, on the determinable-based approach, MI involves indeterminacy in a given state of affairs itself, where the status of a state of affairs as indeterminate is cashed in the holding of a certain pattern of determinable and determinate features.20 This pattern—whereby an object (entity, system) has a determinable feature, but no unique determinate of that determinable feature—may be instantiated by states of affairs that are fundamental and by states of affairs that are derivative. If the state of affairs instantiating the pattern is fundamental, then so will be the associated MI; if the state of affairs instantiating the pattern is derivative, then so will be the associated MI. Reflecting this flexibility, past applications of a determinable-based account have sometimes pertained to fundamental cases of MI (involving certain readings of certain interpretations of QM) and sometimes pertained to derivative cases of MI (involving macro-object boundaries and the open future). By way of contrast, one of the main proponents of a metaphysical supervaluationist approach—namely, Barnes—has argued that this approach is incompatible with MI’s being derivative. More specifically, Barnes (2014) argues that if there is MI, it must exist at the fundamental level, on pain of contradiction. As we observe in our (2021), Barnes’s argument presupposes (as per a metalevel metaphysical supervaluationist approach to MI) that MI involves its being indeterminate which of some range of perfectly determinate options obtains. In

19 One might wonder whether the supervaluationist might aim to provide a non-causal explanation of interference, as Wilson (2020) is himself inclined to do, as somehow reflecting patterns of variation across different worlds. However, Wilson’s non-causal conception aims to accommodate interference across different branches, not within a branch. Interference within a branch would presumably remain a causal affair. 20 Note that here there is no indeterminacy in ‘which’ states of affairs, precise or imprecise, do or do not obtain. Hence it is that, unlike a supervaluationist account of MI, a determinable-based account does not require or invoke propositional or sentential indeterminacy, or associated indeterminacy operators, and as such requires no departures from classical logic or semantics.

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particular, Barnes’s “simple argument that in order for there to be metaphysical indeterminacy at all there has to be indeterminacy in how things are fundamentally” (341) has as a premise that “For some complete description, D, of a way for things to be derivatively, it is indeterminate whether D is true”. This claim is rejected on an object-level determinable-based approach to MI. Correspondingly, Barnes’s argument shows, at best, that any MI of the meta-level, metaphysical supervaluationist variety must be fundamental. So to the extent that Barnes’s argument goes through, it serves also to show that metaphysical supervaluationism cannot accommodate the MI at issue in DEQM.

25.5.4 The Argument from ‘Unfamiliar Properties’ While the EQM strategy of ‘replacing indeterminacy with multiplicity’ was primarily motivated by the aim of providing a non-ad-hoc basis for reconciling Schödinger’s equation with ordinary experience, a secondary motivation was no doubt to avoid commitment to the ‘unfamiliar indeterminate properties’ seemingly represented by superposition states. DEQM provides an attractive means of accomplishing the first aim, but commitment to indeterminate properties remains, on this view. As such, it would be an advantage of an account of MI if it could not only accommodate MI in world nature on DEQM, but do so in a way rendering this MI familiar or in any case intelligible. A determinable-based account has an advantage over a metaphysical supervaluationist account, in appealing to pretheoretically and independently understood notions—a certain pattern of properties of the sort with which we are experientially and theoretically familiar—as opposed to a primitive feature of worldly ‘unsettledness’. As even proponents of a supervaluationist account admit: The conceptual advantage [of a determinable-based account over a supervaluationist account] is this: nobody who understands the machinery of determinates and determinable can fail to understand Wilson when she says that the world is metaphysically indeterminate. She has told you exactly what that means: it is for a certain kind of property to be instantiated without a certain [unique] other kind of property to be instantiated. If you understand what she means by such properties—if you grasp the determinate/determinable distinction—then there is simply no room for not understanding worldly indeterminacy. Our own account, by contrast, makes ineliminable appeal to the notion of indeterminacy when we tell you how the world is. When p is indeterminate, we tell you that either the demands for p’s truth or the demands for p’s falsity are met, it is simply indeterminate which. Someone who is sceptical about the very idea of worldly indeterminacy is of course not going to be helped by this. (Barnes and Cameron 2016, 127–8)

To be sure, proponents of metaphysical supervaluationism aim to fill in their primitive by modeling it along lines familiar from supervaluationist treatments of semantic indeterminacy, with metaphysical indeterminacy reflecting unsettledness not between linguistic precisifications, but between precisificationally possible worlds or states of affairs. Even so, the parallel doesn’t extend so far as to render a primitivist account of MI intelligible. In particular, while it is clear enough how

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semantic indeterminacy might reflect our having not yet decided how to use our language, it is not as clear how MI might reflect the world’s having not yet decided which actual way it is. In the non-quantum context: how could the world be settled (right now) about Mount Everest’s having a determinate boundary, but be unsettled (right now) about which boundary that is? In the quantum context: how could the world be settled (right now) that the particle has gone through exactly one slit, but be unsettled (right now) about which slit that was? Appeals to ‘multiple actualities’ and the like don’t do much to render such claims intelligible, much less familiar. That said, one might wonder if a determinable-based account really does accommodate MI in familiar terms, insofar as this account rejects a traditional assumption about determinables and determinates, according to which an object possessing a determinable property also possesses one and only one—a unique— determinate of that property (at a given level of determination). As discussed in Wilson 2013, however, attention to cases such as that of an iridescent feather (which has the determinable colour but no unique determinate of that determinable) indicate that the uniqueness supposition is arguably too strong, and should be rejected as generally characterizing determinables and determinates.21 In addition, as Wilson goes on to discuss (Sect. III.iv), the slate of traditionally endorsed features of determinables and determinates can be imported without much ado into the more general (and more accurate) understanding of determinables and determinates.22 Correspondingly, it remains that a determinable-based account as involving a certain pattern of instantiation of determinable and determinate properties accommodates MI in terms that are—unlike the primitivist terms of a supervaluationist account— experientially and theoretically familiar.

25.5.5 The Argument from Quantum Modal Realism Our fifth argument elaborates the first, in a way drawing on Wilson’s 2020 suggestion that DEQM provides a basis for Quantum Modal Realism (QMR)—a streamlined, naturalistic heir to Lewis’s classical modal realism. A core tenet of QMR is what Wilson calls Alignment: Alignment: to be a metaphysically possible world is to be an Everett world. (22)

21 For

discussion of other sources of resistance to the uniqueness supposition, see Wilson 2017. example, consider the ‘core’ feature of the determinable/determinate relation, according to which it is a relation of increased specificity different from the conjunct/conjunction and disjunction/disjunct relations. Even if a determinable instance may be multiply determined (in relativized or degree-theoretic fashion) or undetermined, this core feature would characterize each of the actual or counterfactual determinable/determinate relations at issue. It also remains that instances of determinates are necessarily accompanied by instances of all associated determinables. The traditional supposition that the determinates associated with a given determinable may be ordered along one or more ‘determination dimensions’ remains intact. And so on. 22 For

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On QMR, roughly speaking, “for an event to be metaphysically possible is for it to occur in some Everett world, for it to be metaphysically necessary is for it to occur in all Everett worlds, and for it to be actual is for it to occur in our own world” (29). Given QMR, the first argument in this section can be precisified (no pun intended) as follows. Supervaluationist precisifications are maximally precise; hence they are not Everett worlds. It follows, given Alignment, that supervaluationist precisifications are not metaphysically possible worlds. In that case, however, a proponent of QMR cannot appeal to metaphysical supervaluationism as a means of accommodating MI in decoherence-based EQM, for not only will supervaluationist precisifications be inadmissible, they will be metaphysically impossible. To be sure, Wilson does consider impossible worlds as a means of making sense of epistemic and conceptual modalities; but such uses of impossible worlds do not clearly carry over to the explicitly metaphysical case of quantum MI. An additional difficulty is that precisifications are supposed to be admissable, in not determinately misrepresenting reality. As such, an appeal to impossible worlds in a supervaluationist treatment of MI in world nature in DEQM would need to argue that something impossible does not determinately misrepresent something possible (and indeed actual). That seems like a hard row to hoe, to put it mildly.

25.6 Conclusion Let’s sum up the main results of this paper and their significance. First and perhaps most importantly, on a widely endorsed realist interpretation of quantum mechanics—(Decoherence Only) Everettian Quantum Mechanics—there is metaphysical indeterminacy, and in particular, indeterminacy in world nature. This is significant, insofar as quantum indeterminacy has most frequently been located in the less popular orthodox interpretation (as in Darby’s 2010 and Skow’s 2010 discussions; but see Calosi and Wilson (2018)). Second, indeterminacy in world nature in DEQM is derivative, a result which undercuts recent arguments according to which metaphysical indeterminacy must be fundamental. Third, indeterminacy in world nature in DEQM cannot be accounted in metaphysical supervaluationist terms, as yet another case-in-point of the failure of a supervaluationist approach to quantum indeterminacy. Fourth, and by way of contrast, a determinable-based account of metaphysical indeterminacy provides the basis for an illuminating explanation of indeterminacy in the multiverse, as yet another case-in-point of the success of a determinable-based approach to quantum indeterminacy. It is also worth noting that, reflecting that the Eigenstate-Eigenvalue link plays no role in DEQM, these results do not hinge on acceptance of that link; hence they sidestep concerns (as in Fletcher and Taylor 2021) about treatments of quantum indeterminacy relying on that link. All told, then, indeterminacy in world nature in DEQM represents a powerful case study of quantum metaphysical indeterminacy and of the aptitude of a determinable-based account to accommodate such indeterminacy.

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Acknowledgments We would like to thank Valia Allori, Alastair Wilson, and two anonymous referees for helpful comments and suggestions. Calosi acknowledges the support of the Swiss National Science Foundation, Project Number PCEFP1_181088, and Wilson acknowledges the support of the Social Sciences and Humanities Research Council (SSHRC), Fellowship Number 435-2017-1362.

References Barnes, E. (2010). Ontic vagueness: A guide for the perplexed. Noûs, 44, 601–627. Barnes, E. (2014). Fundamental indeterminacy. Analytic Philosophy, 55, 339–362. Barnes, E., & Cameron, R. (2016). Are there indeterminate states of affairs? No. In E. Barnes (Ed.), Current controversies in metaphysics. London: Routledge. Barnes, E., & Williams, J. R. G. (2011). A theory of metaphysical indeterminacy. In K. Bennett & D. W. Zimmerman (eds.), Oxford studies in metaphysics (Vol. 6, pp. 103–148). Oxford: Oxford University Press. Bokulich, A. (2014). Metaphysical indeterminacy, properties, and quantum theory. Res Philosophica, 91, 449–475. Calosi, C., & Mariani, C. (2021). Quantum indeterminacy. Philosophy Compass, 16(4), e12731. Calosi, C., & Wilson, J. M. (2018). Quantum metaphysical indeterminacy. Philosophical Studies, 176, 2599–2627. Calosi, C., & Wilson, J. M. (2021). Quantum indeterminacy and the double slit experiment. Philosophical Studies, 178, 3291–3317. Darby, G. (2010). Quantum mechanics and metaphysical indeterminacy. Australasian Journal of Philosophy, 88, 227–245. Deutsch, D. (1985). Quantum theory as a universal physical theory. International Journal for Theoretical Physics, 24, 1–41. Dewitt, B. (1968). The Everett-Wheeler interpretation of quantum mechanics. In C. DeWitt & J. Wheeler (Eds.), Battelle Rencontres: 1967 Lectures in Mathematics and Physics. New York: W. A. Benjamin. Dewitt, B. (1970). Physics Today 23(9), 30–35. Everett, H. (1957). Relative state formulation of quantum mechanics. Review of Modern Physics, 29, 454–462. Fine, K. (1975). Vagueness, truth and logic. Synthese, 30, 265–300. Fletcher, S. C. & Taylor, D. E. (2021). Quantum indeterminacy and the eigenstate-eigenvalue link. Synthese, 199, 1–32. Gell-Mann, M., & Hartle, J. B. (1990). Quantum mechanics in the light of quantum cosmology. In W. Zurek (ed.), Complexity, entropy and the physics of information. Boston: Addison Wesley. Gell-Mann, M., & Hartle, J. B. (1993). Categories and induction in young children. Physical Review D, 47, 3345. Glick, D. (2017). Against quantum indeterminacy. Thought, 6, 204–213. Lewis, P. J. (2016). Quantum ontology: A guide to the metaphysics of quantum mechanics. Oxford: Oxford University Press. McGee, V., & McLaughlin, B. (1995). Distinctions without a difference. Southern Journal of Philosophy, 33,203–251. Ney, A. (2021). The World in the Wave Function: A Metaphysics for Quantum Physics. New York, NY: Oxford University Press. Rovelli, C. (1996). Relational quantum mechanics. International Journal of Theoretical Physics, 35, 1637–1678. Saunders, S. (1993). Decoherence, relative states, and evolutionary adaptation. Foundations of Physics, 23, 1553–1585.

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Saunders, S. (1994). What is the problem of measurement? The Harvard Review of Philosophy, 4, 4–22. Saunders, S. (1995). Time, decoherence and quantum mechanics. Synthese, 102, 235–266. Saunders, S., Barrett, J., Kent, A., & Wallace, D. (2010). Many worlds?: Everett, quantum theory, & reality. Oxford: Oxford University Press. Skow, B. (2010). Deep metaphysical indeterminacy. Philosophical Quarterly, 60, 851–858. Wallace, D. (2008). Philosophy of quantum mechanics. In D. Rickles (Ed.), The Ashgate companion to contemporary philosophy of physics (pp. 16–98). Burlington: Ashgate Publishing Company. Wallace, D. (2012). The emergent multiverse: Quantum theory according to the Everett interpretation. Oxford: Oxford University Press. Wallace, D. (2019). What is orthodox quantum mechanics? In A. Cordero (Ed.), Philosophers Look at Quantum Mechanics. Springer Verlag. Wilson, J. M. (2013). A determinable-based account of metaphysical indeterminacy. Inquiry, 56, 359–385. Wilson, J. M. (2016). Are there indeterminate states of affairs? Yes. In E. Barnes (Ed.), Current Controversies in Metaphysics. London: Routledge. Wilson, J. M. (2017). Determinables and determinates. Stanford encyclopedia of philosophy. Wilson, A. (2020). The nature of contingency: Quantum physics as modal realism. Oxford: Oxford University Press.

Chapter 26

Fundamentality and Levels in Everettian Quantum Mechanics Alastair Wilson

Abstract Distinctions in fundamentality between different levels of description are central to the viability of contemporary decoherence-based Everettian quantum mechanics (EQM). This approach to quantum theory characteristically combines a determinate fundamental reality (one universal wavefunction) with an indeterminate emergent reality (multiple decoherent worlds). In this chapter I explore how the Everettian appeal to fundamentality and emergence can be understood within existing metaphysical frameworks, identify grounding and concept fundamentality as promising theoretical tools, and use them to characterize a system of explanatory levels (with associated laws of nature) for EQM. This Everettian levels structure encompasses and extends the ‘classical’ levels structure. The ‘classical’ levels of physics, chemistry, biology, etc. are recovered, but they are emergent in character and potentially variable across Everett worlds. EQM invokes an additional fundamental level, not present in the classical levels picture, and a novel potential role for self-location in interlevel metaphysics. When given a modal realist interpretation, EQM also makes trouble for supervenience-based approaches to levels.

26.1 Introduction Quantum physics is dramatically different from classical physics, but it is an open question exactly how deep the differences run. Some approaches to quantum theory – such as Bohmian mechanics allied with a ‘nomic’ reading of the wavefunction (Miller, 2014; Callender, 2014; Esfeld, 2014; Bhogal & Perry, 2017) – retain a picture of fundamental reality which closely resembles a classical fundamental picture of particles or fields. The distinctive novelty of quantum theory is then located in how the fundamental stuff behaves, rather than in what there fundamentally is or in

A. Wilson () University of Birmingham, Birmingham, UK Monash University, Melbourne, VIC, Australia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Allori (ed.), Quantum Mechanics and Fundamentality, Synthese Library 460, https://doi.org/10.1007/978-3-030-99642-0_26

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how the fundamental stuff grounds everything else. By contrast, Everettian quantum mechanics (EQM) revises both our view of fundamental reality and our view of how fundamental reality grounds the non-fundamental. This chapter explores the distinctive role which the notion of fundamentality plays in EQM of the contemporary ‘decoherence-based’ variety, in which quantum theory is understood along scientific realist lines without any collapse of the wavefunction. Everettians including Saunders and Wallace have exploited techniques from decoherence theory (see Crull, 2022) to argue that a space of approximately classical histories can be identified within a suitable, unitarily evolving, universal quantum state. As originally emphasized by Simon Saunders (1993, 1995, 1998), and subsequently by David Wallace (2003, 2010), the emergence of the multiverse of Everett worlds from the universal quantum state has an imprecise and for-allpractical-purposes character. This feature even makes it into the title of Wallace’s authoritative work on the Everett interpretation, The Emergent Multiverse (Wallace, 2012); it is the source of some of the most philosophically interesting features of EQM. Everettians give a highly unfamiliar picture of fundamental reality. It evolves deterministically and encompasses all of the different quantum possibilities rather than corresponding only to one quantum possibility among many. From this alien starting point, Everettians then reconceptualize ordinary scientific reasoning as correctly capturing truths about a non-fundamental subject-matter: the contents of our own Everett world. A distinction with respect to fundamentality between the fundamental quantum state and the emergent multiverse thus plays a central enabling role in the theory: it provides a principled basis for Everettians to use decoherence theory in modelling quantum measurement, it defuses demands for precise individuation criteria for worlds, and it aligns the high-level ontology of Everett worlds with higher-level ontology in science much more generally. If Everett worlds do not need to be seen as fundamental structure within the theory, and can instead be posited as higher-level explanatory structure, then adopting the many-worlds language and concepts for quantum theory is recast as a relatively conservative interpretive move rather than as a decidedly non-conservative revision of the fundamental physics. Saunders calls this point Wallace’s ‘killer observation’ (Saunders, 2010a). In my own recent work on the metaphysics of EQM (A. Wilson, 2020a) I have likewise relied on the emergent nature of the Everett multiverse when developing accounts of modality, probability, causation, moral value, and related notions in the Everettian setting. There is clear need for a serviceable notion of fundamentality if we are to make sense of the decoherence-only Everettian picture. Which accounts of fundamentality can measure up? I will suggest in Sect. 26.2 that Everettians would do well to co-opt some existing account of fundamentality which is both flexible enough to accommodate their novel relative fundamentality claims while still capturing more familiar relative fundamentality relationships in established higher-level sciences. I commend to them the options of grounding and concept fundamentality in particular.

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What consequences does EQM have for the metaphysics of levels? I will describe in Sect. 26.3 how straightforward implementations of an Everettian system of levels extend orthodox views of levels in a way which is analogous to conservative extension of theories in logic and mathematics. A traditional ordering of scientific levels is then recovered, but as an emergent substructure of the overall levels structure rather than as fundamental – and moreover, as an emergent structure which emerges in a dynamical way and may be different across different regions of the multiverse. What consequences does EQM have for the metaphysics of laws? I will suggest in Sect. 26.4 that the system of Everettian levels here outlined is naturally paired with a multi-level law structure, such that the concept of law has a unified character (that of modally strong generalization) which plays out in different ways at different levels of the Everettian hierarchy. Even though laws of Everett worlds are all non-fundamental strictly speaking, we can still make sense of gradations of fundamentality amongst the laws of these worlds. Finally, in Sect. 26.5, I will argue that Everett-specific considerations tell against using the familiar notion of supervenience to capture a levels structure. The more flexible notions of grounding and of concept fundamentality are better suited to model a scenario where physical contingency itself is emergent. The arguments of Sect. 26.5 draw on controversial premises about the interpretation of modality and probability in EQM, which I defend at some length in a recent book (A. Wilson 2020a). The arguments of Sects. 26.2, 26.3 and 26.4, however, rely only on assumptions which are common to most contemporary Everettians.

26.2 Frameworks for Fundamentality in Everettian Quantum Theory Why is an account of fundamentality in EQM needed? Wallace has after all provided an account of the emergent multiverse in terms of Dennett’s notion of real patterns (Wallace, 2003, 2010; Dennett, 1991). From most metaphysicians’ point of view, real patterns are intriguing but underdeveloped; more importantly, in their application to EQM they are embedded within the distinctive metaphysical view of science called ontic structural realism (OSR) which Wallace favours; see also Ladyman and Ross (2007). Wallace’s specific application of OSR to the Everettian setting is directed at the ontology of decoherent worlds. The idea, roughly, is that Everett worlds are dynamically robust patterns in the fundamental quantum state. It is not my intention here either to undermine OSR or to defend it; for a recent critical discussion, see Sider (2020). Instead, in this section I will offer schematic accounts of the emergent multiverse which do not presuppose OSR and which are congenial to a wider range of metaphysical views of science – including orthodox scientific realism, as well as more radically eliminative views of science. I hope that everything I say in what follows will be compatible with OSR, but one of the

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secondary messages of the chapter is that the full package of OSR is not compulsory for Everettians. So: how might we regiment the ontology of decoherence-based EQM using the tools of contemporary metaphysics? Supervenience is a natural place to start; it is familiar to philosophers from 40 years of disputes over the definition of physicalism in the philosophy of mind. Supervenience is modal correlation: the A-facts supervene on the B-facts iff there can be no difference in the A-facts without some difference in the B-facts. While this relationship can hold symmetrically, we can easily define a one-way notion: A one-way supervenes on B iff A supervenes on B and B does not supervene on A. One-way supervenience of all facts on the fundamental physical facts is close to a universal assumption within philosophy of physics, and it is typically presupposed in all discussions of EQM (with the exceptions of outliers such as Albert and Loewer 1988). Supervenience can also provide a simple but effective system of levels characterized in modal terms: levels can be modelled as a partial order generated by the relation of one-way supervenience between subject-matters, with the microphysical level at the base of the supervenience ordering. A contemporary version of this approach is offered by List (2019). Minimalist accounts of intertheoretic relations have often tried to make do with nothing but a supervenience ordering; this was Lewis’ considered approach, for example (Lewis, 1994). But philosophical times change, and increasing dissatisfaction with supervenience as a level-connection relation has come to focus on its perceived explanatory limitations. The story is told (from a dissenting point of view) by Kovacs (2019); but a consensus in many areas of metaphysics has emerged that a notion distinct from supervenience is needed to account for the explanatory role of intertheoretic relations. The holding of that distinct relation would then explain oneway supervenience, but not vice versa. This explanatory objection to supervenience applies equally to its use in characterizing the emergence of the emergent Everettian multiverse, and I think it gives good reason to look beyond supervenience when explicating the Everettian worldview. However, there is an additional reason for Everettians in particular to avoid appeal to supervenience in the present context. Doing so would rule out one of the main prospective philosophical applications of EQM – to the metaphysics of modality – and complicate the project of making sense of objective probability in EQM. This additional reason is the focus of Sect. 26.5. In the remainder of this section I will show how the distinctively Everettian combination of a fundamental universal quantum state with an emergent multiverse of non-fundamental Everett worlds can be captured within two specific metaphysical frameworks for fundamentality: metaphysical grounding and concept fundamentality. Both these frameworks have the advantage that they do not presuppose any particular ontological or metaphysical account of the nature of physical reality; they can be applied to the physical facts regardless of what we may discover or hypothesize about their underlying nature. The theory of metaphysical grounding (henceforth, just ‘grounding’) has proven a popular theoretical tool in recent metaphysics. Much of its appeal stems from its relative theoretical neutrality, which permits comparisons to be made between

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different metaphysical approaches1; key discussions of the notion are Fine (2012), Rosen (2010), Schaffer (2009), Bennett (2017). Grounding permits a logically transparent account of interlevel relations which generalizes to any kind of subjectmatter: facts at one level can ground facts at another, whatever peculiar kinds of properties or logical structures those facts might involve. This feature will be exploited in Sect. 26.3 to relate the very metaphysically heterogenous kinds of levels which are combined into the decoherence-based EQM picture. Grounding is also assigned a constitutive link to explanation. In some approaches to grounding (e.g. Fine, 2012), this link to explanation is assumed as basic, but in others (e.g. Dasgupta, 2015; Schaffer, 2017; A. Wilson, 2018, 2020b) the explanatory element is linked to the presence of mediating principles which systematize the pattern of grounding relationships. The notion of grounding has been applied to physics in a number of recent works (e.g. Schaffer & Ismael, 2020; Hicks MS). Without attempting a survey of these applications, what they typically have in common is the attempt to map a relationship in physics where there is an important asymmetry overlaid on certain background symmetric features. For example, in the context of Noether’s theorem (Hicks’ example), the existence of a suitable symmetry principle and the existence of a corresponding conservation law are interderivable in the presence of certain assumptions about the form of our dynamical theory. Despite this interderivability, consensus holds that symmetries are explanatorily prior to conservation laws. That judgment of explanatory priority can be captured by the idea that symmetries ground conservation principles in theories of the kind to which Noether’s theorem applies. For the applications to physics, the standard logical properties of ground – irreflexivity, transitivity, and anti-symmetry – are typically held fixed, although it has been suggested that anti-symmetry might be weakened to help model quantum entanglement (Calosi & Morganti, forthcoming make a related move, though in the context of essential dependence rather than of grounding). We will not need to tweak any of the standard logical properties of ground for the Everettian application, although in Sect. 26.3.2 I will argue that Everettians would do well to adopt a slightly non-standard approach to the distinction between partial and full grounds. Grounding is not the only interlevel game in town. Another prominent approach to the fundamental in recent metaphysics is Ted Sider’s generalization of Lewisian naturalness (Lewis, 1983), into a more flexible notion of concept fundamentality (Sider, 2011, 2020). Like grounding, concept fundamentality is a framework which is well suited to EQM. Items of all categories – not just properties and relations – can correspond to fundamental concepts. So we have the prospect that Ψ itself, the fundamental quantum state, is a perfectly fundamental concept. The Siderian notion of concept fundamentality is detached from any link to free recombination: there is no automatic presumption that if some particular items of vocabulary correctly capture some aspect of the structure of the world then all possible sentences which can be grammatically formed out of those items of vocabulary correspond to 1

There are limits to this theoretical neutrality; see A. Wilson (2019) for discussion.

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possible ways for the world to be. This keeps the door open to a characterization of the fundamental quantum state of the world in terms of fundamental concepts, and an account of the higher-level structure of Everett worlds and laws thereof in terms of a Sider-style metaphysical semantics (Sider, 2011) specified in terms of the fundamental. A metaphysical semantics is, at a first pass, a specification of what it is for some facts to hold in terms of some other facts which are regarded as conceptually more fundamental. The key difference between grounding theory and concept fundamentality, elaborated by Sider (2011), is that concept fundamentality and the associated notion of metaphysical semantics are naturally read as metaphysically deflationary with respect to the higher level facts: all that the higher-level facts amount to is a certain configuration (encoded by the metaphysical semantics) of the fundamental level. The concept-fundamentality picture can therefore be seen as a generalization of the often-caricatured nihilist metaphysician’s view of the world as ‘simples arranged tablewise’ – now we have ‘fundamentalia configured derivative-wise’, where there is no restriction on the fundamentalia to be simples and no restriction on the configuration to be a spatiotemporal one, and where ‘derivative-wise’ is in principle specifiable using fundamental concepts. Grounding, on the other hand, is naturally interpreted as metaphysically inflationary: when something is grounded in the fundamental, it is not merely a redescription of the fundamental in non-fundamental terms but is something real in its own right, at least partly distinct from its grounds. This conception of ground is clearly articulated by Schaffer (2009). It ties ground closely to the notion of ontological levels; to the extent that the pattern of grounding relations forms a stratified structure, grounding brings with it ontological levels. While the contrast I have just drawn between grounding and conceptfundamentality could of course be contested, it will serve for the purposes of this paper to illustrate the consequences of applying different conceptions of interlevel relations – as merely representational, or as metaphysically substantial – to the context of Everettian levels. In this section I have suggested that both grounding and Siderian concept fundamentality are promising potential candidates for illuminating the levels structure of an Everettian worldview. In the next section, I explore their application to the Everettian scenario, and three distinct types of inter-level relationship which decoherence-based EQM encompasses.

26.3 Explanatory Levels in Everettian Quantum Theory 26.3.1 The Fundamental Level and the Multiverse Level The fundamental quantum state is a strange beast. There are two main ontological accounts offered by Everettians of this state: wavefunction realism and spacetime state realism. Fundamental reality according to wavefunction-realist EQM resides in

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configuration space rather than in three-dimensional space (Albert, 1996; Albert & Ney, 2013; Ney, 2021). Spacetime state realism is unique to the Everettian scenario, and it involves attributing an extraordinary amount of structure to a given spacetime region (Wallace & Timpson, 2010).2 In this chapter I will try to stay neutral on the ontology of the quantum state, and instead focus on some facts about it which are relatively uncontroversial among Everettians. It evolves deterministically according to the unitary/linear Schrödinger equation. It exhausts fundamental physical reality. And it gives rise to an emergent multiverse. The metaphysics of the emergent multiverse is also a disputed matter which I will finesse so far as possible. Wallace’s real pattern criterion has a strongly pragmatist streak: all that it takes for something to be real is that it be useful for someone to track it for some theoretical purpose or other. Saunders (2010b) and A. Wilson (2013, 2020a) have taken this pragmatist line of thought one step further, arguing that the best pattern to extract from the fundamental state is a pattern of diverging, or parallel, worlds – rather than the splitting worlds often envisaged by Everettians. Any pattern-based strategy of this kind is of course fraught with controversy: see Kent 2010, Maudlin 2010, and Hawthorne 2010 for a variety of critiques of the Everettian appeal to ontic structural realism. I will set all of these critiques aside here: if no account of high-level ontology as patterns in the low-level ontology can be sustained, then the version of decoherence-based Everett which this chapter considers is a non-starter. If an Everettian approach to the ontology of decoherent worlds succeeds at least in broad outline, we are still left with a number of puzzles. The decoherence basis (in which we obtain histories strongly peaked around approximately classical evolutions of macroscopic observables) is itself only approximately defined. This immediately gives rise to a corresponding indeterminacy in the space of Everett worlds – both with respect to what each individual world is like, but also (at least if the state space of the cosmos is finite) with respect to how many of them there are. In A. Wilson (2020a) I call the former indeterminacy of world nature and the latter indeterminacy of world number. These indeterminacies are closely (and inversely) linked, since the more worlds there are the more determinate each individual world becomes, up to the (vaguely defined) point at which the worlds become so determinate that decoherence conditions cease to be satisfied and the worlds are no longer dynamically decoupled from each other. Quantum indeterminacy is a huge and difficult topic; for some recent discussions, see J. Wilson (2013), Wolff (2015), Calosi and Wilson (2019). The presence of these unfamiliar forms of imprecision in the space of emergent quasi-classical worlds strongly suggests we are not dealing with a standard case of interlevel relations. This suggestion is substantiated by closer consideration of

2

I have not seen any detailed discussion of the application of spacetime state realism to quantum gravity scenarios which lack fundamental spacetime. The simplest extension of the view would be to transfer what spacetime state realism says about spacetime over mutatis mutandis to whichever fundamental space hosts the fields posited by one’s preferred theory of quantum gravity.

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the metaphysical relationship which Everettians identify between the fundamental ontology and the individual worlds of an Everettian multiverse. Everett worlds are not parts of the fundamental object, the quantum state – rather, Everett worlds are parts of the multiverse, a complex derivative object identified as a distinctive kind of pattern in the fundamental object. The individual Everett worlds are indeterminate in nature, and their mereological fusion is likewise indeterminate, while the underlying fundamental quantum state remains determinate. The Everettian picture of levels, then, incorporates a distinctive holism: the fundamental state is a single object, and multiplicity of Everett worlds is only found at the derivative level. The relationship here is not one of part/whole; there are no interesting mereological relations between fundamental elements as parts and Everett worlds as wholes, or vice versa.3 Wallace instead says that the Everett worlds are ‘instantiated by’ the fundamental state. This observation rules out the application of some influential accounts of interlevel relationships which rely on mereological relations, such as that of Oppenheim and Putnam (1958). However, the flexibility of the two approaches which we have adopted for this chapter allows them to be applied immediately to the emergence of the emergent multiverse. Grounding applies straightforwardly: we obtain a real, grounded, imprecise multiverse. Metaphysical semantics also applies straightforwardly: we have a emergent multiverse language with an imprecise semantics in terms of the precise fundamental language.

26.3.2 The Multiverse Level and the Everett World Level The move from the multiverse level to the Everett world level is a quite different kind of shift from the move from the fundamental level to the multiverse level just discussed. There is no new vagueness introduced at the level of an individual Everett world, over and above the vagueness in world number and world nature already present at the multiverse level. What is introduced instead is the worldcentric perspective: a given system being centred in one specific Everett world, and having a corresponding special relationship with other events centred in same world. The distinctive explanatory power of the world-centric perspective derives from the dynamic decoupling of the worlds, such that they obey approximately classical equations of motion: systems located in one of the worlds have their causal interactions effectively limited to other systems located in the very same world. It might seem peculiar (even misguided) to label the shift from the multiverse perspective to the Everett world perspective a shift in level. After all, we don’t label the shift from considering the whole of my lawn to considering a quarter of the lawn a change in levels – and aren’t individual Everett worlds just parts of the Everett

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The parthood claim might be vindicated by loosening usual assumptions about mereology and its relation to space and time in the manner of Le Bihan (2018); Saunders (2010b) provides axioms for a mereology of state vectors which makes Everett worlds part of the universal quantum state.

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multiverse in the same way that a quarter of my lawn is part of the lawn? But the analogy breaks down immediately: the shift from the multiverse perspective to the individual Everett world perspective is more like the relation between the lawn and my own location on the lawn. Facts about which world is ours are not to be found in the fundamental physics of EQM, or in a third-person-perspective description of the whole emergent multiverse. Rather, adopting the Everett world perspective requires adopting a perspective on the multiverse, a perspective from inside one individual world and one which is occupied in common with all one’s worldmates. What outcome of quantum processes you and I observe is a matter of which Everett world we are both in. The self-locating element of a fact about an outcome of a quantum process, in the Everettian picture, is not something which is in any way determined by the non-self-locating facts about the multiverse. Knowing all there is to know about the multiverse cannot tell us where we are in it, any more than a paper map (no matter how detailed) will tell me where in my environment I am currently located without further supplementation with, e.g., perceptual information. The case is familiar from what philosophers of language call ‘essential indexicals’ – expressions like ‘here’ and ‘now’ which cannot be replaced with the specific places or times they refer to without distortion of meaning. Just as I can know that someone is spilling sugar but not know that this person is myself (this example is from Perry 1979), subsequent to a Stern-Gerlach measurement I can know that there is an xspin-up world and a x-spin-down world but not know which of these two worlds I am in. These self-locating contents are no mere curiosity: for many Everettians they provide the subject-matter for objective probabilities in EQM (Saunders & Wallace, 2008; Saunders, 2010b; Wallace, 2012; A. Wilson, 2013, 2020a; Sebens & Carroll, 2018). There is little by way of consensus in the broader metaphysics of perspectives about how to think about essentially self-locating facts, especially where these facts play important explanatory roles as they do in EQM. Some of the options on the table for understanding them include deflationary approaches, where perspectives are conceived as wholly representational; perspectives are just a mode of presentation of a non-perspectival reality. The options also include more inflationary approaches, including the prospect of perspectival facts (Giere, 2006) or an irreducible fragmentation to reality (Fine, 2005; Lipman, 2015). But it’s clear that Everettians need to give some sort of positive account of the nature of perspectives, given the unique explanatory role that perspectives on the multiverse play in their overall worldview – the role of accounting for our observations of specific outcomes of quantum processes. Grounding can handle the relation between the multiverse and Everett world levels in a distinctive way, if we acknowledge the possibility of partial grounds which cannot be completed into any set of full grounds. The perspectival fact about the outcome of a quantum process is partially grounded in the multiverse, of course – the multiverse determines what the possible outcomes of that process are. But the perspectival fact also includes a self-locating element which is not grounded in the multiverse, and is incomplete without it; so the perspectival fact can be modelled as partially grounded (in the multiverse) without being wholly grounded in it. Which

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world is ours is, in grounding terms, a brute fact. See Leuenberger (2020) for more on partial grounds without any full ground, and Bader (2021) for the connection to bruteness (though neither envisage the Everettian application I am suggesting). Concept fundamentality, and the associated metaphysical semantics, can also be used to account for the in-world perspective. What concept-fundamentalists should say is that some of our concepts – not the wholly fundamental ones, but still relatively fundamental ones, through which we view all of the contingent goings-on that are the regular subject-matter of the sciences – are essentially self-locating/indexical in character. This approach is metaphysically more lightweight than the appeal to grounding, it is recognisably a descendant of Lewis’ treatment of the semantics of centred content (Lewis 1979), and it still allows for a distinctive explanatory force to the Everett-world level facts, given that the concepts they involve are at least relatively concept-fundamental. Accordingly I think this approach is likely to appeal to many Everettians. Some may wish, though, to avoid giving indexicality any role at all in the higherlevel facts. It remains an option for Everettians to avoid bringing indexicality either into the grounding network or into fundamental concepts, and instead to make do with a purely representational account of indexicality. This would be to do without a distinct Everett world level altogether, locating special science levels as above the multiverse level without intermediaries, and would thereby avoid some of the more interesting features of the full levels hierarchy I have been describing. I leave it up to the reader to decide whether that would be a good thing.

26.3.3 The Everett World Level and the Special-Science Levels The classical model of a multi-levelled science consists of a bottom level of fundamental physics, with additional levels corresponding (at least broadly, or perhaps in the ideal limit) to key explanatory disciplines like chemistry, biology, and psychology. I will assume three features will be had by any adequate account of these levels. First, the ordering of levels is partial rather than total: levels related ‘horizontally’ such as economics and geology need not stand in any direct dependence relation. Second, the dependence between levels is asymmetric at least for the most part: the higher level depends on the lower level in an ‘upwards’ fashion.4 Third, dependence between levels is synchronic rather than diachronic. This latter feature is usually taken to rule out a causal understanding of interlevel relations, but it still leaves open a variety of possible views of the interlevel dependence relation: reduction (whether by definitional extension, model construction, or some other method), grounding, elimination of the higher level, identification of the higherlevel as the lower level, composition, essential dependence – and various others.

4

Some acknowledge some higher to lower level ‘downwards’ dependence against a background of mostly upwards dependence: Gillett (2016) offers one such view.

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How does the novel Everettian level structure relate to the familiar ‘classical’ level structure, of the special sciences overlaid on top of physics? It does so by approximating this structure within a limited domain, and extending it beyond that domain. The classical level structure is wholly embedded within the Everettian level picture, as an emergent substructure. The full hierarchy of Everettian levels, including the new fundamental level of the non-contingent quantum state, is therefore something like a semi-conservative extension of the structures of laws envisaged by previous theories of levels. The special science level and Everett world level, taken together, approximate previously envisaged systems of levels; but they should be understood by Everettians as a self-contained and largely autonomous subsystem within a deeper levels hierarchy. The Everettian reconceptualization of classical levels as the higher levels within an enlarged level structure leaves most of their core features intact. That is only to be expected: a theory at the intersection of physics and metaphysics should not have substantive implications for chemistry, for biology, or for the relation between them.5 The upshot is that, as with classical levels, we can use either grounding or concept fundamentality to model the Everettian’s emergent quasiclassical levels. When grounding is applied to the Everettian context, there need be no in-principle distinction between how the Everettian multiverse depends on the fundamental level and how different scientific levels depend on one another within the emergent Everett worlds. The differences between these types of dependence will boil down to the character of the mediating principles linking the levels in each case. Likewise, there is no difficulty in principle in specifying a metaphysical semantics for (say) facts about the frequency of a sound in terms of facts about the underlying oscillations of the air molecules. There is a very extensive discussion to be had about what non-fundamental laws there are and about how quantum mechanics underwrites and enables their operation. However, we will be able to mostly bypass that debate here, since the focus of this chapter is on Everettian quantum theory specifically, and there are few reasons to think that higher-level lawhood will play out differently depending on the interpretation of quantum mechanics that is chosen.6 The Everettian implementation of classical levels does, though, place some constraints on which interlevel relations can be involved in linking levels. In particular, the global chance measure incorporated into Everettian quantum theory is distinctively antagonistic to the type of strong emergence in which the nomic behaviour of complex systems fails to supervene on the nomic behaviour of simpler subsystems. Probabilities for events at the micro-level – and, via physicalist 5 Why disciplines have this autonomy is an interesting question, raised by Fodor (1997) and discussed by Loewer (2009) and Strevens (2012). What matters for present purposes is that they have it. 6 One possible exception is the arrow of time, for which Albert (2000) floats a candidate explanation relying on distinctive features of the Ghirardi-Rimini-Weber theory; however, that explanation would render the arrow of time after all a fundamental phenomenon rather than a high-level phenomenon, so it need not detain us here.

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supervenience, for all events – are fixed by the global chance measure. This apparently leaves no room for emergent laws at higher levels to have any further effect on those probabilities.7 Not all the features of the classical level structure are retained in the emergent levels of EQM: in particular, there is likely to be physical contingency in the levels structure, with very different higher-level phenomena playing out in Everett worlds in different regions of the multiverse. This hypothesis is supported by the apparent extreme sensitivity of physical phenomena in our current cosmological epoch to the exact value of certain ‘fine-tuned’ cosmological parameters. If – as seems quite plausible – even one of these parameters takes its value as the result of a quantum-mechanical process, then there will be Everett worlds in which the parameter took on a different value, and in those Everett worlds there will be very different physical processes ordered into different sorts of levels structures. A potential candidate mechanism is a compactification process that generates a string landscape cosmology; see Susskind (2005). The Everettian scenario of physically contingent levels contrasts with the classical picture, where levels structures are typically regarded as physically – perhaps even metaphysically – non-contingent. That is, any worlds with the same fundamental physical laws as ours (perhaps even any worlds with the same natural kinds as ours) are typically expected to also have the same levels structure. The emergent levels of EQM confound this expectation. Putting the pieces of this section together, we obtain the resulting structure of levels:

7

See Meacham (2014) for arguments of this general form.

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26.4 Levels of Laws in Everettian Quantum Theory To fix the details of an Everettian multiverse, what is needed is a suitable initial quantum state evolving according to the unitary evolution described by the Schrödinger equation. The unitary evolution, and perhaps the initial quantum state too, are obvious candidates for being truly fundamental laws of physics in an Everettian scenario. In A. Wilson (2020a) I speculated that as cosmology progresses the initial quantum state of physical reality will turn out to be a precisely defined pure quantum state, most likely with a high or maximal level of symmetry; this assumption is in line with the expectations of Wallace (2012, Forthcoming). It is important for Everettians to mark the difference between the fundamental quantum state and the emergent multiverse. It is correspondingly important to mark the difference between laws of the fundamental quantum reality and laws of individual Everett worlds. The former are novel in character, without direct analogue amongst laws of classical physics. The latter resemble closely what have traditionally been regarded as fundamental physical laws: constraint laws, force laws, conservation laws, and the like. When contrasted with classical physics, the fundamental laws in a decoherence-based Everettian picture supplement the laws which we had previously regarded as physically fundamental rather than replacing those laws directly. In A. Wilson (2020a), I offer a unified account of the fundamental and nonfundamental laws at work in the Everettian picture, making use of the notion of modally strong generalization: roughly, a generalization which is non-accidentally true. Laws of individual Everett worlds are true generalizations which hold across instances not only in the actual Everett world but also in other Everett worlds. (Fundamental laws are degenerately modally strong, since there is only one fundamental quantum state.) Each individual Everett world, on this account, comes equipped with a set of laws of its own, including both fundamental and nonfundamental laws. Lawsets of individual Everett worlds are in some respects similar to the total lawsets envisaged in one-world interpretations of quantum and classical physics: they assign probabilities (including of 0 and 1) to various histories. But they are importantly dissimilar in other respects. The Schrödinger equation itself will not appear in the laws of individual Everett worlds; that law holds only of physical reality as a whole. Likewise, the initial quantum state of physical reality is not amongst the laws of any individual Everett world, even the fundamental laws of that world. The Schrödinger equation and the initial quantum state may of course still be used by physicists to predict and explain actual events – on the present proposal, it is not only laws of the actual world which can play that predictive and explanatory role.

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This does not rule out a fundamental law of individual Everett worlds taking a boundary-condition form; (Chen, Forthcoming) highlights this possibility. One respect in which the lawsets of individual Everett worlds resemble lawsets of worlds in a one-world theory is that both kinds of lawset give rise to the problem of the arrow of time. From where, we may ask, does the evident temporal asymmetry of the actual world arise? Everettians, of course, can appeal to the initial quantum state and whatever symmetries it has, as an explanation of why the evolution of the universal quantum state is temporally asymmetric. But this account works at the fundamental level, as opposed to at the level of an individual Everett world. Here I suggest that there is a role for a vague Past Hypothesis of each individual Everett world which specifies that world’s macroscopic state at a suitable early time. Chen implements this idea as a constraint on an impure state’s density matrix in an interpretation-neutral setting.8 The resulting quantum Past Hypothesis has the curious feature of being a vague yet very general physical law. Chen suggests that we respond by positing vague fundamental laws, but the quantum modal realist can instead say that it shows that fundamental laws of individual Everett worlds can be vague even while the truly fundamental laws remain wholly precise. Then we would have an emergent vague Past Hypothesis holding within each Everett world as well as a fundamental precise initial pure quantum state of high symmetry: a boundary condition for the whole of physical reality. Here then is a full proposed hierarchy of levels of laws for Everettian quantum theory:

8

This would bring the Everett-world-level explanations of the source of probabilities and of the arrow of time much closer to the ‘Mentaculus’ picture of statistical mechanics associated with David Albert and Barry Loewer (Loewer, 2020); Chen calls his alternative the Wentaculus (Chen, Forthcoming).

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26.5 Novel Features of Everettian Levels Sections 26.3 and 26.4 sketched the application of grounding and concept fundamentality to the complex system of levels and laws which arise in decoherencebased EQM. But could more modest theoretical tools, in particular the more familiar and purely modal notion of supervenience, also do the trick? In this section I will argue that the answer is no: the distinctive nature of physical contingency in EQM precludes the exclusive use of supervenience in modelling Everettian levels. To make my case for the need to go beyond supervenience in modelling fundamentality in EQM, I will need to introduce some further assumptions about the interpretation of EQM – assumptions not shared by all Everettians. Up to this point I have tried to stay neutral on disputed features of decoherence-based EQM, but I now want to focus on the specific consequences of my preferred approach to probability in EQM: quantum modal realism. In a recent book (A. Wilson, 2020a) I argued that EQM furnishes a powerful reductive account of objective contingency. To be possible is to occur in some Everett world; to be necessary is to occur at them all. The core principles of quantum modal realism are: Alignment: to be a metaphysically possible world is to be an Everett world. (ibid. p. 22) Indexicality-of-actuality: Each Everett world is actual according to its own inhabitants, and only according to its own inhabitants. (ibid. p. 22)

Everett worlds then represent alternative possibilities – different ways things objectively could turn out – rather than representing different parts of one single, complicated, possibility. Quantum modal realism renders supervenience hopeless as an account of interlevel dependence within EQM itself. If contingency is a matter of variation across the multiverse, then the fundamental quantum state itself is non-contingent. If the emergent multiverse supervenes on the fundamental state, then there is no possible difference in the emergent multiverse without some possible difference in the fundamental. Since a non-contingent fundamental quantum state cannot be different, nor can the emergent multiverse. And so we lose the one-way nature of the dependence relationship: the fundamental quantum state supervenes on the emergent multiverse and vice versa. This ought to be no surprise: when modality lives wholly inside an Everett multiverse, it can’t also be used to characterise the emergence of that multiverse from something else. What is needed for EQM, it emerges, is an interlevel relation which can hold compatibly with one-way supervenience – and which entails one-way supervenience in cases where there is any modal variation at all – but which can also hold nontrivially in the absence of modal variation. It is also desirable that this relation should be an explanatory relation: we want to be able to explain the higher levels, including the emergence of a multiverse, on the underlying fundamental quantum level. Both of these considerations point towards employing a more substantial metaphysical level-connector framework such as grounding or concept fundamentality. In each

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case, it is supposed that the holding of the relevant grounding relation or the relevant portion of the metaphysical semantics explains why any corresponding relation of one-way supervenience holds. More directly it is supposed that the holding of the ground facts (facts specified in terms of fundamental concepts), itself explains the holding of the grounded facts (facts specified in terms of less-fundamental concepts). Hence the holding of these relations underwrites both the one-way supervenience between levels and the corresponding explanatory asymmetry. Numerous other candidate relations other than grounding and concept fundamentality have of course been suggested in the literature. Several of the relations which J. Wilson (2014) calls ‘small-g’ grounding relations – relations of functional realization, set membership, determinate/determinable relations – are also intended to carry explanatory weight, and hence to be able to explain the holding of supervenience. What is required is that the notion in question should be able to hold in asymmetric patterns even in the presence of symmetric modal dependence – and realization, set membership and determinate/determinable relations all meet this condition. My argument accordingly does not tell against the application of these notions to EQM – it is only directed against those who would try to make do with nothing but supervenience.

26.6 Conclusion An Everettian approach to quantum theory invokes a levels structure which extends previous conceptions of levels by including a level below the fundamental level of previous systems of laws. In extending systems of laws downwards in this way, the distinction between metaphysics and physics becomes blurred, and accordingly there is reason to look for explanatory relations between levels which are at home both in the contingent domain and in the noncontingent domain, and in both physics and metaphysics. Grounding and Siderian concept fundamentality both offer potential metaphysical frameworks which can accommodate an Everettian level structure. As in other domains, these approaches differ in their implications for the metaphysics of the emergent worlds; concept fundamentality lends itself to a deflationary picture where Everett worlds are really just a manner of speaking about the fundamental level of the universal quantum state, while grounding lends itself to a more inflationary picture where Everett worlds are genuine, though grounded, emergent ingredients of reality. Acknowledgements My thanks to the FraMEPhys work-in-progress group (especially Nicholas Emmerson, Joaquim Giannotti, Michael Townsen Hicks, Francis Longworth, John Murphy, Joshua Quirke, Noelia Iranzo Ribera, and Katie Robertson), to Baptiste Le Bihan and Elizabeth Miller for written comments and helpful discussion, to two very constructive referees for this volume, to an audience at the Nature of Quantum Objects conference in Geneva, and to Valia Allori for masterminding the volume. This work forms part of the project A Framework for Metaphysical Explanation in Physics (FraMEPhys), which received funding from the European Research

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Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 757295). Funding was also provided by the Australian Research Council (grant agreement no. DP180100105).

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