Progress and Visions in Quantum Theory in View of Gravity: Bridging Foundations of Physics and Mathematics 3030389405, 9783030389406

Table of contents :
Preface
Contents
Algebraic Quantum Field Theory
1 Introduction
2 Algebraic Quantum Mechanics
2.1 Postulates of Quantum Mechanics
2.2 Algebraic Approach
2.3 States and Representations
3 Case Study: The van Hove Model
4 Algebraic QFT
4.1 Basic Requirements
4.2 Examples
4.2.1 Free Real Scalar Field
4.2.2 Real Scalar Field with External Source (van Hove Encore)
4.2.3 Weyl Algebra
4.2.4 Complex Scalar Field
4.2.5 Dirac Field
4.3 Quasifree States for the Free Scalar Field
4.3.1 Examples
5 The Spectrum Condition and Reeh–Schlieder Theorem
5.1 The Spectrum Condition
5.2 The Reeh–Schlieder Theorem
6 Local von Neumann Algebras and Their Universal Type
7 The Split Property
8 Superselection Sectors
8.1 Representations of Interest in Particle Physics
8.2 Localized Endomorphisms
8.3 Intertwiners and Permutation Symmetry
9 Conclusions
Appendix 1: Some Basic Functional Analysis
Appendix 2: Construction of an Algebra from Generators and Relations
Appendix 3: Fock Space
References
Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts
1 Unifying Quantum Field Theory and General Relativity
2 Overview of Concepts and Mathematical Structures in Theoretical Physics
2.1 The Fabric of Spacetime
2.1.1 Topological Manifolds as Models of Spacetime
2.1.2 Establishing Smooth Structures in Spacetime
2.1.3 Encoding the Lorentzian Geometry of Spacetime
2.2 The Einstein Field Equations
2.3 Quantum Theory in a Classical Spacetime
2.4 Incompatibility of General Relativity and Quantum Field Theory
2.5 A Step Back: Quantum Mechanics in Curved Spacetime
2.5.1 The Dirac Equation in Minkowski Space
2.5.2 The Dirac Equation in Curved Spacetime
3 Conceptual and Mathematical Foundations of Causal Fermion Systems
3.1 Guiding Principles of the Theory of Causal Fermion Systems
3.2 Unified Description of Spacetime and the Objects Therein
3.2.1 The Measure Space (F, B, ρ)
3.2.2 The Causal Action Principle
3.3 The Equivalence Principle
3.4 Principle of Causality
3.5 Local Gauge Principle
3.6 Fermionic Building Blocks
3.7 Microscopic Spacetime Structure
4 Modelling a Lorentzian Spacetime by a Causal Fermion System
4.1 General Construction in Curved Spacetimes
4.2 Physical Significance of the Regularization Operator
4.3 Concrete Example: The Minkowski Vacuum
5 Results of the Theory and Further Reading
References
Quantum Spacetime and the Renormalization Group: Progress and Visions
1 Introduction
2 Functional Renormalization Group: Brief Overview
3 The Asymptotic Safety Scenario for Quantum Gravity
4 Background-Independent Renormalization Group Flows
5 Visions: Bridging the Gap Between Different Approaches to Quantum Gravity
References
Proposal 42: A New Storyline for the Universe Based on the Causal Fermion Systems Framework
1 Introduction
2 Causal Fermion Systems
2.1 Continuum Limit
3 Dynamical
3.1 Thermodynamic Interpretation
3.2 Dynamical Gravitational Coupling
4 Mechanism of Matter Creation
5 Reheating Uncertainty
6 The Story Line
6.1 Slow Roll to Instability
6.2 Dynamical Gravitational Coupling
7 Discussion of the Causal Fermion Systems Framework
7.1 Relevant Results
7.2 Open Questions
8 Conclusion
References
Energy Inequalities in Interacting Quantum Field Theories
1 Introduction
2 QEIs in Integrable Systems at One-Particle Level
3 Special Example: The Ising Model
4 Conclusions
References
Snyder-de Sitter Meets the Grosse-Wulkenhaar Model
1 Introduction
2 Snyder-de Sitter Model
3 Two-Point Function Renormalization
4 Conclusions
References
Fakeons, Quantum Gravity and the Correspondence Principle
1 Introduction
2 Fakeons
3 Quantum Gravity
4 The Dressed Propagators
5 Projection and Classicization
6 The Upgraded Correspondence Principle
6.1 Uniqueness
6.2 Causality
7 Conclusions
References
Implementation of the Quantum Equivalence Principle
1 Introduction
2 Basic Idea in This Paper
3 Background
4 Path from the Path Integral to Extended States for a Particle
5 From the Path Integral to Extended States in Quantum Gravity
6 Quantum Diffeomorphisms
7 Beables
8 Quantum Coordinate Systems
9 Quantum Coordinate Transformations
10 Implementing the QEP
11 General Relativity
11.1 The Problem of Relativistic Gravity
11.2 How Einstein Solved the Problem of Relativistic Gravity
12 Quantum Gravity
12.1 The Problem of Quantum Gravity
12.2 A Proposed Path to a Theory of Quantum Gravity
12.3 How Can We Use the QEP?
12.4 Quantum Manifolds
12.5 Quantum Tensor Fields
13 Questions, Comments and Conclusions
References
The D-CTC Condition in Quantum Field Theory
1 The D-CTC Condition: Bipartite Quantum Systems
2 The D-CTC Condition and Dynamics on CTC Spacetimes
3 Relativistic Quantum Field Theory
4 The D-CTC Condition in QFT on Globally Hyperbolic Spacetimes
5 Discussion
References
Remarks on Matter-Gravity Entanglement, Entropy, Information Loss and Events
1 Introduction
2 A Classic Thought Experiment in Nonrelativistic Quantum Mechanics
3 Environment Decoherence: Pros and Cons
4 Mathematical Interlude
5 The Matter-Gravity Entanglement Hypothesis
6 The Thermal Atmosphere Puzzle and Its Resolution
7 Resolution of the Black Hole Information Loss Puzzle—and of the Second Law Puzzle
8 Open Systems (Cups of Coffee)
9 Events and Time
9.1 Unravelings
9.2 A Different Possible Approach to Events Which Happen (With or Without Resets) and the Symmetry Puzzle
9.3 An Alternative Different Possible Approach to Events Which Happen (Again With or Without Resets)
9.4 Entropy
9.5 Time
References
A Generally Covariant Measurement Scheme for Quantum Field Theory in Curved Spacetimes
1 Introduction
2 System, Probe, and Coupling
3 Induced System Observables
4 Instruments and Change of State
5 A Specific Model
6 Conclusion
References
Understanding ``Understanding''
1 Introduction
2 Modes of Understanding
3 The Unreasonable Effectiveness of Physics in Mathematics
4 The Donaldson Polynomial Invariants
5 Another Perspective on γ0(M)
6 Witten's Topological Quantum Field Theory
7 Seiberg–Witten
8 More Questions
References

Citation preview

Felix Finster Domenico Giulini Johannes Kleiner Jürgen Tolksdorf Editors

Progress and Visions in Quantum Theory in View of Gravity Bridging Foundations of Physics and Mathematics

Progress and Visions in Quantum Theory in View of Gravity

Felix Finster • Domenico Giulini • Johannes Kleiner • J¨urgen Tolksdorf Editors

Progress and Visions in Quantum Theory in View of Gravity Bridging Foundations of Physics and Mathematics

Editors Felix Finster Department of Mathematics University of Regensburg Regensburg, Germany Johannes Kleiner Institute for Theoretical Physics Leibniz University of Hannover Hannover, Germany

Domenico Giulini Institute for Theoretical Physics Leibniz University of Hannover Hannover, Germany ZARM (Center of Applied Space Technology and Microgravity) University of Bremen Bremen, Germany J¨urgen Tolksdorf Max-Planck-Institute for Mathematics in the Sciences Leipzig, Germany

ISBN 978-3-030-38940-6 ISBN 978-3-030-38941-3 (eBook) https://doi.org/10.1007/978-3-030-38941-3 Mathematics Subject Classification (2010): 81-02, 81-06, 81T05, 81T10, 81T17, 81T08, 81T20, 83-02, 83-06, 85-02, 85-06, 82-02, 82-06 © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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 book is published under the imprint Birkhäuser, www.birkhauser-science.com, by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This volume contains a carefully selected cross-section of the talks and ideas presented at the international conference Progress and Visions in Quantum Theory in View of Gravity: Bridging foundations of physics and mathematics which was held at the Max Planck Institute for Mathematics in the Sciences in Leipzig from October 1 to October 5, 2018. This conference was a successor of similar international conferences which took place at the Heinrich Fabri Institute (Blaubeuren) in 2003 and 2005, at the Max Planck Institute for Mathematics in the Sciences (Leipzig) in 2007, and at the University of Regensburg in 2010 and 2014. The basic intention of this series of conferences is to bring together physicists, mathematicians, and philosophers working in foundations of mathematical physics to discuss profound questions in Quantum Theory, Quantum Field Theory, and Gravity. The unification of General Relativity and Quantum Field Theory into one coherent framework is generally considered to be one of the largest open problems in fundamental physics, and research programs which aim at providing such a framework are being pursued with great effort. To date, the lack of experimental access to physical systems which experience high gravitational fields as well as strong quantum effects constitutes the largest obstacle in the development of these approaches, rendering experimental guidance for these research programs almost nonexistent. A promising way to find new experimental constraints in this area is offered by the study of quantum systems in weak gravitational fields as well as Quantum Field Theory beyond perturbation theory. Contrary to general belief, many of the most basic physical questions are still open in this realm. The goal of this conference was to bring together physicists, mathematicians, and philosophers to discuss the conceptual and technical questions at the core of contemporary research related to the interplay between Quantum (Field) Theory and Gravity. The explicit intention of our discussions was to highlight the otherwise often ignored open conceptual problems, to facilitate exchange and cooperation between researchers of the various different disciplines when addressing these

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problems, and eventually to find answers that guide and constrain future theoretical as well as experimental work. In order to motivate open-minded discussions and exchanges, the participants and speakers were asked to present visions, research strategies, and new ideas in foundational physics, in particular concerning foundations of Quantum (Field) Theory and Gravity, instead of reporting on the most recent technical results. Also, new perspectives and strategies concerning experiments and observations at the interface between Gravity and Quantum Theory were particularly encouraged for presentation and discussion. In addition to the talks by invited speakers and participants, a substantial number of slots were reserved to discussion sessions intended to discuss the presented material as well as other pressing questions in the field. The following sessions have been scheduled in parallel groups. 1. Conceptual significance and experimental possibilities of Quantum Field Theory beyond scattering theory 2. Which mathematical and physical structures of Quantum Field Theory are believed to transcend the gravity-quantum unification? 3. The role of observables in Quantum Gravity in light of diffeomorphism invariance and quantization 4. Axiomatization of Quantum Field Theory: Will practical Quantum Field Theory one day be replaced by a proper mathematical framework with axiomatizable foundations? 5. A critical assessment of current approaches to Quantum Gravity concerning unification 6. Theory-experiment wish list: What is most urgently needed by one side from the other in order to make progress? 7. Do we really have a fundamental understanding of ordinary matter? Can Quantum Mechanics be derived from Quantum Field Theory? 8. Does Gravity play a role when considering interpretations or modifications of Quantum Theory? 9. Guiding questions for the next generation 10. Which problems and which opportunities would arise if Gravity were fundamentally classical? 11. Unification wish list: What questions do we hope/expect/demand a fundamental theory to answer? What is a checklist for a new unified theory? 12. What cherished physical principles are you most willing to give up, and why, if Quantum Gravity will not allow them all? 13. Nature of Time (Is time fundamental? What mathematical structure could time have?) 14. What is the meaning of probabilities in physical theories where properties/events do not repeat themselves? 15. What is the microscopic structure of spacetime? If spacetime were fundamentally discrete, what would be the macroscopically observable implications?

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16. Tabletop/low energy experiments for probing gravitational effects in quantum systems 17. Must spacetime singularities be resolved in Quantum Gravity? Will new insights into the causal structure of quantum spacetime shift the focus in studying black holes? 18. To what extent do the rigorous proofs of mathematical results in Quantum Field Theory contribute to our “understanding” of those results? 19. Is Quantum Field Theory in a classical external field (gravitational or electromagnetic) well understood conceptually and mathematically? 20. What is the measurement problem in Quantum Field Theory? 21. What is the significance of a global state? Is global hyperbolicity a necessary condition for consistent Quantum Field Theory? 22. What counts as “understanding” in Quantum Mechanics and Quantum Field Theory? 23. Does gravitational entanglement imply that Gravity is quantum? (And if yes, may it inform specific Quantum Gravity Theories?) 24. Quantum Causal Indefinite Structures and Quantum Field Theory—Mutual inspirations and conflicting concepts 25. New experimental paradigms at the interface of Quantum Mechanics, Quantum Field Theory, and Gravity which are currently controversial or go beyond what has been published so far 26. Ideas for innovative formats of conferences: How can we communicate more efficiently and understand more? 27. Which visions do senior scientists have for the future? What could senior scientists learn from juniors? 28. What you always wanted to ask/know/clarify? We are grateful for support received from Max Planck Institute for Mathematics in the Sciences, the Leibniz University of Hanover, the University of Regensburg, the Research Academy Leipzig, the International Association of Mathematical Physics, the Johannes-Kepler-Research Center for Mathematics, the Collaborative Research Center 1227 Designed Quantum States of Matter (DQ-mat), the Cluster of excellence “QUEST” of the Leibniz University Hanover as well as the Leopoldina National Academy of Sciences. The conference was dedicated to Eberhard Zeidler, who sadly passed away on 18 November 2016. He was the founding director of the Max Planck Institute for Mathematics in the Sciences. With his outstanding knowledge and warm-hearted personality, he helped shape this conference series. Being thankful for all the inspiration he gave us, the organizers aim at keeping alive his scientific visions. Regensburg, Germany Hannover, Germany Hannover, Germany Leipzig, Germany September 2019

Felix Finster Domenico Giulini Johannes Kleiner Jürgen Tolksdorf

Contents

Algebraic Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Christopher J. Fewster and Kasia Rejzner

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Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Felix Finster and Maximilian Jokel

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Quantum Spacetime and the Renormalization Group: Progress and Visions . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Antonio D. Pereira

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Proposal 42: A New Storyline for the Universe Based on the Causal Fermion Systems Framework.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 119 Claudio F. Paganini Energy Inequalities in Interacting Quantum Field Theories . . . . . . . . . . . . . . . . 155 Daniela Cadamuro Snyder-de Sitter Meets the Grosse-Wulkenhaar Model .. . . . . . . . . . . . . . . . . . . . 163 Sebastián A. Franchino-Viñas and Salvatore Mignemi Fakeons, Quantum Gravity and the Correspondence Principle . . . . . . . . . . . . 171 Damiano Anselmi Implementation of the Quantum Equivalence Principle.. . . . . . . . . . . . . . . . . . . . 189 Lucien Hardy The D-CTC Condition in Quantum Field Theory . . . . . . . .. . . . . . . . . . . . . . . . . . . . 221 Rainer Verch Remarks on Matter-Gravity Entanglement, Entropy, Information Loss and Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 233 Bernard S. Kay

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A Generally Covariant Measurement Scheme for Quantum Field Theory in Curved Spacetimes . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 253 Christopher J. Fewster Understanding “Understanding” .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 269 Gregory Naber

Algebraic Quantum Field Theory An Introduction Christopher J. Fewster and Kasia Rejzner

Abstract We give a pedagogical introduction to algebraic quantum field theory (AQFT), with the aim of explaining its key structures and features. Topics covered include: algebraic formulations of quantum theory and the GNS representation theorem, the appearance of unitarily inequivalent representations in QFT (exemplified by the van Hove model), the main assumptions of AQFT and simple models thereof, the spectrum condition, Reeh–Schlieder theorem, split property, the universal type of local algebras, and the theory of superselection sectors. The abstract discussion is illustrated by concrete examples. One of our concerns is to explain various ways in which quantum field theory differs from quantum mechanics, not just in terms of technical detail, but in terms of physical content. The text is supplemented by exercises and appendices that enlarge on some of the relevant mathematical background. These notes are based on lectures given by CJF for the International Max Planck Research School at the Albert Einstein Institute, Golm (October, 2018) and by KR at the Raman Research Institute, Bangalore (January, 2019).

1 Introduction Algebraic Quantum Field Theory (AQFT) is one of two axiomatic programmes for QFT that emerged in the 1950s, in response to the problem of making QFT mathematically precise. While Wightman’s programme [41] maintains an emphasis on quantum fields, AQFT [1, 28], developed initially by Haag, Kastler, Araki and others, takes the more radical step on focussing on local observables, with the idea that fields can emerge as natural ways of labelling some of the observables. Like Wightman theory, its primary focus is on setting out a precise mathematical framework into which all QFTs worthy of the name should fit. This permits one to separate the general study of the structure and properties of QFT

C. J. Fewster () · K. Rejzner Department of Mathematics, University of York, Heslington, York, UK e-mail: [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 F. Finster et al. (eds.), Progress and Visions in Quantum Theory in View of Gravity, https://doi.org/10.1007/978-3-030-38941-3_1

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from the problem of constructing (by whatever means) specific QFT models that obey the assumptions. The early development of AQFT is well-described in the monographs of Haag [28] and Araki [1]. Mathematically, it makes extensive use of operator algebra methods and indeed has contributed to the theory of von Neumann algebras in return. Relevant aspects of operator algebra theory, with links to the physical context, can be found in the monographs of Bratteli and Robinson [3, 4]. AQFT also comprises a lot of machinery for treating specific QFT models, which have some advantages relative to other approaches to QFT. During the last 20 years it has also been adapted to provide rigorous constructions of perturbative QFT, and also of some low-dimensional models, and its overall viewpoint has been particularly useful in the theory of quantum fields in curved spacetimes. A recent edited collection [6] summarises these developments, and the two recent monographs [17, 39] in particular describe the application to perturbation theory, while [31] concerns entanglement measures in QFT. An extensive survey covering some of the topics presented here in much greater depth can be found in [30]. The purpose of these lectures is to present an introduction to AQFT that emphasises some of its original motivations and de-mystifies some of its terminology (GNS representations, spectrum condition, Reeh–Schlieder, type III factors, split property, superselection sectors. . . ). We also emphasise features of QFT that sharply distinguish it from quantum mechanics and which can be seen particularly clearly in the AQFT framework. Our treatment is necessarily limited and partial; the reader is referred to the literature mentioned for more detail and topics not covered here. The idea of algebraic formulations of quantum theory, which we describe in Sect. 2, can be traced back to Heisenberg’s matrix mechanics, in which the algebraic relations between observables are the primary data. Schrödinger’s wave mechanics, by contrast, starts with spaces of wavefunctions, on which the observables of the theory act in specific ways. As far as position and momentum go, and for systems of finitely many degrees of freedom, the distinction is rather inessential, because the Stone-von Neumann theorem guarantees that any (sufficiently regular1) irreducible representation of the commutation relations is unitarily equivalent to the Schrödinger representation. However, the angular momentum operators provide a classic example in which inequivalent physical representations appear, and it is standard to study angular momentum as an algebraic representation problem. However, it was a surprise in the development of QFT that unitarily inequivalent representations have a role to play here, and indeed turn out to be ubiquitous. Section 3 is devoted to the van Hove model, one of the first examples in which this was understood. The van Hove model concerns a free scalar field with an external source, and is explicitly solvable. However, one can easily find situations in which a naive interaction picture approach fails to reproduce the correct solution—a failure that can be clearly ascribed to a failure of unitary equivalence between different representations of the canonical commutation relations (CCRs).

1 To

deal with the technical problems of using unbounded operators.

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After these preliminaries, we set out the main assumptions of Algebraic Quantum Field Theory in Sect. 4. In fact there are many variants of AQFT and we give a liberal set of axioms that can be strengthened in various ways. We also describe how some standard QFT models can be formulated in terms of AQFT. Although we focus on free theories, it is important to emphasise that AQFT is intended as a framework for all quantum field theories worthy of the name, and successfully encompasses some nontrivial interacting models in low dimensions. AQFT distinguishes between two levels of structure: on the one hand, the algebraic relations among observables and on the other, their concrete realisation as Hilbert space operators. The link is provided by the GNS representation theorem (described in Sect. 2.3) once a suitable state has been given. For this reason we spend some time on states of the free scalar field, describing in particular the quasi-free states, which have representations on suitable Fock spaces. These include the standard vacuum state as well as thermal states. The remaining parts of the notes concern general features of AQFT models, where the power of the technical framework begins to come through. Among other things, we prove the Reeh–Schlieder theorem and discuss some of its consequences in Sect. 5, before turning in Sect. 6 to the structure of the local von Neumann algebras in suitable representations and the remarkable result (which we describe, but do not prove) that they are all isomorphic to the unique hyperfinite factor of type III1 . The distinction between one theory and another therefore lies in the way these algebras are situated, relative to one another, within the algebra of bounded operators on the Hilbert space of the theory. Finally, Sects. 7 and 8 discuss the split property and the theory of superselection sectors. Like the theory in Sect. 6, these are deep and technical subjects and our aim here is to present the main ideas and some outline arguments, referring the dedicated reader to the literature. On the subject of literature: in this pedagogical introduction we have tended to give references to monographs rather than original papers, so the reference list is certainly not intended as a comprehensive survey of the field. These notes represent a merger and expansion of lectures given by CJF at the AEI in Golm (October, 2018) and by KR at the Raman Research Institute, Bangalore (January, 2019). We are grateful to the students and organisers of the lecture series concerned. We are also grateful to the organisers of the conference Progress and Visions in Quantum Theory in View of Gravity (Leipzig, October 2018) for the opportunity to contribute to their proceedings.

2 Algebraic Quantum Mechanics 2.1 Postulates of Quantum Mechanics The standard formalism of quantum theory starts with a complex Hilbert space H, whose elements φ ∈ H are called state vectors. (For convenience some basic

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definitions concerning operators on Hilbert space are collected in Appendix 1.) The key postulates of quantum mechanics say that: • Pure states of a quantum system are described by rays in H, i.e. [ψ] := {λψ|λ ∈ C}. Mixed states are described by density matrices, i.e., positive trace-class operators ρ : H → H, with unit trace. • Observables are described by self-adjoint operators A on H. However, the selfadjoint operators corresponding to observables may be a proper subset of the self-adjoint operators on H; in particular, this occurs if the system is subject to superselection rules. The probabilistic interpretation of quantum mechanics2 is based on the idea that one can associate to a self-adjoint operator A and a normalised state vector ψ ∈ H a probability measure μψ,A , so that the probability of the measurement outcome to be within a Borel subset  ⊂ R (for instance, an interval [a, b]) is given by  μψ,A (λ) = ψ|PA ()ψ ,

Prob(A ∈ ; ψ) = 

where PA () is the spectral projection of the operator A associated with , and indeed  → PA () determines a projection-valued measure. The probability measure μψ,A depends only on the ray [ψ] and has support contained within the spectrum of A. The moments of this measure are given by  νn :=

  λn μψ,A (λ) = ψ, An ψ ;

conversely, the moments determine the measure uniquely subject to certain growth conditions. For example, the Hamburger moment theorem [40] guarantees uniqueness provided that there are constants C and D such that |νn | ≤ CD n n!

for all n ∈ N0 .

(1)

Note, however, that there are many examples in quantum mechanics in which the moments grow too fast for the unique reconstruction of a probability measure. Consider, for example, a quantum particle confined to an interval (−a, a) subject to either Dirichlet or Neumann boundary conditions at the endpoints, with corresponding Hamiltonian operators HD or HN respectively. Measurements of the energy in a state ψ ∈ L2 (−a, a) supported away from the endpoints are distributed according to probability distributions μψ,HD and μψ,HN , which differ because the spectra of HD and HN differ. However, they share a common moment sequence because HD and HN agree on state vectors supported away from the endpoints.

2 For simplicity, we restrict to sharp measurements, avoiding the introduction of positive operator valued measures.

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One can combine effects from two physical states described by state vectors ψ1 and ψ2 by building their superposition, which, however depends on the choice of representative state vectors, since the ray corresponding to ψ = αψ1 + βψ2 typically depends on the choice of α and β. However, sometimes the relative phase between the state vectors we are superposing cannot be observed. For example, this occurs if ψ1 is a state of integer angular momentum, while ψ2 is a half-integer angular momentum state. The physical reason for this is that a 2π-rotation cannot be distinguished from no rotation at all. Of course there are self-adjoint operators on the Hilbert space that do sense the relative phase: the point is that these operators are not physical observables. Let us give a brief argument for the existence of such superselection rules when the theory possesses a charge Q, which is supposed to be conserved in any interactions available to measure it. For simplicity, we assume that Q has discrete spectrum. Let ψ be any eigenstate Qψ = qψ and let P be any projection corresponding to a zero-one measurement. After an ideal measurement of P in state ψ returning the value 1, the system is in state P ψ. But as charge is conserved, P ψ must be an eigenstate of Q with eigenvalue q, so QP ψ = qP ψ = P Qψ. We deduce that [Q, P ]ψ = 0 and, as the Hilbert space is spanned by eigenstates of Q, it follows that Q and P commute. Furthermore, Q commutes with every selfadjoint operator representing a physical observable because any spectral projection of such an operator also corresponds to a physical observable. Mathematically, the allowed physical observables are all block diagonal with respect to a decomposition of the Hilbert space H as  H= Hi , i∈I

where I is some index set and the subspaces Hi are called superselection sectors, which would be the charge eigenspaces in our example above. The relative phases between state vectors belonging to different sectors cannot be observed. One of the main motivations behind AQFT was to understand how superselection sectors arise in QFT. We will see that the different superselection sectors correspond to unitarily inequivalent representations of the algebra of observables.

2.2 Algebraic Approach The main feature of the algebraic description of quantum theory is that the focus shifts from states to observables, and their algebraic relations. It is worth pausing

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briefly to consider the motivation for an algebraic description of observables—this is a long story if told in full (see [18]), but one can explain the essential elements quite briefly. The central issue is to provide an operational meaning for the linear combination and product of observables. Let us suppose that a given observable is measured by a certain instrument; for measurements conducted in each particular state, the numerical readout on the instrument is statistically distributed in a certain way, so the observable may be thought of as a mapping from states to random variables taking values in R. Given two such observables, we can form a third, by taking a fixed linear combination of the random variables concerned, restricting to real-linear combinations in the first instance. So there is a clear justification for treating the set of observables as a real vector space. Similarly, we may apply a function to an observable by applying it to the random variables concerned; this may be regarded as repainting the scale on the measuring instruments. Given any two observables A and B it is now possible to form the observable A ◦ B :=

 1 (A + B)2 − A2 − B 2 , 2

(2)

simply by forming linear combinations and squares. This may be regarded as a symmetrised product of A and B. The remaining problem, which is naturally where the hard work lies, is to find appropriate additional conditions under which the vector space of observables can be identified with the self-adjoint elements of a ∗-algebra, so that A ◦ B = AB + BA. We refer the reader to [18]; however, it is clear that observables naturally admit some algebraic structure beyond that of a vector space. The main postulates of quantum theory, in its algebraic form, are now formulated as follows: 1. A physical system is described by a unital ∗-algebra A, whose self-adjoint elements are interpreted as the observables. It is conventional though slightly imprecise to call A the algebra of observables. In many situations we impose the stricter condition that A be a unital C ∗ -algebra. 2. States are identified with positive, normalized linear functionals ω : A → C, i.e. we require ω(A∗ A) ≥ 0 for all A ∈ A and ω(1) = 1 as well as ω being linear. The state is mixed if it is a convex combination of distinct states (i.e., ω = λω1 + (1 − λ)ω2 with λ ∈ (0, 1), ω1 = ω2 ) and pure otherwise. Some definitions are in order. Definition 1 A ∗-algebra (also called an involutive complex algebra) A is an algebra over C, together with a map, ∗ : A → A, called an involution, with the following properties: 1. for all A, B ∈ A: (A + B)∗ = A∗ + B ∗ , (AB)∗ = B ∗ A∗ , 2. for every λ ∈ C and every A ∈ A: (λA)∗ = λA∗ , 3. for all A ∈ A: (A∗ )∗ = A.

Algebraic Quantum Field Theory

7

The ∗-algebra is unital if it has an element 1 which is a unit for the algebraic product (A1 = 1A = A for all A ∈ A) and is therefore invariant under the involution. Unless explicitly indicated otherwise, a homomorphism α : A1 → A2 between two ∗-algebras will be understood to be an algebraic homomorphism that respects the involutions ((αA)∗ = α(A∗ )) and preserves units (α1A1 = 1A2 ). The bounded operators B(H) on a Hilbert space H form a ∗-algebra, with the adjoint as the involution, but there are other interesting examples. Exercise 1 (Technical—for those familiar with unbounded operators.) Given a dense subspace D of a Hilbert space H, let L(D, H) be the set of all (possibly unbounded) operators A on H defined on, and leaving invariant, D, (i.e., D(A) = D, AD ⊂ D) and having an adjoint with D ⊂ D(A∗ ). Then L(D, H) may be identified with a subspace of the vector space of all linear maps from D to itself. Verify that L(D, H) is an algebra with respect to composition of maps and that the map A → A∗ |D is an involution on L(D, H), making it a ∗-algebra. Show also that L(H, H) = B(H). (Hint: Use the Hellinger–Toeplitz theorem [38, §III.5].) The algebra of bounded operators also carries a norm that is compatible with the algebraic structure in various ways. In general we can make the following definitions: Definition 2 A normed algebra A is an algebra equipped with a norm . satisfying

AB ≤ A

B . If A is unital, then it is a normed unital algebra if in addition 1 = 1. If A is complete in the topology induced by · then A is a Banach algebra; if, additionally, A is a ∗-algebra and A∗ = A , then A is a Banach ∗-algebra or B ∗ -algebra. A C ∗ -algebra is a particular type of B ∗ -algebra. Definition 3 A C ∗ -algebra A is a B ∗ -algebra whose norm has the C ∗ -property:

A∗ A = A

A∗ = A 2 ,

∀A ∈ A .

The bounded operators B(H), with the operator norm, provide an important example of a C ∗ -algebra. A useful property of unital C ∗ -algebras is that homomorphisms between them are automatically continuous [3, Prop. 2.3.1], with unit norm. Turning to our second postulate, the role of the state in the algebraic approach is to assign expectation values: if A = A∗ , we interpret ω(A) as the expected value of A if measured in the state ω. At first sight this definition seems far removed from the notion of a state in conventional formulations of quantum mechanics. Let us see that it is in fact a natural generalisation. Suppose for simplicity (and to reduce notation) that A is an algebra of bounded operators acting on a Hilbert space H, with the unit of A coinciding with the unit operator on H. Then every unit vector ψ ∈ H induces a vector state on A by the

8

C. J. Fewster and K. Rejzner

formula ωψ (A) = ψ|Aψ, as is seen easily by computing ψ|1ψ = 1 and ψ|A∗ Aψ = Aψ 2 ≥ 0. Exercise 2 Show that every density matrix (a positive trace-class operator ρ on H with tr ρ = 1) induces a state on A according to ωρ (A) = tr ρA

A ∈ A.

However it is important to realise that, in general, not all algebraic states on A need arise from vectors or density matrices in a given Hilbert space representation. A further important point is that the definition of a state is purely mathematical in nature. It is not guaranteed that all states correspond to physically realisable situations, and indeed a major theme of the subject is to identify classes of states and representations that, by suitable criteria, may be regarded as physically acceptable. Exercise 3 By mimicking the standard arguments from quantum mechanics or linear algebra, show that every state ω on a ∗-algebra A induces a Cauchy–Schwarz inequality |ω(A∗ B)|2 ≤ ω(A∗ A)ω(B ∗ B)

(3)

for all A, B ∈ A. Show also that ω(A∗ ) = ω(A), for any A ∈ A. (Hint: consider the linear combination 1 + αA, for α ∈ C.)

2.3 States and Representations The Hilbert space formulation of quantum mechanics is too useful to be abandoned entirely and the study of Hilbert space representations forms an important part of AQFT. Let us recall a few definitions. Definition 4 A representation of a unital ∗-algebra A consists of a triple (H, D, π), where H is a Hilbert space, D a dense subspace of H, and π a map from A to operators on H with the following properties: • each π(A) has domain D(π(A)) = D and range contained in D, • π(1) = 1|D , • π respects linearity and products, π(A + λB + CD) = π(A) + λπ(B) + π(C)π(D),

A, B, C, D ∈ A, λ ∈ C

• each π(A) has an adjoint with D ⊂ D(π(A)∗ ), whose restriction to D obeys π(A)∗ |D = π(A∗ ).

Algebraic Quantum Field Theory

9

In short, π is a homomorphism from A into the ∗-algebra L(D, H) defined in Exercise 1. Note that every π(A) is closable, due to the fact that π(A)∗ is densely defined. We will also use the shorthand notation (H, π) for a representation (H, H, π). In this case, π is a homomorphism π : A → B(H), and is necessarily continuous if A is a C ∗ -algebra. A representation π is called faithful if ker π = {0}. It is called irreducible if there are no subspaces of H invariant under π(A) that are not either trivial or dense in H. Definition 5 Two representations (H1 , D1 , π1 ) and (H2 , D2 , π2 ) of a ∗-algebra A are called unitarily equivalent, if there is a unitary map U : H1 → H2 which restricts to an isomorphism between D1 and D2 , and U π1 (A) = π2 (A)U holds for all A ∈ A. They are unitarily inequivalent if they are not unitarily equivalent. On a first encounter, algebraic states feel unfamiliar because one is so used to the Hilbert space version. However algebraic states are not too far away from a Hilbert space setting. The connection is made by the famous GNS (Gel’fand, Naimark, Segal) representation theorem. Theorem 1 Let ω be a state on a unital ∗-algebra A. Then there is a representation (Hω , Dω , πω ) of A and a unit vector ω ∈ Dω such that Dω = πω (A) ω and ω(A) =  ω |πω (A) ω ,

∀ A ∈ A.

(4)

Furthermore (Hω , Dω , πω , ω ) are unique up to unitary equivalence. If A is a C ∗ algebra, then, additionally, (i) each πω (A) extends to a bounded operator on Hω ; (ii) ω is pure if and only if the representation is irreducible; and (iii) if πω is faithful [i.e., πω (A) = 0 only if A = 0] then πω (A) = A A . Due to the fact that πω (A) ω is dense in Hω , we say that ω is cyclic for the representation. The existence of a link between purity and irreducibility is easily understood from the following example: if ψ, ϕ ∈ H are linearly independent (normalised) vector states on a subalgebra A of B(H), then the density matrix state ρ = λ|ψψ| + (1 − λ)|ϕϕ| can be realised as the vector state ρ = representation

0 0; see Fig. 1. All double cones are causally convex. • For technical reasons it is often useful to require that each A(O) is in fact a C ∗ algebra, but we need not insist on this, nor even that the A(O) carry any topology. In the C ∗ -case, one would require A(M) to be generated in a C ∗ -sense by the local algebras—technically it is their C ∗ -inductive limit. In particular, the union

O A(O) would be dense in A(M). • Sometimes (particularly in curved spacetimes) it is convenient to allow for local algebras indexed by unbounded (= noncompact closure) regions. • Einstein causality requires elements of algebras of spacelike separated regions to commute. Therefore Fermi fields can only appear in products involving even numbers of fields. We return to this later. • As mentioned, these are minimal requirements for AQFT but do not, by themselves, suffice to distinguish a quantum field theory from other relativistic models. • Nothing has yet been said about Hilbert spaces, or about what algebraic states on the observable algebras are to be regarded as physical; we will discuss these issues later. Note that one can do quite a lot without ever going into Hilbert spaces. For example, let A(M) be the algebra of the free real scalar field [see below], and let ω be a state on A(M). Then the smeared n-point function is Wn (f1 , f2 , . . . , fn ) := ω((f1 )(f2 ) · · · (fn )),

f1 , . . . , fn ∈ C0∞ (M)

and if this is suitably continuous w.r.t. the fi , it defines a distribution Wn ∈ D  (Mn ). Here, (f ) denotes a ‘smeared field’ as will be described shortly. Therefore sufficiently regular states ω define a hierarchy of distributional npoint functions, without ever using a Hilbert space. Here, as elsewhere in these notes, C0∞ (M) denotes the space of smooth, complex-valued functions on M with compact support (i.e., vanishing outside a bounded set).

Algebraic Quantum Field Theory

19

4.2 Examples We continue by giving some specific examples of field theories in the framework of AQFT, drawing out various lessons as we go.

4.2.1 Free Real Scalar Field Consider the field equation ( + m2 )φ = 0

(23)

and let E + and E − be the corresponding retarded and advanced Green operators, i.e., φ = E ± f solves the inhomogeneous equation ( + m2 )φ = f

(24)

and the support of φ is contained in the causal future (+, retarded) or causal past (−, advanced) of the support of f . Also define the advanced-minus-retarded solution operator E = E − − E + and write  E(f, g) =

M

f (x)(Eg)(x) dvolM (x) ,

(25)

where dvolM (x) ≡ d 4 x. The integral kernel is familiar from standard QFT:  E(x, y) = − where k • = (ω, k), ω = Minkowski spacetime.



d 3 k sin k · (x − y) , (2π)3 ω

(26)

k 2 + m2 , and we use standard inertial coordinates on

Exercise 6 Show that E(x, y)|y 0 =x 0 = 0, and ∂y 0 E(x, y)|y 0 =x 0 = δ(x − y). To formulate the quantized field in AQFT, we give generators and relations for the desired algebra A(M), thus specifying it uniquely up to isomorphism. For completeness, a detailed description of the construction is given in Appendix 2. The generators are written (f ), labelled by test functions f ∈ C0∞ (M)—for the moment this just a convenient way of writing them; there is no underlying field (x) to be understood here. The relations imposed, for all test functions f, g ∈ C0∞ (M), are: SF1 SF2 SF3 SF4

Linearity f → (f ) is complex linear Hermiticity (f )∗ = (f ) Field equation (( + m2 )f ) = 0 Covariant commutation relations [(f ), (g)] = iE(f, g)1.

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C. J. Fewster and K. Rejzner

As a consequence of the identities in Exercise 6, SF4 is a covariant form of the equal time commutation relations; a nice way of seeing this directly is to follow the on-shell Peierls’ method [36] to find a covariant Poisson bracket for the classical theory. The axioms we have just stated may be regarded as the result of applying Dirac’s quantisation rule to this bracket, with (f ) regarded as the quantization of the observable  Ff [φ] = f φ dvolM (27) on a suitable solution space to the Klein–Gordon equation. One should check, of course, that the algebra A(M) is nontrivial. It is not too hard [though we will not do this here] to show that the underlying vector space of A(M) is isomorphic to the symmetric tensor vector space C⊕

∞ 

Qn ,

Q = C0∞ (M)/P C0∞ (M),

n=1

where P =  + m2 and  denotes a symmetrised tensor product. Therefore the nontriviality of A(M) reduces to the question of whether the quotient space Q is nontrivial. The latter follows from the properties of the Green operators, which can be summarised in an exact sequence [2] P

E

P

∞ ∞ (M) −→ Csc (M) −→ 0, 0 −→ C0∞ (M) −→ C0∞ (M) −→ Csc

(28)

where the subscript sc denotes a space of functions with ‘spatially compact’ support, which means that they vanish in the causal complement of a compact set. Together with the isomorphism theorems for vector spaces, this gives ∞ (M) : P φ = 0} =: Sol(M). Q = C0∞ (M)/ im P = C0∞ (M)/ ker E ∼ = im E = {φ ∈ Csc

(29) Thus Q is isomorphic to the space of smooth Klein–Gordon solutions with spatially compact support, and is therefore nontrivial. Consequently, A(M) is a nontrivial unital ∗-algebra. Now define, for each causally convex open bounded O ⊂ M, the algebra A(O) to be the subalgebra of A(M) generated by elements (f ) for f ∈ C0∞ (O), along with the unit 1. Then it is clear that, if O1 ⊂ O2 , then A(O1 ) ⊂ A(O2 ). Then properties A1, A2 are automatic. Next, because E(f, g) = 0 when f and g have causally disjoint support (as Eg is supported in the union of the causal future and past of supp g) it is clear that all generators of A(O1 ) commute with all generators of A(O2 ); hence A3 holds. Next, let ρ ∈ P0 . Then the Poincaré covariance of  + m2 and E can be used to show that the map of generators α(ρ)(f ) = (ρ∗ f ), where (ρ∗ f )(x) = f (ρ −1 (x)), is compatible with the relations and extends to a well-

Algebraic Quantum Field Theory

21

defined unit-preserving ∗-isomorphism α(ρ) : A(M) → A(M) .

(30)

Clearly α(ρ) maps each A(O) to A(ρO); as we also have α(σ ) ◦ α(ρ) = α(σ ◦ ρ), condition A4 holds. Finally, let O1 ⊂ O2 such that O1 contains a Cauchy surface of O2 . Then any solution φ = Ef2 for f2 ∈ C0∞ (O2 ) can be written as φ = Ef1 for some f1 ∈ C0∞ (O1 ). An explicit formula is f1 = P χφ where χ ∈ C ∞ (O2 ) vanishes to the future of one Cauchy surface of O2 contained in O1 and equals unity to the past of another (since O1 contains a Cauchy surface of O2 , it actually contains many of them). Then (f2 ) = (f1 ), which implies that A(O2 ) = A(O1 ) as required by A5. In fact this whole construction can be adapted to any globally hyperbolic spacetime, which is the setting in which (28) was proved [2]. Here, we recall that a globally hyperbolic spacetime is a time-oriented Lorentzian spacetime containing a Cauchy surface.

4.2.2 Real Scalar Field with External Source (van Hove Encore) Let ρ ∈ D  (M) be a distribution that is real in the sense ρ(f ) = ρ(f ), and let φρ ∈ D  (M) be any weak solution to ( + m2 )φρ = −ρ. The AQFT formulation of the real scalar field with external source ρ is given in terms of the same algebras A(O) as in the homogeneous case. The only difference is that we define smeared fields ρ (f ) = (f ) + φρ (f )1, where (f ) are the generators used to construct A(M) by SF1–SF4, and observe that they obey the algebraic relations vH1 vH2 vH3 vH4

f → ρ (f ) is complex linear ρ (f )∗ = ρ (f ) ρ (( + m2 )f ) + ρ(f )1 = 0 [ρ (f ), ρ (g)] = iE(f, g)1

which are the relations that would be obtained from Dirac quantisation of the classical theory. Two remarks are in order: • We see that the algebra is not so specific to the theory in hand—this is a general feature of AQFT: what is more interesting is how the local algebras fit together and how the elements can be labelled by fields.

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C. J. Fewster and K. Rejzner

• All the difficulties encountered in Sect. 3 appear to have vanished. Actually, they have been moved into the question of the unitary (in)equivalence of representations of the algebra A(M). The separation between the algebra and its representations makes for a clean conceptual viewpoint.

4.2.3 Weyl Algebra We return to the real scalar field and note two formal identities: first, that  ∗ ei(f ) = e−i(f ) = ei(−f )

(31)

if f is real-valued; second, from the Baker–Campbell–Hausdorff formula, that ei(f ) ei(g) = ei(f )+i(g)−[(f ),(g)]/2 = e−iE(f,g)/2 ei(f +g) .

(32)

As there is no topology on A(M) we cannot address any convergence questions, so these are to be understood as identities between formal power series in f and g. We may also note that (f ) and E(f, g) depend only on the equivalence classes of f and g in C0∞ (M)/P C0∞ (M) ∼ = Sol(M). Moreover, the exponent in (32) is related to the symplectic product on the space of real-valued Klein–Gordon solutions, SolR (M) ∼ = C0∞ (M; R)/P C0∞ (M; R) by σ ([f ], [g]) = E(f, g).

(33)

These considerations motivate the definition of a unital ∗-algebra, generated by symbols W([f ]) labelled by [f ] ∈ C0∞ (M; R)/P C0∞ (M; R) and satisfying relations mimicking (31) and (32). In fact, any real symplectic space (S, σ ) determines a unital ∗-algebra, generated by symbols W(φ), φ ∈ S and satisfying the relations: W1 W(φ)∗ = W(−φ)  W2 W(φ)W(φ  ) = e−iσ (φ,φ )/2W(φ + φ  ). It is a remarkable fact that this algebra can be given a C ∗ -norm and completed to form a C ∗ -algebra in exactly one way (up to isomorphism) [4]. This is the Weyl algebra W(S, σ ). In our case of interest S = C0∞ (M; R)/P C0∞ (M; R), the symplectic form is given by (33), and we will denote the corresponding Weyl algebra by W(M) for short. As before, we can form local algebras by defining W(O) as the C ∗ -subalgebra generated by W([f ])’s with supp f ⊂ O and O being any causally convex open bounded subset of M. Exercise 7 In a general Weyl algebra W(S, σ ), prove that W(0) = 1, the algebra unit.

Algebraic Quantum Field Theory

23

It is worth pausing to examine the explicit construction of the Weyl algebra W(S, σ ). Consider the (inseparable) Hilbert space H = 2 (S) of square-summable sequences a = (aφ ) indexed by φ ∈ S, and define 

(W(φ  )a)φ = e−iσ (φ ,φ)/2 aφ+φ  .

(34)

Obviously the W(φ)’s are all unitary. The Weyl algebra W(S, σ ) is the closure of the ∗-algebra generated by the W(φ)’s in the norm topology on B(H), equipped with the operator norm. Exercise 8 Check that (34) implies W(φ) = W(−φ)∗ and 

W(φ)W(φ  ) = e−iσ (φ,φ )/2 W(φ + φ  ). Exercise 9 Let ∈ 2 (S) be the sequence (δφ,0 ), where  δφ,0 =

1 φ=0 0 φ = 0.

If φ = φ  , show that W(φ) and W(φ  ) are orthogonal and deduce that

W(φ) − W(φ  ) = 2. This shows that there are no nonconstant continuous curves in the Weyl algebra. A corollary of the exercise is that one cannot differentiate λ → W([λf ]) within the Weyl algebra in the hope of recovering a smeared field operator, nor can we exponentiate i(f ) within the algebra A(M) to obtain a Weyl operator. The heuristic relationship W([f ]) = ei(f ) does not hold literally in either of these algebras. Exercise 10 Show that the GNS representation of the (abstract) Weyl algebra over symplectic space (S, σ ) induced by the tracial state ωtr (W(φ)) = δφ,0 coincides with the concrete construction of a representation on H = 2 (S) given earlier, with the GNS vector tr = (δφ,0 ), i.e., ωtr (A) =  tr |A tr .

4.2.4 Complex Scalar Field The algebra of the free complex scalar field C(M) may be generated by symbols (f ) (f ∈ C0∞ (M)) subject to the relations CF1 Linearity f → (f ) is complex linear CF2 Field equation (( + m2 )f ) = 0 CF3 Covariant commutation relations [(f ), (g)] = 0 and [(f )∗ , (g)] = iE(f¯, g)1.

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C. J. Fewster and K. Rejzner

It is also usual to write  (f ) := (f¯)∗ , so that f →  (f ) is also complexlinear. This algebra admits a family of automorphisms ηα given on the generators by ηα ((f )) = e−iα (f ),

(35)

corresponding to a global U (1) gauge symmetry of the classical complex field. Exercise 11 Check that (35) extends to a well-defined automorphism for each α ∈ R. Show also that there is an isomorphism between C(M) and the algebraic tensor product A(M) ⊗ A(M) of two copies of the real scalar field algebra (with the same mass m) given on generators by 1 (f ) → √ (r (f ) ⊗ 1 + i1 ⊗ r (f )) , 2

f ∈ C0∞ (M),

where we temporarily use r (f ) to denote the generators of A(M). In this sense, the complex field is simply two independent real scalar fields. We may identify local algebras C(O) in the same way as before, and within each of these, identify the subalgebra Cobs (O) consisting of all elements of C(O) that are invariant under the global U (1) gauge action. These are the local observable algebras. The reader may notice that the real scalar field admits a global Z2 gauge symmetry generated by (f ) → −(f ), and that the theory of two independent real scalar fields with the same mass admits an O(2) gauge symmetry, of which the U (1) symmetry corresponds to the SO(2) subgroup. Why, then, do we not restrict the observable algebra of the real scalar field, or further restrict the observable algebras of the complex field? The answer is simply that these are physical choices. The U (1) gauge invariance of the complex scalar field is related to charge conservation, while the additional Z2 symmetry in the isomorphism O(2) ∼ = U (1)  Z2 corresponds to charge reversal. If the theory is used to model a charge that is conserved in nature, but for which states of opposite charge can be physically distinguished, then the correct approach is to proceed as we have done.

4.2.5 Dirac Field One can proceed in a similar way to define an algebra F(M) with generators (u) and  + (v) labelled by cospinor and spinor test functions respectively, and with relations abstracted from standard QFT (this is left as an exercise). However, the resulting local algebras F(O) do not obey Einstein causality—if u and v are spacelike separated then (u) and  + (v) anticommute. Of course we do not expect to be able to measure a smeared spinor field by itself, and what one can do instead is consider algebras generated by second degree products of the spinor and cospinor

Algebraic Quantum Field Theory

25

fields, labelled by (co)spinor test functions in supported in O. The resulting algebras A(O) then obey the axioms A1–A5. Meanwhile, the algebras F(O) obey A1, A2, A4, A5 and a graded version of A3. We describe them as constituting local field algebras, to emphasise the fact that they contain elements carrying the interpretation of smeared unobservable fields. We will come back to the discussion of F(O) and their relation to A(O) in Sect. 8 on superselection sectors.

4.3 Quasifree States for the Free Scalar Field We have seen how local algebras for the free scalar field may be constructed. As emphasised in Sect. 2.2, however, this is only half of the data needed for a physical theory: we also need some states, and (for many purposes) the corresponding GNS representations. These are nontrivial problems in general: one needs to fix the value of ω(A) and verify the positivity condition that ω(A∗ A) ≥ 0 for every element A ∈ A(M); furthermore, while the GNS representation is fairly explicit, it evidently involves a lot of work to do it by hand. Fortunately, in the case of free fields, there is a special family of quasifree states where quite explicit constructions can be given. In particular, these states are determined by their two-point functions and all the conditions required of a state can be expressed in those terms. Moreover, the eventual GNS Hilbert space is a Fock space, and the smeared field operators in the representation may be given by explicit formulae. It is important to note that these include representations that are unitarily inequivalent to the representation built on the standard vacuum state. Once again, many of the arguments we will use carry over directly to curved spacetimes. Let W be any bilinear form on C0∞ (M) obeying W (f, g) − W (g, f ) = iE(f, g),

∀f, g ∈ C0∞ (M)

(36)

and which induces a positive semidefinite sesquilinear form on Sol(M) by the formula w(Ef, Eg) = W (f , g),

∀f, g ∈ C0∞ (M).

In particular, W must be a weak bisolution to the Klein–Gordon equation so that w is well-defined. Exercise 12 Check that the positivity condition W (f¯, f ) ≥ 0 implies that W (f, g) = W (g, ¯ f¯). Also derive the Cauchy–Schwarz inequality | Im w(φ, φ  )|2 ≤ w(φ, φ)w(φ  , φ  ),

φ, φ  ∈ Sol(M).

(37)

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C. J. Fewster and K. Rejzner

Under the above conditions, it may be proved (cf. Prop 3.1 in [33]) that there is a complex Hilbert space H and a real-linear map K : SolR (M) → H so that K SolR (M) + iK SolR (M) is dense in H and KEf |KEgH = W (f, g),

f, g ∈ C0∞ (M; R).

(38)

(These structures are unique up to unitary equivalence.) The full construction is essentially explicit, but slightly involved. However, there is a particularly simple and interesting case, arising when the Cauchy–Schwarz inequality (37) is saturated, i.e., | Im w(φ, φ  )|2 =1   φ  =0 w(φ, φ)w(φ , φ ) sup

for all 0 = φ ∈ SolR (M),

(39)

where the supremum is taken over nonzero elements of SolR (M). This occurs if and only if the quasifree state constructed below is pure. Under these circumstances, H is first defined as a real Hilbert space by completing SolR (M) in the norm φ w = w(φ, φ)1/2 with inner product φ1 |φ2 w = Re w(φ1 , φ2 ). Due to (39), H carries an isometry J defined so that Im w(φ1 , φ2 ) = φ1 |J φ2 w ,

φi ∈ H,

and which fulfils the conditions J 2 = −1, J † = −J . Hence J is a complex structure on H, and we can make H into a complex Hilbert space in which multiplication by i is (annoyingly) implemented by −J and the inner product is φ1 |φ2 H = φ1 |φ2 w + iφ1 |J φ2 w . (See Appendix 1 for more details.) The map K is just the natural inclusion of SolR (M) in H and it is clear that K SolR (M) is dense in H (hence K SolR (M) + iK SolR (M) is also dense). Verification of (38) is left as an exercise. Returning to the general case, it may be shown that there is a state on A(M) given, as a formal series in f , by   ω ei(f ) = e−W (f,f )/2 ,

f ∈ C0∞ (M; R).

(40)

Expanding each side of (40) in powers of f , and equating terms at each order, all expectation values of the form ω((f )n ) are fixed. Arbitrary expectation values may then be formed using multilinear polarisation identities (see e.g. [42] and references therein) and linearity. It may be shown that all odd n-point functions vanish, while  ω((f1 ) · · · (f2n )) = W (fs(e) , ft (e) ), G∈G2n e∈G

Algebraic Quantum Field Theory Fig. 2 An example graph in G8

27

1

2

3

4

5

6

7

8

where G2n is the set of directed graphs with vertices labelled 1, . . . , 2n, such that each vertex is met by exactly one edge and the source and target of each edge obey s(e) < t (e). An example for n = 4 is given in Fig. 2. Another characterisation of the n-point functions is that all the truncated n-point functions vanish for n = 0, 2. This type of state is described as quasifree. Exercise 13 Using (40), show that ω((f )2n+1 ) = 0 for all n ∈ N0 , while ω((f )2n ) = (2n − 1)!! W (f, f )n ,

∀n ∈ N0 .

Deduce that the sequence νn = ω((f )n ) satisfies the growth conditions in the Hamburger moment theorem. Therefore these are the moments of at most one probability measure—which is of course a Gaussian probability distribution. Although we have called ω a state, it is not yet clear that it has the required positivity property. This is most easily justified by noting that there is an explicit Hilbert space representation of A(M) containing a unit vector whose expectation values match those of ω. To be specific, the Hilbert space is the bosonic Fock space F(H) =

∞ 

Hn

(41)

n=0

over H, on which the field is represented according to the formula πω ((f )) = a(KEf ) + a ∗ (KEf ),

f ∈ C0∞ (M; R),

(42)

and πω ((f )) := πω ((Re f )) + iπω ((Im f )) for general complex-valued f ∈ C0∞ (M). Here a(φ) and a ∗ (ψ) are the annihilation and creation operators on the Fock space which obey the CCRs [a(φ), a ∗ (ψ)] = φ|ψH 1,

φ, ψ ∈ H,

(43)

on a suitable dense domain in F(H) (note that a is antilinear in its argument, and that a(φ) = a ∗ (φ)∗ ). Readers unfamiliar with the basis-independent notation used here should refer to Appendix 3.

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We will not give a detailed proof of the claims ((40)–(43)), which would require consideration of operator domains. At the level of formal calculation, however, it is easily checked that these operators do indeed lead to a representation of A(M). Exercise 14 Verify formally that f → πω ((f )) is C-linear, obeys the field equation in the sense πω ((Pf )) = 0, and is hermitian in the sense that πω ((f ))∗ = πω ((f )). For the CCRs, we compute [πω ((f )), πω ((g))] = (KEf |KEgH − KEg|KEf H ) 1 = (W (f, g) − W (g, f )) 1 = iE(f, g)1, using (43), (38), and (36). Finally, it may be verified that the Fock vacuum vector ω satisfies  ω |πω ((f1 )) · · · πω ((fn )) ω  = ω((f1 ) · · · (fn )). Consequently, ω is seen to be a vector state on A(M) and the quadruple (F(H), Dω , πω , ω ) is its GNS representation, where the dense domain Dω consists of finite linear combinations of finite products of operators a ∗ (KEf ) acting on ω . The ‘one-particle space’ H may be interpreted as follows. By (42), we see that πω ((f )) ω = a ∗ (KEf ) ω = (0, KEf, 0, . . .) ∈ F(H), which is an eigenstate of N with unit eigenvalue. Elements of H can be identified with (complex linear combinations of) vectors generated from ω by a single application of the field. Due to assumption SF3, these vectors may be identified with certain complex-valued solutions to the field equation, which may be regarded as wavepackets of ‘positive frequency modes’ relative to a decomposition induced by the choice of quasifree state.

4.3.1 Examples We describe two important examples. The Minkowski vacuum state8 ω0 is a quasifree state with two-point function  W (f, g) =

8A

ˆ d 3 k fˆ(−k)g(k) , 2ω (2π)3

where k • = (ω, k),

 g(k) ˆ =

d 4 x eik·x g(x),

general definition of what a vacuum state should be will be given in Definition 7.

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and ω = |k|2 + m2 .9 The corresponding one-particle space is H = L2 (Hm+ , dμ), where Hm+ is the hyperboloid k · k = m2 , k 0 > 0 in R4 , and the measure of S ⊂ Hm+ is   d 3 k χS (k) 1 k∈S χS (k) = μ(S) = (2π)3 2ω 0 otherwise. The map K : SolR (M) → H is KEg = g| ˆ Hm+ , which already has dense range—as mentioned above, this signals that the vacuum state is pure. Consequently, the vacuum representation π0 is given by π0 ((g)) = a(g| ˆ Hm+ ) + a ∗ (g| ˆ Hm+ ) ,

g ∈ C0∞ (M; R),

which may be written in more familiar notation using sharp-momentum annihilation and creation operators obeying (7) using  a(g| ˆ Hm+ ) =

d 3k 1 √ g(k)a(k), ˆ (2π)3 2ω

a ∗ (g| ˆ Hm+ ) =



d 3k 1 ∗ ˆ √ g(k)a (k) . (2π)3 2ω

Recalling that g(k) ˆ = g(−k) ˆ for real-valued g, we retrieve the field with sharp position as the operator-valued distribution  π0 ((x)) =

 d 3k 1  −ik·x ∗ ik·x a(k)e . √ + a (k)e (2π)3 2ω

Our second example is the thermal state of inverse temperature β, with two-point function    ˆ d 3 k 1 fˆ(−k)g(k) fˆ(k)g(−k) ˆ Wβ (f, g) = . + βω (2π)3 2ω 1 − e−βω e −1 Here, Hβ = H ⊕ H with H as before, and (cf. [32]) (Kφ)(k) (Kφ)(k) (Kβ φ)(k) = √ ⊕√ 1 − e−βω eβω − 1

9 There is a notational conflict with the symbol used to denote states but it will always be clear from context which is meant.

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(remember that Kβ only has to be real-linear!). The range of Kβ E is not dense in Hβ , but its complex span is, reflecting the fact that the thermal states are mixed. Exercise 15 Check that the analogue of (38) holds for Hβ , Kβ and Wβ . A nice feature of the algebraic approach is that, while the representations corresponding to the vacuum and thermal states are unitarily inequivalent, they can be treated ‘democratically’ as states on the algebra A(M). There are many other quasifree states; indeed one can start with any state and construct its ‘liberation’, the quasifree state with the same two-point function. All quasifree representations carry a representation π˜ ω of the Weyl algebra as well, so that π˜ ω (W([f ])) = eiπω ((f )) and also   1 d W([λf ]) πω ((f )) = . i dλ λ=0 Remember that these relationships cannot hold literally either in A(M) or W(M), but here we see that they do hold in (sufficiently regular) representations. Summarising, the quasifree states provide a class of states for which explicit Hilbert space representations may be given with the familiar Fock space structure, and in which the Weyl operators and smeared field operators are related as just described.

5 The Spectrum Condition and Reeh–Schlieder Theorem In this section we begin to draw general conclusions about the properties of QFT in the AQFT framework, proceeding from the basic axioms and additional requirements that will be introduced along the way. We shall emphasise features that distinguish QFT from quantum mechanics. The starting point is a more detailed discussion of the action of Poincaré transformations.

5.1 The Spectrum Condition Assumption A4 required that the Poincaré group should act by automorphisms of A(M). An important question concerning Hilbert space representations of the theory is whether or not these automorphisms are unitarily implemented, i.e.,

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whether there are unitaries U (ρ) on the representation Hilbert space such that π(α(ρ)A) = U (ρ)π(A)U (ρ)−1 . In such cases, we say that the representation is Poincaré covariant. As we now show, the GNS representation of a Poincaré invariant state ω is always Poincaré covariant. Here, Poincaré invariance of ω means that ω(α(ρ)A) = ω(A),

∀A ∈ A(M), ρ ∈ P0

(written equivalently as α(ρ)∗ ω = ω for all ρ ∈ P0 , with the star denoting the dual map). Of course, the same question can be asked of any automorphism or automorphism group on a ∗-algebra. Theorem 2 Let α be an automorphism of a unital ∗-algebra A. If a state ω on A is invariant under α, i.e., α ∗ ω = ω, then α is unitarily implemented in the GNS representation of ω by a unitary that leaves the GNS vector invariant. Any group of automorphisms leaving ω invariant is unitarily represented in Hω . Proof Observing that ω(α(A)∗ α(A)) = (α ∗ ω)(A∗ A) = ω(A∗ A), we see that α maps the GNS ideal Iω to itself. Therefore the formula U [A] = [α(A)] gives a well-defined map U from the GNS domain Dω to itself, which fixes the GNS vector ω = [1] and is obviously invertible (consider α −1 ). Now U [A]|U [B] = ω(α(A)∗ α(A)) = ω(A∗ A) = [A]|[B], so U is a densely defined invertible isometry, and therefore extends uniquely to a unitary on Hω . The calculation πω (α(A))[B] = [α(A)B] = [α(Aα −1 (B))] = U [Aα −1 (B))] = U πω (A)[α −1 B] = U πω (A)U −1 [B] shows that U implements α. If β is another automorphism leaving ω invariant, let V be its unitary implementation as above. Then U V [A] = [α(β(A))] = [(α ◦ β)(A)] shows that U V implements α ◦ β.

 

Among other things, this result may be applied to states that are translationally invariant. Thermal equilibrium states, for example, are spatially homogeneous but

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not Poincaré invariant because they have a definite rest frame. Although typical states are not translationally invariant, many interesting states are in a suitable sense local deviations from such invariant states. Even so, not all invariant states are physically acceptable. One way of doing narrowing the field is to require the spectrum condition: Definition 6 Let ω be a translationally invariant state so that the unitary implementation of the translation group U (x) is strongly continuous, i.e., the map x → U (x)ψ is continuous from R4 to Hω for each fixed ψ ∈ Hω , where x = (x 0 , . . . , x 3 ) ∈ R4 . By Stone’s theorem, there are four commuting self-adjoint operators P μ such that (lowering the index using the metric) μ

U (x) = eiPμ x . To any (Borel) subset  ⊂ R4 we may assign a projection operator E() corresponding to the binary test of whether the result of 4-momentum measurement P μ would be found to lie in . The assignment  → E() is a projection-valued measure, and in fact one can write  μ U (x) = eipμ x dE(p• ). The state ω is said to satisfy the spectrum condition if the support of E lies in the closed forward cone V + = {p• ∈ R4 : pμ pμ ≥ 0, p0 ≥ 0}, i.e., supp E ⊂ V + . This is sometimes expressed by saying that the joint spectrum of the momentum operators P μ lies in V + . An important consequence of the spectrum condition is that the definition of U (x) can be extended to complex vectors x ∈ R4 + iV + , with analytic dependence on x: to be precise, U (x) is strongly continuous on R4 + iV + and holomorphic on R4 + i V + , where V + = intV + . One can check that the usual vacuum and thermal states of the free field obey the spectrum condition. More generally, we will make the following definition: Definition 7 A vacuum state10 is a translationally invariant state obeying the spectrum condition, whose GNS vector is the unique translationally invariant vector (up to scalar multiples) in the GNS Hilbert space. The corresponding GNS representation is called the vacuum representation.

10 The term ‘vacuum’ is used in various ways by various authors, differing, for example, on whether Poincaré invariance is required as well. Somewhat remarkably, there is an algebraic criterion on the state that implies translational invariance, the spectrum condition and (if the state is pure) that the there is no other translationally invariant vector state in its GNS representation. See [1].

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5.2 The Reeh–Schlieder Theorem We come to some general results that show how different QFT is from quantum mechanics. For simplicity, suppose that the theory is given in terms of C ∗ -algebras. Let be the GNS vector of a state obeying the spectrum condition (we drop the ω subscripts). Suppose the theory obeys the following condition: A6 Weak additivity For any causally convex open region O, π(A(M)) is contained in the weak closure11 of BO , the ∗-algebra generated by the algebras π(A(O + x)) as x runs over R4 . Weak additivity asserts that arbitrary observables can be built as limits of algebraic combinations of translates of observables in any given region O (as one would expect in a quantum field theory). In combination with the spectrum condition it has a striking consequence. Our proof is based on that of [1]. Theorem 3 (Reeh–Schlieder) Let O be any causally convex bounded open region. Then (a) vectors of the form A for A ∈ π(A(O)) are dense in H; (b) if A ∈ π(A(O)) annihilates the vacuum, A = 0, then A = 0. Part (a) says that is cyclic for every local algebra; part (b) that it is also separating. The existence of a cyclic and separating vector on a (von Neumann) algebra is the starting point of Tomita–Takesaki theory (see, e.g. [3]). These are quite remarkable statements: if a local element of A(O) corresponds to an operation that can be performed in O, it seems that we can produce any state of the theory up to arbitrarily small errors by operations in any small region anywhere. To give an extreme example: by making a local operation in a laboratory on earth one could in principle modify the state of the theory to one approximating a situation in which there is a starship behind the moon, if such things can be modelled by the theory (e.g., as a complicated state of the standard model). It is an expression of how deeply entangled states in QFT typically are. Proof (a) Suppose to the contrary that the set of vectors mentioned is not dense. Then it has a nontrivial orthogonal complement, so there is a nonzero vector  ∈ H such that |A  = 0,

∀A ∈ π(A(O)).

Now let O1 be a slightly smaller region with O1 ⊂ O. For any n ∈ N and any Q1 , . . . , Qn ∈ π(A(O1 )) we have U (x)Qi U (x)−1 ∈ π(A(O)) for sufficiently

sequence of operators converges in the weak topology, w − lim An = A, if ψ|An ϕ → ψ|Aϕ for all vectors ψ and ϕ.

11 A

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C. J. Fewster and K. Rejzner

small |x|, whereupon |U (x1 )Q1 U (x2 − x1 )Q2 U (x3 − x2 ) · · · U (xn − xn−1 )Qn  = 0

(44)

for sufficiently small x1 , . . . , xn . By what was said above, the function F : (ζ1 , . . . , ζn ) → |U (ζ1 )Q1 U (ζ2 )Q2 · · · U (ζn )Qn  is a continuous function on (R4 )n extending to an holomorphic function in (R4 + iV + )n ⊂ (C4 )n , whose boundary value on (R4 )n moreover vanishes in some neighbourhood of the origin. The ‘edge of the wedge’ theorem [41, 44] implies that F vanishes identically in its domain of holomorphicity, which means that the boundary value also vanishes identically. This means that (44) holds for all xi , or put another way, that |BO  = 0 where BO is the algebra generated by the algebras π(A(O1 + x)) for x ∈ R4 . By weak additivity, we now know that |π(A(M))  = 0. But ω is cyclic, so  = 0. This proves the first assertion. For the second, suppose that A ∈ π(A(O1 )) annihilates . Choose O2 causally disjoint from O1 , and note that for each ψ ∈ H and all B ∈ π(A(O2 )) we have A∗ ψ|B  = ψ|AB  = ψ|BA  = 0 using Einstein causality. By the first part of the theorem, A∗ ψ is orthogonal to a dense set and therefore vanishes; as ψ ∈ H is arbitrary we have A∗ = 0 and hence A = 0.   Corollary 1 All nontrivial sharp local binary tests (with possible outcomes ‘success’ or ‘failure’) have a nonzero success probability in the vacuum state. (All local detectors exhibit ‘dark counts’). Proof A binary test can be represented by a projector P , with ‘failure’ corresponding to its kernel. Suppose P ∈ π(A(O)) is a projector with vanishing vacuum expectation value, i.e., a zero success probability. Then

P 2 = P |P  =  |P  = 0, so P = 0 and hence P = 0 by the Reeh–Schlieder theorem (b).

 

Corollary 2 For every pair of local regions O1 and O2 there are vacuum correlations between A(O1 ) and A(O2 ) (assuming dim H ≥ 1). Proof For suppose to the contrary that there are states ωi on A(Oi ) such that  |π(A1 )π(A2 )  = ω1 (A1 )ω2 (A2 )

Ai ∈ A(Oi ).

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By setting A1 = 1, and then repeating for A2 , this implies  |π(A1 )π(A2 )  =  |π(A1 )  |π(A2 ) 

Ai ∈ A(Oi )

Fixing A2 and letting A1 vary in A(O1 ), the Reeh–Schlieder theorem gives π(A2 ) =  |π(A2 )  ,

A2 ∈ A(O2 )

which contradicts the Reeh–Schlieder theorem (a) unless dim H = 1.

 

The correlations indicated by Corollary 2 become small at spacelike separation due to the cluster property (a general feature of vacuum states in AQFT), which implies  |π(A1)π(α(x)A2 )  →  |π(A1 )  |π(A2 )  as x → ∞ in spacelike directions, with exponentially fast convergence if the theory has a mass gap, i.e., σ (P · P ) ⊂ {0} ∪ [M 2 , ∞) for some M > 0. See [1, §4.3– 4.4]. The Reeh–Schlieder theorem does not present a very practical method for constructing starships.

6 Local von Neumann Algebras and Their Universal Type So far, we have encountered AQFTs given in terms of ∗-algebras or C ∗ -algebras. The theory is considerably enriched when expressed in terms of von Neumann algebras. Consider any net of local C ∗ -algebras O → A(O) obeying A1–A5. Given any state ω on A(M), form its GNS representation. For any open bounded causally convex O ⊂ M we can define an algebra Mω (O) := πω (A(O)) , namely, the double commutant of the represented local algebra. Recall that the commutant is defined for any subalgebra B of the bounded operators B(H) on Hilbert space H by B = {A ∈ B(H) : [A, B] = 0 ∀B ∈ B}. A basic result asserts that the algebra Mω (O) is also the closure of πω (A(O)) in the weak topology of Hω —as such, and because it contains the unit operator and is stable under the adjoint, Mω (O) is a (concrete) von Neumann algebra. Exercise 16 Show that for any subalgebra B of B(H) one has B ⊂ B and B = B . Therefore Mω (O) = Mω (O).

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There is a good rationale for this construction. The norm topology of the C ∗ -algebra (which coincides with the norm topology B(Hω ) in a faithful representation) is quite stringent: we have already seen that there are no nonconstant continuous curves in the Weyl algebra. For example, a Weyl generator differs from all its nontrivial translates by operators of norm 2. The situation is different in the weak topology: if translations are implemented in a strongly continuous way then matrix elements of an observable change continuously as the observable is translated. This motivates the weak topology as a better measure of proximity than the norm topology. Note also that Einstein causality implies that the commutant πω (A(O)) contains all local algebras πω (A(O1 )) where O1 is causally disjoint from O. As πω (A(O)) = πω (A(O)) = Mω (O) , we see that also Mω (O1 ) ⊂ Mω (O) . It is natural to interpret Mω (O) in terms of (limits of) observables localised in the causal complement of O and further natural to assume that there are no nontrivial observables that can be localised both in O and its causal complement. This motivates an assumption that Mω (O) ∩ Mω (O) = C1, that is, that the local algebras are factors, so called because an equivalent statement is that Mω (O) ∨ Mω (O) = B(H), i.e., Mω (O) and its commutant generate [in the sense of von Neumann algebras] the full algebra of bounded operators on H. The connection to von Neumann algebras permits a vast body of technical work to be brought to bear on AQFT, and indeed some of it was spurred by developments in AQFT. One of the most striking examples is the 1985 result of Fredenhagen [24] that for physically reasonably states ω of reasonable theories obeying the AQFT axioms, every Mω (O) is a type III 1 factor. Shortly afterwards, Haagerup [29] proved that there was a unique hyperfinite type III1 factor. A further paper of Buchholz, D’Antoni and Fredenhagen [9] pointed out the local algebras of QFT are hyperfinite, therefore fixing them uniquely up to isomorphism. Remarkably, the local algebras themselves are completely independent of the theory! Therefore the distinction between different theories lies purely in the ‘relative position’ of the local algebras within the bounded operators on Hilbert space. To appreciate these results and their consequences, we need to delve into the classification of von Neumann factors. Type I factors are easily defined and familiar. Definition 8 A von Neumann factor M on Hilbert space H is of type I if there is a unitary U : H → H1 ⊗ H2 , for some Hilbert spaces Hi , so that M = U ∗ (B(H1 ) ⊗ 1H2 )U . The type is further classified according to the dimension of H1 . One could say that type I factors are the natural playground of quantum mechanics: in an obvious way, they are the observables of one party in a bipartite system. The result mentioned above indicates that QFT prefers type III for its local algebras.

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Definition 9 A von Neumann factor M on an infinite-dimensional separable Hilbert space H is of type III if every nonzero projection E ∈ M may be written in the form E = W W ∗,

for some W ∈ M obeying W ∗ W = 1H .

(It is then the case that every two projections E and F in M can be written as E = W W ∗ , F = W ∗ W for some W ∈ M.) Type III1 is a further subtype whose definition would take us too far afield, but it should already be clear that types I and III are quite different. The foregoing definitions are quite technical in nature. However the proof that the local algebras have type III is founded on physical principles, namely that (a) the theory should have a description in terms of quantum fields, (b) that the n-point functions of these fields have a well-behaved scaling limit at short distances and (c) that there are no local observables localised at a point, other than multiples of the unit. In more detail, the assumptions are: (a) There is a dense domain D ⊂ Hω and a linear map φ from real-valued test functions to symmetric operators f → φ(f ) defined on D and obeying φ(f )D ⊂ D, so that φ(f ) has a closure affiliated12 to Mω (O) for any O containing supp f . Furthermore, the n-point functions of the fields φ(f ) define distributions Wn (f1 , . . . , fn ) =  ω |φ(f1 ) · · · φ(fn ) ω . (b) Defining scaling maps on test functions by (βp,λ f )(x) = f ((x − p)/λ), there should be a positive monotone function ν so that the scaling limit n-point functions Wns.l. (f1 , . . . , fn ) = lim ν(λ)n Wn (βp,λ f1 , . . . , βp,λ fn ) λ→0+

exist and satisfy the (vacuum) Wightman axioms [41]. One may think of this theory as living in the tangent space at p; it is not usually the same theory as the one we started with.  (c) It is required that O#p Mω (O) = C1, where the intersection is taken over all local regions containing the point p. Assuming condition (a), we say that ω has a regular scaling limit at p ∈ M if the conditions (b) and (c) hold. Fredenhagen’s result [24] can now be stated precisely.

is, in the polar decomposition φ(f ) = U |φ(f )| of the closure φ(f ) of φ(f ), the operator U and any bounded function of |φ(f )| belong to Mω (O).

12 That

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Theorem 4 Let O be a double-cone. If the state ω has a regular scaling limit at some point of the spacelike boundary of O (i.e., its equatorial sphere) then Mω (O) has type III1 . Conversely, if any local von Neumann algebra of a double-cone has type other than III1, then at least one of the assumptions (b), (c) must fail at every point on its spacelike boundary, or (a) fails. One application of this argument has been to prove that ‘SJ states’ of the free field on double-cone regions do not extend to Hadamard states on a larger region—typically because the stress-energy tensor diverges as the boundary is approached [21]. We will say a little more about this below. Various key distinctions between quantum mechanics and QFT can be attributed to the differences between factors of types I and III, and the fact that the local algebras of QFT are typically type III. To conclude this section, we collect some properties of type III factors and their consequences for QFT, gathered under some catchy slogans. We assume that the Hilbert space on which the type III factors act is infinite-dimensional and separable. All Eigenvalues Have Infinite Degeneracy No type III factor can contain a finiterank projection; for if E is finite rank, then writing E = W W ∗ for an isometry W , we see that W ∗ EW = W ∗ W W ∗ W = 1H is also finite rank, contradicting our assumption on the dimension of H. It follows that no self-adjoint element of a local algebra M(O) can have a finite-dimensional eigenspace, which is far removed from the situation of elementary textbook quantum mechanics. In particular, if is the vacuum vector, then the projection |  | belongs to no local algebra M(O), nor to any commutant Mω (O) [because the commutant of a type III factor is also of type III]. Local States Are Impure Any state ω on A(M) restricts in a natural way to a state ω|A(O) on each local algebra A(O). What sort of state is it? To start, we note the simple: Lemma 1 Suppose ω is a pure state on a C ∗ -algebra A. Then the von Neumann algebra M = πω (A) in the GNS representation of ω is a type I factor. Proof The GNS representation of a pure state is irreducible. So the commutant πω (A) consists only of multiples of the unit, and M = B(H), which is evidently type I.   In the light of this result, it may not be so much of a surprise that: Theorem 5 Under the hypotheses of Theorem 4, the algebra πω|A(O) (A(O)) is of type III. Therefore ω|A(O) is not pure, nor is it a normal state in the GNS representation of a pure state of A(O). Furthermore, in the case where Mω (O) is a factor, the previous statements hold if ω is replaced by any state in its folium. Proof This requires a few standard results from von Neumann theory and can be found as Corollary 3.3 in [21].  

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The SJ states mentioned above induce pure states and therefore it follows that they cannot be induced as restrictions of states on A(M) with good scaling limit properties. See [21]. Local Experiments Can Be Prepared Locally Suppose E is any projection in

M(O) and let ω be any state on the C ∗ -algebra formed as the closure of O M(O) in B(H). By the type III property, we may write E = WW∗ for some isometry W ∈ M(O). Then the modified state (Exercise: check that it is a state!) ωW (A) := ω(W ∗ AW ) obeys ωW (E) = ω(W ∗ W W ∗ W ) = 1 so the yes/no test represented by E is certainly passed in state ωW . On the other hand, if A ∈ M(O1 ) where O1 is causally disjoint from O, then A ∈ M(O) and so ωW (A) = ω(W ∗ AW ) = ω(W ∗ W A) = ω(A). So by changing the state in this way we can prepare a state with a desired property in our lab without changing the rest of the world. For a brief survey of this and other features of type III factors and their relevance to QFT, see [45, 46].

7 The Split Property The fundamental postulates of relativistic physics entail that two spacelike separated laboratories should not be able to communicate. Consequently, an experimenter situated in one of these laboratories should be able to conduct experiments independently of the actions (or even the presence) of an experimenter in the other region. The Einstein causality condition A3 reflects this idea, because any observable from one local algebra will commute, and be simultaneously measurable, with any observable from the other. However, this is far from being the only way in which the two regions must be independent. For example, the two observers should be able to prepare their experiments for measurement independently, too. In quantum mechanics this independence is modelled by assigning each of the two local systems their own Hilbert space, H1 and H2 , on which the observables in the two laboratories respectively act. If the experimenters prepare states ψi , the global

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state is then taken to be ψ1 ⊗ ψ2 on the tensor product H1 ⊗ H2 . This section describes the split property which provides conditions under which a similar level of independence may be established in QFT. First, we introduce some terminology: two von Neumann algebras M1 and M2 acting on a Hilbert space H are said to form a split inclusion if there is a type I von Neumann factor N so that M1 ⊂ N ⊂ M2 . By Definition 8, this means that there is a unitary U : H → H1 ⊗ H2 with N = U ∗ (B(H1 ) ⊗ 1H2 )U for some Hilbert spaces H1 and H2 . The commutant of N is easily described: N = U ∗ (1H1 ⊗ B(H2 ))U and it follows from the split inclusion that M1 ⊂ U ∗ (B(H1 ) ⊗ 1H2 )U,

M2 ⊂ U ∗ (1H1 ⊗ B(H2 ))U.

Suppose that states ω1 and ω2 are given on M1 and M2 that may be expressed in terms of density matrices ρi on Hi , so that ω1 (A) = tr((ρ1 ⊗ 1H2 )U AU ∗ ),

ω2 (B) = tr((1H1 ⊗ ρ2 )U BU ∗ )

for A ∈ M1 , B ∈ M2 . Then there is an obvious joint state ω(C) = tr((ρ1 ⊗ ρ2 )U AU ∗ ) with the property (Exercise: prove it!) that ω(AB) = ω1 (A)ω2 (B) = ω(BA),

for all A ∈ M1 , B ∈ M2 .

With a little more technical work it can be shown that this construction is possible whenever ωi are given as density matrices on H (see e.g. [19] for an exposition). Returning to QFT, we make the following definition. Definition 10 A net O → M(O) of von Neumann algebras has the split property if, whenever O1 ⊂ O2 , the inclusion M(O1 ) ⊂ M(O2 ) is a split inclusion. Note that we only require a split inclusion when O2 contains the closure of O1 , so that there is a ‘safety margin’ or collar around O1 within O2 . We can also make the less restrictive assumption that the split inclusion holds when this safety margin has a minimum size, in which case one speaks of the distal split property. If the split property holds, and O1 and O3 are spacelike separated pre-compact regions whose closures do not intersect, then we may certainly find a precompact neighbourhood O2 of O1 within the causal complement of O3 . Then

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M(O1 ) ⊂ M(O2 ) is split and M(O3 ) ⊂ M(O2 ) . It follows that experimenters in O1 and O3 are able to independently prepare and measure observables of the field theory. Moreover, there is an isomorphism of von Neumann algebras ∼ ¯ M(O1 )⊗M(O 3 ) = M(O1 ) ∨ M(O3 ) extending the map A⊗B → AB, where the left-hand side is a spatial tensor product of von Neumann algebras and the right-hand side is the von Neumann algebra generated by sums and products of elements in M(O1 ) and M(O3 ). In situations where the Reeh–Schlieder theorem applies, and there is a vector that is cyclic and separating for all causally convex bounded regions, the split inclusions have more structure and are called standard split inclusions. A deep analysis by Doplicher and Longo [11] shows, among many other things, that there is a canonical choice for the type I factor appearing in standard split inclusions. The split property is enjoyed by free scalar fields of mass m ≥ 0 and the observable algebra for the Dirac field, but also for certain interacting models in 1+1dimensions (see [34] for a survey and exposition). It is intimately connected to the way in which the number of local degrees of freedom available to the theory grows with the energy scale. These are expressed technically in terms of various nuclearity conditions, which we will not describe in detail here (but see [19] for discussion and a relation to yet a further topic—Quantum Energy Inequalities). Instead, we limit ourselves to some examples involving a theory comprising countably many independent free scalar fields of masses mr (r ∈ N). It may be shown, for instance, that this theory has the split property if the function G(β) :=

∞

e−βmr /4

r=1

is finite for all β > 0 and grows at most polynomially in β −1 as β → 0+ . This is the case if mr = rm1 , for instance. On the other hand, if mr = (2d0)−1 log(r + 1) for some constant d0 > 0 then the series defining G(β) diverges for β ≤ 8d0 . Further analysis [10, Thm 4.3] shows that the split property fails in this situation, but that the distal split condition holds provided that the ‘safety margin’ is sufficiently large. For concentric arrangements of double cones of base radii r and r+d, splitting fails if d < d0 and succeeds if d > 2d0 . The overall message is that a (well-behaved) tensor product structure across regions at spacelike separation can only be expected in well-behaved QFTs and with a safety margin between the regions. It turns out that the split property is closely related both to the existence of well-behaved thermal states (absence of a Hagedorn temperature) and to whether the theory satisfies quantum energy inequalities— see [19] for discussion and original references. Intuitively, the reason for this is

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that there is a cost associated with disentangling the degrees of freedom between the two regions. For example, a joint state ‘glued together’ from two states given on their local algebras might be expected to have a higher energy than the sum of the two original energies, with the excess in energy being higher if the ‘gluing’ has to take place over a smaller gap between the regions. If the number of degrees of freedom available grows excessively with the energy scale, it becomes impossible to achieve the gluing with a finite excess energy, or possible only if the regions are sufficiently distant from one another.

8 Superselection Sectors We come back to the discussion of superselection sectors, mentioned in Sect. 2.1. In the algebraic viewpoint on quantum theory, superselection sectors correspond to a class of unitarily inequivalent representations of the algebra of observables. One of the major structural results in algebraic quantum field theory [12–16] was to show that these sectors are related to irreducible representations of some compact Lie group G (the global gauge group). The key idea of algebraic QFT is that all the relevant information is contained in the net of observables, and from this net one can construct an algebra of fields F(O), which then contains non-observable objects, e.g. smeared Dirac fields ψ(f ) (see Sect. 4.2). This algebra is uniquely fixed by the net and it carries the action of the gauge group G, so that local algebra A(O) consists of elements of F(O) invariant under G. The reconstruction of both F and G is achieved through the DHR (Doplicher–Haag–Roberts) analysis [13–16] together with the Doplicher–Roberts reconstruction theorem [12]. The brief exposition in Sects. 8.1–8.3 is based on [25]. As motivating example, we consider the complex scalar field, whose algebra C(M) was described in Sect. 4.2. Its vacuum representation π is given by π((g)) = a(g| ˆ Hm+ ) ⊗ 1 + 1 ⊗ a ∗ (g| ˆ Hm+ ) ˆ Hm+ ) ⊗ 1 + 1 ⊗ a(g| ˆ Hm+ ) π( (g)) = a ∗ (g| for real-valued g [with (g) = (Re g) + i(Im g) in general, compare with the real scalar field in Sect. 4.3], on F(H) ⊗ F(H), where H = L2 (Hm+ , dμ) and a ∗ , a are creation and annihilation operators on the Fock space F(H). The charge operator is Q=N ⊗1−1⊗N, where N is the number operator on F(H) (see Appendix 3 for definition). The theory has a global U (1) gauge symmetry generated by Q: eiαQ π((f ))e−iαQ = e−iα π((f )),

α ∈ R,

(45)

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which implements the automorphisms ηα described earlier, and the Fock space decomposes into charged sectors F(H) ⊗ F(H) =



Hq ,

q∈Z

each Hq being the eigenspace of Q with eigenvalue q. Assuming that Q is conserved in all interactions available to the observer, the argument described in Sect. 2.1 shows that the observables of the theory should commute with Q and be block-diagonal in this decomposition. Clearly, this is not the case for the smeared fields π((f )) or π( (f )), which are consequently unobservable. On the other hand, operators of the form π( (f )(f )) is gauge-invariant, as is (any smearing of) the Wick product of the field π( (x)) with π((x)). The main point of interest for us is that the local observable algebras Cobs (O) are gauge-invariant by definition and therefore π(Cobs (O)) consists of block-diagonal operators. More than that, we can see that there are representations πq of Cobs(O) on each Hq , given by πq (A) = Pq π(A)Pq , where Pq is the orthogonal projector onto Hq within F(H)⊗F(H). In particular, π0 is a representation in H0 , the charge-zero sector, which contains the vacuum vector ⊗ ∈ F(H) ⊗ F(H) and will be called the vacuum sector for the observables. We now have a whole family of representations of the algebras Cobs (O) on different Hilbert spaces. It is easy to argue (at least heuristically) that these are mutually unitarily inequivalent, because the charge operator itself can be regarded as a limit of local observables, i.e., local integrals of the Noether charge density associated with the global gauge symmetry. Therefore, if there were a unitary U : Hq → Hq  obeying πq (A) = U πq (A)U −1 for all A ∈ Cobs (O) and all local regions O, it would follow that the charge operators in the two representations should be equivalent under the same mapping, giving q  1Hq  = qU 1Hq U −1 = q1Hq  which can only happen in the case q  = q. The physical distinction between different sectors is precisely that they have different charge content, and the central insight of the DHR programme is that the relevant charges might be gathered in some local region, outside which the distinction can be, as it were, gauged away. An extra charged particle behind the moon ought not to change our description of particle physics on earth.13 As we will describe, this physical insight allows a remarkable reversal of the process we have just followed: instead of starting with

13 This picturesque statement applies to confined charges rather than those with long range interactions such as electric charges. We discuss this issue briefly later on.

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an algebra of unobservable fields transforming under a global gauge group and obtaining from it a collection of unitarily inequivalent representations of the algebra of observables, one starts from a suitable class of representations of the observable algebras and attempts to reconstruct the unobservable fields and the a unifying gauge group.

8.1 Representations of Interest in Particle Physics In the first step of DHR analysis we want to single out a class of representations relevant for the study of superselection sectors. We focus on theories without long range effects, for example strong interactions in hadron physics. Loosely speaking, the representations of interest correspond to states that are generated from the vacuum by (possibly unobservable) local field operators. The intuition is that these states have different global charges (e.g. QCD charges) but which are localized in compact regions. Let O → A(O) be a net of C ∗ -algebras and let A be the quasilocal algebra (to reduce clutter, we write A(M) ≡ A in this section). Let π0 be the vacuum representation (assumed here to be faithful). We are interested in reconstructing the field algebras F(O) realized as von Neumann algebras of operators on some common Hilbert space Ht ot , i.e. F(O) ⊂ B(Ht ot ). We want them to satisfy (among others) the following properties: F1 F(O) carries a strongly continuous representation of the (covering group) of the Poincaré group and there exists a unique vector ∈ Ht ot (up to phase) that is invariant under this action. F2 There exists a compact group G (the gauge group) and a strongly continuous faithful representation U of G in Ht ot such that the automorphism A → αg (A) = AdU (g) (A) = U (g)AU (g)−1 obeys αg (F(O)) = F(O) ,

U (g) = ,

and the U (g)’s commute with the representation of the Poincaré group mentioned above. On general grounds, the Hilbert space Ht ot decomposes into superselection sectors as:   σ ⊗ Hσ , H U (g) = Uσ (g) ⊗ 1Hσ , (46) Ht ot = σ

σ

σ , Uσ ) where the sum is taken over equivalence classes σ of unitary irreps (H  of G, with representation space Hσ , and Hσ reflects the multiplicity with which these representations appear (including the possibility that Hσ has zero dimension, in which case σ does not appear).

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F3 Reconstructing the observables: Ht ot carries a representation π of A(O) and π(A(O)) = F(O) ∩ U (G) = {A ∈ F(O), αg (A) = A, ∀g ∈ G}. By construction, this representation decomposes w.r.t. (46) as π=



1H σ ⊗ πσ

σ

where each πσ is a representation of A(O) on Hσ . In particular, the representation corresponding to the trivial representation of G should coincide with π0 , the vacuum representation. The task of reconstructing the field algebra therefore amounts to determining the relevant representations πσ , and corresponding (irreducible) representations Uσ of G, which can then be assembled to form Ht ot . It may be shown that the representations πσ of interest are those which satisfy the following criterion (due to DHR [15]): Definition 11 A representation π of a net of C ∗ -algebras O → A(O) is a DHR representation if it is Poincaré covariant and the following holds:   π A(O ) ∼ = π0 A(O )

(47)

for all double-cones14 O, where O is the causal complement of O and ∼ = means unitary equivalence, i.e. there exists a unitary operator between the appropriate Hilbert spaces V : H0 → Hπ such that V π0 (A) = π(A)V for all A ∈ A(O ). In the above definition, the region O is unbounded, so A(O ) is not given a priori in the specification of the theory. It is defined to be the C ∗ -algebra generated by all local algebras A(O1 ) for bounded O1 ⊂ O . The intertwining unitary V has the interpretation of a charge-carrying field. To get some intuition about these objects, consider the example of a complex scalar field mentioned at the beginning of this section. Consider a test function f ∈ C0∞ (M; R) with  (f ) = 0. Firstly, we note that the charge is related to the phase of the field π( (f )), so in order to extract an intertwiner V , we consider the polar decomposition π( (f )) = Vf |π( (f ))|, where the overline denotes an operator closure. The partial isometry Vf may be taken to be unitary because π( (f )) is normal; acting on vectors in the vacuum sector H0 , it has the property q q Vf π0 (A) = πq (A)Vf , where A ∈ C(M) and πq is the representation with charge q ∈ Z. We see that Vf creates a single unit of charge and the support of f determines the region, where the charge is localized. If f is supported in a double cone O and

14 The

criterion can be weakened so as to refer to sufficiently large double-cones.

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C. J. Fewster and K. Rejzner q

−q

we take f1 supported inside another double cone O1 , then Vf1 Vf transports q units of charge from O to O1 . There exists a generalization of the DHR framework to the situation, where charges are not localized in bounded regions, but rather in cone-like regions (spacelike cones formed as a causal completions of spacial cones). This is relevant if one wants to apply this analysis to theories with long-range interactions, e.g. quantum electrodynamics (QED). It is expected that the electron is a charged particle with this type of localization. The corresponding version of the DHR construction has been developed by Buchholz and Fredenhagen (BF analysis) in [7] and the full analysis of superselection sectors for QED has been achieved in [8]. We may now turn things around and phrase our problem as follows: starting with the abstract algebra of observables and its vacuum representation we want to classify the equivalence classes of its (irreducible) representations satisfying (47). Following the literature [13–16, 28], we call each of these equivalence classes a superselection sector (also called charge superselection sectors, whereupon the labels σ are referred to as charges, though they need not be numbers). σ ) is a one-dimensional representaA special case is the situation where (Uσ , H tion. Remarkably, such simple sectors are distinguished by the following property of πσ : Definition 12 We say that a representation π satisfies Haag duality, if π(A(O )) = π(A(O)) ∩ π(A) ,

(48)

for any double-cone O. If π is irreducible then the intersection with π(A) is superfluous, whereupon one also has π(A(O )) = π(A(O)) .

(49)

(Exercise!) σ is one-dimensional One can show that πσ satisfies Haag duality if and only if H [13]. In particular, the vacuum sector is always simple; however, if the global gauge group is nonabelian, there will necessarily be some non-simple sectors. Exercise 17 If A is a C ∗ -algebra with Hilbert space representations π1 and π2 on Hilbert spaces H1 and H2 , define the representation (π1 ⊕π2 )(A) = π1 (A)⊕π2 (A) on H1 ⊕ H2 . Compute the commutant (π1 ⊕ π2 )(A) and double commutant (π1 ⊕ π2 )(A) within B(H1 ⊕ H2 ), which may be regarded as consisting of 2 × 2 ‘block matrices’ of operators, and compare the results with π1 (A) ⊕ π2 (A) .

8.2 Localized Endomorphisms We now want to introduce some algebraic structures on the space of representations of interest, following closely the exposition presented in [25]. Our standing

Algebraic Quantum Field Theory

47

assumptions are that the vacuum representation π0 is faithful and irreducible, satisfies Haag duality, and that the local algebras π0 (A(O)) are weakly closed, π0 (A(O)) = π0 (A(O)), so they are actually von Neumann algebras.15 Together with Haag duality, this gives π0 (A(O )) = π0 (A(O))

(50)

for every double-cone O. Let us fix a double cone O. Consider a representation π of A satisfying the DHR criterion with a unitary V : H0 → Hπ implementing the equivalence in (47) for O, and define a representation π˜ on H0 by π(A) ˜ = V −1 π(A)V ,

A ∈ A.

Take O1 ⊃ O and A ∈ A(O1 ), B ∈ A(O1 ). Since A(O1 ) ⊂ A(O ) and π˜ = π0 on A(O ) (by the DHR criterion), we have ˜ = π([B, ˜ A]) = 0 , [π0 (B), π(A)] so π(A) ˜ ∈ π0 (A(O1 )) = π0 (A(O1 )), where the last assertion follows from Haag duality (50). We conclude that π(A(O ˜ ˜ (A) ⊂ π0 (A), as 1 )) ⊂ π0 (A(O1 )), so π local operators are dense in A. Since π0 is faithful, there exists an endomorphism ρ of the abstract C ∗ -algebra A with ρ = π0−1 ◦ π. ˜ There is an obvious equivalence relation on endomorphisms: ρ1 ∼ ρ2 ⇔ ρ1 = ι ◦ ρ2 ,

(51)

for some inner automorphism ι, i.e. one that can be written as ι(A) = AdU (A) = U AU −1 for some unitary U ∈ A. Endomorphisms ρ obtained from representations satisfying the DHR criterion for a given O have the following properties [25]: LE1 Localised in O: ρ(A) = A, A ∈ A(O ). LE2 Transportable: ∀O1 , O2 with O ∪ O1 ⊂ O2 , there is a unitary U ∈ A(O2 ) with AdU ◦ ρ(A) = A, A ∈ A(O1 ), i.e. for every region O1 there exists an endomorphism equivalent to ρ under (51) that is localized in O1 . LE3 ρ(A(O1 )) ⊂ A(O1 ), ∀O1 ⊃ O. In fact, this is a consequence of LE1 and LE2 by an argument similar to show π(A(O ˜ 1 )) ⊂ π0 (A(O1 )) above. Changing the perspective, we can use the properties above as defining properties for the following class of endomorphisms of A:

weak closure does not hold, one can apply this discussion to the net O → M0 (O) := π0 (A(O)) .

15 If

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Definition 13 Given a double-cone O, let (O) be the set of all transportable endomorphisms of A localised in O, i.e. those satisfying LE1 and LE2 (and . hence LE3). We define  = O (O). It can be shown that the equivalence classes /∼ (where ∼ is given by (51)) are in one to one correspondence with unitary equivalence classes of representations satisfying (47) (i.e. with superselection sectors). /∼ is equipped with a natural product (given by the composition of endomorphisms of A), which induces a product on the space of sectors, namely: . [π1 · π2 ] = [π0 ◦ ρ1 ρ2 ] ,

(52)

. where πi = π0 ◦ ρi , i = 1, 2. Exercise 18 Show that the composition of representations introduced in (52) is well defined and the resulting representation satisfies the DHR criterion. It can also be shown (see e.g. [25]) that the · product of two representations that are Poincaré covariant and satisfy the spectrum condition also has these two features. The space of sectors equipped with the composition product is a semigroup with the vacuum sector as the identity. One can verify that simple sectors correspond to morphisms ρ that are in fact automorphisms of A, i.e. ρ(A) = A. Hence the space of simple sectors equipped with · is a group. Transportability of endomorphisms LE2 is crucial for the DHR analysis, since it allows us to “move morphisms around”. First we establish the following: Proposition 1 Endomorphisms ρ are locally commutative, i.e. for O1 ⊂ O2 and ρi ∈ (Oi ) we have ρ1 ρ2 = ρ2 ρ1 . ˆ i, O ˜ i , i = 1, 2 with Proof Fix an arbitrary double cone O and choose double cones O  ˆ ˆ ˜ ˜ ˜  (see Fig. 3). the following properties: Oi ⊂ O , Oi ∪ Oi ⊂ Oi , i = 1, 2 and O1 ⊂ O 2 ˜ i ) such that AdU ◦ρi = ρˆi ∈ (O ˆ i ), LE2 implies that there exist unitaries Ui ∈ A(O i i = 1, 2. Hence for any A ∈ A(O) we have ρi (A) = AdUi∗ ◦ ρˆi (A) = AdUi∗ (A), as ˆ 1 and O ˆ 2 and we have also used LE3. Hence O is causally disjoint from both O ρ1 ρ2 (A) = ρ1 ◦ AdU2∗ (A) = Adρ1 (U2∗ ) ◦ AdU1∗ (A) = AdU2∗ U1∗ (A) = AdU1∗ U2∗ (A) = ρ2 ρ1 (A) , where we used LE1 to conclude that ρ1 (U2∗ ) = U2∗ , while U1∗ U2∗ = U2∗ U1∗ follows from A3. We can repeat the same reasoning for any double cone O, so ρ1 ρ2 = ρ2 ρ1 .   We can now use this result to show that the product of representations is commutative, i.e. [π1 · π2 ] = [π2 · π1 ]. Let ρ1 , ρ2 ∈ (O) and take two spacelike separated double cones O1 , O2 . By LE2 there are morphisms ρ˜i , i = 1, 2 localized in Oi and

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Fig. 3 Configuration of doublecones in the proof of Proposition 1

ˆ1

˜1

˜2 1

2

ˆ2

unitaries Ui , i = 1, 2 such that ρ˜i = AdUi ◦ ρi . It follows by Proposition 1 that ρ2 ρ1 = Adρ2 (U1∗ )U2∗ ◦ ρ˜2 ρ˜1 = Adρ2 (U1∗ )U2∗ ◦ ρ˜1 ρ˜2 = Adε(ρ1 ,ρ2 ) ◦ ρ1 ρ2 , with the unitary ε(ρ1 , ρ2 ) = ρ2 (U1∗ )U2∗ U1 ρ1 (U2 ) ∈ A .

(53)

This proves the equivalence of ρ2 ρ1 and ρ1 ρ2 , so the commutativity of the product of representations follows. Hence the space of all sectors, equipped with · is an abelian semigroup and the space of all simple sectors equipped with · is an abelian group. The unitary operator ε(ρ1 , ρ2 ) that we have discovered here is called the statistics operator. It depends only on ρ1 , ρ2 and not on ρ˜i nor the choice of Ui , i = 1, 2. Let ερ ≡ ε(ρ, ρ). In spacetime dimension d > 2 we have ερ2 = id. For simple sectors π0 ◦ ρ 2 is irreducible, so ερ is a multiple of 1 and in d > 2 this implies that ερ = ±1. Physically this corresponds to the alternative between Fermi and Bose statistics. For general sectors in d > 2 one can also have para-statistics or even infinite statistics. In lower dimensions (d ≤ 2), ερ is instead related to representations of the braid group. This kind of behaviour appears also for conelocalized charges (BF analysis), but in this case the braided statistics appears already in d ≤ 3. More details relating ερ to statistics will be given in the next section. The physical interpretation of the product of representations can be understood as follows [15]. Let ω0 be the vacuum state and ρi ∈ (Oi ), i = 1, 2. We have the two states ωi = ω0 ◦ ρi , which are vector states in their respective representations π = π0 ◦ ρi ; indeed, one suitable vector is the vacuum vector that is the GNS vector induced by ω0 . Clearly, ω = ω0 ◦ ρ1 ρ2 is a vector state in representation [π1 ·π2 ], with the same representing vector. Let O1 be spacelike to O2 . Then ω looks like the vector state ω0 ◦ ρ1 with respect to observations in O2 and like ω0 ◦ ρ2 for observations in O1 . Hence, taking the product [π1 · π2 ] has a physical interpretation of composing two vector states that are localized “far apart”.

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8.3 Intertwiners and Permutation Symmetry As stated in the previous section, states ω0 ◦ ρk , k = 1, . . . , n obtained from transportable localized morphisms ρk ∈  are interpreted as vector states corresponding to localized charged particles and composition of morphisms describes creating several such charges in spacetime. Now we want to understand how one may create these charges in a given order. As the state ω0 ◦ ρ1 . . . ρn is independent of the ordering of the ρk ’s, this must be done by permuting a family of vectors all of which represent this state. In order to do this, we need some notation. Definition 14 Let ρ, σ ∈ . Define the space of intertwiners between ρ and σ by16 . (σ, ρ) = {T ∈ A| σ (A)T = T ρ(A), ∀ A ∈ A} . We can now define some algebraic operations on intertwiners. Let S ∈ (τ, σ ) and T ∈ (σ, ρ) be intertwiners. We can compose them to obtain ST ∈ (τ, ρ) and can also define the adjoint T ∗ ∈ (ρ, σ ). Moreover, there is a natural product between intertwiners in different spaces. Let T1 ∈ (σ1 , ρ1 , ) and T2 ∈ (σ2 , ρ2 ). We define T1 × T2 ∈ (σ1 σ2 , ρ1 ρ2 ) as . T1 × T2 = T1 ρ1 (T2 ) = σ1 (T2 )T1 (using the fact that T1 ∈ (σ1 , ρ1 ) for the last equality). The ×-product is associative, and distributive with respect to the composition of intertwiners (i.e., their product in A). Next we discuss localization properties of the intertwiners. Consider T ∈ (σ, ρ), where ρ is supported in O1 and σ in O2 . We call O2 the left support and O1 the right support of the intertwiner T . If A ∈ A(O1 ) ∩ A(O2 ), then the support and intertwining properties imply T A = T ρ(A) = σ (A)T = AT so T is bilocal, in the sense that T ∈ (A(O1 ) ∩ A(O2 )) (here the commutant is taken in A). In the case O1 = O2 = O, we have T ∈ A(O) by Haag duality (50) in the vacuum representation and the assumption that π0 is faithful. We call two intertwiners causally disjoint if their right supports lie space-like to each other and the same holds for their left supports.

16 In [15, 16] the intertwiners are denoted by T ≡ (σ |T |ρ), to emphasize to which space they belong.

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Proposition 2 Let Ti ∈ (σi , ρi ), where ρi , σi ∈ (Oi ), i = 1, 2, with O1 and O2 spacelike separated (i.e. T1 and T2 are causally disjoint), then T1 × T2 = T2 × T1 . Proof To see this, note that Ti ∈ A(Oi ), so T1 × T2 = T1 ρ1 (T2 ) = T1 T2 = σ2 (T1 )T2 = T2 × T1 .   The statistics operator ερ is also an intertwiner and it has a nice expression in terms of products of other intertwiners. Exercise 19 Let ρ1 , ρ2 ∈ (O) and ρ˜i = Ui ρi Ui−1 , i = 1, 2, as in the construction leading to (53). Show that ε(ρ1 , ρ2 ) ∈ (ρ2 ρ1 , ρ1 ρ2 ) and ε(ρ1 , ρ2 ) = (U2∗ × U1∗ )(U1 × U2 ) . Clearly, ερ ∈ (ρ 2 , ρ 2 ), which means that ερ commutes in A with every element of ρ 2 A, ερ ∈ (ρ 2 A) , and therefore π0 (ερ ) commutes with all observables in the representation π0 ◦ ρ 2 . Now take Ti ∈ (σi , ρi ), where ρi , σi ∈ , i = 1, 2. One can also easily check (Exercise) that ε(σ1 , σ2 )(T1 × T2 ) = (T2 × T1 )ε(ρ1 , ρ2 ) ,

(54)

so in particular, for σ1 = σ2 and ρ1 = ρ2 , we have εσ (T1 × T2 ) = (T2 × T1 )ερ . Hence ε changes the order of factors in ×-product. Exercise 20 Prove (54). You will need to use Proposition 2. (n)

Using ερ , we can construct a representation ερ of the braid group for each ρ. The braid group Bn with n strands is the group generated by ς1 , . . . ςn−1 with relations: ςi ςj = ςj ςi ,

if |i − j | > 2 ,

ςi ςi+1 ςi = ςi+1 ςi ςi+1 . . Let ςi , i ∈ {1, . . . , n − 1} be a generator of Bn , then we set ερ(n) (ςi ) = ρ i−1 ερ . The following exercise shows that this is indeed a representation of the braid group. Exercise 21 Show that ερ ρ(ερ ) = ε(ρ 2 , ρ) and use this together with (54) to prove ερ ρ(ερ )ερ = ρ(ερ )ερ ρ(ερ ). In d > 2, this also gives us a representation of the permutation group Sn (as ερ2 = 1). In the more detailed analysis that follows we will focus on

52

C. J. Fewster and K. Rejzner (n)

d > 2 and refer the reader to [27] for the general case. Let ερ (P ) denote the representative of P ∈ Sn . More generally, we may also define ε(ρ1 , . . . , ρn ; P ) ∈ (ρP −1 (1) . . . ρP −1 (n) , ρ1 . . . ρn ) by ε(ρ1 , . . . , ρn ; P ) := U ∗ (P )U (e) , where (in analogy with the n = 2 case) ρ˜i = Ui ρi Ui−1 are auxiliary morphisms localized in spacelike separated regions Oi , i = 1 . . . , n and we use the notation U (P ) := UP −1 (1) × · · · × UP −1 (n) . One can easily check (Exercise) that ε(ρ, . . . , ρ; P ) = ερ(n) (P ). Note that in n = 2 case we implicitly have ε(ρ1 , ρ2 ; τ ) ≡ ε(ρ1 , ρ2 ), where τ is the transposition of 1 and 2. Property (54) generalizes to: ε(σ1 . . . , σn ; P )T (e) = T (P )ε(ρ1 , . . . , ρn ; P ) , where Ti ∈ (σi , ρi ) i = 1, . . . , n. If σi = σ , ρi = ρ for all i = 1, . . . , n, we have εσ(n) (P )T (e) = T (P )ερ(n) (P ) . To understand better the physical interpretation of ερ(n) (P ), fix a morphism ρ and consider a family of intertwiners Uk ∈ (σk , ρ), where σk are morphisms localized in spacelike separated regions Ok , k = 1, . . . , n. Let ∈ H0 be the vacuum vector. Clearly, π0 (Uk∗ ) |π0 (ρ(A))π0 (Uk∗ )  =  |π0 (Uk ρ(A)Uk∗ )  =  |π0 (σk (A))  = ω0 ◦ σk (A) , so ω0 ◦ σk is a vector state on ρA with the distinguished vector π0 (Uk∗ ) . Now consider vectors of the form P = π0 (U ∗ (P )) ,

P ∈ Sn ,

which can be interpreted as a product of n state vectors with identical charge quan(n) tum numbers but with an ordering determined by P . The operator ερ (Q) changes the order of factors in this vector [15]. To see this, recall that Q → ερ(n) (Q) ∈ A is a unitary representation of Sn , whereupon P = π0 (ερ(n) (P )U ∗ (e)) and π0 (ερ(n) (Q))P = π0 (ερ(n) (Q)ερ(n) (P )U ∗ (e)) = π0 (ερ(n) (QP )U ∗ (e)) = QP Each vector P induces the same state on ρ n A, namely ω0 ◦ σ1 . . . σn . Therefore, (n) the action of εQ is analogous to permutations of the wave functions of n identical particles in quantum mechanics, which also leaves expectation values of observable quantities unchanged.

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In the next step one introduces the notion of a conjugate sector. Physically, the relation between a sector and its conjugate is that of having a charged particle localized in some compact region versus having the corresponding antiparticle localized in that region. We have already mentioned that sectors can be equipped with the structure of a semigroup and simple sectors form a group. In the latter case, the conjugate sector is just given by the group inverse and ερ = ±1, so we have the simple fermion/boson alternative. More generally, to obtain a left inverse to a given ρ, we want to find a map φ : A → A with φ(ρ(A)Bρ(C)) = Aφ(B)C ,

φ(A∗ A) ≥ 0 ,

φ(1) = 1 .

Note that φ on ρ(A) can be set as ρ −1 . We then use the Hahn-Banach theorem to extend the state ω0 ◦ ρ −1 on ρ(A) to A. Let (π, H, ) be the corresponding GNS ˜ We triple and define an isometry V : H0 → H, by means of V A = π ◦ ρ(A) . ∗ then define the left inverse of the given sector ρ as φ(A) = V π(A)V . The left inverse is used to study the representations of the permutation group in d > 2 (or the braid group in low dimensions). We note that φ(ερ )ρ(A) = φ(ερ ρ 2 (A)) = φ(ρ 2 (A)ερ ) = ρ(A)φ(ερ ) , which for irreducible ρ implies φ(ερ ) = λρ 1 , where λρ is the statistics parameter and it characterizes the statistics of the sector ρ. One finds that the allowable values for this parameter are: • λ = d1 , d ∈ N giving para Bose statistics of order d; • λ = − d1 , d ∈ N is giving para Fermion statistics of order d; • λ = 0, giving infinite statistics. In DHR theory, d is called the statistical dimension.17 To obtain the sector ρ conjugate to a given sector [ρ], one shows (under appropriate, physically motivated assumptions, including the finite statistics λ = 0 [15]) that π satisfies the DHR criterion, so there exists a morphism ρ ∈ (O) such that π = π0 ◦ ρ and an isometry R such that ρρ(A)R = RA. In fact, in the language of category theory, localized morphisms and intertwiners form a symmetric (or braided in d < 2) monoidal category, where the monoidal structure is given by ×. The existence of conjugate sectors (unique up to equivalence) means that the category is rigid.

17 Mathematically, as discovered by Longo in [35], d is in fact the square root of the Jones index of the inclusion ρ(A(O)) ⊂ ρ(A(O )) , i.e. it quantifies how badly Haag duality is broken in the given sector.

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The reconstruction theorem of Doplicher and Roberts [12] allows one to reconstruct the field net F(O) and the gauge group G from the above data completing the programme set out in Sect. 8.1. Abstractly, firstly they show the equivalence of the DHR category to a category of representations of some compact group and then reconstruct the group from that category (this step is a version of the Tannaka–Krein duality), together with the algebra of fields on which this group acts.

9 Conclusions In these notes, we have summarized some important aspects of quantum field theory in the algebraic formulation of Haag and Kastler, focusing on the features that make it very different from quantum mechanics. One such feature is the existence of inequivalent representations. This was illustrated by the case study of the van Hove model in Sect. 3, and further emphasized in Sect. 8, where inequivalent representations corresponding to different charges were discussed. In both situations, one can see that all the physical information can be recovered from the abstract net of algebras, so it is more advantageous to think of the net rather than the collection of Hilbert space representations as the fundamental object. The axioms for the net have been formulated in Sect. 4.1, followed by some simple examples in Sect. 4.2. The connection between the algebraic and the Hilbert space centred approaches can be made through the choice of an algebraic state. In particular, for QFT on Minkowski spacetime,18 one can consider the distinguished Poincaré invariant state, the vacuum; we have also described the more general class of quasi-free states that can be specified by their two-point functions and have Fock space representations. In Sect. 5, we showed that QFT in the vacuum representation has some peculiar features that make it very different from quantum mechanics, the most dramatic being the Reeh-Schlieder theorem. A net of C ∗ -algebras together with a state induces a net of von Neumann algebras via weak completion. In Sect. 6 we pointed out that the type of von Neumann algebras that arise in QFT (type III) is very different from the type characteristic for quantum mechanics (type I). We discussed the main consequences of this fact in Sect. 7, in the context of independence of measurements by spacelike observers, in the guise of the split property. To close, let us emphasize that AQFT, although very different from quantum mechanics, is not to be regarded as disjoint from “traditional” QFT as presented in standard textbooks. To the contrary, it is a framework that allows one to derive and study common structural and conceptual features of QFT, which then become realized in physical, experimentally testable models.

18 Here we focused only on QFT on Minkowski spacetime, but the algebraic approach also easily generalizes to curved spacetimes [5, 22, 26].

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Appendix 1: Some Basic Functional Analysis A general reference for this brief summary is [38]. Recall that a Hilbert space H is a complex inner product space, with an inner product ·|· that is linear in the second slot √ and conjugate-linear in the first, and for which the associated norm

ψ := ψ|ψ is complete, i.e., all Cauchy sequences converge. The Hilbert space is separable if it has a finite or countably infinite orthonormal basis, and inseparable otherwise. A linear operator A : H1 → H2 between Hilbert spaces H1 and H2 is bounded . iff A = sup x 1 =1 Ax 2 is finite, where we use the subscript to denote the Hilbert space norm concerned. For maps between Hilbert spaces, boundedness and continuity are equivalent properties. In this case A has an adjoint A∗ : H2 → H1 , with the defining property ϕ|Aψ2 = A∗ ϕ|ψ1 ,

(55)

for all ψ ∈ H1 , ϕ ∈ H2 . Several interesting classes of bounded operator may be defined: a bounded operator A : H → H is self-adjoint if A = A∗ ; while A is a projection if A = A∗ = A2 (more strictly, this defines an ‘orthogonal projection’ but we follow common usage in simply saying ‘projection’). A bounded operator U : H1 → H2 is unitary if U ∗ U = U U ∗ = 1, and a partial isometry if U ∗ U and U U ∗ are projections. Every bounded operator A : H → H has a unique polar decomposition A = U |A| such that U is a partial isometry with ker U = ker A, and |A| is a positive operator such that |A|2 = A∗ A. Here, a self-adjoint operator A : H → H is said to be positive if ψ|Aψ ≥ 0 for all ψ ∈ H. For obvious reasons |A| is called the positive square root of A∗ A. An operator A on H is said to be of trace class if α eα | |A|eα  is finite, where eα is some orthonormal basis of H; in this case, the trace tr A := α eα |Aeα  is finite and independent of the basis used to compute it. If A is a partially defined linear map between Hilbert spaces H1 and H2 , we denote its domain of definition within H1 by D(A). We typically only consider the situation where D(A) is dense. If sup{ Ax 2 : x ∈ D(A), x 1 = 1} is finite, then A can be extended by continuity to a unique bounded operator from H1 to H2 ; otherwise, A is described as an unbounded operator. The adjoint A∗ of a densely defined unbounded operator A is again defined through (55) and D(A∗ ) is the set of all ϕ ∈ H for which this definition makes sense: ϕ ∈ D(A∗ ) if and only if there exists η ∈ H such that ϕ|Aψ = η|ψ holds for all ψ ∈ D(A), whereupon we write A∗ ϕ = η. An unbounded densely defined operator A is called self-adjoint if ϕ|Aψ = Aϕ|ψ for all ϕ, ψ ∈ D(A) and in addition D(A∗ ) = D(A). Any operator is completely described by its graph (A) := {(x, Ax) ∈ H × H : x ∈ D(A)}.

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One says that A is a closed operator if (A) is a closed subset of H×H with respect to the norm of H ⊕ H; it is closable if (A) has a closure that is the graph of some (closed) operator A, which is naturally called the closure of A. All densely defined operators with densely defined adjoints are closable and their adjoints are closed, so in particular self-adjoint operators are closed. The polar decomposition extends to closed operators. The spectrum σ (A) of a (bounded or unbounded) operator A on a Hilbert space H is the set of z ∈ C for which A − z1 fails to have a bounded two-sided inverse. In particular, eigenvalues lie in the spectrum but not every spectral point is an eigenvalue. The spectrum of a self-adjoint operator is real, σ (A) ⊂ R, while the spectrum of a unitary operator lies on the unit circle in C. Finally, suppose that H is a real Hilbert space (i.e., a real inner product space with a complete induced norm). To distinguish real spaces from complex ones in this appendix, we write the real inner product with round brackets and the adjoint with a dagger. A complex structure on H is a linear map J : H → H obeying J 2 = −1, J † = −J . Then we may convert H into a complex Hilbert space by adding two structures: first, the operation of multiplication by a complex scalar, C × H # (z, ψ) → (Re z)ψ − (Im z)J ψ ∈ H, in which sense multiplication by i is implemented by −J (this convention is annoying but avoids a proliferation of minus signs elsewhere in the main body of the text), and second, a sesquilinear inner product ψ|ϕ = (ψ, ϕ) + i(ψ, J ϕ). Exercise 22 Check that H, with these additional structures, is indeed a complex Hilbert space.

Appendix 2: Construction of an Algebra from Generators and Relations Several algebras encountered in Sect. 4.2 were presented in terms of generators and relations. Here, we give more details on how an algebra may be constructed in this way, taking the real scalar field as our example. • First consider the free unital ∗-algebra U containing arbitrary finite linear combinations of finite products of the (f )’s and (f )∗ ’s and unit 1. • Construct a two-sided ∗-ideal I in U generated by the relations. Thus I contains all finite linear combinations of terms of the form A((f )∗ − (f ))B

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as A and B range over U and f ranges over C0∞ (M), and similar terms obtained from the other relations, and all terms obtained from these by applying ∗. Recall that a two-sided ∗-ideal is a subspace of the algebra that is stable under multiplication by algebra elements on either side, and under the ∗-operation. • The algebra A(M) is defined as the quotient U/I, namely, the vector space quotient, equipped with product and ∗-operations so that [A][B] = [AB],

[A]∗ = [A∗ ],

1A(M) = [1U ].

One may check that the fact that I is a two sided ∗-ideal guarantees that these operations are well-defined (independent of the choice of representatives). • For future reference: let B be another algebra obtained as a quotient B = V/J, where J is a two-sided ∗-ideal in a unital ∗-algebra V. Then any function mapping the (f ) into V extends uniquely to a unit-preserving ∗homomorphism from U to V, and induces a unit-preserving ∗-homomorphism from A(M) to B, provided every element of I is mapped into J, i.e., the map on generators is compatible with the relevant relations.

Appendix 3: Fock Space Let H be a complex Hilbert space. As usual, Hn denotes the n’th symmetric tensor power of H, with H0 = C, whereupon the bosonic Fock space over H is F(H) =

∞ 

Hn .

n=0

Thus, a typical Fock space vector is a sequence  = (n )n∈N0 , where n ∈ Hn is called the n-particle component of . In particular the Fock vacuum vector is = (1, 0, . . .), and the number operator N is defined by (N)n = nn

n ∈ N0

 2 2 on the domain of all  ∈ F(H) for which ∞ n=0 n n < ∞. We will describe the annihilation and creation operators and the number operator on F(H) in the basis-free notation used in Sect. 4.3. See Refs. [4, §5.2.1] and [37, §X.7] for more details. In this framework, each ψ ∈ H labels annihilation and creation operators a(ψ) and a ∗ (ψ) on F(H), with ψ → a(ψ) being antilinear, and ψ → a ∗ (ψ) := a(ψ)∗ being linear in ψ. These operators are unbounded and have to be defined on suitable dense domains within F(H), be taken as the domain of N 1/2 , i.e., those ∞ which may 2  ∈ F(H) for which n=0 n n < ∞. The annihilation operator acts (on

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vectors in the domain) by (a(ϕ))n =

√ n + 1n+1 (ϕ)(n+1 )

where n+1 (ϕ) : H⊗(n+1) → H⊗n is defined by n+1 (ϕ)(ψ1 ⊗ · · · ⊗ ψn+1 ) = ϕ|ψ1 ψ2 ⊗ · · · ⊗ ψn+1 and restricts to a map H(n+1) → Hn . It follows in particular that a(ϕ) = 0 for all ϕ ∈ H. One may check that the adjoint operators obey (a ∗ (ϕ))0 = 0,

(a ∗ (ϕ))n+1 =

√ n + 1Sn+1 (ϕ ⊗ n ),

n ∈ N0

where Sn+1 is the orthogonal projection onto H(n+1) in H⊗(n+1) . Acting on vectors  in the domain of N, the canonical commutation relations hold in the form [a(ψ), a ∗ (ϕ)] = ψ|ϕH ,

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and vectors obtained by acting with sums of products of a ∗ (ϕ) operators on are dense in F(H). This presentation of the annihilation and creation operators may seem unfamiliar to those who prefer their annihilation and creation operators to look more like ai and aj∗ . But let (ei ) be any orthonormal basis for H and define ai = a(ei ), ai∗ = a(ei )∗ = a ∗ (ei ). Then the CCRs become [ai , aj∗ ] = ei |ej H 1 = δij 1 (understood as acting on a suitable domain) and of course ai = a(ei ) = 0, which provides a set of annihilation and creation operators labelled by a discrete index. At least formally (because infinite sums of unbounded operators should be handled with care) a(ψ) =

i

ψ|ei ai ,

a ∗ (ϕ) =

ei |ϕai∗ . i

The advantage of the basis-independent approach is that it does not give any basis a privileged status, and avoids the need for infinite series of the type just given if changing basis, for example. The number operator can also be related to the annihilation and operators in the basis-independent form—see [4, §5.2.3]. It is also common in QFT to use annihilation and creation operators indexed by a continuous momentum variables, in situations where H is a space of square-integrable functions of momentum. This is essentially a matter of using a continuum-normalised ‘improper basis’ for H, but one should be aware that, while a(k) does define an (unbounded) operator, it is sufficiently poorly-behaved that it

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does not have a densely defined operator adjoint. Nonetheless, any normal-ordered string a ∗ (k 1 ) · · · a ∗ (k m )a(k 1 ) · · · a(k n ) can be given meaning as a quadratic form, that is, defining its matrix elements as |a ∗ (k 1 ) · · · a ∗ (k m )a(k 1 ) · · · a(k n )   := a(k m ) · · · a(k 1 )|a(k1 ) · · · a(k n )   on suitable vectors ,   ∈ F(H). For more on this viewpoint, see [37, §X.7].

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Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts Felix Finster and Maximilian Jokel

Abstract We give an elementary introduction to the theory of causal fermion systems, with a focus on the underlying physical ideas and the conceptual and mathematical foundations.

1 The Challenge: Unifying Quantum Field Theory and General Relativity One of the biggest problems of present-day theoretical physics is the incompatibility of Quantum Field Theory and General Relativity. While the standard model of elementary particle physics provides a quantum field theoretical description of matter together with its electromagnetic, weak and strong interactions down to atomic and subatomic scales, General Relativity applies to gravitational phenomena on astrophysical or cosmological scales. Just as the standard model of elementary particle physics is well-confirmed by high-precision measurements, also the theoretical predictions of General Relativity agree with the experimental results to high accuracy. Nevertheless, when combining Quantum Field Theory and General Relativity on very small length scales, these theories become mathematically inconsistent, making physical predictions impossible. The fact that combining Quantum Field Theory with General Relativity leads to inconsistencies, although each theory by itself provides excellent theoretical predictions, allows for different possible conclusions: While a convinced elementary particle physicist will refer to the overwhelming triumph of Quantum Field Theory

F. Finster () · M. Jokel Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany e-mail: [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 F. Finster et al. (eds.), Progress and Visions in Quantum Theory in View of Gravity, https://doi.org/10.1007/978-3-030-38941-3_2

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and postulate the existence of a gravitational exchange particle, namely the graviton, thus forcing General Relativity into the setting of the standard model of elementary particle physics, a dedicated relativist, on the other hand, will question the mathematical formalism of Quantum Field Theory and instead refer to the aesthetics and mathematical clarity of the differential geometric approach to General Relativity. Undecided physicists, who are convinced of the concepts of both Quantum Field Theory and General Relativity, may argue that, instead of incorporating one theory in the other, one should try to find a new theory which reproduces both Quantum Field Theory and General Relativity in suitable limiting cases. Physicists skeptical of both theories will bring into play alternative approaches such as string theory or the theory of loop quantum gravity which are based on completely new assumptions. Due to the lack of experimental evidence, most alternative approaches are mainly based on personal preferences and paradigms. They involve ad-hoc assumptions which are often detached from the well-established physical principles which were developed based on physical experiments. Since there are many ways to introduce new assumptions ad hoc, it is questionable whether these approaches will turn out to be successful. Therefore, we prefer to proceed differently as follows: We begin with a detailed and honest review of the concepts and principles which form the basis of Quantum Field Theory and General Relativity. Afterward, we select those principles which we consider to be essential (clearly, this is a subjective choice). Then we combine these principles in a novel mathematical setting, referred to as causal fermion systems. Working exclusively with the objects in this setting, we postulate new physical equations by formulating the so-called causal action principle. The causal action principle gives rise to additional objects and structures in space-time together with equations describing their dynamics. In this way, we obtain a new physical theory with predictive power.

2 Overview of Concepts and Mathematical Structures in Theoretical Physics Following the above outline, this section is devoted to a review of the concepts and ideas, common beliefs as well as selected mathematical structures and objects used in contemporary theoretical physics. To sharpen the view for the few really fundamental principles underlying our present understanding and mathematical description of nature, we have decided to take a bird’s-eye perspective rather than a high-resolution examination of sophisticated mathematical constructions.

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2.1 The Fabric of Spacetime Before Einstein’s Special Theory of Relativity, physicists thought of space as being the geometric background in which physical processes take place while time evolves. With this concept of space and time in mind, nobody could imagine that space itself might change while time evolves or—even more—that space-time as a whole participates in the physical interactions. After more than one hundred years of studying Einstein’s Theory of Relativity, however, our understanding of space and time has changed completely. Nowadays, we are used to the fact that spacetime and its matter content cannot be considered independently, but rather form an inseparable unity interwoven by mutual interactions. This unity is sometimes referred to as the fabric of spacetime. We now review the necessary concepts to capture and cast this intuitive notion in a formal mathematical framework as provided by differential geometry. In order to make this paper accessible to a broad readership, we also recall basic definitions which are clearly familiar to mathematicians.

2.1.1 Topological Manifolds as Models of Spacetime At the most basic level, namely without considering any additional structures, spacetime is nothing but a set of points which locally—that is within the tiny snippet of the universe which is accessible to our everyday experience—looks like the familiar, three-dimensional Euclidean space. Including time as a fourth dimension naturally leads to the idea to model the fabric of spacetime as a four-dimensional topological manifold.

Definition 2.1 (Topological Manifold) A topological manifold of dimension d is a second-countable, topological Hausdorff space (M, O) which at every point p ∈ M has a neighborhood which is homeomorphic to an open subset of Rd . Here O denotes the family of all open subsets of M. The reason why we do not consider a completely structureless set rather than the tuple (M, O) consisting of a set equipped with a topology, is needed in order to have a notion of continuity.

2.1.2 Establishing Smooth Structures in Spacetime By modelling spacetime as a four-dimensional topological manifold, we have already implemented some of our knowledge about the general structure of our Universe. In order to describe smooth functions in spacetime and to be able to

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do calculus, one important ingredient is still missing and calls for the following definition: Definition 2.2 (Smooth Compatibility of Coordinate Charts) Let (M, O) be an d-dimensional topological manifold together with two coordinate charts (U, ϕ) and (V , ψ) such that the open sets U, V ⊆ Rn satisfy U ∩ V = ∅. The composition of the coordinate functions given by ψ ◦ ϕ −1 : ϕ(U ∩ V ) → ψ(U ∩ V ) is called transition map from ϕ to ψ. Two coordinate charts (U, ϕ) and (V , ψ) are smoothly compatible if the transition map ψ ◦ ϕ −1 is a diffeomorphism.

The definition of smoothly compatible coordinate charts allows us to introduce the notion of smooth atlases which in turn prepares the ground for defining smoothness of functions on manifolds.

Definition 2.3 (Smooth Atlas) Let {(Ui , ϕi )}i∈I with I ⊆ N be a family of charts of a topological manifold (M, O) with open sets Ui ⊆ R n . The family {(Ui , ϕi )}i∈I of charts is called atlas, if the open sets Ui cover M. If in addition any two charts in the atlas are smoothly compatible, the atlas is referred to as smooth atlas. A topological manifold equipped with a smooth atlas A is referred to as a smooth manifold. We can now specify what we mean by smoothness of functions on a manifold. Definition 2.4 (Smooth Functions on Manifolds) Let (M, A) be a smooth manifold. A function f : M → R on the manifold is called smooth if for every chart (U, ϕ) ∈ A the function f ◦ ϕ −1 is smooth in the sense of functions being defined on open subsets of Rd .

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2.1.3 Encoding the Lorentzian Geometry of Spacetime From our everyday life within a small snippet of the universe, we are used to the properties of three-dimensional Euclidean space, especially its vector space character. In order not to loose these familiar and useful properties when modelling spacetime as a differentiable manifold, one introduces a vector space structure at every single point of the manifold. In order to avoid the idea that spacetime is embedded in some higher-dimensional ambient space, we must work with an intrinsic characterization which only makes use of the already defined concepts of coordinate charts and smooth functions. Definition 2.5 (Derivations and Tangent Space) Let (M, A) be an ddimensional smooth manifold and p an element of M. A linear map Xp : C ∞ (M, R) → R is called derivation at p ∈ M if it satisfies the Leibniz product rule ∀f, g ∈ C ∞ (M, R) : Xp (fg) = f (p)Xp (g) + g(p)Xp (f ) The set of all derivations at p ∈ M forms a vector space under the operations (X + Y )p (f ) = Xp (f ) + Yp (f ) (αX)p (f ) = αXp (f ) which is referred to as the tangent space Tp M at p ∈ M.

It can be shown that the tangent space is a d-dimensional real vector space. In order to better understand the similarities between the differential geometric formulation of Einstein’s General Theory of Relativity and the theory of causal fermion systems in the further course of this article, we shall introduce the bundle formulation. Definition 2.6 (Tangent Bundle and Vector Fields) Let (M, A) be an ddimensional smooth manifold with tangent spaces Tp M at all points p ∈ M. The tangent bundle T M is defined as the disjoint union of the tangent spaces Tp M at all points p ∈ M T M :=



{p} × Tp M

p∈ M

(continued)

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Definition 2.6 (continued) (endowed with the coarsest topology which makes the bundle charts continuous). A continuous function X ∈ C 0 (M, T M) is called vector field if it satisfies the condition ∀p ∈ M : X(p) := Xp ∈ Tp M

Having defined tangent spaces, we are ready to add our knowledge about the geometric structure of spacetime to our model. In the familiar Euclidean geometry, the geometry is retrieved by computing lengths and angles between vectors. These quantities are encoded in a scalar product, being a positive definite bilinear form gp : Tp M × Tp M → R . In Special Relativity, the geometry is described again by a bilinear form, which however is no longer positive definite, but instead has signature (1, 3): Definition 2.7 (Lorentzian Manifold) Let (M, A) be a d-dimensional smooth manifold with tangent bundle T M. A function g : T M × T M → R is called Lorentzian metric if the restriction gp : Tp M × Tp M → R is a bilinear, symmetric and smooth mapping g(X, Y ) : M → R

  p → g(X, Y ) (p) := gp (Xp , Yp )

of signature (1, 3). A smooth manifold (M, A) equipped with a Lorentzian metric is referred to as Lorentzian manifold (M, g).

The Lorentzian signature implies that the inner product gp (ξ, ξ ) of a tangent vector ξ ∈ Tp M with itself can be positive or negative. This gives rise to the following notion of causality. A tangent vector ξ ∈ Tp M is said to be ⎧ ⎨ timelike if gp (ξ, ξ ) > 0 spacelike if gp (ξ, ξ ) < 0 ⎩ lightlike if gp (ξ, ξ ) = 0 .

(1)

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Lightlike vectors are also referred to as null vectors, and the term non-spacelike refers to timelike or lightlike vectors. The spacetime trajectory of a moving object is described by a curve γ (τ ) in M (with τ an arbitrary parameter). We say that the spacetime curve γ is timelike if the tangent vector γ˙ (τ ) is everywhere timelike. Spacelike, null and non-spacelike curves are defined analogously. Then the usual statement of causality that nothing can travel faster than the speed of light can be formulated as follows: Causality: Information can be transmitted only along non-spacelike curves.

2.2 The Einstein Field Equations After these preparatory considerations, we are now ready to formulate and investigate the significance of the Einstein field equations which are at the heart of General Relativity. They take the form Ric −

1 R g +  g = 8πκ T , 2

where Ric is the Ricci tensor, R is scalar curvature,  is the cosmological constant, κ is the gravitational coupling constant, and T is the energy-momentum tensor. These equations can be derived from an action principle. More precisely, metrics which solve the Einstein equations are critical points of the Einstein–Hilbert action  EH

S

=

 1    R − 2 + Lmatter dμ M (x) . M 16πκ

(2)

The Einstein equations relate the curvature of spacetime (on the left side) to the matter distribution described by the energy-momentum tensor (on the right side). Combining the field equations with the equations of motion for the matter fields (like the geodesic equation, the Dirac equation, etc.), one gets a coupled system of partial differential equations. This coupled system can be understood in simple terms by the popular phrase that matter tells spacetime how to curve, and spacetime tells matter how to move. Taking up the comparison between the brain-mind-relationship and the interplay of spacetime and physical processes therein, the Einstein field equations characterize this interrelation. In a similar way as our thinking shapes the brain structures which in turn have an influence on our thoughts, also the objects existing and processes happening in spacetime deform spacetime, which has a back effect on physical processes. Einstein’s revolutionary insight that spacetime together with its matter and energy content form an inseparable unity, is one of the cornerstones which the theory of causal fermion systems is built on.

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2.3 Quantum Theory in a Classical Spacetime The second groundbreaking discovery in the twentieth century besides General Relativity was Quantum Theory. The insight that certain physical quantities take discrete rather than continuous values revolutionized our understanding of Nature. This discovery triggered the development of Quantum Mechanics which is the appropriate framework to study the quantum behaviour of a single particle or a constant finite number of particles. Although the framework of Quantum Mechanics is appropriate to describe even arbitrarily large quantum systems of a fixed number of particles, it is in principle incapable to formalize processes involving a varying number of quantum particles. This limitation is overcome in Quantum Field Theory, where a quantum state can be a superposition of components involving a varying and arbitrarily large number of particles. Relativistic Quantum Field Theory is usually formulated in Minkowski space, disregarding the gravitational field (see for example [2, 3, 29]). The fact that in Quantum Field Theory one deals with arbitrarily large number of particles which can have arbitrarily large momenta can be understood as the reason why divergences occur in the perturbative description. The renormalization program provides a systematic computational procedure for dealing with these divergences. The suchrenormalized Quantum Field Theory makes excellent physical predictions which have been confirmed experimentally to high precision. Nevertheless, it is often criticized that the renormalization program lacks a foundational justification. Also, it is not quite satisfying that the theory is well-defined only to every order in perturbation theory. But the perturbation series does not need to converge. Also, it is not clear whether there exists a mathematically meaningful non-perturbative formulation of Quantum Field Theory. Most methods of Quantum Field Theory also apply to Quantum Field Theory in a fixed curved spacetime (see for example [5] and the references therein). In other words, one considers Quantum Fields in the background of a classical gravitational field without taking into account the backreaction of the quantum fields to classical gravity.

2.4 Incompatibility of General Relativity and Quantum Field Theory Quantum field theory in a classical spacetime has the shortcoming that classical and quantum objects coexist in a way which is conceptually not fully convincing. It would be desirable to describe all the objects on the same footing by “unifying” the theories. However, there is no consensus on how this “unification” should be carried out, or even on what “unification” should mean. Nevertheless, most physicists agree that serious problems arise, no matter which approach for “unification” is taken. In

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order not to take sides, we here merely list some of the most common arguments pointing towards the difficulties: • The simplest method is to start from the Heisenberg Uncertainty Principle p x ≥ h2¯ , which states that position and momentum of a point particle in quantum mechanics can be determined simultaneously only up to a fundamental uncertainty given by Planck’s constant h¯ . In Quantum Field Theory, similar uncertainty relations hold for the field operators and the associated canonical momentum operators. In particular, acting with the local field operator φ(x) on the vacuum state, the quantum state is localized in space, meaning that there is a large momentum uncertainty. This also gives rise to a large uncertainty in the corresponding energy. Intuitively speaking, we thus obtain large “fluctuations” of energy in a small spatial region. In General Relativity, on the other hand, high energy densities lead to the formation of black holes. Therefore, combining the principles of General Relativity and Quantum Field Theory in a naive way leads to the formation of microscopic black holes, implying that the concept of a spacetime being “locally Minkowski space” breaks down. The relevant length scale for such effects is the Planck length P ≈ 1.6 × 10−35 m. • The renormalization program only applies to a class of theories called renormalizable. It turns out that applying the canonical quantization methods to Einstein’s gravity, the resulting theory is not renormalizable. This shows that quantizing gravity with the present methods of perturbative Quantum Field Theory is not a fully convincing concept. • It is sometimes argued that the problem of “unification” is rooted in shortcomings of present Quantum Field Theory. Indeed, the ultraviolet divergences of Quantum Field Theory suggest that the structure of spacetime should be modified for very small distances. A natural length scale for such modifications is given by the Planck length. In this way, the problem of the ultraviolet divergences seems to be intimately linked to gravity. Therefore, in order to resolve these problems, one should modify the structure of spacetime on the Planck scale.

2.5 A Step Back: Quantum Mechanics in Curved Spacetime In order to avoid the just-described problems which arise when “unifying” General Relativity with Quantum Field Theory, it is a good idea to take a step back and return to the familiar and well-understood grounds of one-particle quantum mechanics. Indeed, formulating quantum mechanics in curved spacetime does not lead to any conceptual or technical problems. We now review a few basic concepts, which will also be our starting point for the constructions leading to causal fermion systems.

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2.5.1 The Dirac Equation in Minkowski Space In non-relativistic quantum mechanics, a particle is described by its Schrödinger wave-function ψ(t, x). It has a probabilistic interpretation, meaning that its absolute square |ψ(t, x)|2 is the probability density for the particle to be at the position x ∈ R3 . A relativistic generalization of the Schrödinger equation is the Dirac equation. In this case, the wave function ψ has four complex components, which describe the spin of the particle. In flat Minkowski space, the Dirac equation takes the form 

iγ k

 ∂ − m ψ(x) = 0 , ∂x k

(3)

where x = (t, x) ∈ M is a point of Minkowski space, m is the rest mass, and the so-called Dirac matrices γ k are 4 × 4-matrices which are related to the Lorentzian metric by the anti-commutation relations 2 g j k 1 = {γ j , γ k } ≡ γ j γ k + γ k γ j . The Dirac spinors at every spacetime point are endowed with an indefinite inner product of signature (2, 2), which we call spin scalar product and denote by ≺ψ|φ,(x). To every solution ψ of the Dirac equation we can associate a vector field J by J k = ≺ψ | γ k ψ, , referred to as the Dirac current. For solutions of the Dirac equation, this vector field is divergence-free. This is referred to as current conservation. Current conservation is closely related to the probabilistic interpretation of the Dirac wave function. Indeed, as a consequence of current conservation, for a solution ψ of the Dirac equation, the spatial integral  (ψ|ψ) := 2π

R3

≺ψ | γ 0 ψ,(t, x) d 3 x

is time independent. Normalizing the value of this integral to one, its integrand gives the probability density of the particle to be at position x.

2.5.2 The Dirac Equation in Curved Spacetime In curved spacetime, the Dirac equation is described most conveniently using vector bundles. Similar to the tangent bundle in Definition 2.6, the spinor bundle is obtained by “attaching” a vector space Sp M to every spacetime point, SM =



{p} × Sp M .

p∈ M

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But now, the vector space Sp M, the so-called spinor space, is a four-dimensional complex vector space. This vector space is endowed with an indefinite inner product of signature (2, 2) which, just as in Minkowski space, we refer to as the spin scalar product and denote by ≺.|.,p : Sp M × Sp M → C . At each spacetime point p, the Dirac wave function ψ takes a value in the corresponding spinor space Sp M. The Dirac operator D takes the form D := iγ j ∇j , where ∇j is a connection on the spinor bundle, and the Dirac matrices are related to the Lorentzian metric again by the anti-commutation relations {γ j (p), γ k (p)} = 2 g j k (p) 1Sp M . The Dirac equation in curved spacetime reads (D − m) ψ = 0 . On solutions of the Dirac equation, one has the scalar product  (ψ|φ)m :=

N

≺ψ | ν j γj φ,p dμN (p) ,

(4)

where ν is the future-directed normal on the Cauchy surface N , and dμN is the induced measure. For mathematical completeness, we point out that we always assume that spacetime is globally hyperbolic, so that Cauchy surfaces exist. Moreover, in order for the integral in (4) to be finite, we restrict attention to wave functions of spatially compact support (i.e. to wave functions whose restriction to any Cauchy surface have compact support). Due to current conservation, the scalar product (4) is independent of the choice of the Cauchy surface. Choosing ψ = φ as a unit vector, the integrand of the above scalar product again has the interpretation as the quantum mechanical probability density.

3 Conceptual and Mathematical Foundations of Causal Fermion Systems The theory of causal fermion systems is a novel approach to fundamental physics which is built on our conviction that, in order to resolve the incompatibility of General Relativity and Quantum Field Theory described above, one should

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modify the geometric structure of spacetime on microscopic scales. Having already surveyed our current way of modelling the fabric of spacetime and quantum wave functions therein, we now introduce the conceptual foundations of the theory of causal fermion systems.

3.1 Guiding Principles of the Theory of Causal Fermion Systems Following Einstein’s celebrated insight that “one cannot solve problems with the same level of thinking that created them,” the theory of causal fermion systems does not try to force obviously incompatible concepts into an already existing setting, but instead provides a new mathematical framework which is inspired by carefully selected concepts from contemporary theoretical physics. The main guiding principles of the theory of causal fermion systems are the following ideas: • Unified Description of Spacetime and the Objects Therein The General Theory of Relativity impressively demonstrates that seemingly disparate concepts such as the motion of matter and the metric tensor structure of spacetime are closely related and cannot be considered independent of each other. This surprising insight illustrates the geometric character, high degree of complexity and interconnectedness of the Universe. The interdependence of matter distributions and the shape of spacetime which reacts on local changes as a whole, strongly suggests to take a unified point of view when developing new physical theories. This fundamental conviction is implemented in the theory of causal fermion systems in that spacetime, together with all objects therein (such as particles, fields, etc.), are determined dynamically as a whole by minimizing the so-called causal action. • Equivalence Principle In General Relativity, the equivalence principle is implemented mathematically by working with geometric objects on a Lorentzian manifold. In particular, the Einstein–Hilbert action is diffeomorphism invariant. Allowing for a nontrivial microscopic structure, in the setting of causal fermion systems spacetime does not necessarily need to be a smooth manifold. Consequently, instead of diffeomorphisms, one must allow for more general transformations of spacetime. The causal action is invariant under these more general transformations, thereby generalizing the equivalence principle. • Principle of Causality The principle of causality plays a crucial role in our understanding of the structure of physical interactions in spacetime. A guiding conception in the development of causal fermion systems was that the causal structure of spacetime is not give a-priori, but that it is determined dynamically when solving the physical equations. For a better comparison, we recall that in General Relativity, the causal structure is encoded in the Lorentzian metric (as explained after (1)).

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Therefore, when varying the metric in the Einstein–Hilbert action (2), also the causal structure changes. Only after a critical point of the Einstein–Hilbert action has been found, the corresponding metric determines the causal structure of spacetime. Similarly, in the theory of causal fermion systems, the causal structure of spacetime is determined only after a critical point of the causal action has been found. The principle of causality is implemented in the form that points with spacelike separation are not related to each other in the Euler–Lagrange equations corresponding to the causal action principle. • Local Gauge Principle In classical electrodynamics, the local gauge principle means the freedom A → A + d in changing the electromagnetic potential A by the derivative of a scalar function . This observation was the starting point for the development of gauge theories, which have been highly successful in describing all the interactions in the standard model. In Quantum Theory, local gauge transformations correspond to generalized local phase transformations of the wave functions ψ(x) → U (x) ψ(x) , where U (x) is an isometry on the fibres of the spinor bundle. The theory of causal fermion systems incorporates this principle in that the causal action is invariant under such local transformations. • Microscopic Spacetime Structure The ultraviolet divergences in Quantum Field Theory suggest that one should modify the microscopic structure of spacetime. In order to include these microscopic features of spacetime, the theory of causal fermion systems does not assume physical spacetime to be continuous down to smallest scales, but instead allows for a nontrivial, possibly discrete microstructure of spacetime. • Fermionic Building Blocks From high energy physics we have a quite clear and consistent picture of the elementary building blocks of Nature which is formalized in the Standard Model of Particle Physics. In particular, we know that the fundamental matter particles are fermions while the forces are mediated by bosons. Inspired by Dirac’s concept that in the Minkowski vacuum a whole “sea” of fermions is present, we consider the fermions as being more fundamental, whereas bosons appear in our approach merely as a device to describe the interaction of the fermions. Causal fermion systems evolved in the attempt to combine the above principles in a simple and compact mathematical setting. In the following sections we enlarge on each of the guiding principles and explain how they are formalized within the mathematical framework of the theory of causal fermion systems.

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3.2 Unified Description of Spacetime and the Objects Therein The basic conceptual idea underlying the theory of causal fermion systems consists in the belief that a successful unified theory must provide a unified description of the Universe in the sense that it does not treat spacetime separate from its matter and energy content. This central conception is inspired by the inseparable unity of spacetime and its matter-energy content as described by Einstein’s field equations. In much the same way as the Einstein–Hilbert action singles out those metric tensors which are critical points of the action and declares them to be the physically admissible choices, also the theory of causal fermion systems is based on a variational principle. Before we can formulate such a variational principle, we give the general definition of a causal fermion system and explain it afterward.

Definition 3.1 (Causal Fermion System) A causal fermion system of spin dimension n ∈ N is a triple (H, F, ρ) consisting of the following three mathematical structures: 1. H is a complex, separable Hilbert space (H, ·|·H ). 2. F is the subset of the Banach space (L(H), · ) comprising all self-adjoint operators on H of finite rank, which—counting multiplicities—have at most n ∈ N positive and at most n ∈ N negative eigenvalues. 3. ρ is a positive Borel measure ρ : B → R+ 0 ∪ {∞} on F (where B is the σ -algebra generated by all open subsets of F). The measure ρ is referred to as the universal measure.  

The connection of this definition to physics is not obvious. In order to convey a better, more intuitive understanding of this definition, let us have a detailed view on the individual ingredients. The structure of a complex Hilbert space (H, ·|·H ) is a commonly used structure both in mathematics as well as in theoretical physics and should therefore need no further explanation. In contrast to this, the set F as well as the measure ρ—although familiar to mathematicians—are not commonly used in theoretical physics. In order to make the theory of causal fermion systems easier accessible to interested physicists, we now explain these structures in greater detail. 3.2.1 The Measure Space (F, B, ρ) In contrast to what one might expect from the ordering in the above definition, the central structure of a causal fermion system is not the Hilbert space H itself but rather the measure ρ. Measures appear in physics mainly as integration measures,

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like for example the measure dμ = d 3 x in the three-dimensional integral  R3

f (x) dμ(x)

(of a, say, continuous and compactly supported function f ). In mathematics, the measure μ is a mapping which to a subset ⊂ R3 associates its volume,  μ : → μ( ) :=

d 3x .

A central conclusion from measure theory is that it is mathematically not sensible to associate a measure to every subset of R3 . Instead, one must distinguish a class of sufficiently “nice” subsets as being measurable. The measurable sets form a σ algebra, meaning that applying any finite or countable number of set operations on measurable sets gives again a measurable set. Here it suffices to always work with the Borel algebra, defined as the smallest σ -algebra where all open sets are measurable. A difference to usual integration measures is that the universal measure ρ is a measure on linear operators. The starting point is the Banach space L(H) of all bounded linear operators on H together with the operator norm   

x := sup xu H  u H = 1 .

(5)

The set F is by definition a subset of L(H). We point out that F is not a subspace of LH, because linear combinations of operators in F will in general have rank greater than 2n. But, being a closed subset of L(H), it is a complete metric space with the distance function d : F × F → R+ 0 ,

d(x, y) := x − y .

We remark that F is not a manifold, even if H is finite-dimensional. However, the subset of all operators of maximal rank    Freg := x ∈ F  x has rank 2n is dense in F and indeed a smooth manifold of dimension   dim Freg = 4n dim H − n . (for details see [27, Proposition 2.4.4]). Since in physical applications the dimension of H is very large, F should be regarded as a subset of L(H) of very high dimension.

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In order to define a measure ρ on this set of operators, we must construct a σ -algebra. The simplest choice (which also covers all cases of present physical interest) is to take the Borel algebra B, i.e. the σ -algebra generated by all open subsets of F, with respect to the topology induced by the operator norm (5). The measure ρ makes it possible to integrate a continuous (or Borel) function f : F → R+ 0 over F,  F

f (x) dρ(x) ∈ [0, ∞] .

All familiar concepts from integration theory in R3 also apply here. However, one should keep in mind that we integrate over a set of operators of the Hilbert space (in other words, the integration variable x is operator-valued).

3.2.2 The Causal Action Principle We are now in the position to define the causal Lagrangian and the causal action. For any x, y ∈ F, the product xy is an operator of rank at most 2n. We denote its xy xy non-trivial eigenvalues counting algebraic multiplicities by λ1 , . . . , λ2n ∈ C (more xy xy specifically, denoting the rank of xy by k ≤ 2n, we choose λ1 , . . . , λk as all the xy xy non-zero eigenvalues and set λk+1 , . . . , λ2n = 0).

Definition 3.2 (Causal Lagrangian and Causal Action) The causal Lagrangian is a function defined as L : F × F → R+ 0

(x, y) → L(x, y) :=

2n 1  xy   xy 2 λi − λj . 4n i,j =1

(6)  xy  xy where λi  denotes the absolute values of the eigenvalues λi of the operator product xy. The causal action is obtained by integrating the Lagrangian with respect to the universal measure,  S(ρ) := L(x, y) dρ(x) dρ(y) . F×F

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Having defined the causal action, we can introduce the variational principle, which is the core of the theory of causal fermion systems:

Definition 3.3 (Causal Action Principle and Constraints) The causal action principle is to minimize S by varying the universal measure under the following constraints: volume constraint :

ρ(F) = const

(7)

tr(x) dρ(x) = const

(8)



trace constraint :

F

2n

 boundedness constraint :

T(ρ) :=

F×F

xy |λi |

2

i=1

dρ(x) dρ(y) ≤ C .

(9)

Here C is a given parameter (and tr denotes the trace of a linear operator on H). The constraints are needed in order to obtain a well-posed variational principle without trivial minimizers. Although the mathematical structure of the causal action principle can be understood from general considerations (as will be outlined below), its detailed form is far from obvious. It is the result of many computations and long considerations, which we cannot review here. Instead, we note that the causal action was first proposed in [7, Section 3.5], based on considerations outlined in [7, Sections 5.5 and 5.6]. The significance of the constraints became clear in the later mathematical analysis [9]. Note that the universal measure ρ is the basic object in the theory of causal fermion systems. It is a unified theory in the sense that all spacetime structures are encoded in and must be derived from this measure. In other words, the measure ρ describes our universe as a whole. This explains the name universal measure.

3.3 The Equivalence Principle Let (H, F, ρ) be a causal fermion system of spin dimension n which minimizes the causal action, respecting the constraints.

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Definition 3.4 (Spacetime) Spacetime M is defined as the support of the universal measure, M := supp ρ ⊂ F .

Here the support of a measure is defined as the complement of the largest open set of measure zero, i.e.    ⊂ F  is open and ρ( ) = 0 . supp ρ := F \ Thus the space-time points are symmetric linear operators on H. On M we consider the topology induced by F (generated by the sup-norm (5) on L(H)). Moreover, the universal measure ρ|M restricted to M can be regarded as a volume measure on space-time. This makes space-time to a topological measure space. Let  : M → M be a homeomorphism of spacetime. Given a Borel set ⊂ F, the preimage −1 ( ∩M) is a Borel set of M. Therefore, we can define a new Borel measure ρ˜ on F by ρ( ) ˜ := ρ(−1 ( ∩ M)). This is the so-called push-forward measure denoted by ρ˜ = ∗ ρ . The causal action as well as all the constraints are invariant under the transformation M → (M) ,

ρ → ρ˜ .

This invariance generalizes the diffeomorphism invariance of General Relativity. In this sense, the equivalence principle is implemented in the theory of causal fermion systems.

3.4 Principle of Causality For any x, y ∈ M, the product xy is an operator of rank at most 2n. Exactly as defined at the beginning of Sect. 3.2.2, we denote its non-trivial eigenvalues xy xy (counting algebraic multiplicities) by λ1 , . . . , λ2n .

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Definition 3.5 (Causal Structure) The points x, y ∈ M are said to be ⎧ ⎪ spacelike separated ⎪ ⎪ ⎪ ⎨ timelike separated ⎪ ⎪ ⎪ ⎪ ⎩ lightlike separated

xy

if all the λj have the same absolute value xy

if the λj are all real and do not all have the same absolute value otherwise .

This “spectral definition” of causality indeed gives back the causal structure of Minkowski space or a Lorentzian manifold in the corresponding limiting cases (for more details see Sect. 4 below). At this stage, one sees at least that our definition of the causal structure is compatible with the Lagrangian in the following sense. Suppose xy that two points x, y ∈ F are spacelike separated. Then the eigenvalues λi all have the same absolute value, implying that the Lagrangian vanishes. Working out the corresponding Euler–Lagrange equations (for details see [23]), one finds that pairs of points with spacelike separation again drop out. This can be seen in analogy to the usual notion of causality where points with spacelike separation cannot influence each other. In this sense, the principle of causality is built into the theory of causal fermion systems.

3.5 Local Gauge Principle The fact that spacetime points of a causal fermion system are operators in F gives rise to additional structures. In particular, there is an inherent notion of spinors and wave functions, as we now explain.

Definition 3.6 (Spin Spaces) For every x ∈ M we define the spin space Sx by Sx = x(H); it is a subspace of H of dimension at most 2n. On Sx we introduce an inner product ≺.|.,x by ≺.|.,x : Sx × Sx → C ,

≺u|v,x = −u|xvH ,

(10)

referred to as the spin scalar product.

Since x has at most n positive and at most n negative eigenvalues, the spin scalar product is an indefinite inner product of signature (px , qx ) with px , qx ≤ n (for textbooks on indefinite inner product spaces see [4, 28]). In this way, to every spacetime point x ∈ M we associate a corresponding indefinite inner product

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Sy M

Fig. 1 The spin spaces

Sx M Hilbert space

Fig. 2 The physical wave function

SyM

u (y)

u

Sx M u (x)

space (Sx , ≺.|.,x ). If the signature of the spin spaces is constant in spacetime, we thus obtain the structure of a topological vector bundle (for more details in this direction see [20]). However, in contrast to a vector bundle, all the spin spaces are subspaces of the same Hilbert space H; see Fig. 1. The vectors in H can be represented as wave functions in spacetime: Definition 3.7 (Physical Wave Function) For a vector u ∈ H one introduces the corresponding physical wave function ψ u as ψu : M → H ,

ψ u (x) = πx u ∈ Sx ,

where πx : H → Sx denotes the orthogonal projection on the subspace Sx ⊂ H.

This definition is illustrated in Fig. 2. A local gauge principle becomes apparent once we choose basis representations of the spin spaces and write the wave functions in components. Denoting the signature of (Sx , ≺.|.,x ) by (px , qx ), we choose a pseudo-orthonormal basis (eα (x))α=1,...,px +qx of Sx , i.e. ≺eα (x)|eβ (x),x = sα δβα with s1 = . . . = spx = 1 and spx +1 = . . . = spx +qx = −1. Then a physical wave function ψ u can be represented as ψ u (x) =

p x +qx α=1

ψ α (x) eα (x)

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with component functions ψ(x)1 , . . . , ψ(x)px +qx . The freedom in choosing the basis (eα ) is described by the group U(px , qx ) of unitary transformations with respect to an inner product of signature (px , qx ). This gives rise to the transformations eα (x) →

p x +qx

U −1 (x)βα eβ (x)

and

ψ α (x) →

β=1

p x +qx

U (x)αβ ψ β (x)

β=1

(11) with U ∈ U(px , qx ). As the basis (eα ) can be chosen independently at each spacetime point, one obtains local gauge transformations of the wave functions, where the gauge group is determined to be the isometry group of the spin scalar product. The causal action is gauge invariant in the sense that it does not depend on the choice of spinor bases. This connection becomes clearer if the Lagrangian is expressed in terms of the physical wave functions. This can be accomplished as follows.

Definition 3.8 (Kernel of the Fermionic Projector) For any x, y ∈ M we define the kernel of the fermionic projector P (x, y) by P (x, y) = πx y|Sy : Sy → Sx

(12)

This definition is illustrated in Fig. 3. We remark that this definition harmonizes with the definition of the spin scalar product (10) in the sense that the kernel of the fermionic projector is symmetric with respect to the spin scalar product, ≺u | P (x, y) v,x = −u | x P (x, y) vH = −u | xy vH = −πy x u | y vH = ≺P (y, x) u | v,y (where u ∈ Sx and v ∈ Sy ). Taking the trace of (12) in the case x = y, one finds that tr(x) = TrSx (P (x, x)) (where tr and TrSx are the traces on H and the spin space, respectively), making it possible to express the integrand of the trace constraint (8) in terms of the kernel of the fermionic projector. In order to also express the eigenvalues of the operator xy in terms of the kernel of the fermionic Fig. 3 The kernel of the fermionic projector

Sy M

Sx M Sx M

P(y, x)

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projector, we introduce the closed chain Axy as the product Axy = P (x, y) P (y, x) : Sx → Sx .

(13)

Computing powers of the closed chain, one obtains Axy = (πx y)(πy x)|Sx = πx yx|Sx ,

(Axy )p = πx (yx)p |Sx .

Taking the trace, one sees in particular that     p TrSx (Axy ) = tr (yx)p = tr (xy)p (where the last identity simply is the invariance of the trace under cyclic permutations). As a consequence,1 the eigenvalues of the closed chain coincide with xy xy the non-trivial eigenvalues λ1 , . . . , λ2n of the operator product xy. This makes it possible to express both the Lagrangian (6) and the integrand of the boundedness constraint (9) in terms of Axy . The main advantage of working with the kernel of the fermionic projector is that the closed chain (13) is a linear operator on a vector xy xy space of dimension at most 2n, making it possible to compute the λ1 , . . . , λ2n as the eigenvalues of a finite matrix. The kernel of the fermionic projector can be expressed in terms of the physical wave functions as follows. Choosing an orthonormal basis (ei ) of H and using the completeness relation as well as (10), one obtains for any φ ∈ Sy P (x, y) φ = πx y|Sy φ =



πx ei ei |y φH = −

i



ψ ei (x) ≺ψ ei (y) | φ,y ,

i

showing that P (x, y) is indeed composed of all the physical wave functions, i.e. in a bra/ket notation |ψ ei (x), ≺ψ ei (y)| . (14) P (x, y) = − i

1 More precisely, since all our operators have finite rank, there is a finite-dimensional subspace I of H such that xy maps I to itself and vanishes on the orthogonal complement of I . Then the nontrivial eigenvalues of the operator product xy are given as the zeros of the characteristic polynomial of the restriction xy|I : I → I . The coefficients of this characteristic polynomial (like the trace, the determinant, etc.) are symmetric polynomials in the eigenvalues and can therefore be expressed in terms of traces of powers of Axy .

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Finally, choosing again bases (eα (x))α=1,...,px +qx of the spin spaces, the kernel P (x, y) is expressed by a (px + qx ) × (py + qy )-matrix. According to (11), this matrix behaves under gauge transformations as P (x, y)αβ →

p y +qy x +qx p γ =1

δ γ  U (x)αγ P (x, y)δ U (y)∗ β ,

δ=1

where the star denotes the adjoint with respect to the spin scalar product. Since U (y) ∈ U (px , qx ) is unitary with respect to the spin scalar product, the gauge transformation at y drops out when forming the closed chain, i.e. (Axy )αβ

→

p x +qx

δ γ  U (x)αγ (Axy )δ U (x)∗ β .

γ ,δ=1

Since U (x) ∈ U (px , qx ) is unitary, the eigenvalues of the closed chain do not depend on the choice of the gauge. This explains in particular why the Lagrangian is invariant under local gauge transformations of the physical wave functions. Such computations were helpful for formulating the causal action principle (for details see [7, Chapter 3]).

3.6 Fermionic Building Blocks In the above formulas, the physical wave functions play a dominant role. Indeed, according to (14), the ensemble of all these wave functions determines the kernel of the fermionic projector, which, forming the closed chain and computing its eigenvalues, gives rise to all the quantities needed in the causal action principle. In this way, the causal variational principle can be formulated directly in terms of the ensemble of all physical wave functions. Minimizing the causal action amounts to finding an “optimal” configuration of the physical wave functions. In other words, the causal action principle can be understood as a variational principle which determines the collective behavior of all physical wave functions. As will be worked out in detail in Sect. 4 below, in concrete examples the physical wave functions go over to solutions of the Dirac equation. More specifically, describing the Minkowski vacuum as a causal fermion system (see Sect. 4.3), the ensemble of all physical wave functions correspond to all the negative-frequency solutions of the Dirac equation. In this way, Dirac’s original concept of the Dirac sea is realized. The fact that Dirac wave functions describe fermionic particles is the motivation for the name “causal fermion system.”

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3.7 Microscopic Spacetime Structure In the theory of causal fermion systems, spacetime defined as the support of the universal measure ρ (see Definition 3.4) does not need to be a differentiable manifold. Instead, it could be discrete on a microscopic scale or could have another nontrivial microstructure. Exactly as explained above for the causal structure, also the microscopic structure of spacetime is not given a-priori, but it is determined dynamically by the causal action principle. The analysis of simple model examples reveals that minimizing measures of the causal action principles are typically discrete (for details see [1, 26] or the survey in [15, Section 3]). Although it is an open problem whether these discreteness results also hold for general causal fermion systems, these results suggest that the concept of smooth spacetime structures should be modified on small scales, typically thought of as the Planck scale. The theory of causal fermion systems provides a mathematical setting in which such generalized spacetimes can be described and analyzed.

4 Modelling a Lorentzian Spacetime by a Causal Fermion System 4.1 General Construction in Curved Spacetimes We return to the setting of the Dirac equation in curved spacetime in Sect. 2.5. We now explain how to describe this spacetime by a causal fermion system. We denote the Hilbert space of solutions of the Dirac equation with the scalar product (4) by (Hm , (.|.)m ) (more precisely, we take the completion of all smooth solutions with spatially compact support). Next, we choose a closed subspace H ⊂ Hm of the solution space of the Dirac equation. The induced scalar product on H is denoted by .|.H . There is the technical difficulty that the wave functions in H are in general not continuous, making it impossible to evaluate them pointwise. For this reason, we need to introduce an ultraviolet regularization, described mathematically by a linear regularization operator

R : H → C 0 (M , S M ) .

We postpone the discussion of the physical significance of the regularization operator to Sect. 4.2. Mathematically, the simplest method to obtain a regularization operator is by taking the convolution with a smooth, compactly supported function on a Cauchy surface or in spacetime (for details see [25, Section 4] or [12, Section §1.1.2]). Given R, for any space-time point p ∈ M we consider the bilinear form bp : H × H → C ,

bp (ψ, φ) = −≺(Rψ)(p)|(Rφ)(p),p .

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This bilinear form is well-defined and bounded because R maps to the continuous wave functions and because evaluation at p gives a linear operator of finite rank. Thus for any φ ∈ H, the anti-linear form bp (., φ) : H → C is continuous. By the Fréchet-Riesz theorem, there is a unique χ ∈ H such that bp (ψ, φ) = ψ|χH for all ψ ∈ H. The mapping φ → χ is linear and bounded, giving rise to the following linear operator: Definition 4.1 (Local Correlation Operator) For any p ∈ M, the local correlation operator F (p) on H is defined by the relation (ψ | F (p) φ) = −≺(Rψ)(p)|(Rφ)(p),p

for all ψ, φ ∈ H .

(15)

Taking into account that the inner product on the Dirac spinors at p has signature (2, 2), the local correlation operator F (p) is a symmetric operator on H of rank at most four, which (counting multiplicities) has at most two positive and at most two negative eigenvalues. Varying the space-time point, we obtain a mapping F : M → F ⊂ L(H) , where F denotes all symmetric operators of rank at most four with at most two positive and at most two negative eigenvalues. Finally, we introduce the universal measure

dρ := F∗ dμ M

(16)

as the push-forward of the volume measure on M under the mapping F (thus ρ( ) := μ M (F −1 ( ))). We thus obtain a causal fermion system (H, F, ρ) of spin dimension two. We close with a few comments on the underlying physical picture. The vectors in the subspace H ⊂ Hm have the interpretation as those Dirac wave functions which are realized in the physical system under consideration. If we describe for example a system of one electron, then the wave function of the electron is contained in H. Moreover, H includes all the wave functions which form the so-called Dirac sea (for an explanation of this point see for example [10]). According to (15), the local correlation operator F (p) describes densities and correlations of the physical wave functions at the space-time point p. Working exclusively with the local correlation operators and the corresponding push-forward measure ρ means in particular that the geometric structures are encoded in and must be retrieved from the physical wave functions. Since the physical wave functions describe the distribution of matter in space-time, one can summarize this concept by saying that matter encodes geometry. Going one step further, one can also say that matter and geometry form an inseparable unity.

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4.2 Physical Significance of the Regularization Operator The regularization operator requires a detailed explanation. We first convey the underlying physical picture. The regularization operators should leave the wave functions unchanged on macroscopic scales (i.e. scales much larger than the Planck length). Thus on macroscopic length scales, the Dirac equation still holds, giving agreement with the common physical description. However, on a microscopic scale ε, which can be thought of as the Planck scale, the regularization may change the wave functions completely. As a consequence, also the universal measure ρ in (16) is changed, which means that the microscopic structure of spacetime is modified. Therefore, in contrast to the renormalization program in Quantum Field Theory, in the theory of causal fermion systems the regularization is not just a technical tool, but it realizes our concept that we want to allow for a nontrivial microstructure of spacetime. With this in mind, we always consider the regularized quantities as those having mathematical and physical significance. Different choices of regularization operators realize different microscopic spacetime structures. This concept immediately raises the question how the “physical regularization” should look like. Generally speaking, the regularized spacetime should look like Lorentzian spacetime down to distances of the scale ε. For distances smaller than ε, the structure of space-time may be completely different, in a way which cannot be guessed or extrapolated from the structures of Minkowski space. Since experiments on the length scale ε seem out of reach, it is completely unknown what the microscopic structure of space-time is. Within the theory of causal fermion systems, the above question could be answered in principle by minimizing the causal action over all possible regularization operators. However, this approach turns out to be very difficult and at present is out of reach (for a first step in this direction see [8]). In view of these difficulties, the only available method is the so-called method of variable regularization: Instead of trying to determine the microstructure experimentally or with mathematical analysis, the strategy is to a-priori include all conceivable regularizations and, with hindsight, to eliminate those which are in conflict with well-established physical facts. The remaining regularizations which comply with all experimental constraints should be treated as equally admissible, because at present there is no criterion to distinguish between different choices or to favor one regularization over the others. For the method of variable regularization to be sensible and to retain the predictive power of the theory, the detailed form of the microstructure must have no influence on the effective physical equations which are valid on the energy scales accessible to experiments. More precisely, the picture is that the general structure of the effective physical equations should be independent of the microstructure of spacetime. Values of mass ratios or coupling constants, however, may well depend on the microstructure (a typical example is the gravitational constant, which is closely tied to the Planck length). In more general terms, the unknown microstructure of spacetime should enter the effective physical equations only by a finite (hopefully small) number of free parameters, which can then be taken as

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empirical free parameters of the effective macroscopic theory. In [12] it was shown that these conditions are indeed satisfied.

4.3 Concrete Example: The Minkowski Vacuum We now make the construction of Sect. 4.1 more explicit by working out the example of the Minkowski vacuum with the simplest possible regularization. We proceed in the following steps: • Choosing the Hilbert Space of All Negative-Frequency Solutions Our starting point are the plane-wave solutions of the Dirac equation in Minkowski space (3), which we write as 1 ψpa± (x) = e∓iωt +ip·x χpa± 3 (2π)

with

ω = ω( p ) :=

! | p | 2 + m2 .

Here the spinor χpa± solves the algebraic equation (γ k pk − m1)χpa± = 0 , where (pk ) = (ω, p ) denotes the four-momentum. Negative-frequency wave packets of the form  ψf (x) :=

R3

ψpa− (x) f (p ) d 3 p

with

f ∈ C0∞ (R3 , C)

(17)

span a subspace of Hm . We choose the Hilbert space H of the causal fermion system as the closure of this subspace. This choice realize the concept of the Dirac sea vacuum. • Constructing the Local Correlation Operators The simplest method to choose regularization operators consists in inserting a convergence-generating factor e−εω into the wave packet (17), i.e.   Rψf (x) :=

 R3

e−εω ψpa− (x) f (p ) d 3 p .

Now we can define the local correlations operators by (15) and construct the universal measure according to (16). We thus obtain a causal fermion system (H, F, ρ). In this example, one can compute the objects of the causal fermion system explicitly (for details see [12, Section 1.2]). One finds that in the limit ε . 0, the inherent structures of the causal fermion system go over to the usual objects and

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relations in Minkowski space. More specifically, mapping a point p ∈ M to the corresponding local correlation operator F (p) gives a one-to-one correspondence between Minkowski space M and the spacetime M := supp ρ of the causal fermion system. Moreover, the causal structure of Definition 3.5 gives back the causal structure of Minkowski space, and the spin space Sx of Definition 3.6 can be identified with the space of Dirac spinors Sp M. Under these identifications, the physical wave functions of Definition 3.7 agree with the regularized Dirac wave functions of negative frequency.

5 Results of the Theory and Further Reading Let us explain in which sense and to which extent the goal of unifying Quantum Field Theory and General Relativity has been achieved. Causal fermion systems provide a mathematically consistent theory which gives General Relativity and Quantum Theory as limiting cases. The causal action principle has well-defined minimizers in the case of a finite-dimensional Hilbert space and finite total volume (see [9]; more general cases are presently under investigation). The reason why the inconsistencies of Quantum Field Theory and General Relativity as described in Sect. 2.4 as well as the divergences of Quantum Field Theory disappear is that we modified the structure of spacetime on the Planck scale. In more technical terms, in a causal fermion system one works with the regularized objects. Thus we consider the regularized objects as the fundamental physical objects. This concept could be implemented coherently because the causal action principle is formulated purely in terms of these regularized objects. Causal fermion systems are a unified theory in the sense that spacetime and all objects therein are described by a single object: the universal measure. The causal action principle singles out those measures which describe physically admissible spacetimes. The Euler–Lagrange equations corresponding to the causal action principle describe the spacetime dynamics. Clearly, in this short review we could only cover certain aspects of the theory from a particular perspective. Therefore, in order to help the interested reader to get a more complete picture, we now outline a few other directions and give references for further study. For other review articles with a somewhat different focus we refer to [10, 17, 21]. (a) A causal fermion system also provides topological (topological spinor bundle) and geometric objects (parallel transport and curvature). We refer the interested reader to [16, 20] or the introduction [14]. (b) The limiting case ε . 0, when the ultraviolet regularization is removed, is worked out in detail in [12]. In this limiting case, the so-called continuum limit, the causal action principle gives rise to the interactions of the standard model and gravity, on the level of classical bosonic fields interacting with a secondquantized fermionic field.

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(c) An important concept for more recent developments are surface layer integrals, which generalize surface integrals to the setting of causal fermion systems. Symmetries of causal fermion systems give rise to conservation laws which can be expressed in terms of surface layer integrals [22]. (d) Another concept which has turned out to be fruitful for the analysis of the causal action principle are linearized solutions [23]. Similar to linearized gravitational waves, linearized solutions can be understood as linear perturbations of the measure ρ which preserve the Euler–Lagrange equations of the causal action principle. As shown in [23, 24], linearized solutions come with corresponding conserved surface layer integrals, in particular the symplectic form and the surface layer inner product. (e) Generally speaking, the conservation laws for surface layer integrals give rise to objects in space which evolve dynamically in time. This concept was worked out for linearized solutions in [6], where it is proven under general assumptions that the Cauchy problem for linearized solutions is well-posed and that the solutions propagate with finite speed. (f) A first connection to Quantum Field Theory has been made in [11], however based on the classical field equations obtained in the continuum limit. Deriving Quantum Field Theory as a limiting case of causal fermion systems without referring to the continuum limit is a major objective of present research: The perturbation theory for the universal measure is worked out in [13]. For interacting bosonic fields, the constructions in [19] give rise to a description of the dynamics in terms of a unitary time evolution on bosonic Fock spaces. The generalization of these constructions to include fermionic fields is currently under investigation [18]. Acknowledgements We would like to thank the participants of the conference “Progress and visions in quantum theory in view of gravity” held in Leipzig in October 2018 for fruitful and inspiring discussions. We would like to thank Christoph Langer, José M. Isidro, Claudio Paganini and the referee for helpful comments on the manuscript.

References 1. L. Bäuml, F. Finster, H. von der Mosel, D. Schiefeneder, Singular support of minimizers of the causal variational principle on the sphere. Calc. Var. Partial Differ. Equ. 58(6), 205 (2019). arXiv:1808.09754 [math.CA] 2. J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964) 3. J.D. Bjorken, S.D. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965) 4. J. Bognár, Indefinite Inner Product Spaces (Springer, New York, 1974). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78 5. R. Brunetti, K. Fredenhagen, in Quantum Field Theory on Curved Backgrounds, ed. by C. Bär, K. Fredenhagen. Quantum Field Theory on Curved Spacetimes. Lecture Notes in Physics, vol. 786 (Springer, Berlin, 2009), pp. 129–155. arXiv:0901.2063 [gr-qc] 6. C. Dappiaggi, F. Finster, Linearized fields for causal variational principles: existence theory and causal structure. Meth. Appl. Anal. (2020, to appear). arXiv:1811.10587 [math-ph]

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7. F. Finster, The Principle of the Fermionic Projector. Studies in Advanced Mathematics, vol. 35 (American Mathematical Society, Providence, 2006). hep-th/0001048, hep-th/0202059, hepth/0210121, AMS/IP 8. F. Finster, On the regularized fermionic projector of the vacuum. J. Math. Phys. 49(3), 032304 (2008). arXiv:math-ph/0612003 9. F. Finster, Causal variational principles on measure spaces. J. Reine Angew. Math. 646, 141– 194 (2010). arXiv:0811.2666 [math-ph] 10. F. Finster, A formulation of quantum field theory realizing a sea of interacting Dirac particles. Lett. Math. Phys. 97(2), 165–183 (2011). arXiv:0911.2102 [hep-th] 11. F. Finster, Perturbative quantum field theory in the framework of the fermionic projector. J. Math. Phys. 55(4), 042301 (2014). arXiv:1310.4121 [math-ph] 12. F. Finster, The Continuum Limit of Causal Fermion Systems. Fundamental Theories of Physics, vol. 186 (Springer, Berlin, 2016). arXiv:1605.04742 [math-ph] 13. F. Finster, Perturbation theory for critical points of causal variational principles. Adv. Theor. Math. Phys. (2020, to appear). arXiv:1703.05059 [math-ph] 14. F. Finster, Causal fermion systems: a primer for Lorentzian geometers. J. Phys. Conf. Ser. 968, 012004 (2018). arXiv:1709.04781 [math-ph] 15. F. Finster, Causal fermion systems: discrete space-times, causation and finite propagation speed. J. Phys. Conf. Ser. 1275, 012009 (2019). arXiv:1812.00238 [math-ph] 16. F. Finster, A. Grotz, A Lorentzian quantum geometry. Adv. Theor. Math. Phys. 16(4), 1197– 1290 (2012). arXiv:1107.2026 [math-ph] 17. F. Finster, A. Grotz, D. Schiefeneder, in Causal Fermion Systems: A Quantum Space-Time Emerging from an Action Principle, ed. by F. Finster, O. Müller, M. Nardmann, J. Tolksdorf, E. Zeidler. Quantum Field Theory and Gravity (Birkhäuser, Basel, 2012), pp. 157–182. arXiv:1102.2585 [math-ph] 18. F. Finster, N. Kamran, Fermionic Fock space dynamics for causal fermion systems (in preparation) 19. F. Finster, N. Kamran, Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles (2018). arXiv:1808.03177 [math-ph] 20. F. Finster, N. Kamran, Spinors on singular spaces and the topology of causal fermion systems. Mem. Am. Math. Soc. 259(1251), v+83 (2019). arXiv:1403.7885 [math-ph] 21. F. Finster, J. Kleiner, Causal fermion systems as a candidate for a unified physical theory. J. Phys. Conf. Ser. 626, 012020 (2015). arXiv:1502.03587 [math-ph] 22. F. Finster, J. Kleiner, Noether-like theorems for causal variational principles. Calc. Var. Partial Differ. Equ. 55:35(2), 41 (2016). arXiv:1506.09076 [math-ph] 23. F. Finster, J. Kleiner, A Hamiltonian formulation of causal variational principles. Calc. Var. Partial Differ. Equ. 56:73(3), 33 (2017). arXiv:1612.07192 [math-ph] 24. F. Finster, J. Kleiner, A class of conserved surface layer integrals for causal variational principles. Calc. Var. Partial Differ. Equ. 58:38, 34 (2019). arXiv:1801.08715 [math-ph] 25. F. Finster, M. Reintjes, A non-perturbative construction of the fermionic projector on globally hyperbolic manifolds I—space-times of finite lifetime. Adv. Theor. Math. Phys. 19(4), 761– 803 (2015). arXiv:1301.5420 [math-ph] 26. F. Finster, D. Schiefeneder, On the support of minimizers of causal variational principles. Arch. Ration. Mech. Anal. 210(2), 321–364 (2013). arXiv:1012.1589 [math-ph] 27. F. Finster, J. Kleiner, J.-H. Treude, An introduction to the fermionic projector and causal fermion systems (in preparation). https://www.dropbox.com/s/4g0nh4nxxcb9175/intro-public. pdf?dl=0 28. I. Gohberg, P. Lancaster, L. Rodman, Indefinite Linear Algebra and Applications (Birkhäuser, Basel, 2005) 29. M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, 1995)

Quantum Spacetime and the Renormalization Group: Progress and Visions Antonio D. Pereira

Abstract The quest for a consistent theory which describes the quantum microstructure of spacetime seems to require some departure from the paradigms that have been followed in the construction of quantum theories for the other fundamental interactions. In this contribution we briefly review two approaches to quantum gravity, namely, asymptotically safe quantum gravity and tensor models, based on different theoretical assumptions. Nevertheless, the main goal is to find a universal continuum limit for such theories and we explain how coarsegraining techniques should be adapted to each case. Finally, we argue that although seemingly different, such approaches might be just two sides of the same coin.

1 Introduction The construction of the Standard Model (SM) of particle physics under the perturbative and continuum quantum-field theoretic dogmas led to a successful theory beyond dispute. The detection of the Higgs [1] and the absence of new physics at the LHC so far have crowned the SM as a very accurate quantum description of the fundamental interactions but gravity. However, there are some puzzles which are not addressed by the SM as, e.g., neutrino masses and dark matter. Moreover, the SM is not a fundamental quantum field theory valid up to arbitrarily short scales due to the existence of a Landau pole being thus valid up to some cutoff scale [2–5]. The classical dynamics of spacetime is successfully described by General Relativity (GR). The recent direct detection of gravitational waves emitted by black holes binary systems [6] provides a new arena to test the dynamics of the

A. D. Pereira () Instituto de Física, Universidade Federal Fluminense, Niterói, RJ, Brazil Institute for Theoretical Physics, University of Heidelberg, Heidelberg, Germany e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 F. Finster et al. (eds.), Progress and Visions in Quantum Theory in View of Gravity, https://doi.org/10.1007/978-3-030-38941-3_3

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gravitational field in the strong regime, where its non-linear effects cannot be disregarded.1 So far, GR has shown to be a very good description of it. Given the extraordinary success of those theories, the most natural attitude in order to build a quantum theory of the gravitational interaction is to apply the perturbative quantization techniques of continuum quantum field theory (QFT) to GR. This is the program which is by now referred to as perturbative quantum gravity. As it is known, such a theory is not perturbatively renormalizable meaning that infinitely many terms are needed to cancel ultraviolet (UV) divergences of the theory. Such terms come with arbitrary coefficients which are fixed by experimental data. Therefore, the underlying QFT is not predictive. Nevertheless, the UV divergences can be regularized by the introduction of a UV cutoff and the theory can provide sensible results up to the cutoff scale, since higher-order terms are suppressed by the UV-cutoff. However, at the cutoff scale, such a suppression is lost together with predictivity. For the description of very short distances, a fundamental theory (UVcomplete) is necessary. At this point, many different routes can be taken. In fact, the existence of several different approaches to quantum gravity [8] is due to very different underpinning theoretical assumptions. The lack of experimental data as well as clear “quantumgravity observables” makes the task of ruling out quantum-gravity models much more subtle. On the other hand, it is even difficult, sometimes, to find a common language for different approaches such that it is obviously clear whether the same physical quantity is being computed from different perspectives. Irrespective of the chosen approach, a theory of quantum gravity aims at making sense of the following path integral,2 Z=

   D(geometries) eiSgrav ,

(1)

T

where the functional integral is performed over all geometries, Sgrav is the classical or microscopic action encoding the gravitational interaction and the discrete sum is over different topologies. This is just a pictorial representation of the quantumgravity path integral and should be adapted to each approach accordingly. Moreover, although we do not write explicitly, the path integral might also include matter degrees of freedom and depending on the approach one wants to follow, those are fundamentally important for the consistency of the quantum theory of gravity. In this contribution, we will review two seemingly different perspectives for the definition of (1). The first one takes the continuum QFT point of view but instead of insisting on the standard perturbative quantization algorithm, looks for a quantumscale invariant regime driven by an interacting fixed-point in the renormalization group flow. Due to scale invariance, one can zoom in up to arbitrarily short distances

1 See also the very recent results from the Event Horizon Telescope in [7], in very good agreement with GR predictions. 2 Of course, this assertion is restricted to path-integral approaches to quantum gravity.

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without running into divergences. This perspective has witnessed intense progress in the last few years and is known as asymptotically safe quantum gravity (ASQG). The other viewpoint we will review can be seen as a generalization of the matrix model program for two-dimensional quantum gravity to higher dimensions. It goes under the name of tensor models. In this approach, the path integral for quantum gravity is regularized by a lattice-regularization procedure and the integral over geometries is replaced by an integral over tensors which are dual to the building blocks which discretize a given (pseudo-)manifold. The gluing of such building blocks comes with the combinatorially non-trivial interactions of such models. Ultimately, one looks for a continuum limit where the number of building blocks goes to infinity while their volume shrinks to zero and is able to generate a structure which resembles our Universe at large distances. These approaches take very different routes in their formulation of a consistent description of quantum spacetime. Nevertheless, it is conceivable that despite very different in their construction, they are equivalent, in the sense that they provide the same physics. As an analogy, one can take Yang-Mills theories. In the deep infrared (IR), where the theory becomes strongly coupled, one can use MonteCarlo simulations to compute correlation functions at the non-perturbative regime. Alternatively, continuum methods as, e.g., functional methods can be employed to reproduce such results. Of course, each method has its own advantages and pitfalls, but being different formulations of the same physics, they can be used conveniently depending on the problem of interest. The very construction of each aforementioned approach depends on the existence of a well-defined continuum limit. In ASQG, the scale-invariant regime allows for a consistent removal of the UV-cutoff introduced to regularize the path integral. In tensor models, the continuum limit is related to the very large number of building blocks used to discretize geometry. Therefore, we look for a mathematical tool which allows us to probe different scales of the theory. Our goal is to find a coarse-graining toolbox which can be applied to quantum gravity in its different formulations. Our choice is the functional renormalization group (FRG), a flexible and powerful framework which allows for a practical implementation of the Wilsonian renormalization program to QFTs. Despite of being formulated very differently, the FRG can be adapted to ASQG as well as to tensor models as we will review later on. Universal quantities, such as critical exponents, can be computed with the FRG and an explicit comparison can be established for ASQG and tensor models. It allows an explicit comparison between the universality classes such theories would fall in. This consists in a very first step towards the establishment of a possible connection between different approaches to quantum gravity. In the case that those theories belong to the same universality class, many important issues can be enlightened. As previously explained in the case of Yang-Mills theories, such an equivalence would allow us to use the strengths of each approach to converge towards a consistent picture of the quantum microstructure of spacetime. This article is organized as follows: In Sect. 2 we provide a concise review about the FRG, fixed points and critical exponents. In Sect. 3, a short description about the asymptotic safety scenario for quantum gravity is provided together with a brief

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summary of some recent progress in the field. In Sect. 4, the tensor models are introduced with a view towards quantum gravity. The main focus is the construction of a practical tool for the discovery of continuum limits for such models. We argue that the FRG can be adapted to this setting and serve as an exploratory tool for continuum limits. In Sect. 5 we summarize our visions regarding the use of renormalization group tool to quantum gravity and advocate that it can be a key element to bridge the gap between different approaches. Consequently, different strengths of different approaches can be combined in a description of the quantum structure of spacetime.

2 Functional Renormalization Group: Brief Overview Consider a quantum field theory defined by the Euclidean path integral,  Z[J ] =

[Dϕ]UV e−S[ϕ]+ϕ·J ,

(2)

with ϕ being the field content of the theory (not necessarily a simple scalar field), J are the sources coupled to ϕ and ϕ · J denotes the contraction of all indices (spacetime and internal) of the field with the source. The parameter UV corresponds to a UV cutoff3 introduced to make the functional measure welldefined. The Wilsonian renormalization perspective incorporates quantum effects by evaluating (2) not at once, but step by step. Using standard QFT on flat spacetime, this can be phrased as the integration of modes with a given momentum shell by shell. The path integral is completely computed by integrating all modes. For each integration step, the result can be expressed in terms of an effective action which takes into account the effects of the integrated modes. Iterating this procedure from UV to zero should be completely equivalent to calculate the full path integral at once. The FRG [9–15] implements the Wilsonian idea in a smooth way, by modifying the path integral through the introduction of a regulator, as follows,  Zs [J ] =

1

[Dϕ]UV e−S[ϕ]+ϕ·J − 2 ϕ·Rs ·ϕ .

(3)

A quadratic term on the fields is introduced with a kernel Rs . The field configurations are organized according to the parameter s (at this level, this is completely general) and the kernel Rs implements the suppression of all field configurations which are labeled by a value smaller than s. In other words, the quadratic term introduced is such that the path integral is evaluated only over a shell from UV to s. In this sense, the parameter s plays the role of a IR cutoff. For s → UV , 3 This is very general and is not necessarily a cutoff in energy. As we will see in the case of tensor models, this parameter is associated to the size of the tensor being thus a dimensionless parameter.

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Fig. 1 Shape of the regulator kernel Rs for a fixed value of s

all configurations are suppressed while for s → 0, all configurations are taken into account corresponding thus to the complete evaluation of the path integral. The smooth suppression can be implemented by any function Rs provided that for any s  such that s  < s, Rs (s  /s) > 0 and Rs (s  /s) = 0 for s  /s sufficiently large. Pictorially, the regulator kernel Rs should have the form displayed in Fig. 1 for a fixed value of s. It is very simple to show that the effective average action s [φ] defined by 1 s [φ] = supJ (TrJ · φ − lnZs ) − φ · Rs · φ , 2 with φ = ϕJ , satisfies the exact flow equation, # "

−1 1 δ 2 s [φ] s∂s s [φ] = Tr + Rs s∂s Rs . 2 δφ 2

(4)

(5)

By definition, s=UV = S and s=0 = , with  being the full effective action of the underlying QFT. Hence, the effective average action s interpolates between the classical/microscopic action S and the effective action  which takes all the quantum fluctuations into account. Equation (5) is known as the FRG equation, flow equation or Wetterich equation. It is exact and has a very simple structure, being thus very useful for practical purposes. In fact, it has the same structural form of the one-loop contribution to the effective action in perturbation theory, a fact which is exactly preserved due to the quadratic form of the regulator. The flow equation can be seen as an alternative to the path integral: Instead of performing the functional integral in order to assess the quantum action which generates all the building blocks of a given QFT, one can solve the flow equation with the initial condition being given by the classical action. Therefore, in general, solving the flow equation exactly is as difficult as evaluating the path integral completely and approximations are needed. Nevertheless, let us assume that we are able to compute the effective average action. It is expanded on a basis of (quasi-)local operators O(φ) of the fields φ as s [φ] =

i

g¯i (s)Oi (φ) ,

(6)

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with g¯ i (s) being called the running coupling constant associated to Oi (φ). Typically, this basis is infinite-dimensional.4 Plugging (6) in the flow equation (5) yields s∂s s [φ] =

(s∂s g¯i (s))Oi (φ) .

(7)

i

  By expanding the righthand side of Eq. (5) on the same basis Oi (φ) , one can read off the coefficients s∂s gi (s) = β¯g i . Such coefficients are known as beta functions of the dimensionful couplings. This leads to a system of infinitely many coupled equations which, typically, is not autonomous. The system can be, sometimes, made autonomous by rescaling all the couplings with an appropriated power of s, i.e., g¯i (s) = s [g¯ i ] gi (s) .

(8)

  The number g¯ i is known as the canonical dimension of the coupling g¯ i and g i (s) are called dimensionless couplings. They parameterize an infinite-dimensional space called theory space. In standard QFTs formulated on a background spacetime, the canonical dimension is immediately extracted by dimensional analysis (mass dimension associated to each couplings). However, as explained before, the parameter s might be more abstract than a momentum scale and even dimensionless. In such a case, we fix the canonical dimensions by demanding that the resulting system of equations obtained from the flow equation is autonomous. Thence, the beta functions of the dimensionful couplings are expressed as $ % β¯g i = g¯ i s [g¯i ] gi (s) + s [g¯i ] βg i ,

(9)

with βg i = s∂s g i being the beta functions of the dimensionless couplings (which we will simply refer to as beta functions). The first term on the righthand side of (9) is associated to the canonical dimension of the corresponding coupling while the second term arises due to the non-trivial dependence on the parameter s, i.e., due to quantum effects. Ultimately, we are interested in a UV completion for quantum gravity. A UV complete theory is such that all dimensionless couplings are finite at s → UV → ∞. A sufficient condition for finiteness is that the system of beta functions admits a fixed point, i.e., a point in theory space g ∗ = g1∗ , g2∗ , g3∗ , . . . (with finite values for all couplings) where all beta functions vanish, βg i (g ∗ ) = 0 , ∀i. At the fixed point, the running of the couplings with s ceases and the theory reaches a scale-invariant regime [16]. At this point, the limit s → UV → ∞ can be safely taken.

4 When integrating the modes step by step in the path integral, one realizes that all terms compatible with the symmetries of the theory are generated in the effective action—apart from anomalies which we ignore in this article.

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g(s)

99

g(s)

s

s

Fig. 2 On the left plot, a representation of an asymptotically free coupling. In this case, the fixed point occurs at vanishing coupling. The plot on the right shows an asymptotically safe coupling which attains a non-trivial but finite value in the UV

A particularly well-known case of fixed point is the asymptotically free one. In this case, for arbitrarily large values of energy, the couplings tend to zero, i.e., they reach vanishing value at the fixed point. However, one could think of a fixed point at non-vanishing (non-trivial) values.5 This would correspond to a well-defined UV completion of the theory. In this case, we refer to the theory as asymptotically safe. We refer to Fig. 2 for a pictorial representation of asymptotically free and safe couplings. While asymptotically free fixed points are well described in perturbation theory, non-trivial fixed-points can be impossible to be accessed within this framework. In fact, if the location of the non-Gaussian fixed point is too far from the origin of the theory space, then the couplings can assume large values, which invalidates the perturbative assumption. As such, non-perturbative techniques are required. Technically, this is the reason behind the difficulty to probe non-Gaussian fixed points, in general. Fortunately, although the FRG requires an approximation to be useful at the practical level, different schemes which go beyond the perturbative expansion are possible. Hence, even within approximations, the FRG can be a valuable tool to probe interacting fixed points. Among the different approximation methods to the FRG, the one we will be mostly interested in this article consists in truncating the infinite-dimensional theory space to a subspace (it does not need to be finite dimensional necessarily). Within this subspace, an ansatz for the effective average action is proposed and the righthand side of the flow equation can be evaluated. The result will typically leak from the original subspace and a projection rule must be defined. Thus, the flow is projected to the truncated subspace and one can immediately read off the beta functions. If no external input is given, the quality of the truncation is tested by enlarging the subspace and testing the stability of the results. The more convergent the results are for arbitrary enlargements of the truncation, the better the truncation is. As can be guessed, if no guiding principle is discovered, setting a “good” truncation is not a simple task and requires several consistency tests.

5 Oftenly, the asymptotically free fixed point is called Gaussian or non-interacting fixed point and the asymptotically safe one, non-Gaussian or interacting fixed point.

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Let us assume that we managed to find an interacting fixed point for a given QFT. A natural question which arises is: Since the theory space is, in general, infinite dimensional, do we actually need to provide infinitely many boundary conditions to set up a renormalization group trajectory which hits the fixed point? The  answer is: typically, no. In order to argue that, let us consider the fixed point g ∗ = g1∗ , g2∗ , . . . and the linearized flow around it, i.e., s∂s (gi − gi∗ ) =

∂βi   (gj − gj∗ ) ≡ Mij (gj − gj∗ ) ,  ∂gj g=g ∗ j

(10)

j

where Mij is known as stability matrix. It can be diagonalized by a suitable change of coordinates zi = j Sij (gj − gj∗ ) leading to s∂s zi = λi zi ,

(11)

with λi being the eigenvalues of Mij . It is convenient to define the renormalization group “time” t = ln s/s0 , with s0 being a reference value. Then, Eq. (11) can be solved, zi (t) = Ci eλi t ,

(12)

where Ci are integration constants. Changing back to the original coordinates, Eq. (12) is expressed as gi (s) = gi∗ +



Vij eλj t ,

(13)

j

with Vij being constants. Defining the so-called critical exponents θi = −λi , Eq. (13) becomes gi (s) =

gi∗

+

j

Vij

s s0

−θj .

(14)

From Eq. (14), one sees that if θj > 0, then the flow approaches to the fixed point in the UV (s → UV → ∞) irrespective of the value of Vij . Hence, for those directions the Vij are free parameters. Towards the infrared (decreasing values of s), the distance between the coupling gi and gi∗ grows. Hence, we say that θj > 0 defines a UV attractive/IR repulsive direction. Such positive critical exponents define the so-called relevant directions. On the contrary, if θj < 0, the flow of the corresponding coupling is driven away from the fixed-point value. In this case, in order to hit the fixed point, the coefficient Vij must vanish. Such directions are the so-called irrelevant directions and are UV repulsive/IR attractive. If the critical exponent vanishes, then the direction is called marginal. In this case, one needs to go beyond the linearized order to check whether the direction is marginally relevant,

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Fig. 3 The surface represents the critical surface associated to the fixed point. Trajectories which hit the fixed point (the red ones ) span the surface—defined by the UV attractive/relevant directions. The green trajectory is an example of a deviation to a UV-repulsive direction

irrelevant or exactly marginal. Towards the IR, relevant directions are associated with free parameters while irrelevant ones are not. Therefore, we should fix those free parameters with experimental data. Consequently, if the number of relevant directions is finite, then a finite number of experiments should be performed to fix all free parameters. This ensures that the theory is predictive. The relevant directions define a hypersurface in theory space known as critical surface. Predictivity is achieved if the hypersurface is finite dimensional. We refer to Fig. 3 for a cartoon representing the hypersurface in theory space. If the fixed point is trivial (Gaussian), then the number of relevant directions is dictated by the number of couplings with positive and zero (depending whether it is marginally relevant or not) canonical dimension in usual QFTs. In the language of perturbation theory, the critical surface is spanned by the power-counting renormalizable couplings.6 However, for a non-Gaussian fixed point, the situation is more complicated. In fact, canonically relevant directions can turn to irrelevant ones at the fixed point and vice-versa due to the interacting nature of the fixed point. If the fixed-point is near-Gaussian, i.e., if its location is sufficiently close to the Gaussian fixed point, then it is expected that the canonical dimension works as a reasonably good guidance for the dimensionality of the critical surface. In this case, perturbation theory should be applicable despite of the interacting nature of the fixed point. In summary, the existence of a fixed-point of the renormalization group flow which features a finite number of relevant directions prevents the couplings of the theory to diverge and ensures predictivity. Since this property might be probed just

6 More

precisely, it is spanned by canonically and marginally relevant couplings.

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beyond perturbation theory, one might want to call such a theory non-perturbatively renormalizable. We will simply call such a theory as an asymptotically safe theory.

3 The Asymptotic Safety Scenario for Quantum Gravity In this section, we present a very brief overview about the asymptotic safety paradigm in the context of quantum gravity [17–22]. As is well known, the perturbative quantization of GR as a QFT for the metric leads to a (perturbatively) non-renormalizable theory. In this case, the path integral is formally expressed as7  Z=

  Dgμν e−SEH ,

(15)

with SEH being the Einstein–Hilbert action, given by SEH =

1 16πGN



√ d4 x g (2 − R) ,

(16)

where  and GN denote the cosmological and Newton constants. The quantization usually employs the background field method [20], where a fiducial metric g¯μν is introduced. Then, the complete fluctuating metric is split as gμν = g¯μν + hμν ,

(17)

where hμν stands for the quantum fluctuations about the fixed metric g¯μν . This is usually called “linear split” of the metric. Although the separation between background and fluctuation is linear, such a choice has some drawbacks as, for instance, it does not preserve the signature of the metric if hμν is allowed to fluctuate widely. Different choices of parametrization as, e.g., the exponential parametrization, circumvent some of those issues. See, e.g. [23–31] for discussions on different choices of parametrization for the quantum fluctuations in perturbative quantum gravity as well as in asymptotic safety. However, typically, this comes with a non-linear separation between background and fluctuation. In perturbation theory, the fluctuation hμν is taken to be small. Being an integral over metrics, we need to “gauge fix” it in order to perform a functional integral over geometries. This can be achieved by the usual Faddeev–Popov procedure. This entails the introduction of a gauge-fixing term together with Faddeev–Popov ghosts. It is not the scope of this article to explain such things in details. For a recent pedagogical introduction to that, we refer to [20]. Introducing all that, we can perform a standard 7 We restrict ourselves to the Euclidean path integral. The choice is for technical reasons. We should emphasize that establishing whether what is going to be presented remains valid in the Lorentzian setting is still a challenging open question.

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calculation of the divergences order by order in perturbation theory. In [32, 33], the one-loop divergences were computed for the Einstein–Hilbert action without and with cosmological constant. The logarithmic divergence contains terms that are not part of the microscopic action and they are proportional to background curvature squared terms. Nevertheless, if matter is absent, those terms can be dealt with on-shell. If we disregard the cosmological constant, then the theory is Ricci flat and the one-loop divergence vanishes on-shell. If a non-vanishing cosmological constant is included, then the Ricci tensor is related to the cosmological constant and the one-loop divergence can be absorbed by a redefinition of . This means that the theory, i.e., quantum GR is one-loop renormalizable if matter is absent. In the presence of matter, however, the on-shell condition produces terms which are proportional to the energy-momentum tensor. Such a structure is absent in the microscopic action and, therefore, the theory is not renormalizable at one-loop. If one insists on pure gravity, the natural next step is to check whether divergences will pop up at two-loops. In [34], it was shown that a non-vanishing divergence, μν λρ αβ proportional to R¯ λρ R¯ αβ R¯ μν exists. This term is not present in the bare action and a counterterm with free coefficient should be added in order to absorb this divergence. At every order, in perturbation theory, new counterterms with arbitrary coefficients will be needed to control divergences and this spoils completely the predictivity power of the theory. In fact, this was not an unexpected result due to power-counting arguments, but it was logically possible that such terms could arise with vanishing coefficient. Explicit calculations show that this is not the case. This fact does not imply that there is a fundamental incompatibility between GR and QFT as is oftenly spread. In fact, one can treat quantum GR as an effective field theory, meaning that it is valid up to some ultraviolet cutoff. Calculations performed below such a scale are perfectly valid. At the cutoff scale, the theory breakdown and begs for a UV completion. Nevertheless, quantum gravitational corrections within such a framework are possible to be computed and the most iconic result consists on the evaluation of quantum corrections to the Newtonian potential, see [35]. The perturbative non-renormalizability of GR can be interpreted in several different ways. One possibility is to assume that the classical action, the Einstein– Hilbert action, should be replaced by another action which renders a perturbatively renormalizable QFT. A celebrated example is higher-derivative gravity introduced in [36]. On top of the Einstein–Hilbert terms, Rμν R μν and R 2 contributions are added to the classical Lagrangian. The underlying quantum theory is perturbatively renormalizable and, under certain circumstances, asymptotically free. This would characterize a UV-complete theory amenable to perturbation theory. Nonetheless, this theory contains a ghost state in its spectrum, a fact that might jeopardize unitarity. In fact, this issue is subject of a long-standing debate and different strategies to circumvent such a problem have been explored. We refer to [37–40] for a very recent account on the unitarity issue in higher-derivative gravity with curvature squared terms (and to [41] for a recent review on curvature-squared higher-derivative gravity) and [42] for models with higher-order curvature terms.

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Thence, the quest for a UV complete theory of quantum gravity within the standard perturbative QFT toolbox is still an active research field. Another possibility is to verify whether quantum gravity features a non-trivial ultraviolet fixed point along the renormalization group flow. Following the discussion we have made in the previous section, this would entail a “non-perturbatively renormalizable” QFT which is free of divergences and predictive, provided that the number of relevant directions at the fixed point remains finite. Hence, quantum gravity would be asymptotically safe. Such a possibility was pointed out and elaborated, for the first time, by Weinberg in [43]. However, it was in [44], where the FRG equation was adapted to quantum gravity, that a toolbox was developed for practical verifications of this scenario. The background metric g¯ μν defines a momentum scale by the eigenvalues of the background Laplacian and the field configurations can be organized in momentum shells. This sets up a local coarsegraining procedure and the step by step functional integral is performed by the introduction of a regulator quadratic in the fluctuation fields hμν as discussed in the previous sections, i.e.,  Sk =

μναβ ¯ 2 d4 x g¯ hμν Rk (∇ )hαβ .

(18)

μναβ ¯ 2 (∇ ) (not to be confused with curvature) ensures that The regulator kernel Rk only modes with momentum (squared) larger than k 2 are integrated out. The quadratic structure of (18) ensures that the flow equation will have the simple oneloop form as in Eq. (5). In [44], a simple choice for the effective average action k was made. It is known as the Einstein–Hilbert truncation and consists in assuming that the effective average action is written as

kEH [gμν , g¯μν ] =

1 16πGk

 d4 x

√

g(2k − R) + Sgf + Sgh ,

(19)

where Sgf and Sgh are suitable gauge-fixing and Faddeev–Popov terms. The parameters Gk and k are the running and dimensionful Newton and cosmological constants. Another feature regarding Eq. (19) is that the effective average action is written as a functional of the full8 gμν and background metrics g¯μν independently. This is due to the introduction of a gauge-fixing term and the regulator (18). Those terms treat the full metric and the fluctuation hμν in such a way that they do not appear just as g = g¯ + h. Consequently, the renormalization group flow will generate further terms which do not respect the linear split of the metric and, therefore, it should be projected to a symmetric subspace by demanding that the effective average action satisfies suitable Ward identities. This is way beyond the

8 This

notation is not accurate. In fact, the metric gμν that enters as the argument of the effective average action corresponds to the expectation value of the metric gμν that appears in definition of the path integral (15). We employ the same name for both for simplicity.

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scope of the present discussion and we refer to, e.g. [21] and references therein for further details. The conceptual ingredient we wanted to introduce with this discussion is that although the introduction of an auxiliary background g¯μν has brought up the possibility to define a momentum scale and thus a local coarsegraining procedure, it came with the cost of breaking/deforming the split symmetry between background and fluctuation fields. Thence, background independence, a fundamental requirement in a theory of quantum gravity, is not manifest and should be explicitly verified. Remarkably, in the pioneering work [44], a non-trivial fixed point, the Reuter fixed point, was found in the bidimensional truncated theory space parametrized by the dimensionless Newton and cosmological constants. It features two-relevant directions. Concretely, see, e.g. [45], the beta functions for the dimensionless Newton and cosmological constants, gk ≡ k 2 Gk and λk = k −2 k are, respectively, βg = 2gk −

gk2 11 − 18λk + 28λ2k , 3π (1 − 2λk )2 − 1+10λk gk 12π

βλ = −2λk +

107−20λ gk 3 − 4λk − 12λ2k − 56λ3k + 12π k gk . 6π (1 − 2λk )2 − 1+10λk gk

(20)

12π

The fixed-point value is λ∗ ≈ 0.2 and g ∗ ≈ 0.7 and the critical exponents form a complex conjugate pair with real part of order one. The existence of such a fixed point was robustly verified in much more sophisticated truncations, where higher order curvature terms were took into account. The inclusion of more terms can be done in several different directions in theory space. One example is to consider higher powers of the Ricci scalar R or non-polynomial functions of R. Explicit calculations with polynomials up to R 70 , see [46] show not only that the existence of the fixed point is stable but also it is quantitatively apparently convergent, i.e., its value does not change significantly under improvements of the truncations. Furthermore, the near-canonical scaling probed by the critical exponents suggest that such a fixed point is near-perturbative, see [47]. Extensions of the truncations to different directions in theory space were also performed, namely, the inclusion of Rμν R μν terms and, notably, the Goroff-Sagnotti term. For an incomplete list of works we refer to [26, 46, 48–70]. Remarkably, the number of relevant directions seems to saturate at around three a fact that ensures that the theory is predictive. Of course, a formal proof of the existence of such a fixed point is desirable, but this seems to be an extremely challenging problem. The more realistic strategy is to find evidence for the existence of the fixed point and, so far, all results point towards that. Very schematically, the mechanism which drives asymptotic safety is the compensation between the canonical dimension term in the beta functions (first terms on the righthand side of both equations in (20)) and the non-trivial quantum corrections. Such corrections are typically non-vanishing away from the free-theory fixed point.

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Although pure gravity has shown several different evidence for the existence of a suitable non-trivial fixed point in the renormalization group flow, there is the possibility that matter degrees of freedom, when taken into account, induce contributions to the gravitational beta functions and destroy the existence of the fixed point. Hence, a crucial question to be answered in the asymptotic safety program is whether observed matter degrees of freedom are compatible with the fixed point structure. Fortunately, since the asymptotic safety scenario is described in terms of usual continuum QFT language, coupling matter fluctuations is straightforward. In the recent years, many different works explored that and revealed that the fixed point structure is compatible with the SM matter content. Such an achievement opens the door for the formulation of a consistent description of quantum gravity as a QFT which is compatible with our knowledge about the matter content of our Universe. See [19, 21, 47, 71–96] for some works on that. Conversely, the impact of quantumgravity fluctuations to the matter sector can be also analyzed. In fact, the assumption of the existence of an asymptotically safe fixed point in the gravitational couplings leads to several interesting effects in the matter couplings and, in particular, can provide a prediction of the Higgs and top quark masses [96–99], resolve the Landau pole in the SM [100–102] and explain the gauge hierarchy problem, see [103]. Of course, all this consists a promising picture towards the construction of a quantum theory of gravity (and matter). However, the challenging task of proving the existence of the fixed point as well as how all this fits to a Lorentzian setting and the fate of background independence are important open questions that need to be addressed in this approach. Besides the important recent work done within the continuum formulation of quantum gravity, we advocate in this article that a potential very fruitful route is to use different methods such as lattice simulations or different formulations to probe such properties. The interplay between different frameworks by combining different strengths of each of them could provide new key insights and avoid technical complications that can hamper a convincing check of the aforementioned properties. In the next section, we briefly explain how to define the path integral for quantum gravity using tensor models a how this could be, eventually, connected to asymptotic safety.

4 Background-Independent Renormalization Group Flows Since a formal mathematical proof of the existence of suitable fixed point in the renormalization group flow is beyond our current capabilities, it is extremely desirable (and necessary) to find evidence for its existence from different perspectives. Furthermore, as we mentioned, there are challenges that might be easier to tackle employing a different framework. A concrete example is background independence. In the QFT for metrics, an explicit background is chosen for concrete calculations and the statement of background independence should be phrased as the fact that no special role is played by the choice of background. Nevertheless, the explicit check of that is far from trivial and the issue whether the resulting quantum theory

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is really background independent is open. On the other hand, if the path integral over geometries could be evaluated with no reference to a background, then a comparison between the resulting quantum theories would be possible and, if they are physically equivalent, background independence would be established for the asymptotic safety scenario. One possibility is to discretize the path integral in a suitable way, providing a lattice-like regularization, and perform the sum over discretized geometries. The physical content of the theory would then be available in the continuum limit where the lattice spacing shrinks to zero. Of course, it is not clear whether the continuum limit would be well-behaved giving rise to extended four-dimensional geometries at large distances. This program has a successful story in two dimensions. There, the sum over geometries and topologies is encoded in matrix models [104]. They correspond to statistical models of random matrices φab of size N  whose interactions are dual to building blocks of geometry. For instance, if one considers a cubic matrix model defined by the partition function  Z∼

[dφ] e

− 12 Trφ 2 + √g  Trφ 3 N

,

(21)

the interaction is dual to a triangle as is shown in Fig. 4. The continuum limit should not depend on the choice of the building blocks, i.e., we could have chosen squares instead of triangles as the building blocks, for example. Hence, we aim at finding a universal continuum limit. As is intuitive to understand, the Feynman diagrams of this theory correspond to gluing vertices and, consequently, triangles (or different building blocks). Therefore, the Feynman expansion correspond to a sum of triangulated surfaces. It can be organized by the topology of the resulting manifold, the so-called 1/N  -expansion, and due to tools particular to two dimensions, this can be completely evaluated. In this case, in order to have contributions from all different topologies, a continuum limit is obtained by tuning the coupling g to a critical value while the matrix model size N  is taken to infinity. This is the so-called double-scaling limit [105–108]. It is defined by N  (g − g∗ )5/4 = const Fig. 4 Each dashed line is associated to a matrix. They are dual to the edges of a triangle. Feynman diagrams correspond to gluing such vertices with propagators. Geometrically, this corresponds to glue the triangles along their edges

Trφ3 ∼

(22)

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with N  → ∞ and g → g∗ . Intuitively, N  counts the number of degrees of freedom (the size of the matrix) and the continuum limit is obtained by going from finitely to infinitely many degrees of freedom. Equation (22) has a particularly interesting interpretation when rewritten as g = g∗ + #(N  )−4/5 ,

(23)

where # stands for a constant. This equation has the same structure of Eq. (14) if N  is taken as the renormalization group parameter. The critical value g∗ represents thus a fixed point in this abstract renormalization group flow on the matrix size and Eq. (23) is just the linearized flow around such a fixed point. This serves as an inspiration to set up a coarse-graining procedure on the size of the matrix. The Wilsonian perspective would then be realized by integrating out rows and columns of the matrices step by step as discussed in [109] and in [110] in the context of the FRG. In summary, one could discover the double-scaling limit as a fixed-point in the renormalization group flow. Such a coarse-graining procedure does not rely on the introduction of a background which sets up a reference scale. Inhere, the perspective is not the usual one where we flow from “fast modes” (or high momentum modes) to “slow modes” (low momentum modes) but rather, we flow from many degrees of freedom to fewer degrees of freedom. In this sense, this is a background-independent coarse-graining procedure. For an extensive discussion on that, we refer to [111]. Having understood how to sum over geometries (and topologies) in two dimensions and how to take a universal continuum limit, the natural step forward is the generalization to higher dimensions. However, in this case, more limited tools are available. One possibility to perform an accurate calculation is to evaluate the discretized partition function employing Monte-Carlo numerical simulations. Such a program has evolved to the so-called (Causal) Dynamical Triangulations ((C)DT) approach to quantum gravity, see [112–116]. Several strong indications point towards a continuum limit which features an extended, four-dimensional geometry at large distances. More refined numerical calculations are the source to establish more robust indications towards the desirable continuum limit. On the other hand, one could try to generalize the matrix model to higher dimensions. This leads to the so-called tensor models [117–123]. The Feynman diagrams for such models are dual to discrete geometries and the Feynman expansion corresponds to a sum over geometries and topologies. As before, the fundamental building blocks of geometry correspond to the interacting vertex of random tensors of size N  . For a rank-d tensor, each index is associated to a (d − 2)-subsimplices of a (d − 1)simplex, i.e., to the edges of a triangle for rank-3 tensor model. The interaction in this case corresponds to the gluing of such triangles along their edges. In contrast to matrix models, the Feynman expansion of tensor models could not be organized in a 1/N  expansion in its original formulation. Nevertheless, a class of tensor models introduced in [124–129] with a particular type of interactions feature a 1/N  -expansion leading to the possibility of probing the continuum limit for such models analytically at least for leading order in 1/N  .

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The particular class of tensor models we mentioned are the so-called (un)colored models [121]. In these models, the interactions are subject to a O(N  )⊗d (U (N  )⊗d ) symmetry for real (complex) tensors. This means that under the transformation Ta1 ...ad → Ta1 ...ad =



Oa(1) . . . Oa(d) Tb1 ...bd , 1 b1 d bd

(24)

b1 ...bd (n)

the tensor model action is left invariant, where Oab are orthogonal matrices (the case of complex tensors is completely analogous). Such a symmetry restricts how tensors Ta1 ...ad (which do not have any symmetry under index permutation) should be contracted: the first index of a tensor should contract with the first index of another tensor, the second index should contract with the second index and so on. In complete analogy to matrix models, it is possible to set up a coarsegraining procedure based on the size of the tensor. The Wilsonian renormalization is realized by integrating out “layers” of the tensors step by step and suitable universal continuum limits can be discovered by finding fixed points along the renormalization group flow. This is a subject which has witnessed growing interest in the last years not only for pure tensor models [111, 130, 131], but also, group field theories, see [132–147]. See also [148, 149] for discussions on the application of coarse-graining techniques to causal sets, another discrete approach to quantum gravity. As discussed in [111, 130, 131], we can adapt the FRG to tensor models. In fact, all the derivation of the flow equation outlined in Sect. 2 is valid for tensor models provided that the parameter s is identified with the parameter which measures the size of the tensor N. Explicitly, the flow equation for tensor models is expressed as 1 ∂t N [T ] = N∂N N [T ] = Tr 2

"

∂ 2 N [T ] + RN δa1 b1 . . . δad bd ∂Ta1 ...ad ∂Tb1 ...bd

#

−1 ∂t R N

,

(25) where RN is the regulator kernel. Hence, beta functions for the tensor model couplings can be derived. Since the theory is background independent and the coarse-graining parameter N is dimensionless, the assignment of canonical dimensions to the couplings is performed by requiring that the system of beta functions is autonomous. This can be achieved by demanding that the canonical dimensions are such that the beta functions can be expanded in powers of 1/N. For sufficiently large N, the leading order terms form an autonomous system. In [131] a systematic search of fixed points was performed for a rank-3 real tensor model. Different classes of fixed points were found: one class which features dimension reduction, i.e., at the fixed point, the tensor model can be reduced to a matrix model and, therefore, cannot be associated to three-dimensional quantum gravity and other class which does not reduce to a matrix model. The computation of the critical exponents which characterize the continuum limit associated with the second class of fixed point led to two relevant directions with values of order

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one and a third relevant direction with value close to 0.4. These were obtained in a relatively simple truncation for the effective average action meaning that such numbers come with systematic errors and a check of apparent convergence of the results is definitely needed. Nevertheless, these numbers are compatible with those obtained for the critical exponents associated to the fixed point obtained in the asymptotic safety program in three dimensions [64]. It is definitely too early to state that such fixed points belong to the same universality class and that tensor models and asymptotic safety could be completely different point of views of the same physics. Nevertheless, this opens an exciting possibility to explicitly compare critical exponents in more refined calculations and check whether this premature conclusion could be indeed realized. Results obtained in the simplest truncation for rank-4 models were also reported in [111].

5 Visions: Bridging the Gap Between Different Approaches to Quantum Gravity The asymptotic safety scenario for quantum gravity as well as tensor models were discussed as different perspectives to make sense of the path integral for quantum gravity. They intend to perform the path integral using different mathematical tools. In the first case, the “sum over geometries” is performed by a gauge-fixed sum over metrics. This is achieved, in practice, by introducing an auxiliary background metric and summing over the quantum fluctuations (not necessarily small) around it. On the other hand, tensor models provide a lattice-like discretization of geometries. In both cases, a continuum limit which allows for the removal of a cutoff (in the case of asymptotic safety, a UV cutoff in momentum space and in tensor models, the lattice spacing) is needed. This is translated to the existence of fixed points in the renormalization group flow. We have discussed how the FRG can be a versatile tool, adapted to both cases, to discovery such fixed points. It allows for the calculation of critical exponents which characterize the universality class of the underlying continuum limit. As pointed out, there are preliminary indications that those theories could belong to the same universality class. If this is really true, then they correspond to different mathematical formulations of the same physical theory, i.e., they lead to the same observables. Although it is very challenging to show this is really the case, the application of the FRG to these theories is relatively simple and more refined results could be obtained in the near future establishing a closer relation between them or revealing that they are incompatible. We advocate that such an interplay is crucial for further developments in those fields. As a concrete example, the equivalence between asymptotically safe quantum gravity and tensor models could establish background independence of the Reuter fixed point since in tensor models, no preferred background is employed. Conversely, probing phenomenological aspects such as quantum-gravitational effects in black holes [150–153], cosmology

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[154, 155] and matter is much easier in the usual QFT for metrics setting. We should exploit the advantages of each formulation to answer different questions. Renormalization group techniques are extremely useful and important in many different areas of physics. This article intended to briefly comment on some aspects of the renormalization group in quantum gravity. It could work as a unifying tool which brings together different approaches to quantum gravity giving a coherent and consistent picture of quantum spacetime. Our vision is that such cross-fertilization between different approaches mediated by the renormalization group will bring key new insights to our understanding of the quantum microstructure of spacetime. Acknowledgements I would like to thank Astrid Eichhorn for many inspiring discussions on the topic and the organizers of the conference “Progress and Visions in Quantum Theory in View of Gravity: Bridging foundations of physics and mathematics” in Leipzig (2018) for the invitation. This work was supported by the DFG through the grant Ei/1037-1.

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Proposal 42: A New Storyline for the Universe Based on the Causal Fermion Systems Framework Claudio F. Paganini

Abstract Based on preliminary results from the Causal Fermion Systems framework regarding the matter-antimatter asymmetry in the universe, I propose a novel story line for the universe that would, if correct, resolve a number of problems in cosmology. First and foremost, the here-presented arguments suggest to identify cold dark matter as third generation (anti-)neutrino mass-eigenstates ν3 . Furthermore, the proposal suggests a new look at the problem of initial conditions. Last but not least, the proposal also provides a new angle on the cosmological constant.

1 Introduction To this day we lack both a theory of quantum gravity and a fundamental theory of physics that manages to unify general relativity and the standard model of particle physics. Various candidates for quantum gravity/unified theory such as String Theory, Loop Quantum Gravity, Asymptotic Safety, Non-Commutative Geometry. . . have either failed to put forward falsifiable predictions or have not even succeeded in obtaining a fully consistent reproduction of our current models in an appropriate limit. There are some other approaches of which I want to highlight the work of Padmanabhan [23–31]. They arrive at a relationship between the CMB fluctuations and today’s value of the cosmological constant. That series of work was started with [23], for my considerations here it were especially the ideas in [29] that served as an inspiration. A lot of the ideas that I will discuss in the present work are directly inspired and thus as far as I can judge compatible with the work of Padmanabhan, however and that is an important remark, they are not reliant on those

C. F. Paganini () Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany Albert Einstein Institute, Max Planck Gesellschaft, Potsdam, Germany e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 F. Finster et al. (eds.), Progress and Visions in Quantum Theory in View of Gravity, https://doi.org/10.1007/978-3-030-38941-3_4

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ideas. I will point out, where appropriate, how the picture painted in Padmanabhan’s work can be fit into what I will present here. Despite some of the ideas being conceptionally close, they often come with a slight twist, at least at first sight there doesn’t seem to be a neat one-to-one correspondence in the pictures (at least not in their underlying motivations). A detailed investigation of how the two pictures relate will however be postponed to later work. The present work is based on ideas originating from the Causal Fermion Systems (CFS) framework, a novel approach to unification of the Standard Model of particle physics and General Relativity. A summary of the current state of knowledge on the framework can be found in [15] for a proper introduction geared towards physicists see [14, 16] for a introduction geared more towards mathematicians see [12]. I will include an introductory section to the CFS framework to make this paper sufficiently self contained. In my paper I will discuss how the CFS framework can give us a new look at the universe and how it might resolve some of the fundamental outstanding problems, and provide a new angle on others. As the ideas are heavily reliant on the CFS framework I will include a discussion on open questions in the framework and the physical interpretation, to the best of my current understanding. The ideas put forward in the present paper are based on preliminary results regarding the matter anti-matter asymmetry in the universe, in the context of the CFS framework [17]. If the here presented picture is correct, we could identify cold dark matter as third generation (anti-)neutrino mass-eigenstates. The approach, if correct, would not only explain the origin of the matter-/anti-matter asymmetry, but the origin of all matter and radiation. Let me emphasize the novel part of the content here: The core of this paper is a new story line of the universe based on a new approach to unification. The story line does not only address a number of conceptional issues in today’s models but also arrives at a prediction for the composition of Cold Dark Matter inside the Standard Model. Note that most pieces of the story line for the universe presented here, are not entirely new. In fact most of the common pictures (inflation, big bang. . . ) partially fit into the picture presented here. What is novel here is the fact that they are put together in a slightly different and conceptionally more consistent way. The modifications of the Friedmann equations that are introduced have at most a weak theoretical motivation, however in interplay with the CFS mechanism for matter creation they lead to an instability mechanism that can induce a transition from a high  phase to a low  phase. According to [29] such an instability could explain the arrow of time. Overview of the Paper The paper is organized in the following way. In Sect. 2 I will collect the fundamental notations and ideas of the CFS framework. In Sect. 3 I will present a possible modification to the Friedmann equations that will facilitate the desired instability mechanism. In Sect. 3.2 I will discuss a slightly different modification that, on universal scales, has a equivalent effect but might be better motivated by the CFS framework. In Sect. 4 I discuss the preliminary results regarding the matter and radiation creation mechanism derived in [17]. In Sect. 5 I will discuss how the matter

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creation mechanism leads to an uncertainty in the reheating time. In Sect. 6 I will discuss how the pieces fit together to form a coherent story line for the universe. Finally in Sect. 7 I will discuss some of the open questions regarding the CFS framework.

2 Causal Fermion Systems As I assume most readers to be unfamiliar with the CFS framework I will here give a quick overview of the most important concepts and how they relate to well known structures. This section will be brief. For a proper introduction geared towards physicists see [14, 16] for a introduction geared more towards mathematicians see [12]. For a overview of the state of the art of the CFS framework see [15]. All the material in this section can be found in these sources. In Sect. 7 I will come back to the CFS framework and discuss some questions regarding the interpretation and open questions associated with it. Remark 1 When reading into the CFS framework for the first time, it is advisable to the reader to take all the concepts she knows from well established theories, put them in a box, and only get them out again, once she understands the abstract structures in themselves and is ready to start investigating the relations of concepts in the CFS framework to established theories. The abstract definition of a CFS consists of three objects: a Hilbert space H, a suitably chosen subset F of the linear operators on the Hilbert space L(H) and a measure ρ that lives on F. Definition 1 (Causal Fermion System) Let (H, .|.H ) be a Hilbert space. Given a parameter n ∈ N (“spin dimension”) we set & F := x ∈ L(H) with the properties: x is self-adjoint and has finite rank x has at most n positive

' and at most n negative eigenvalues

and ρ a measure on F (“universal measure”) From now on when I talk about a CFS I will always refer to a triple (ρ, F, H). If we have a CFS we can simply define a spacetime to be given by M := suppρ. With that definition we get that spacetime points are linear operators on H.1 In the following 1 I will come back to that

in Sect. 7, where I will discuss a possible link between the CFS formalism and the Events, Trees, Histories (ETH) interpretation of quantum mechanics [20].

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we want to introduce a causal structure on the spacetime M. For that we have to investigate the relationship between two points x, y ∈ M. The operator product xy xy x·y ∈ L(H) in this case has non-zero, in general complex, eigenvalues λ1 , . . . , λ2n . It is important to remark that F is not a group under operator composition. Hence for points x, y ∈ F their product x·y is not necessarily in F but it still has finite rank ≤ 2n. Now using the eigenvalues of the operator product of x, y we can define a causal structure on F. Definition 2 (Causal Structure) The points x, y ∈ F are called ⎧ spacelike separated ⎪ ⎪ ⎪ ⎪ ⎨ timelike separated ⎪ ⎪ ⎪ ⎪ ⎩ lightlike separated

xy

xy

if |λj | = |λk | for all j, k = 1, . . . , 2n xy

xy

if λ1 , . . . , λ2n are all real xy xy and |λj | = |λk | for some j, k otherwise

In my opinion it is one of the core strength of the CFS framework that one can define consistent pairwise causality relations without appeal to any form of metric. For completeness I mention here that one can in fact write down a functional that can be used to define a time direction between two points. The following notation will be used x(H) ⊂ H πx : H → H

subspace of dimension ≤ 2n orthogonal projection on x(H)

where πx is the projection operator on the image of the operator x. With this definition we can introduce the time direction functional C:M ×M →R,

  C(x, y) := i tr y x πy πx − x y πx πy

(1)

which leads us to define a time direction in the following way. Definition 3 (Time Direction) For timelike separated points x, y ∈ M, 

y lies in the future of x if C(x, y) > 0 y lies in the past of x

if C(x, y) < 0

In general the so defined time direction is not transitive. Therefore it is currently unclear whether this functional can be used to make any interesting global statements about causal ordering. See [10] for recent progress with respect to that question.

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Now we come to core of the CFS framework: The causal action principle. Lagrangian L[Axy ] =

2n 1  xy xy 2 |λi | − |λj | ≥ 0 4n

 Action

S=

(2)

i,j =1

F×F

L[Axy ] dρ(x) dρ(y) ∈ [0, ∞]

(3)

A physical system is then given by a measure ρ which minimizes the causal action S. To obtain the abstract CFS Euler–Lagrange equations we vary the action with respect to ρ under the following constraints volume constraint:

ρ(F) = const

(4)

tr(x) dρ(x) = const

(5)



trace constraint: 

 boundedness constraint:

F×F

F 2n

2

xy |λi |

dρ(x) dρ(y) ≤ C

(6)

i=1

C determines regularization scale ε The volume constraint is rather natural and guarantees that the minimizing ρ is non trivial. The other two constraints are of more technical nature. The technical definition of a minimizer is then given by Definition 4 (Minimizer) ρ is a minimizer if S[ρ] ˜ − S[ρ] ≥ 0 for all ρ˜ with |ρ˜ − ρ| < ∞

and

(ρ˜ − ρ)F = 0.

Here, |ρ˜ − ρ| denotes the total variation. Using the notation  L(x, y) dρ(y)

(x) = M

the abstract CFS Euler–Lagrange equations are then given by Lemma 1 (Euler–Lagrange Equations) |M = inf  =: c. F

(7)

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Hence for a measure to be a minimizer the function (x) needs to be constant for all x ∈ suppρ and as a consequence the action S is a multiple of the volume ρ(F). For most practical purposes the CFS Euler–Lagrange equations are linearized around a given minimizer.

2.1 Continuum Limit To make sense of the discussions to follow I need to briefly introduce how these structures relate to the classical notions in the continuum limit. Note again, what I will present here is at most a sketch, for a more detailed elaborations consult the sources mentioned before. To represent Minkowsky space as a minimizer to the causal action principle we begin by looking at solutions to the free Dirac equation (iγ k ∂k − m) ψ = 0

(8)

together with the usual scalar product  ψ|φ =

t =const

(ψγ 0 φ)(t, x) dx

(9)

where γ 0 is the Dirac matrix and ψ = ψ † γ 0 is the adjoint spinor. One then chooses the Hilbert space to be all the negative energy solutions to Eq. (8) and defines the local correlation operator via ψ|F (x)φ = −ψ(x)φ(x)

∀ψ, φ ∈ H.

(10)

Where one uses the fact that every bilinear form on a Hilbert space can be expressed via the scalar product and a linear operator. Note that the operator F (x) is selfadjoint and has by definition finite rank ≤ 4 and at most 2 positive and 2 negative eigenvalues. Therefore we have that F (x) ∈ F ⊂ L(H) for spin dimension n = 2. The local correlation operator thus gives a mapping from the spacetime points into F. We can then use the push-forward measure  ρ( ) :=

F −1 ( )

  d 4 x = μ F −1 ( )

(11)

to obtain all the structures required by the definition of the CFS. Now I wasn’t quite honest here, as of course if we take the Hilbert space to be the solutions to (8) with negative energy which are in L2 (R3 ) then (10) is generically ill-defined. Therefore to make the mapping well-defined we need to introduce a

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regularization ψε = Rε (ψ)

(12)

where Rε is the regularization operator—for example a convolution with a suitable kernel—such that ψ" is smooth and we have that ψ = lim Rε (ψ).

(13)

ε→0

This gives us the regularized local correlation operator ψε |Fε (x)φε  = −ψε (x)φε (x)

∀ψ, φ ∈ H.

(14)

This is now a regularized CFS and the continuum limit is then defined as taking ε → 0 in a suitable sense. It can be shown that in this limiting case the Dirac sea vacuum is a critical point of the causal action in a well-defined mathematical sense. In this sense, the Dirac sea drops out of the action, and only perturbations to the Dirac sea can be observed as matter and physical fields. In a way you can consider the Dirac sea to be the stage on which the world plays. An analogy I like to make is the following: You can think of the Dirac sea as the geography of a country while the actual physical world, i.e. the spacetime, is given by the travel time distance between any combination of two points in the country. It is obvious that the travel time distance between two places can change dramatically (for example when a new tunnel is opened) without the geography changing at all. A similar picture can be employed with respect to the Hilbert space, i.e. the Dirac sea in the CFS framework.2 I will only briefly mention one example here to establish a picture on how to think about some concepts in the context of the CFS framework. We now want to vary the vacuum minimizer. So let us introduce an external field as a vector potential B into the Dirac equation. (i∂ / + B − m)ψ = 0 .

(15)

Now if we regularize and demand that the CFS Euler–Lagrange equation be satisfied in the limit ε → 0 in a suitable sense, we get as a result, that the vector potential has to satisfy Maxwell’s equation. Now this result can be viewed from two different sides: either the presence of the Maxwell field deforming the Dirac sea, or a collective deformation of the Dirac sea giving rise to a Maxwell field as an effective description and thus only the Dirac sea being fundamental and the Maxwell field being an emergent object. My favourite analogy at this point is the following: Suppose we would go for a walk at the beach,

2 Such analogies are of course always to be taken with the necessary grain of salt. They are intended to help the reader gain a rough intuition on the role different objects play in the framework.

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and for some weird reason I get interested in the shape of your feet. Now I will obtain the same information whether I ask you to show me your feet or whether I have a look at the collective behaviour of the sand grains where you just stepped a minute before. It is important to stress that in the CFS framework electro-magnetic fields have no fundamental existence but are merely a simplified description of a phenomenon arising from the collective behaviour of all states in the Dirac sea. In fact the same is true for all bosonic fields in the CFS framework (including gravity). In summary the Dirac sea encodes all information about the physical world.

3 Dynamical  Running or decaying vacuum models have been studied in recent years, see for example [2, 9, 11, 19, 34, 35, 38] and some are indeed compatible with recent Planck data [38]. Also the set of ideas put forward by Padmanabhan [23–31] are based on the idea of a transition between two “cosmological energy levels”, i.e. a high  phase and a low  phase, however to my best knowledge a proper mechanism describing the why and the how of this shift in “cosmological energy levels” seems to be absent from his work. It is the idea of a transition between “cosmological energy levels” that I will rely on in this paper. For the time being I will assign the mechanism to some sort of running vacuum model. At the current stage I have only a very weak theoretical motivation3 as to why I would want to consider the particular modification of the Friedman equation other than the fact that it provides the desired mechanism I need in Sect. 6. One might ask whether it makes sense to modify gravity on the level of the Friedmann equations. This can indeed make sense if the modification is an emergent phenomenon from the global behaviour of the universe in some underlying framework. In this case modifying the Friedman equation can indeed make sense, as the FRW coordinate system is special, as it can actually be determined by observations in the universe, see [29] for a discussion. Therefore emergent phenomena can indeed enter on this level in the formulation of gravity. For the sake of the argument I will assume here the simplest possible running/decaying vacuum model. Hence we take the Friedmann equation H2 = R a −4 + M a −3 +  H02

(16)

3 The weak theoretical motivation mentioned here comes from my limited understanding of the CFS framework and how the objects therein might be interpreted. In Sect. 3.2 I will discuss a different point of view which is superficially equivalent but not so commonly considered. This point of view might be more readily compatible with the CFS framework and seems to be favorable with respect to conceptional issues as well.

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where H is the Hubble expansion rate, R the radiation density, M the matter density and  the energy density related to the cosmological constant. I now impose that the dynamics of  are given as a function of H in a similar fashion to the modifications in [38]  = 3H 2.

(17)

Note that I leave the Friedmann equations unchanged beyond the fact that I allow for  to depend on the Hubble expansion rate H . In the de Sitter Universe, hence when R = M = 0 we have ds 2 = −dt 2 + a(t)2 dσ 2

(18)

a(t) = eH t

(19)

with

where σ is the flat metric in R3 and a(t) is the scale function. We have that the relation (17) is satisfied. If we write de Sitter space in stationary coordinates instead



 −1 r 2 r 2 2 dr 2 + r 2 d 2n−2 . ds = − 1 − dt + 1 − 3 3 2

(20)

we get that the cosmological horizon rCH is located at  rCH =

1 3 = .  H

(21)

In a slight abuse of notation i will in the following refer to de-Sitter as a “stationary” solution to the system of equations given by (16) and (17).4 In the following I will use the radius of the causally interacting region rCH =

3 2 rCH

(22)

to parametrize the dynamics of  instead of (17) as it is more natural to the way of thinking about the universe I want to introduce later on. If we consider the relation

4I

consider this to be a sensible notation as for any two times ta and tb the hypersurfaces can be mapped into each other isometrically hence there is no change that could be observed by an outside observer. In the absence of matter or other perturbations the notion that de-Sitter space is exponentially expanding is absolutely meaning less. It only acquires meaning once you put test matter (or any group of test bodies for that matter) there, tracing out a geodesic foliation with their proper time functions.

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as functions in terms of the scale function a(t) then we get the following derivative 6 drCH d =− 3 . da rCH da

(23)

One sees immediately that the change in  is more dramatic in the early universe CH when rCH is small and drda is non vanishing due to the fact that the universe is either matter or radiation dominated. In the late universe on the other hand rCH is CH large and particularly in the  dominated era drda is close to zero hence the change in  should be small. This gives an intuition for why the compatibility of this sort of running vacuum models with Planck data reported in [38] comes at no surprise. On the conceptional level it would be necessary to check the following conjecture. Conjecture 1 Let ρR (a(t))|0 ≥ 0, ρM (a(t))|0 ≥ 0 and (a(t))|0 > 0 be initial data at some time t = t0 (or equivalently at some value of the expansion parameter a = a0 ) to the system of equations consisting of (16) and (22). Then we have that lim ρR = lim ρM = 0

t →∞

t →∞

and

lim (a(t)) = ∞ > 0.

t →∞

(24)

Here I denote with ρM the physical energy density of matter and with ρR the physical energy density of radiation in a homogenous universe. In colloquial terms that conjecture states that if the universe evolves under Eqs. (16) and (22) it will asymptotically approach to a de-Sitter universe. In Fig. 1 you find a sketch of this expected behaviour of the various densities.

Fig. 1 The graph shows a sketch of the expected future development in “time” (horizontal axis) of the energy densities (vertical axis) associated with radiation (red), matter (blue) and  (black) under the Friedman equations with a running vacuum (22) model. Significant modification to the value of  are only expected in the very early universe. The standard sequence of having a radiation dominated phase that would go over to a matter dominated phase and then ending in a  dominated phase is not affected. I used tnow to roughly indicate where in the evolution of the universe we are today and tCMB to roughly indicate where the CMB decoupled in this picture

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3.1 Thermodynamic Interpretation This section is intended to show that the above modification of the equations could indeed have a deeper theoretical motivation. It is however quite speculative and not fully expanded. Thus this section can be ignored by anybody just interested in the larger picture. For black holes one can define an entropy via the surface area A in dimensionless form SBH =

Ahor c3 Ahor = 2 4Gh¯ 4LP

(25)

where LP stands for the Planck length, G for Newton’s constant, h¯ for Planck’s constant and c is the speed of light. Analogous we can define a cosmological entropy for de-Sitter space by SC =

ACH c3 ACH = 4L2p 4Gh¯

(26)

2 and hence with Eq. (21) where the surface area is given by 4πrCH

ACH =

12π 

(27)

and thus SC =

3π 3πc3 . = 2 h¯ G LP 

(28)

With this expression one could in fact use =

3π L2P SC

(29)

as a definition for the running vacuum with a slight further modification =

L2P (SC

3π . + SM + SR )

(30)

Here SM stands for the entropy associated with matter and SR stands for the entropy associated with radiation inside the interacting region. Equation (30) is a slight variation on (29) which I will not get further into. Both modifications (29) and (30) satisfy the condition that if coupled with (16) the de-Sitter universe, i.e. de Sitter

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space is a stationary solution. In fact in this case (29) and (30) are identical because SR = SM = 0.5 It is interesting to note that with this interpretation entropy would play an active role in the evolution. In a sense given Conjecture 1 is true and we interpret the running vacuum as suggested by the form of Eq. (29), we would have that the presence of entropy itself leads to the following conjecture for the second law of thermodynamics. Conjecture 2 Suppose the universe evolves according to Eq. (16) coupled to (29) than we have that d(SC + SM + SR ) ≥0 dt

(31)

d(SC + SM + SR ) =0 dt

(32)

holds on universal scales, and

is only true in the stationary case of de-Sitter space where ρM = ρR = 0 and hence SM = SR = 0. Now how does this all connect to the CFS framework? I have to be honest that from this point of view it is extremely vague and based on the following open question:

Thermodynamic Interpretation of CFS Despite all the results that have come from the CFS framework (see [15] for an overview) and despite how well it fits in the limiting case with the well established theories, its major weakness, in my opinion, is the absence of a clear physical interpretation of the variation principle 2 that lies at the heart of the formalism. One possible avenue towards such a physical interpretation is looking at the mechanism behind the continuum limit where spacetime points are represented by local correlation operators 10. Thus the minimization of the action principle tries to reduce the correlation in the spacetime which one could try to interpret in an entropic sense as trying to maximize the amount of disorder in the spacetime. (continued)

5 In fact for the considerations later on the only thing we need is for the modification to allow for de-Sitter space as a solution and for the instability mechanism to arise in the right form, all without modifying the late time behaviour too much. The choice made here is simply to demonstrate that such modifications in fact exist and [38] shows that the compatibility with observations can be arranged.

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If the above scenario with a dynamical cosmological constant can be realized as a Causal Fermion System, then Conjecture 2, if true, would support this interpretation.

In the following section I will present a short discussion of an alternative modification of the Friedmann equations that superficially seems very similar to what I presented in this section but seems to fit more naturally in the CFS framework.

3.2 Dynamical Gravitational Coupling This section mainly serves to discuss some very general points about an alternative modification of the Friedmann equations. I include it here because at least intuitively it seems to lead to a very similar evolutionary behaviour of the universe as in Sect. 36 however it is more compatible with the CFS framework. For more details see upcoming work by Finster and Röken [18]. This section is not relevant for the understanding of the general picture that I will present in the following. I will usually refer to the modification in Sect. 3 which have been more widely considered in the literature. However I will reference this section whenever the two modifications are conceptionally different. If we look at the right hand side of the Friedmann equation (16) we see that R a −4 + M a −3 ∝ G and  ∝ . Hence if we are only interested in the question whether the universe is matter, radiation or  dominated, we can simply divide the whole equation by G which gives us a rescaling of H0 and the last term on the right hand side being  ∝ /G Hence if instead of letting G be constant and  ∝ H 2 we assume  to be constant but the gravitational constant G to change with G ∝ 1/H 2 , we arrive at the same conclusion (without considerations of back reactions due to a possibly different evolution of H ) for the question whether the universe is matter, radiation or 

6 This

claim has not been checked beyond the rough sketch of arguments that I will present here.

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dominated in a certain period. In the period between the CMB and now I would expect the difference between the two modifications to be small and hence it is only really the change of that ratio that is relevant for the global dynamics in that period. See for example [21] for a discussion of the compatibility of a dynamical gravitational constant with observations. In the period where one would expect a rapid change in either  or G in these modifications, the issue might be different. Of course I completely ignored here that by dividing the whole equation by G assigning the dynamics to G instead of  would also modify the left hand side. In particular a small change in G between the CMB time and now might easily fix the H0 tension between local observations [36] and CMB measurements by the Planck collaboration [33].7 It is important to note, that also with this modification—a dynamical gravitational constant—de-Sitter space is a stationary solution to the Friedmann equations. This is simply due to the fact that in this case we have ρM = ρR = 0

(33)

and thus the value of G plays absolutely no role. This modification is much more natural to the CFS framework as we have G ∝ L2P and LP is essentially the regularization length scale ε mentioned in Sect. 2.8 The detailed dynamics of such a dynamical gravitational coupling are however dependent on the choice of regularization. This leads to the following open question.

Correct Choice of Regularization For the CFS framework to make sense one needs to make a choice of a particular regularization. This leads of course to a whole array of questions on how to choose the correct regularization. It is important however to note that this dependence on the choice of regularization is not a fundamental problem, as long as we can choose it coherently for all pieces of the story line, as we do actually have a preferred coordinate system in the universe, see cf. [29]. Locally when one sets up a experiment in the lab, implicitly one prepares the experiment such that its local surrounding looks like a vacuum. Thereby one essentially chooses a local reference frame and its corresponding regularization when setting up the experiment.

7 This by itself is of course no feat as introducing an additional functional degree of freedom allows one to fit almost anything. 8 Note, that this suggests, that for the evolution of the universe only the ratio of the  and L2 is P relevant. This in a sense resolves the problem that there are two fundamental energy scales in the universe, as this ratio is dynamical in both scenarios. Hence only one of them sets a fundamental scale.

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Further we have the generic open question for this section

Derivation of Correct Modification The modifications discussed here are only a few of many possibilities. The choice made here is just to illustrate a point. Eventually one needs to be able to derive an adequate modification on the level of the Friedmann equation or in a more general setting from fundamental principles. Otherwise one performs just phenomenology. The considerations in this paper might serve as a guideline for the search of the correct modification.

4 Mechanism of Matter Creation In the following section I will discuss some preliminary results from my upcoming work with Finster [17]. It is important to note that many details are still not fully worked out, so anything I discuss here has to be taken with a grain of salt. Where the functional dependence is unclear I will indicate so by adding a # to the parameters. The core objective of [17] was to work out the origin of the matter-/anti-matter asymmetry in the context of the CFS framework. However I will argue here and in the following section that, provided the mechanism holds true, we achieve much more than that. In fact, the picture that emerges is that all matter and radiation in the universe could be a direct result of the matter-/anti-matter asymmetry. I will discuss the mechanism behind that observation in the present section and how we arrive from this considerations at a dark matter candidate in the next Sect. 5. It is probably fair to mention that obtaining a matter-/anti-matter asymmetry from the CFS framework by itself is an interesting and important result. This result is not entirely surprising however, given that the CFS framework has a built-in asymmetry how it treats the matter and anti-matter states, by choosing the Dirac sea as the relevant Hilbert space. Assuming the Dirac sea to be real and the regularization ε > 0 there is a finite number of states in the Dirac sea in a box or a finite density of states in the Dirac sea in R3+1 . Now if you change the regularization or enlarge the box, you will get a change in the number of states required to fill the Dirac sea. If you do both simultaneously it can be arrange that the two effects cancel out, you get a dynamical gravitational constant but no change in particles. On the other hand, if things play our right you will essentially get a sea level shift, either up or down. As long as the effect is of the right order it does not matter whether we are in a situation with a sea level shift up, i.e. particles created with minimum kinetic energy, or down, i.e. antiparticles created with zero kinetic energy. This last bit is important to understand. The sea level shifting from the Dirac sea configuration will fill the particle or empty the anti-particle states starting from the one with zero kinetic energy successively

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filling/emptying the states with higher and higher kinetic energy. This observation will play a central role in the considerations of the next section. The CFS Euler–Lagrange equation (CFS EL, Lemma 1) play a crucial role in the mechanism of matter creation described in [17]. Finster’s work in [15] shows that to leading order in ε, the CFS EL reproduce the classical field equations. Now to obtain the matter/anti-matter asymmetry we need to consider next to leading order corrections to the CFS EL. Hence all effects will come with a pre-factor of the regularization length scale ε ∝ LP . This suggests that the asymmetry can be seen as a quantum gravitational effect. The details will be presented in [17]. The only things relevant for my present discussion are the following. The mechanism currently seems to obey the pattern that a Friedmann-Robertson-Walker spacetime to leading order in the CFS EL maintains particle number conservation. Only once we add perturbations and solve the CFS EL to first order we get a non-zero effect. (a) Homogenous FRW + zeroth order solution to the CFS EL ⇓ particle number conservation

(b) Homogenous FRW + first order solution to the CFS EL ⇓ particle number conservation

(c) FRW+ perturbations + zeroth order solution to the CFS EL ⇓ particle number conservation

(d) FRW+ perturbations + first order solution to the CFS EL ⇓ change in particle number So the matter/anti-matter asymmetry (or as I will argue in the next section, the existence of all matter and radiation) in the universe arises dynamically due to an interaction between vacuum fluctuations and next to leading order, i.e. quantum gravity, corrections to the field equations. The CFS formulation that leads to the standard model consists of 8 fermionic sectors, 6 for the quarks (for each colour of each type one), one for the charged leptons and one for the neutrinos. Each sector contains exactly three generations i.e. the electron, the muon and the tau all live in the same sector. To get the correct results for the Standard Model out of the CFS

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framework, the violation of the chiral symmetry in the neutrino sector—i.e. that all observed neutrinos to this day have been lefthanded—has to be built in by hand. We make the following assumptions: for every fermionic generation the naked mass of the quarks and the charged lepton are identical,9 but the naked mass of the neutrino is different. This assumption is crucial as I will explain below. Along the notation in [15] I will use β ∈ {1, 2, 3} to label the fermionic generations. So for every generation we have the naked mass of the quarks and leptons denoted by mβ and the difference between mβ and the naked mass of the neutrino denoted by mβ . The density of particles created then obeys the following formula [17] dρpart icles = ε ∗ N(mβ , mβ , #) ∗ (3q1,β + 3q2,β + lβ + α(mβ , #) ∗ νβ ) dt

(34)

where q1,β , q2,β denote the two quarks in a generation, lβ stands for the charged lepton in a generation and νβ denotes a particle in the (anti-)neutrino masseigenstate of that generation.10 The factor ε ∗ N(mβ , mβ , #) is the number density of particles created per volume per unit of time11 and α(mβ , #) is a dimension less function that describes how many neutrinos are created in relation to the quarks and charged leptons. I will now make some assumptions on the functions N and α which are mostly based in preliminary results of our upcoming work [17]. As mentioned in the introduction to this section I use the # symbol to indicate possible further functional dependencies which are unclear in the current state of the work on [17]. I will here assume that N also depends on the amplitude of the perturbations which I will denote with A as well as on the value of the cosmological constant , hence N = N(mβ , mβ , A, ). For α I will consider no further dependencies, hence α = α(mβ ). Preliminary results for [17] suggest that for mβ = 0 we have α(0) = 1 and N(mβ , 0, A, ) = 0. Therefore the mechanism crucially requires for the naked mass of the neutrinos to be different from the naked mass of the quarks and the charged leptons. In the following I will therefore assume that mβ = 0 and treat α as a free parameter which one can adjust at will. In principle it can also be negative which would mean that a corresponding number of anti-neutrinos is created.

9 This leaves the effective masses of the quarks and charged leptons to be explained. Except for the tau, the effective masses of the quarks and their charged leptonic counterparts are within an order of magnitude. Recent calculations show that the binding energy in the strong force can account for almost all the mass in the neutrons and protons, see for example [40]. In that light I consider it plausible that the difference in effective masses within a generation can be accounted for by self-interaction effects. 10 Here I would like to emphasize that it is the mass-eigenstates that matter for the particle creation and not the eigenstates of the weak interaction. 11 “Time” here has to be considered a place holder for an appropriate parameter that tracks the history of the universe.

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For N the preliminary calculations suggest that for mβ1 < mβ2 we have that N(mβ1 ) 1 N(mβ2 ).

(35)

I will further assume that N is linear in A and that we have a strong suppression of the effect with declining . Hence that for 1 > 2 we have N(1 ) 2 N(2 ).12 For the time being the reason to choose these additional conditions is solemnly motivated by the fact that they resolve conceptional issues. Whether they are compatible with/can be derived from the fundamental framework remains to be seen. It is conceivable that the functional dependencies will turn out to be different and hence the story line presented in Sect. 6 will need to be adapted. In the next section I will discuss the consequence of Eqs. (34) and (35).

5 Reheating Uncertainty From here on for the rest of the paper I will assume that the matter creation will happens instantaneously. Hence after the creation event we have the following matter density ρM,βg for each generation of fermions ρM,βg (t0 ) = ε ∗ N(mβ , mβ , #) ∗ (3q1,β + 3q2,β + lβ + α(mβ , #) ∗ νβ ).

(36)

For now I will also assume that the kinetic energy at creation time is vanishing in comparison to the mass energy.13 First, I want to start this section with two thought experiments that will serve to put the later parts into context. Remark 2 (Thought Experiment 1) If an atom is coupled to an electro-magnetic field the probability for an atom to transition between two states is identical in both directions, i.e. for emission and absorption. So if I put an excited atom into a perfect cavity, it will decay, emit a photon and then sit around. The photon will bounce around and eventually be reabsorbed by the atom. You can play the movie forward and backward and everything is fine. Now suppose we play this game in the universe where we place the mirror to reflect the emitted photon one billion light years away. All technicalities aside, even if the photon manages to return to the atom it will never be reabsorbed because it has redshifted due to the expansion. The expansion of the universe therefore acts as a dispersion mechanism. Without which quantum 12 One way to motivate this is to look at the stationary holographic equipartition in [28]. Where an equality between the gravitational degrees of freedom in the bulk and the boundary is stated for stationary spacetimes, hence Nbulk = Nboundary . With (26) this leaves Nbulk ∝ −1 . Together with Eq.√ (21) this leaved the density of degrees of freedom in a spacelike hypersurface to be N/V ∝  which decreases with decreasing . It is conceivable that this could reduce the rate of particles created. However for the present work we will just impose this reduction as a condition. 13 Remember that in Sect. 4 I discussed the fact that the newly created particles fill the kinetic states starting from the one with the least momentum.

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mechanical systems would not tend towards the ground state even if coupled to electro-magnetic fields. The second thought experiment is more closely related to the actual process I will be discussing in this section. However the first one ties into it as well. The following thought experiment is based on the observation that “free” energy (i.e. energy not bounded as mass) has a different effect on the evolution of the universe whether it is bounded in a particle or present as radiation. This can be easily seen from Eq. (16). Remark 3 (Thought Experiment 2) Suppose you have a universe with one sort of atoms that have a rest mass of m1 and one excited state that contains E2 >> m1 c2 in energy. Now suppose all the atoms are excited in the beginning and they are entangled (so the excited states decay simultaneously) then the expansion history of the universe looks completely different depending on how long the excited state takes to decay. Before the decay the expansion is matter dominated, after the decay the expansion is radiation dominated and eventually goes back to matter dominated. So now we are in the curious situation where we have a universe for which we know the expansion history only in a probabilistic sense even though we treat gravity completely classical. This second thought experiment brings us to one of the many uncertainties regarding the reheating process originating from the matter creation mechanism introduced in Sect. 4. Now first I need to discuss how we arrive at a reheating scenario from the considerations in the previous section. Here the condition (35) plays a crucial role. We will assume that m3 > m2 > m1 and therefore we get that N(m3 ) 2 N(m2 ) 2 N(m1 ) and thus for all practical purposes we can effectively assume N(m2 ) = N(m1 ) = 0 and thus all matter being created in the third generation of fermions i.e. ρM,3g (t0 ) = εN(m3 , m3 , A, )(3t + 3b + 1τ + α(m3 )ν3 ).

(37)

This leaves us exactly in the scenario of the second thought experiment. For simplicity we will assume here an instant reheating at t = trh ≥ t0 . In this context, reheating means a decay of the tops, bottoms and taus into first generation fermions plus an amount of leftover “free” energy, i.e. radiation ρM,3g (trh ) =

ρ(t0 )a(t0 )3 = ρM (trh ) + ρR (trh ). a(trh )3

(38)

Here ρM (trh ) represents the energy density bounded in the mass after reheating, i.e. after the decay, and ρR (trh ) represents the energy density set free as radiation in the course of the reheating process. Now it is clear that the actual reheating process in this scenario is a very complicated process and precise computations will require the insight of people more proficient in high energy physics than myself. So I will present only some very simple calculations here and I will have to leave the precise treatment to future research.

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For a first approximation I will assume that in the initial state, all the energy is stored in the mass of the third generation fermions. That means that there are no composite third generations particles and hence no component in the energy density coming from the binding energy. Further I will assume that in the final state after the reheating there is no binding energy stored in the first generation quarks. In this case we would have ρR (trh ) ≈ εN(m3 , m3 , A, , trh )(3t + 3b + 1τ )

(39)

ρM (trh ) ≈ εN(m3 , m3 , A, , trh )(3u + 3d + 1e + α(m3 )ν3 ) + ρνμ ,νe

(40)

where ρνμ ,νe represents the collection of all hot muon neutrinos and electron neutrinos that are created in the reheating process. If we ignore the uncertainty connected to that term for a moment we would get that ρR (trh ) ≈ 105 ρM (trh )

(41)

To me it is at the moment absolutely unclear how much binding energy there is in the initial state before the reheating, however it is clear that the baryonic energy density in the universe is far greater than just the mass of the up and down quarks. From the simple calculations above we would get an equal number of neutrons and protons.14 However we know from observations that the neutron to proton ratio at the neutrino decoupling time t = tν is close to 1/6 and hence we have

ρM (tν ) ≈ εN(m3 , m3 , A, , tν )

12 2 12 mp + mn + me + αmν3 7 7 7

 + ρνμ ,νe (42)

Let us ignore ρνμ ,νe for the moment. Further assume that neutrons and protons have the same mass. Given that for the moment we can treat α as a free parameter, in this scenario the third generation (anti-)neutrino mass-eigenstates make up all the Cold Dark Matter in the universe.15 This leaves us with 85% ρCDM |α|mν3 = . = 15% ρB 2p

(43)

) we take ρM (trh ) ∝ (1p + 1n + 1e + αν3 ) + ρνμ ,νe the ratio ρρMR (t(trh ≈ 102 drops significantly. rh ) Adding binding energy to the third generation fermions prior to reheating would increase that ratio on the other hand. 15 A more precise formulation would be that this scenario is compatible with a universe where third generation (anti-)neutrino mass-eigenstates make up Cold Dark Matter. If other observations would in fact rule out third generation (anti-)neutrino mass-eigenstates this would unfortunately not suffice to kill the narrative presented here as we can—at present—simply adjust α to the bounds given by those experiments. In that case however the scenario presented here would of course fail to account for Cold Dark Matter, which would weaken its case. We hope to be able to constrain α from fundamental considerations in future research. 14 If

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The mass of the proton is roughly 940 MeV the upper bound for the mass mν3 is roughly 15 MeV which gives us a lover bound for |α| |α| ≥ 710.

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Hence if the picture discussed here were to be true we should expect to find roughly 500 third generation (anti-)neutrino mass-eigenstates for every neutron and proton in the universe. I would be surprised if that abundance had no observational effect. I will come back to that in Sect. 8.16 If the picture presented in this section is true than the entire physical content of the universe is initially created in equal numbers of tops and bottoms, albeit the tops dominating due to their mass.17 For the moment I have to leave two minor and three major questions in this section open.

Radiation to Mass Energy Ratio Deriving a precise ratio of the energy density of radiation and matter after the reheating phase might prove to be a very challenging problem. Note that if one manages to derive sharp relative values for ρM (trh ) and ρR (trh ) this would (a)4 allow to fix ρeq = ρρM(a) 3 from first principles. This ratio plays a central role R in the considerations of Padmanabhan [29] and is in fact a time-independent, observable quantity in the universe.

Connected to that is the first minor questions:

Correct Neutrino Decoupling From observations we get that the proton to neutron ratio is 6:1 at neutrino decoupling time. From the particles created in the beginning a decay strictly within the fermionic sectors (plus neutrinos and radiation) would lead to a proton to neutron ratio of 1:1. So the question is, whether in the scenario discussed here, the universe can get hot enough, such that it leads to full neutrino coupling (in the first generation of fermions), for a sufficient amount of time. Hence the number of protons and neutrons emerging in the right ratio along standard considerations when the universe cools due to the subsequent expansion.

16 Note that for example such an abundance would render the total matter content in the universe having a strong chirality asymmetry. 17 If this picture is true and the universe was created by some sort of god, she certainly must have a good sense of humor.

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The second major question is the question of whether the third generation (anti-)neutrino mass-eigenstates will in fact remain un-thermalized in the reheating process.

(Missing) Thermalization of Tau Neutrinos At first glance the assumption that the third generation (anti-)neutrino masseigenstates remain largely unaffected by the reheating process seems plausible to me. The fermions of their own generation which they are most likely to interact with should be at low kinetic energies and it seems plausible that the reheating is too quick for neutrino oscillations to play a significant role before the neutrino decoupling time. Here the result in [37] might be important, as it states that neutrino oscillations occur at a different rate in a dense neutrino gas. At least intuitively this seems to be the situation in the phase we are interested here.

Further what I expect to be a rather minor question.

(Hot) Muon and Electron Neutrinos So far I have swept the term ρνμ ,νe under the rug. This term describes the energy density of the (hot) muon and electron neutrinos which are generated in the reheating process. If this term would turn out to be of significant magnitude it would have two effects: (1) it would shift the radiation to mass energy ratio (2) it would constitute a significant (hot) fraction of the Dark Matter sector.

And last but not lest the elephant in the room:

Can Neutrinos Constitute Dark Matter There are two constraints on the role of Neutrinos as Dark Matter in current observations. One originates form CMB observations and the constraints on hot dark matter obtained there from. I believe that those bounds can not be applied to the here presented scenario based on the following considerations: For this to work I need to assume that the third generation (anti-)neutrino mass-eigenstates will indeed pass the phase of reheating largely unaffected and will not be thermalized. This implies, that the third generation (anti-)neutrino mass-eigenstates remain in the kinetic state they were created in. Hence they remain approximately in a Fermi ground-state distribution where the particles are in the configuration with the lowest possible kinetic energy allowed by the Pauli principle. Hence the εN(m3 , m3 , A, , tν )αmν3 part of ρM (tν ) would constitute ultra cold Dark Matter while only the (continued)

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neutrinos that make up ρνμ ,νe constitute hot dark matter. Hence bounds on hot Dark Matter do not constrain the contribution of the third generation (anti-)neutrino mass-eigenstates in the scenario presented here. The second constraint originates from the Gunn-Tremaine bound for fermionic Dark Matter in galaxies[39]. It rules out neutral leptons with a mass mlept on  1 MeV as dark matter in galaxies based on an argument regarding the available phase space volume and the Pauli principle. Current direct mass bounds on the third generation neutrino mass which I found in the literature are mν3 ≤ 15.5 MeV. Which in principle still allows for third generation (anti-)neutrino mass-eigenstates to constitute Dark Matter in the galactic setting. Particularly since the third generation (anti-)neutrino mass-eigenstates would still be approximately in a Fermi ground-state for the galaxy (much similar to the electrons being in a Fermi ground-state when a star reaches the Chandrashekar limit).18 However there are indirect limits from neutrino oscillations that place a much lower bound on mν3 , which is well below the Gunn-Tremaine bound. There is one potential way to evade these indirect limits. In the scenario described here, we always observe neutrino oscillations in a dense neutrino gas and hence the considerations in [37] might play a role and the effective neutrino oscillations would be substantially altered from vacuum neutrino oscillations. Hence the indirect bounds might be too low.

What is feeding into all these questions is the fact that at present I completely neglected the particles created in the first and second generation of fermions. This has to be kept in mind when thinking about the questions above. In the following section I will discuss how the two previous sections can be fit together into one coherent story line for the evolution of the universe.

6 The Story Line In this section I will present a possible story line for the universe based on the discussions in Sects. 3–5. In [29] Padmanabhan argues that an arrow of time can emerge in a system with time symmetric equations if a suitable instability is present. This story line here is inspired by the ideas in the series of papers by Padmanabhan which was started with [23]. Most importantly I will consider the history of the universe to be that of a system going from a high energy (i.e.  = high ) to a low energy (i.e.  = low ) state. In contrary to the work of Padmanabhan however, I have the preliminary form of a mechanism that can describe the why and the how 18 This might actually have consequences for the initial evolution of supermassive black holes. However this has to be left to future work.

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of this transition. In Sect. 3 I discussed the fact that de-Sitter space is a stationary vacuum solution to the modified equations. Now if we allow for fluctuations in the metric and bring in the considerations from Sects. 4 and 5 one can see that the deSitter space with high becomes unstable. The key to the instability lies in Eq. (23). Before the fluctuations we have  = high and ρM = ρR = 0.19 As soon as the fluctuations set in the mass density ρM,3g of fermions in the third generation becomes non-vanishing and through the Friedmann equation (16) this leads to a change in the Hubble rate and a change in the radius rH of the causally interacting region. As soon as this leads to a decrease in the vacuum energy density more rapid than all other components on the right hand side of (16), i.e. if 6 drCH 4 d =− 3  − 5, da a rCH da

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the instability is triggered. In particular the vacuum energy density has to decay quicker than the matter density ρM ∼ a −3 and the radiation density ρR ∼ a −4 . Hence the system becomes matter or radiation dominated which increases drCH da substantially, decreasing the value of  even faster. The influence of the vacuum energy then becomes rapidly negligible and the universe goes from a high dominated phase to a matter/radiation dominated phase. This process continues 3 until the radius of the interacting region has expanded so far, that the factor 1/rCH 20 becomes dominant and slows the decay of . Once the instability is triggered the matter content ρM,3g decays as 1/a 3 until reheating time t = trh . As before we assume instant reheating for simplicity. After trh the radiation content and matter content evolve according to the standard picture with slight modifications due to the residual dynamic of . Eventually matter and radiation disperse and we are left with a de Sitter vacuum again albeit at much lower value of  = low . A schematic

19 It is important to realize that this period can in principle go on forever back in time. In fact if one considers time to be an emergent phenomenon describing a rate of change, such as was advocated by many discussants at the workshop “Progress and Visions in Quantum Theory in View of Gravity” at the Max Planck for Mathematics in Science in Leipzig in October 2018, then in this phase the concept of time ceases to make sense, because there is no such thing as change. Note that “past incompleteness” results such as [6] typically rely on geodesics, which implies the existence of a test particle. What I suggest here is that everything, all matter, radiation and even vacuum fluctuations have a creation time, i.e. their past directed trajectory is de facto incomplete. For radiation it is the reheating time, for particles the creation time and for fluctuations the point when they shrink below the Planck length. Let us extrapolate back in time. Let us consider the last fluctuation outgrowing the interacting region before creation time. We can trace that back to when it was smaller than Planck scale, hence when it disappeared. Nothing that happened before that moment plays any role for our universe today. One can therefore consider the pure, empty de Sitter space as an idealized past boundary condition. With the spacetime completely void of any structure an external observer could not tell the passage of time for the physical system she is looking at. On the level of the physical system any result that depends on a notion of test particles, i.e. geodesics, has no meaning if we assume the spacetime to be truly empty de Sitter space. 20 The future asymptotic behaviour is content of Conjecture 1.

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Fig. 2 The graph shows a sketch of the expected development in “time” (horizontal axis) of the energy densities (vertical axis) associated with radiation (red), matter (blue) and  (black) under the Friedman equations with a running vacuum model (22). The creation of matter in the third fermion generation and its evolution between the instability time t0 and the reheating time trh is sketched in green. I decided to use a separate color due to all the uncertainties associated with that phase. Also because it breaks up into matter and radiation at reheating time. The universe starts of in a high dominated phase, goes over into a short matter dominated phase right after the instability then becomes radiation dominated after reheating. Eventually it transitions to matter dominated and back to low dominated as in the usual picture. I used tnow to roughly indicate where in the evolution of the universe we are today and tCMB to roughly indicate where the CMB decoupled in this picture

representation of the evolution of the different energy components in the universe for this scenario can be found in Fig. 2. Note, although the scenario is inspired by the work of Padmanabhan, it is not reliant on it. In particular it is not clear, whether the here-presented scenario would lead to the “Cosmic Information” requirement, ⎛ ⎞ 3/2 ρhigh 4 1 ⎠ = 4π Ic = ln ⎝ 1/2 9π 27 ρ ρeq

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low

postulated by Padmanabhan, to be satisfied. In general it would be nice to have some sort of conservation law that links the different phases of the evolution of the universe.21 The following questions need further consideration:

21 It would be interesting to see whether the set of idea developed by Padmanabhan [23–31] can be linked in to the discussion presented here. That would allow to interpret Ic as a sort of conservation law for the transition. For this to be possible however a necessary requirement would be that Ic = 4π is true in the limit where the difference between the two energy levels goes to zero i.e. high = low , hence de Sitter space.

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Dynamical  and Nucleosynthesis After the instability is triggered the value of  decreases rapidly. However no matter how quick that happens it might still have a sizable effect on the processes in the early universe. This phase between the instability and the CMB time in particular also contains the phase of nucleosynthesis. Whether and how this process would be altered and whether it would still be compatible with observations, remains an open question.

Initial Condition What fixes the value of high ?

Trigger of Instability The scenario discussed above ignores two crucial point: • Why does the instability trace out a geodesic hypersurface (and thus lead to an observable Friedman reference frame in the universe)? • How does the instability trigger over a large enough region simultaneously (note that this involves regions that are non-interacting at the time) Essentially these are the questions associated with any inflationary scenario. Here of course the assumption to have no fluctuations until the instability is triggered makes no sense. But that condition was just introduced for the sake of the argument. In practice a detailed study of the interplay between fluctuations, matter creation and the trigger of the instability would be necessary.

On top of the questions mentioned here are of course all the ones mentioned in the sections before.

6.1 Slow Roll to Instability The problem regarding the initial conditions might be circumnavigated by taking into considerations that an empty de-Sitter space without fluctuations can never exist. Hence having the initial high state sitting around for an arbitrary amount of time before the fluctuations kick in is unrealistic. What one could assume instead

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is that we start of with  = 3/L2P and hence rCH = LP .22 In the beginning, any matter that is created is dispersed so fast that it is almost neglectable on the right CH hand side of (16). This leaves drda so small that despite rCH being very small the instability condition (45) is not satisfied. Therefore the value of  decreases slowly until it reaches a value small enough (albeit still very high), that the matter creation can exceed the dispersive effect and the instability is triggered.

6.2 Dynamical Gravitational Coupling A slightly different scenario occurs when we consider a dynamical gravitational constant (and hence a dynamical regularization) and a fix cosmological constant. In this case we would end up with a transition from a de Sitter vacuum with a small gravitational constant (i.e. small regularization length) to a de-Sitter vacuum with a large gravitational constant (i.e. a large regularization length). The picture should be again similar to before: • • • • •

De Sitter vacuum (G = Glow ) Fluctuations create matter Instability triggered (almost) FRW evolution back to De-Sitter vacuum (G = Ghigh )

Now there is something particular about the initial and final state: As they are void of any matter or radiation the gravitational constant doesn’t matter and since we chose  to be constant we arrive at the conclusion that the physical size of the interacting region in the initial and the final state is identical. But wait, don’t we know that in FRW the size of the interacting region increases? So how can the physical size of the interacting region be the same in the beginning and the end? The following explanation seems plausible: For the matter interaction the instability time slice at creation time t = t0 acts as an initial data surface. Therefore the horizon for the matter interaction vanishes at that time and expands from there. The vacuum fluctuations prior to t0 are correlated across the entire interacting region and imprint on the matter distribution at creation time. However the matter interaction only starts once matter is created. In a sense that merges the Big Bang with the inflationary picture. As discussed before: If we assume the absence of fluctuations prior to t0 then in empty de Sitter space the notion of time becomes void of meaning. Physical time as a measurement of the rate of change comes into existence when the fluctuations start and matter is created. Of course vacuum fluctuations always exist and they trace out the pre t0 “inflationary” de Sitter phase.

22 It seems to me that this would ultimately be the highest possible value for  that makes any physical sense.

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This resolves two of the conceptional issues from the dynamical  consideration: • The size of the interacting region before the instability is identical to the size in the asymptotic future. That means, that an observer can never see any region beyond what was already interacting before the instability triggered. Therefore it is natural that the entire region of the CMB that we can see is correlated because it was in fact one interacting region before the instability triggered. Hence the vacuum fluctuations would naturally be correlated across the entire region. • It is sufficient (and plausible) for the instability to be triggered simultaneously across the interacting region. Hence across the maximal domain that we will ever be able to observe. • It seems conceivable that the vacuum fluctuations would be “homogenous” across the interacting region, and would hence trace out a geodesic hypersurface. Therefore the instability would also trace out a geodesic hypersurface, essentially giving us the FRW reference frame that we observe for the CMB. See Fig. 3 for an illustration of this scenario. One observation though that is somewhat worrisome is that the difference in the Gravitational constant in the beginning and the end, i.e. the different regularizations LP ,begin < LP ,end , imply SC,begin > SC,end .

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This can be seen as the gravitational degrees of freedom being transferred to the matter degrees of freedom.23 Those degrees of freedom then eventually leave the interacting region and thus carry away the degrees of freedom associated with them. A rough sketch of this scenario for the evolution of the universe is given in Fig. 4. As a remark on the side, it is interesting to note, that with a fix value of  we get in a sense a fix physical domain that is sufficient to describe everything we will ever be able to observe. Together with a finite regularization this should leave us with a finite volume, finite Hilbert space CFS. In this setting the variational problem (2) is known to be well-posed. The following questions need further consideration:

Instability Mechanism The biggest open question in this scenario is the question, how the instability mechanism could be triggered. It is plausible, that the same mechanism that leads to matter creation in the CFS framework, also leads to a change in the regularization and hence in Newtons coupling constant.

23 At the moment this is in fact my best guess for how the instability mechanism could work in this scenario.

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Fig. 3 Here you see a schematic picture of the evolution of the universe for the scenario with a dynamical gravitational coupling. The evolution starts at the bottom and ends at the top. The sketch shows the region around an observer living on the central world line. The fine black lines indicate the geodesic spread in the spacetime. The thick black lines on the left and the right mark the boundary of the interacting region to the past and the future. The red dashed line indicates the matter creation time t0 . The blue line depicts the boundary of the interacting domain for matter created on the central world line at t0 . The green line depicts the backwards light cone of the observer on the central world line at different times (purple dots). The pink dotted line indicates the rough location of the CMB. From the bottom up there is the first  dominated phase (G = Glow ) then we have the creation (or “Big Bang”) time t0 . This is followed by a short matter dominated phase before reheating. After that we have the normal sequence of radiation dominated-, matter dominated-,  dominated evolution (G = Ghigh ) as in standard cosmology. The sketch is intended to help the reader gain an intuition why the observer on the central world line can never see beyond the region that was causally interacting prior to t0 . Further the rough sketch for the interacting domain for matter shows how the standard FRW picture fits in

Suppression of Particle Creation As the matter creation mechanism comes in with the regularization length scale the dynamical gravitational coupling with increasing G (i.e. dynamical regularization) would in the present form enhances the effect as time progresses. This question needs careful investigation as this would obviously lead to problems with observations.

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Fig. 4 The graph shows a sketch of the expected development in “time” (horizontal axis) of the energy densities (vertical axis) associated with radiation (red), matter (blue) and  (black) under the Friedman equations with a dynamical gravitational constant (second vertical axis on the right). The evolution of the gravitational constant is freely made up to fit the picture and is here drawn in purple. The creation of matter in the third fermion generation and its evolution between the instability time t0 and the reheating time trh is sketched in green. I decided to use a separate color due to all the uncertainties associated with that phase. Also because it breaks up into matter and radiation at reheating time. The universe starts of in a  dominated phase with a small gravitational constant G, goes over into a short matter dominated phase right after the instability then becomes radiation dominated after reheating. Eventually it transitions to matter dominated and back to  dominated as in the usual picture but now with a gravitational constant significantly bigger than in the initial  dominated phase. I used tnow to roughly indicate where in the evolution of the universe we are today and tCMB to roughly indicate where the CMB decoupled in this picture

7 Discussion of the Causal Fermion Systems Framework Given the little consideration the CFS framework has received in the literature so far, I feel compelled to include at this point a discussion section of the most interesting results derived from it. This serves to demonstrate that the considerations in the present paper are part of a much larger picture. I will also include some new view points and how the framework might be linked to other considerations. Lets us first introduce a distance function along the lines of [13]: Definition 5 (Distance Function) We will refer to l(x, y)2 = (|xy|)−1/3 as the distance between two points.  xy Here |xy| = 4i=1 |λi | is just the sum of the absolute values of the eigenvalues of the operator xy. Note that |xy| has dimension [length−6 ] and therefore (|xy|)−1/3 has dimension [length2 ]. The main motivation to introduce this distance function is to establish another relation to the considerations in [28]. Note that in the CFS framework a point x is timelike separated from itself. However taking Definition 3 serious it lies neither in its own future nor its own past because by definition

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C(x, x) = 0 and thus we have no time direction. However one expects the following to hold Conjecture 3 (Zero Point Distance) In any minimizer we have that l(x, x)2 ∼ " 2 . Thereby the distance function would satisfy the modification made by Padmanabhan on the geodesic distance function when deriving the mesoscopic effects of internal degrees of freedom of gravity (i.e. mesoscopic effects of quantum gravity). Alternatively one could consider the following causal distance function: σ (x, y) = (L[Axy ])−2/3. This is just a simple redefinition of the Lagrangian (2) but now one can say that instead of minimizing the amount of correlation within a spacetime, minimizing the action, maximizes the point wise causal distance in the spacetime. The causal distance function has the property that σ (x, y) < ∞ for x, y causally separated and σ (x, y) = ∞ for x, y spacelike separated. The action can then be stated as:  S= σ (x, y)−3/2 dρ(x) dρ(y). (48) F×F

The link to Padmanabhan’s considerations would then be less clear. Another set of thoughts that one might try to relate to the CFS formalism, is the Events, Trees, Histories (ETH) interpretation of Quantum Mechanics, developed and propagated by Fröhlich [20]. The intuitive approach would be to interpret F as the set of all possible physical events over a Hilbert space. Correspondingly the support of the measure M would then identify those events that are actually realized in a physical system. We are currently investigating whether these two frameworks can indeed be connected in a meaningful way. It seems in general that the CFS framework would be compatible with any sort of relational interpretation of quantum mechanics.

7.1 Relevant Results I will here just state some of the results derived from the CFS framework that are the most interesting ones from a physical point of view in my opinion. They can be found in [15]. • Emergent spacetime and bosonic fields including field equations. • The CFS framework works only if there are at least 3 generations of fermions. • The gauge group of the Standard Model as well as the mixing matrices can be derived.24

24 It is unclear how rigid the framework is with that respect. For example, it is unclear whether one could accommodate for axions, by adding additional fermionic sectors.

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• The emergent spacetime has exactly one timelike direction. The maximum extension of the Clifford Algebra is five dimensional (i.e. if we reserve one dimension for chirality there are at maximum 3 spacelike dimensions). • When mapping a spacetime with fields into the CFS framework, taking a unitary transformation U of the resulting measure ρ gives us another minimizer ρ. ˜ A convex combination then has a smaller action. This is referred to as microlocal mixing. This procedure has an effective description in spacetime as a Fock space, i.e. as the second quantized description of the fields in the spacetime. In that sense, second quantization seems to emerge from the minimizing of the causal action principle! • The vacuum polarization graph comes out finite with the identical value as from renormalization. Given the complexity of the CFS framework and the small number of people proficient in it, I would advice a mild amount of precaution despite all these claims being backed up by peer reviewed papers.

7.2 Open Questions I will now mention three of the serious open questions that I am aware of regarding the CFS framework.

Missing Higgs The Higgs field has not yet been worked out. According to Finster, however this is work in progress and looks promising.

What Does It Tell You? So far from my point of view this is one of the biggest questions I have with regard to the framework. Its central object i.e. the action principle including all constraints, has no simple physical interpretation.

What Is Necessary for the Construction? It is unclear, which is the minimal set of observational facts, that one has to build into the framework by hand, to recover everything we know and love about the universe.

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8 Conclusion In the present work I discussed how the matter creation mechanism derived from the CFS framework paired with an adequate modification of gravity can lead to a coherent story line for the universe. Most importantly I discussed how the matter creation mechanism allows for Cold Dark Matter to be made up of third generation (anti-)neutrino mass-eigenstates with minimal kinetic energy. At this point I want to stress how essential the last point is. Despite the many gaps in the argument it is clear that the proposed Dark Matter candidate enters with the right kinematic. On top of that, the scenarios presented here would also resolve the problem of the matter/anti-matter asymmetry. Furthermore the scenario suggests a new angle on the issue of the initial “singularity” and the cosmological constant. Given, that the Cold Dark Matter candidate is within the Standard Model, I would expect this prediction to be testable with current technology or maybe even currently already available data. In the following list I collect some results that might be worth checking against this new prediction. • A residual creation of 3rd generation fermions and their decay could create a distinct fingerprint of high energy radiation in the late time universe. • Can the observed annual modulation in events in the DAMA/LIBRA detector [5] be explained with third generation (anti-)neutrino mass-eigenstates? The DAMA/LIBRA experiment observes an annual modulation in events below the 6 keV level in a NaI(Tl) scintillation detectors. This modulation is consistent with the signal one would expect based on the changing relative velocity of earth with respect to the galactic halo as it orbits the sun. • Does the chirality asymmetry in the Dirac sea or in the dense third generation (anti-)neutrino mass-eigenstate background have an effect on the results of a pear shaped nucleus reported in [8]? • One very interesting experimental result is the Bs0 and B 0 Meson decay into muon pairs investigates at the ATLAS Experiment at CERN [3]. This result is particularly interesting, as the mesons in question are made up of a bottom quark and a lighter partner. Hence an interaction with the dense third generation (anti)neutrino mass-eigenstate background might play a role. • In [22] a slight deviation from the Standard Model predictions is reported in an nuclei decay. They calculate the mass of the involved additional particle to be around 17 MeV which is roughly in the ball park of the bounds for mν3 . However if I understand right they suggest a bosonic particle of that mass, which would of course be incompatible with an interaction with neutrinos. • Neutrino experiments such as T2K [4] and MiniBooNE [1] might be sensitive to a dense third generation (anti-)neutrino mass-eigenstate background. • Observations show different lifetimes for neutrons whether they are kept in a bottle [32] or measured in a beam. Though it doesn’t seem very likely that this were connected to a dense third generation (anti-)neutrino mass-eigenstate background.

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• The absorption spectrum detected by EDGES [7] suggests that the gas in the early universe was colder than previously assumed. The fact, that the third generation (anti-)neutrino mass-eigenstate background in the here-presented scenario is created with minimal kinetic energy, might explain this observation if the gas in the early universe was able to loose enough kinetic energy to the cold third generation (anti-)neutrino mass-eigenstate background, i.e. if the third generation (anti-)neutrino mass-eigenstates can act as a heat sink. An important question is whether the all the background third generation (anti-)neutrino mass-eigenstates can exchange kinetic energy with the gas, or only the fraction that, due to neutrino oscillation, is in the electron neutrino weak interaction state. • It is unclear how a dense gas of fermionic particles effects early gravitational collapse in general and the growth rate of the first black holes in collapsing gas clouds specifically. These are all experiments and observations that I am aware of, that might be affected when taking a dense third generation (anti-)neutrino mass-eigenstate background into consideration.25 There might of course also be some experiments that fit Standard Model predictions perfectly well, that would be affected and could help to falsify the prediction of a dense third generation (anti-)neutrino mass-eigenstate background. As already discussed in the introduction the paper does not produce a complete theory, but it should be considered as a backbone for further investigations. The 15 open questions mentioned throughout the paper hopefully give a comprehensive picture of the gaps that remain to be filled. To conclude I think it is needless to say that a lot more research is necessary to put the ideas presented here on firm theoretical ground. Acknowledgements I would like to thank Felix Finster, Todd Oliynyk, Emmanuel Saridakis, Erik Curiel, Calum Robertson, Markus Strehlau, Mark Bugden, Marius Oancea and Isha Kotecha for listening patiently to my clumsy explanations in the early stage of the development of these ideas. I would like to thank Alice Di Tucci for explaining to me the debate surrounding the initial singularity and pointing me to relevant sources. I would like to thank Thanu Padmanabhan and Hamsa Padmanabhan for helpful discussions during my visit in Zurich. This work was supported by the Australian Research Council grant DP170100630. I am indebted to Ann Nelson, who brought the problem with the Gunn-Tremaine bound to my attention.

25 I will discuss some of the consequences for some of these experiments in more detail in an upcoming paper.

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Energy Inequalities in Interacting Quantum Field Theories Daniela Cadamuro

Abstract The classical energy conditions, originally motivated by the PenroseHawking singularity theorems of general relativity, are violated by quantum fields. A reminiscent notion of such conditions are the so called quantum energy inequalities (QEIs), which are however not known to hold generally in quantum field theory. Here we present first steps towards investigating QEIs in quantum field theories with self-interaction.

1 Introduction One of the fundamental observables both in quantum and in classical field theory is the stress-energy tensor T αβ . It has a special role in general relativity, as the Einstein equations link the curvature of spacetime to the distribution of matter throughout it. Certain positivity conditions on the stress-energy tensor (e.g., the so called “weak energy condition”), which are fulfilled by many classical matter fields, imply severe constraints on exotic spacetime geometries. They also enter the Penrose and Hawking singularity theorems [19], positive mass theorems [22], and Hawking’s chronology protection results [18], among many others. However, in quantum field theory (QFT) these energy conditions are violated; the energy density can have negative expectation values. This raises the question whether the energy conditions on the matter in the assumptions of the singularity theorems are compatible with quantum matter, or whether quantum fields allow the existence of “exotic” spacetimes like time machines, wormholes, warp drives. To exclude these scenarios, there must be constraints to the extent in which quantum fields can cause negative energy densities. These constraints are called “Quantum Energy Inequalities” (QEIs). First investigated by Ford [15], they are reminiscent of

D. Cadamuro () Universität Leipzig, Institut für Theoretische Physik, Leipzig, Germany e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 F. Finster et al. (eds.), Progress and Visions in Quantum Theory in View of Gravity, https://doi.org/10.1007/978-3-030-38941-3_5

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classical energy conditions, suggesting that the singularity theorems can still hold for realistic matter [10]. These inequalities formally go over into the Averaged Null Energy Condition (ANEC) [14] when the stress-energy tensor is averaged over a null geodesic. The ANEC has recently received considerable attention in the context of holography due to its relation with the quantum information carried by black hole horizons (see, e.g., [6, 7, 17]). Violations of the classical energy conditions must exist in any quantum field theory, since the vacuum expectation value of the energy density is supposed to vanish. For free quantum fields, it is straightforward to construct examples of states where the energy density is locally negative. For free scalar bosons this property does not arise by evaluating the energy density in fixed n particle states, but by considering superpositions, for example, of the vacuum and a two-particle state. Namely, local negativity of the energy density appears as a quantum interference effect, and does not occur in the classical regime. As another example of its intimate relation to quantum effects, consider the Casimir effect where an attractive force is generated between infinitely extended parallel plates in the vacuum. One can explicitly compute the stress-energy tensor of the electromagnetic field and find that the energy density between the plates is negative, and depends on the inverse distance between the plates. In a quantum field theory with interacting particles the situation is more involved. One finds that in certain models with interacting bosons, the energy density can be negative also in states of fixed particle number. Namely, in the class of lower dimensional quantum integrable models, one can find one-particle expectation values with locally negative energy density [2, 5]. Hence, negativity of the energy density is more profound in theories with interaction, and existence of QEIs is far from obvious and poses a challenging question. Integrable models are a special class of 1 + 1-dimensional QFTs, where the twoparticle scattering process characterizes the theory completely. They are constructed as an inverse scattering problem, specifically, given a function S2 as a mathematical input, one constructs the corresponding QFT having this two-particle scattering function. There are several simplifications when considering integrable theories. First of all, unlike most interacting field theories they can be represented on a Fock space, with the interacting vacuum given by the usual Fock vacuum, allowing for explicit computations. Further, they are amenable to a treatment in a non-perturbative setting [21], avoiding to deal with formal power series whose convergence is generally unknown. Physically, they are toy models for interaction, but share interesting common features with interacting theories in higher dimensions. For example, the nonlinear O(N) sigma models [1] are linked to experimentally realizable situations in condensed matter systems. They can also be regarded as simplified analogues of four-dimensional nonabelian gauge theories, in as much as they share crucial features with them, including renormalizability, asymptotic freedom, and the existence of instanton solutions.

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Then there are models that can support a representation of a gauge group or a group of internal symmetries, for example, the 1+1 dimensional SU (N)-symmetric models (such as the chiral Gross-Neveu and principal chiral models) which are of importance in physics as toy models of quantum field theory with relations to both gauge and string theory, and have been analyzed both in general and in the limit N → ∞. Finally, the Ising model is widely known also for its counterpart in statistical mechanics in the context of lattice spin systems. Therefore, integrable systems provide a “landscape” of possible interactions, where one may hope to obtain hints for the abstract conditions that underlie the phenomenon of QEIs. Mathematically, QEIs are lower bounds for the smeared energy density, , T 00 (g 2 ) = dt g 2 (t)T 00 (0, t) of the form ϕ, T 00 (g 2 )ϕ ≥ −cg ϕ 2

(1)

for all suitably regular state vectors ϕ and all real-valued test functions g, where cg > 0 is a constant depending only on g. However this inequality may not hold in all physical applications, e.g., in the non-minimally coupled scalar field in a curved spacetime only a weaker form of this inequality can hold [13], where the constant cg in Eq. (1) may depend on the total energy of the state. QEIs of the form (1) have been proved for the linear scalar field, linear Dirac field, linear vector field, both on flat and curved spacetime, the Rarita-Schwinger field, and for 1 + 1 dimensional conformal fields (see [9] for a review). Weaker forms of quantum inequalities have been proved for certain “classically positive” expressions in [3], but without a clear relation to the energy density. Only recently, a state-independent QEI has been established for the massive Ising model [5], which represents the first result to our knowledge of a QEI in a selfinteracting situation. Partial results have been obtained later in a larger class of “scalar” integrable systems, including the sinh-Gordon model [2]. In the next sections we will summarize the results of [5] and [2].

2 QEIs in Integrable Systems at One-Particle Level In this section we summarize some results on QEIs in integrable systems with one species of scalar bosons. Some of these, for example, the sinh-Gordon model [16], can be derived from a classical Lagrangian, and a candidate for the energy density can be computed directly from the Lagrangian. Other models in this class, for example the generalized sinh-Gordon model in Table 1 of [2], are not associated with a Lagrangian, and it is therefore not a priori clear what one should regard as the stress-energy tensor.

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Our first task is therefore to find an intrinsic characterization of the stress-energy tensor T αβ from the generic properties of this observable, and independent of the specific form of scattering matrix. We focus our attention on one-particle matrix elements of T αβ , which is generically given as an integral kernel operator:  ϕ, T αβ (g 2 )ψ =

dθ dη ϕ(θ )F αβ (θ, η)ψ(η),

(2)

where ϕ, ψ are vectors in the single-particle space L2 (R). The generic integral kernel F αβ is restricted by various properties of the stress-energy tensor: locality of the field T αβ (t, x), symmetry of the tensor T αβ , covariance under Poincaré transformations and spacetime reflections, the continuity equation (∂α T αβ = 0), , and the fact that the (0, 0)-component of the tensor integrates to the Hamiltonian ( dx T 00 (t, x) = H ). We can show that these requirements are necessary and sufficient for F αβ to have the form [2, Prop. 3.1]: αβ F αβ (θ, η) = Ffree (θ, η) P (cosh(θ − η))Fmin (θ − η + iπ) g2 (μ cosh θ − μ cosh η), ./ 0 =:FP (θ−η)

(3) where the individual factors are as follows: P is a real polynomial with P (1) = 1, αβ μ > 0 is the mass of the particle and ∼ denotes Fourier transform. Ffree stands for the expression of the “canonical” stress-energy tensor of the free Bose field, αβ Ffree (θ, η)

μ2 = 2π



  1  cosh2 θ+η + η) 2 2 sinh(θ   . 1 2 θ+η 2 sinh(θ + η) sinh 2

(4)

Fmin is the so called minimal solution of the model [20], which encodes the dependence on the scattering function S2 in a unique way. For the free field, Fmin (ζ ) = 1 and for the Ising model, Fmin (ζ ) = −i sinh ζ2 ; for the sinh-Gordon model, see [16]. The function FP in Eq. (3) now determines the negativity of the energy density and (non-)existence of QEIs. Negative Values of the Energy Density If there is a θP ∈ R such that |FP (θP )| > 1, then there exists a one-particle state ϕ ∈ L2 (R) and a real-valued Schwartz function g such that ϕ, T 00 (g 2 )ϕ < 0 [2, Prop. 4.1]. As one can see from the above examples of Fmin , this is fulfilled in the case of the Ising and sinh-Gordon models, but not for the free field if P = 1. In fact, it is known that for free bosons the one-particle energy density is positive. Existence of QEIs The existence of QEIs is determined by the behaviour of FP for large arguments. Namely, if |FP (ζ )| ≤ c cosh Re ζ in a small strip around the

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real axis and with a constant 0 < c < 12 , then ∀ϕ ∈ D(R) :

ϕ, T 00 (g 2 )ϕ ≥ −cg ϕ 2

(5)

for all real-valued Schwartz function g. That is, a state-independent QEI exists at least at one-particle level [2, Thm. 5.1]. If, on the other hand, FP (θ ) ≥ c cosh θ for c > 12 and for large θ , then an inequality of the type (5) cannot hold [2, Proposition 4.2]. This is a no-go theorem on existence of QEIs. Form of the Energy Density We can see from the above that requiring a QEI to exists restricts the form of the energy density, namely the choice of P , sometimes fixing it uniquely. In particular, in the Ising model, where Fmin (θ + iπ) grows like cosh θ2 at large values of θ , a QEI holds if and only if P ≡ 1. Instead, in the free and sinh-Gordon models, Fmin converges to a constant for large θ , thus a QEI can hold only if deg P = 0, 1. This means that we are left with the choice P (x) = (1 − α) + αx

with α ∈ R, |α| 0, − < 0: a fakeon plus has a positive width, while a fakeon minus has a negative width. For ± → 0± we get ¯ m ¯ ± )] ∼ πZδ(p2 − m ¯ 2 ). lim Im[∓ZG+ (p, m,

± →0±

(4.2)

In the case of a physical particle, we would find exactly the same result, which means that if we just watch the decay products of a fakeon, we have the illusion of a true particle. As said, the resummation (4.1) is legitimate only if p2 − m2 is large enough. With physical particles, analyticity allows us to reach the peak straightforwardly. However, fakeons just obey regionwise analyticity, so we must be more careful. Indeed, the resummation misses the contact terms δ(p2 − m2 ), δ  (p2 − m2 ), etc. In general, the sum of such contact terms plus (4.2) gives σ πZδ(p2 − m ¯ 2)

(4.3)

for ± → 0± , where σ = 1, 0, −1 in the case of a physical particle, a fakeon and a ghost, respectively. Formula (4.3) tells us that if we try to detect the fakeons “on the fly”, we do not see anything. With a physical particle, instead, what we obtain from the indirect observation, given by formula (4.2), coincides with what we obtain from the direct observation, given by formula (4.3). Finally, in the case of a ghost, we have the illusion of a particle if we observe its decay products, but get an absurdity (a “minus one particle”), when we try and observe it on the fly. The properties just outlined appear to justify the name “fakeon”, or fake particle. The fakeon can only be virtual, so the only way to reveal it is by means of the interactions it mediates. Since χμν is a fakeon minus, its width χ is negative. In the case of the GFF theory, we find [9] χ = −C

m3χ 2 MPl

,

C=

1 (Ns + 6Nf + 12Nv ), 120

(4.4)

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where Ns , Nf and Nv are the numbers of (physical) scalars, Dirac fermions (plus one half the number of Weyl fermions) and gauge vectors, respectively. We are assuming that the masses of the matter fields are much smaller than mχ , otherwise we have to include mass-dependent corrections. Note that the graviton and the fakeons do not contribute to χ . In the GSF theory there is another contribution due to φ, which depends on mφ . The negative width is a sign that microcausality is violated. However, it is not the only way such a violation manifests itself, as we explain in the next section.

5 Projection and Classicization The generating functional (gμν , φ, χμν , ) of the one-particle irreducible correlation functions can be formally projected by integrating out the fakeons, using the fakeon prescription. This operation gives the physical  functional. In some sense, the fakeons can be viewed as auxiliary fields with kinetic terms. For simplicity, consider an unprojected  functional (ϕ, χ), where ϕ denotes the physical fields and χ denotes the fakeons. Solve the fakeon field equations δ(ϕ, χ)/δχ = 0 by means of the fakeon prescription and denote the solutions by χ. Then the physical, or projected,  functional pr is pr (ϕ) = (ϕ, χ). Since the fakeons are not asymptotic states, at the level of the functional integral it is sufficient to set their sources Jχ to zero:  Zpr (J ) =

    [dϕdχ] exp iS(ϕ, χ) + i J ϕ = exp iWpr (J ) ,

so pr (ϕ) is the Legendre transform of Wpr (J ). Indeed, the unprojected formulas  (ϕ, χ) = −W (J, Jχ ) + ϕ=

δW (J, Jχ ) , δJ

 Jϕ +

Jχ χ,

χ=

δW (J, Jχ ) , δJχ

J =

δ(ϕ, χ) , δϕ

Jχ =

δ(ϕ, χ) , δχ

turn into the projected ones  pr (ϕ) = −Wpr (J ) +

J ϕ,

ϕ=

δWpr (J ) , δJ

J =

δpr (ϕ) , δϕ

when Jχ = 0. In the classical limit, the fakeon prescription and the fakeon projection simplify. In particular, the average continuation plays no role, because there is no loop

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integral, so we can take (2.9) as it stands, which gives the Cauchy principal value: 1 p 2 − m2 =P 2 . (p2 − m2 )2 + E 4 p − m2

(5.1)

To illustrate the projection in a simple case, consider the higher-derivative Lagrangian LHD =

m 2 (x˙ − τ 2 x¨ 2 ) + xFext (t), 2

(5.2)

where x is the coordinate, m is the mass and τ is a real constant. The unprojected equation of motion is mK x¨ = Fext ,

K = 1 + τ2

d2 , dt 2

while the projected equation reads [17] 1 mx¨ = P Fext = K



∞ −∞

du

sin(|u|/τ ) Fext (t − u). 2τ

(5.3)

We see that the external force is convoluted with an oscillating function, so the future (u < 0) contributes as well as the past. This is how the violation of microcausality survives the classical limit. As for the classicization of quantum gravity in four dimensions, the unprojected field equations derived from (3.2) are % 1 κ 2 $ 3κφ μν μν R μν − g μν R = ˜ κφ , ) + f Tφ (g, ˜ φ) + Tχμν (g, χ) , e f Tm (ge 2 ζ (5.4) for the metric tensor, and −

1 −g˜

∂μ



 m2   κe3κφ μν κφ φ eκφ − 1 eκφ = Tm (ge −g˜ g˜ μν ∂ν φ − ˜ , )g˜μν , κ 3ζ

1 δSχ (g, χ) μν μν = e3κφ f Tm (ge ˜ κφ , ) + f Tφ (g, ˜ φ), √ −g δχμν

(5.5)

√ μν from the variations of φ and χμν , where TA (g) = −(2/ −g)(δSA (g)/δgμν ) are the energy-momentum tensors (A = m, φ, χ) and f = det g˜ ρσ / det gαβ . The fakeon projection of the GSF theory is obtained by solving the second line of (5.5) by means of the classical fakeon prescription, i.e. the Cauchy principal value, and inserting the solution χμν  into the other two equations. In the GFF theory, we

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have to solve both Eqs. (5.5) by means of the classical fakeon prescription and insert the solutions φ, χμν  into (5.4). The projected equations can also be obtained from the finalized classical actions GSF SQG (g, φ, ) = SH (g) + Sχ (g, χ) + Sφ (g, ¯ φ) + Sm (ge ¯ κφ , ), GFF (g, ) = SH (g) + Sχ (g, χ) + Sφ (g, ¯ φ) + Sm (ge ¯ κφ , ), (5.6) SQG

respectively, where g¯μν = gμν + 2χμν . The interim, unprojected actions (3.1) and (3.2) are local, while the finalized actions (5.6) are nonlocal. These properties remind us of the gauge-fixed actions, which are local, but unprojected, and become nonlocal (with most types of gauge-fixing conditions), once the Faddeev-Popov ghosts and the temporal and longitudinal components of the gauge fields are projected away. However, there is an important difference between the fakeon projection and the gauge projection, since the former acts on the initial, final and intermediate states [|a, |b and |n in formula (2.5), respectively], but not on the virtual legs inside the diagrams, while the latter also acts on the virtual legs. Thus, the gauge-trivial modes completely disappear, while the fakeons leave an important remnant, which is the violation of causality at energies larger than their masses. The masses of the fakeons are free parameters. If their values are sufficiently smaller that the Planck mass, we may be able to detect the violation of microcausality in the foreseeable future. Moreover, formulas (5.6) show that the violation of microcausality survives the classical limit. As said, the fakeon prescription is not classical, but emerges from the loop corrections. The projected actions (5.6) must be understood perturbatively, since the parent quantum field theory that generates them is formulated perturbatively. Thus, the classicization is also perturbative and shares many features with the quantum theory it comes from, like the impossibility to write down “exact” field equations and the important roles played by asymptotic series and nonperturbative effects [18]. As far as we know, this backlash of the quantization on the classical limit is unprecedented. The nonrelativistic limit can be taken after the fakeon projection and, possibly, the classicization. The fakeon propagator tends to the real part of the usual quantum mechanical kernel. Note that both fakeons and antifakeons contribute. For an analysis of nontrivial issues concerning the nonrelativistic limit of quantum field theory, see Ref. [19].

6 The Upgraded Correspondence Principle In this section we summarize the lessons learned from the previous ones in connection with the correspondence principle and extend them to quantum field theories of particles and fakeons in arbitrary spacetime dimensions.

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Unitarity The unitarity requirement is unmodified, but better understood, since it makes room for both particles and fakeons. Locality The locality assumption must be upgraded, in the sense that it applies to the interim classical action. The finalized classical action is generically nonlocal, like the S matrix and the generating functional  of the one-particle irreducible diagrams. Proper Renormalizability The renormalizability requirement, applied to the interim classical action, must be formulated more precisely, since the usual notions are too generic. We must demand proper renormalizability, which is a refinement of strict renormalizability. It means that the gauge couplings (including the Newton constant) must be dimensionless (with respect to the power counting governing the ultraviolet behaviors of the correlation functions), while the other physical parameters must have nonnegative dimensions in units of mass. The standard model does show that the gauge couplings have this particular status among the couplings, so quantum gravity should conform to that. We regard the three principles just listed as the cornerstones of the correspondence principle of quantum field theory, and in particular quantum gravity. If we remove the locality assumption, for example, we must guess the S matrix or the  functional directly, which are infinitely arbitrary. So doing, we have no way to determine the theory exhaustively, since, as stressed in the introduction, when we explore the infinitesimal world we cannot make infinitely many observations in a finite amount of time and/or without disturbing the system. If we remove unitarity, we open the way to the presence of ghosts, which leads to absurd behaviors. If we renounce renormalizability, then we can just be satisfied with the nonrenormalizable, low-energy theory of quantum gravity, obtained from the Einstein-Hilbert Lagrangian plus the counterterms turned on by renormalization [20]. In addition to the three basic requirements, we must include fundamental symmetries, like Lorentz invariance, general covariance and gauge invariance. Other properties are important, but not so much as to elevate them to fundamental principles. Among those, we mention causality and analyticity, which are downgraded to macrocausality and regionwise analyticity, respectively.

6.1 Uniqueness Is the resulting correspondence principle sufficient to point to a unique theory? Various signals, like the arbitrariness of the matter sector of the standard model, tell us that this might be a utopian goal. However, we do have uniqueness in quantum gravity and a sort of uniqueness in form of the gauge interactions. Let us start from flat space. In every even spacetime dimensions d  4 the correspondence principle made of unitarity, locality, proper renormalizability and Lorentz invariance determines the gauge transformations [21] and the form of the

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interim classical action, which reads  $ % √ 1 d a SYM = − P(d−4)/2(D 2 )F aμν + O(F 3 ) , dd x −g Fμν 4

(6.1)

a denotes the field strength, D is the covariant derivative, P (x) is a where Fμν n real polynomial of degree n in x and O(F 3 ) are the Lagrangian terms that have dimensions smaller than or equal to d and are built with at least three field strengths and/or their covariant derivatives. The quadratic terms have been simplified by means of Bianchi identities and partial integrations. As per proper renormalizability, the gauge coupling is dimensionless. The coefficients of the polynomial P(d−4)/2 must satisfy suitable restrictions. In particular, after projecting away the gaugetrivial modes, the massless poles of the propagators must have positive residues and must be quantized as physical particles. The other poles must have squared masses with nonnegative real parts. The poles with negative or complex residues, as well as those with positive residues but complex masses, must be quantized as fakeons. Finally, the poles with positive residues and nonvanishing real masses can be quantized either as fakeons or physical particles. If we also demand microcausality, i.e. forbid the presence of fakeons, the set of requirements implies that the spacetime dimension d must be equal to four. Then the action is the Yang-Mills one,

SYM

1 =− 4



√ a F aμν . d4 x −gFμν

(6.2)

Although the interim classical actions (6.1) are essentially unique, i.e. they contain finite numbers of independent parameters, we emphasize that the gauge group remains free, as long as it is unitary and (together with the matter content) satisfies the anomaly cancellation conditions (which are other consequences of unitarity). In other words, the correspondence principle fails to explain why the gauge group of the standard model is the product of the three simplest groups, U (1), SU (2) and SU (3), instead of anything else. For example, we cannot say why factors such as SU (13), SU (19), etc., are absent. It also fails to predict the matter content of the theory that describes nature. Indeed, we are allowed to enlarge the standard model at will, to include new massive particles and/or massive fakeons, as long as they are heavy enough (to have no contradiction with experimental data) and the anomaly cancellation conditions continue to hold. The ultimate theory of nature could even contain infinitely many matter fields. In this respect, the correspondence principle is almost completely powerless. So far, every attempt (grand unification, supersymmetry, string theory and so on) to relate the matter content to the gauge interactions, beyond the anomaly cancellation conditions, has failed. Probably, this is a sign of the fading correspondence. Nevertheless, quantum gravity turns out to be more unique than any other theory. Indeed, its local symmetry (invariance under diffeomorphisms times local

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Lorentz invariance) is unique and the requirements of unitarity, locality, proper renormalizability and general covariance lead to the unique interim classical actions (3.1)–(3.2) in four dimensions. We also have solutions in even dimensions d  4. Their interim classical actions read  $ √ 1 d SQG = − 2 dd x −g 2C + ζ R + Rμν P(d−4)/2(D 2 )R μν 2κ %  + RP(d−4)/2 (D 2 )R + O(R 3 ) , (6.3) where Pn and Pn denote other real polynomials of degree n, while O(R 3 ) are the Lagrangian terms that have dimensions smaller than or equal to d, built with at least three curvature tensors and/or their covariant derivatives. The free propagators must satisfy the same requirements listed above and be quantized as explained. If we relax proper renormalizability into simple renormalizability, then we lose most uniqueness properties, because there exist infinitely many superrenormalizable theories of quantum gravity and gauge fields with fakeons in every spacetime dimensions d, with interim actions equal to (6.1) and (6.3), but polynomials Pn and Pn having degrees n > (d − 4)/2. Summarizing, the upgraded correspondence principle is made of unitarity locality proper renormalizability

(6.4)

together with fundamental symmetries and the requirements of having no massless fakeons and finitely many fields and parameters. The combination (6.4) implies quantum gravity coupled to gauge and matter fields in four dimensions, with interim classical actions (3.1)–(3.2). With respect to the version of the correspondence principle that is successful in flat space, the only upgrade required by quantum gravity amounts to better understand the meanings of the principles themselves, renounce analyticity in favor of regionwise analyticity and settle for macrocausality instead of full causality. As we wanted at the beginning, the final solution is as conservative as possible. The gravitational interactions are essentially unique, the Yang-Mills interactions are unique in form and the matter sector remains basically unrestricted.

6.2 Causality Renouncing causality in quantum field theory is not a big sacrifice, because we do not have a formulation that corresponds to the intuitive notion [22]. What we have are off-shell formulations, such as Bogoliubov’s definition [23], which also implies

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the Lehmann-Symanzik-Zimmermann requirement that fields commute at spacelike separated points [24]. The crucial issue is that it is not possible to accurately localize spacetime points by working with relativistic wave packets that correspond to particles that are on shell. This is more or less the reason why microcausality has not been treated as a fundamental principle in quantum field theory so far, maybe in anticipation that it was going to be renounced eventually. We could even say that the fate of causality was sealed from the birth of quantum field theory: quantum gravity just delivered the killing blow. For a more detailed discussion on these topics, see [25].

7 Conclusions Various signals suggest that the correspondence between the macroscopic environment where we live, which shapes our thinking, and the microscopic world is doomed to become weaker and weaker as we explore smaller and smaller distances. The impossibility to predict the gauge group and the matter content of the theory of nature, as well as the fates of determinism and causality are signs that our predictive power is fading away. We have to cope with the fact that nature is not arranged to be understood or explained by us humans to an arbitrary degree of precision. The ultimate theory of the universe may look infinitely arbitrary to us. At the same time, the success of quantum field theory and the recent progresses in quantum gravity give us reasons to believe that we might still have a few interesting things to say before declaring game over. In this paper we have studied the properties of quantum field theory of particles and fakeons in various dimensions. We have seen that the correspondence principle that worked successfully for the standard model admits a natural upgraded version that accommodates quantum gravity. It is encoded in the requirements of unitarity, locality of the interim classical action and proper renormalizability. The upgraded principle actually leads to an essentially unique theory of quantum gravity in every even dimensions greater than 2. In four dimensions, a fakeon of spin 2 and a scalar field are enough to have both unitarity and renormalizability. Causality breaks down at energies larger than the fakeon masses. The classical limit shares several features with the quantum theory it comes from, such as the impossibility to write exact field equations. Our experience teaches us that determinism and causality dominate at large distances. On the other hand, when we explore smaller and smaller distances, we see a gradual emergence of “freedom”, first in the form of quantum uncertainty, then in the forms of acausality and lack of time ordering. These facts suggest that the universe is radially irreversible, i.e. irreversible in the sense of the relative distances. When we move from the large to the small distances we see a pattern, pointing from the absolute lack of freedom to what we may call asymptotic anarchy.

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12. R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes. J. Math. Phys. 1, 429 (1960). https://doi.org/10.1063/1.1703676; M. Veltman, Unitarity and causality in a renormalizable field theory with unstable particles, Physica 29, 186 (1963). https://doi.org/10. 1016/S0031-8914(63)80277-3 13. For details, see for example M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, 1995), Chapter 7, section 3 14. U.G. Aglietti, D. Anselmi, Inconsistency of Minkowski higher-derivative theories. Eur. Phys. J. C 77, 84 (2017). https://doi.org/10.1140/epjc/s10052-017-4646-7, 16A2 Renormalization.com and arXiv:1612.06510 [hep-th] 15. F.A. Berends, R. Gastmans, Quantum electrodynamical corrections to graviton-matter vertices. Ann. Phys. 98, 225 (1976). https://doi.org/10.1016/0003-4916(76)90245-1 16. K.S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16, 953 (1977). https://doi.org/10.1103/PhysRevD.16.953; for the one-loop beta functions of the theory, see [10] and I.G. Avramidi, A.O. Barvinsky, Asymptotic freedom in higher derivative quantum gravity. Phys. Lett. B 159, 269 (1985); see also N. Ohta, R. Percacci, A.D. Pereira, Gauges and functional measures in quantum gravity II: higher-derivative gravity. Eur. Phys. J. C 77, 611 (2017). arXiv:1610.07991 [hep-th]; for recent applications to phenomenological problems and the beta functions in the presence of matter, see A. Salvio, A. Strumia, Agravity. J. High Energy Phys. 06, 80 (2014). https://doi.org/10.1007/JHEP06(2014)080 and arXiv:1403.4226 [hep-ph]; A. Salvio, A. Strumia, Agravity up to infinite energy. Eur. Phys. C 78, 124 (2018). https://doi.org/10.1140/epjc/s10052-018-5588-4, arXiv:1705.03896 [hep-th] 17. D. Anselmi, Fakeons, microcausality and the classical limit of quantum gravity. Classical Quantum Gravity 36, 065010 (2019). https://dx.doi.org/10.1088/1361-6382/ab04c8, 18A4 Renormalization.com and arXiv:1809.05037 [hep-th] 18. D. Anselmi, Fakeons and the classicization of quantum gravity: the FLRW metric. J. High Energy Phys. 04, 61 (2019). https://dx.doi.org/10.1007/JHEP04(2019)061, 19A1 Renormalization.com and arXiv:1901.09273 [gr-qc] 19. T. Padmanabhana, Obtaining the non-relativistic quantum mechanics from quantum field theory: issues, folklores and facts. What happens to the antiparticles when you take the nonrelativistic limit of QFT? Eur. Phys. J. C 78, 563 (2018). https://doi.org/10.1140/epjc/s10052018-6039-y 20. For a convenient way to organize the action, see for example, D. Anselmi, Properties of the classical action of quantum gravity. J. High Energ. Phys. 05, 028 (2013). https://doi.org/10. 1007/JHEP05(2013)028, 13A2 Renormalization.com and arXiv:1302.7100 [hep-th] 21. J.M. Cornwall, D.N. Levin, G. Tiktopoulos, Uniqueness of spontaneusly broken gauge theories. Phys. Rev. Lett. 30, 1268 (1973). https://doi.org/10.1103/PhysRevLett.30.1268 22. In this respect, a particularly illuminating discussion on causality can be found in G. ’t Hooft and M. Veltman, Diagrammar, CERN report CERN-73-09, §6.1 23. N.N. Bogoliubov, D.V. Shirkov, Introduction to the Theory of Quantized Fields (Interscience Publishers, New York, 1959) 24. H. Lehmann, K. Symanzik, W. Zimmermann, On the formulation of quantized field theories — II. Nuovo Cimento 6, 319 (1957). https://doi.org/10.1007/BF02832508 25. D. Anselmi, The correspondence principle in quantum field theory and quantum gravity. 18A5 Renormalization.com, PhilSci 15287, OSF preprints d2nj7, Preprints 2018110213 and hal-01900207

Implementation of the Quantum Equivalence Principle Lucien Hardy

Abstract The quantum equivalence principle says that, for any given point, it is possible to find a quantum coordinate system with respect to which we have definite causal structure in the vicinity of that point. It is conjectured that this principle will play a similar role in the construction of a theory of Quantum Gravity to the role played by the equivalence principle in the construction of the theory of General Relativity. To actually implement the quantum equivalence principle we need a suitable notion of quantum coordinate systems—setting up a framework for these is the main purpose of the present paper. First we introduce a notion of extended states consisting of a superposition of terms (labeled by u) where each term corresponds to a manifold, Mu , with fields defined on it. A quantum coordinate system consists of an identification of points between some subsets, Ou ⊆ Mu , of these manifolds along with a coordinate, x, that takes the same value on those points identified. We also introduce a notion of quantum coordinate transformations (which can break the identification map between the manifolds) and show how these can be used to attain definite causal structure in the vicinity of a point. We discuss in some detail how the quantum equivalence principle might form a starting point for an approach to constructing a theory of Quantum Gravity that is analogous to way the equivalence principle is used to construct General Relativity.

1 Introduction The equivalence principle of Einstein can be stated in the following way The Equivalence Principle: For any given point it is possible to find a coordinate system with respect to which we have inertial behaviour in the vicinity of that point.

L. Hardy () Perimeter Institute, Waterloo, ON, Canada e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 F. Finster et al. (eds.), Progress and Visions in Quantum Theory in View of Gravity, https://doi.org/10.1007/978-3-030-38941-3_8

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We can think of inertial behaviour as being that physical behaviour associated with non-accelerating coordinate systems in the absence of gravity. The equivalence principle says that we also have this behaviour generally (even when there is a gravitational field) albeit only locally in the vicinity of a point. The power of the equivalence principle stems from the fact that it forms a bridge between pre-general relativistic physics (in which we can work in a global inertial reference frame) and General Relativity (in which we have a local inertial reference frame at each point, but not a global one). In Sect. 11 we look in more detail at the conceptual structure within which General Relativity is obtained and the equivalence principle’s role in this (see Fig. 1 in particular). In a theory of Quantum Gravity we expect to have indefinite causal structure (as we will have something like a quantum superposition of different solutions for the metric). If we take definite causal structure to be analogous to inertial behaviour then this suggests the following principle which, I hope, will play a similar role to the equivalence principle but in guiding us to a theory of Quantum Gravity. The Quantum Equivalence Principle: For any given point it is possible to find a quantum coordinate system with respect to which we have definite causal structure in the vicinity of that point.

To implement the quantum equivalence principle we will need to find an appropriate notion of quantum coordinate systems. In Sect. 12 we consider a possible conceptual structure for obtaining Quantum Gravity (see Fig. 2 in particular) that is analogous to that of General Relativity and look at how the quantum equivalence may play an analogous role to that of the equivalence principle in General Relativity. The quantum equivalence principle, which I proposed in [24], was strongly motivated in the first place by the work of Giacomini, Castro-Ruiz, and Brukner on quantum reference frames [13]. This provides a notion of quantum reference frames at a given time that can correspond to a superposition of other frames of reference (at that given time). This principle is supported by the work by Guerin and Brukner on causal reference frames [15], and related work by Oreshkov [38]. In these works it is shown how physically equivalent circuit representations can have event A localized in time while event B is delocalized or vice versa. This can be regarded as a discrete example of the quantum equivalence principle. In the present work we seek a notion of quantum coordinates systems that is analogous to the notion of coordinate systems used in General Relativity. In General Relativity a coordinate system labels the points in some region of the manifold. This labeling is conceptually prior to the introduction of the metric and so the coordinate system knows nothing of space, time, and causal structure as such. Hence, we need to go beyond the ideas of Giacomini et al. which operate at a given time. We also expect these coordinates to be continuous. Hence we also need to go beyond the discrete circuit framework of Guerin and Brukner and of Oreshkov. The notion of quantum coordinate systems proposed here is continuous and, like the classical notion of coordinate systems, conceptually prior to space, time, and causal structure.

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Many authors have proposed quantum equivalence principles of various sorts [8–10, 26, 28, 33, 40] that are quite different from the one given above (generally these are still concerned with inertial motion of some sort). Interesting work on the conceptual role of an appropriate (though undefined) quantum equivalence principle might play as a bridge from Quantum Field Theory to Quantum Gravity has been undertaken by Pipa et al. [43]. There has also been research on applying Einstein’s equivalence principle to Quantum Theory. This is a vast subject but see [3, 44, 47, 49] for some work along this direction. A nice philosophical discussion on this is given by Okon and Callender [37].

2 Basic Idea in This Paper The basic idea to implement the quantum equivalence principle is as follows. First, note that a classical description of a physical situation in General Relativity is given by specifying values of a set, , of tensor fields (corresponding to matter fields and the metric field) at each point in some manifold. We can represent this classical description as u = {(, p) : ∀p ∈ M} A different physical situation may have a different manifold, M. To indicate this, we will use the rather clumsy notation, Mu . The u symbol, when it appears as a subscript, is regarded as a label. Thus, we have u = {(, p) : ∀p ∈ Mu } We write |u] to describe the situation u (we could have written |u but we need our |u] notation to do some extra things that make it worthy of new notation). Next we argue from the path integral approach that the object  |] =

Du cu |u]

(which we call an extended state) contains the necessary physical information (actually we consider more general objects called extended A-states but for this overview it is sufficient to talk about extended states). A quantum coordinate system is given by providing an identification between points in different Ou ⊆ Mu for different u and then providing a coordinate, x, for these points. This identification is pure gauge and has no physical significance in itself. We can imagine covering all points in {Mu } by a set of quantum charts employing quantum coordinate systems in this fashion (forming a quantum atlas). A quantum diffeomorphism can change the identification mapping between the Ou . By applying a quantum diffeomorphism that keeps some point, x, fixed we can

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“rotate” the solutions for the different u so that the lightcones associated with the metrics align (we can also get the conformal factors to match). In so doing the causal structure at x becomes definite. The idea of identifying points across different manifolds in a quantum superposition has been independently arrived at by Ding Jia [30, 31] with different motivations in mind. In particular, points are identified that share some common features (such as fields taking a given value). Here we do not impose this constraint but rather regard the identification as pure gauge (and different identifications are equally valid). Applications Jia considers for his identification structure are to ensure that all terms in the superposition have the same boundary conditions in a Feynman path integral and also to help in collecting terms together in a useful way when evaluating these path integrals. Another difference with the present approach is that Jia includes the metric but not the matter fields in his (Mα , gα ) terms that are superimposed and have points identified. The matter fields are dealt with through the path integral.

3 Background In [17–19] I set up a general operational framework (the causaloid framework) for the purpose of studying indefinite causal structure. This was in the spirit of the general probability theories framework [5, 16]. Since then two teams, Chiribella, D’Ariano, Perinotti, and Valiron (CDPV) [11] working initially in Pavia, and Oreshkov, Costa, and Brukner (OCB)[39] working initially in Vienna have developed operational frameworks based on quantum operators for studying indefinite causal structure. All three approaches (the causaloid and the operator based approaches) have in common that they are linear in probability and that they associate state-like objects with what are effectively arbitrary shaped regions of space-time. The operator tensor framework [21, 22], which is a development of the causaloid approach to Quantum Theory, can also be applied to indefinite causal structure (see [25] for a treatment of OCB’s approach). CDPV proposed the idea of a quantum switch—an explicit way to obtain indefinite causal structure. And OCB proposed some inequalities which, if violated, demonstrate indefinite causal structure. Recent work (already mentioned in Sect. 1) by Oreshkov [38] and also Guerin and Brukner [15] introduces a new perspective on indefinite causal structure. The field of indefinite causal structure has grown considerably in recent years and there are many more papers beyond those cited above. Robert Oeckl’s general boundary formalism (both the original amplitude based version [35] and the more recent positive formalism [36]) also connects with the work on indefinite causal structure. Some of the ideas in this paper come from my operational formulation of General Relativity [23]. This is a reformulation of General Relativity as an operational probabilistic theory motivated by the problem of Quantum Gravity (a theory of Quantum Gravity will likely be both operational and probabilistic). In particular, the representation of classical configurations as u = {(, p) : ∀p ∈ Mu } was

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introduced in that paper (though they were denoted by  rather than u). The notion of “chartable space” as a place where different manifolds can live was introduced there and would find application to the ideas presented here. The field of quantum reference frames goes back to the 1967 with a paper by Aharonov and Susskind [2] on charge superselection, followed in 1984 with a paper by Aharonov and Kaufherr [1] entitled “Quantum frames of reference”. A second wave of activity happened a dozen years ago (see the review paper [6]) using more modern notions from quantum information. Most recently the above mentioned paper [13] by Giacomini et al. has reignited the field. Vanrietvelde et al. [45, 46] have explored how to switch between reference frames via a “perspective-neutral” framework. Höhn has explored how quantum reference frames may impact on the study of Quantum Gravity [27]. The present paper is an elaboration on ideas I first presented in [24] where the quantum equivalence principle was first stated and, further, the idea that we might pursue a route to Quantum Gravity analogous to Einstein’s development of General Relativity was considered. We will make much use of the path integral due originally to Feynman. The path integral has frequently been invoked for looking at the problem of Quantum Gravity (see [48] for recent perspectives) forming the basic motivation for the causal sets approach [7], the spin foam approach [42], the dynamical triangulations approach [4], the Euclidean gravity approach [14] as well as various cosmological models [12].

4 Path from the Path Integral to Extended States for a Particle To get started consider the simple case of the path integral for a single particle moving in one dimension. In this case the amplitude for starting at yi at time ti and arriving at yf at time tf is given by Feynman’s path integral 1 Z



Du eiS/h¯

(1)

u = {(y(t), t) : ti ≤ t ≤ tf }

(2)

u∈V [yi ,ti ,yf ,tf ]

where

This is a path starting at time ti and ending at time tf . We will usually denote such paths by u but we will also use v and w for these paths when we need extra symbols. Du is a measure over such paths so we can perform a functional integral. V [yi , ti , yf , tf ] is the set of paths that start at (yi , ti ) and end at (yf , tf ). Z is a normalization factor. S is the action given by integrating the Lagrangian along a

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path u ∈ V [yi , ti , yf , tf ]  S[yi , ti , yf , tf ] =

u∈V [yi ,ti ,yf ,tf ]

dtL(t)

(3)

In the expression (1) we have integrated over paths and lost information accrued along each individual path. We will consider a series of mathematical objects which contain successively more information until we obtain an object that is useful for the purposes we have in mind. As a first step, consider the object,  %   V [ti ,tf ] = S

u∈V [ti ,tf ]

Du

1 iS/h¯ |u] e Z

(4)

where |u] is a way to refer to the path u in a manner analogous to the way the notation |x refers to the position x (we will elaborate on this notation later). Now we sum over all paths starting (at any position) at time ti and ending at time tf . We can construct the path integral in (1) from the object in (4) as follows  u∈V [yi ,ti ,yf ,tp ]

Du eiS/h¯ =

 v∈V [yi ,ti ,yf ,tp ]

 %  (t ,t ) Dv [v S i f

(5)

where we put  Du δ(v − u)f (u) = f (v)

[v|u] = δ(v − u) where

(6)

u

The object in (4) keeps the information accrued along each individual path. Further, it is a sum over all paths for the given time interval, not just those starting at yi and ending at yf . However, it still contains a sum—the integral over the Lagrangian to obtain the phase. This means (1) that we have lost information about the contribution to the phase coming from each infinitesimal section of the path (i.e. about the Lagrangian at each point) and (2) we have committed to a particular time interval for each term. Hence, we consider the object  %   V L =

Du u∈V

1 Lu (Nu ) |u] Z

(7)

We call this an extended L-state. Here N is a time interval (more generally, it may be the union of a number of time intervals), u = {(y, t) : t ∈ Nu }

(8)

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and Lu (Nu ) = {(L, t) : t ∈ Nu }

(9)

When u appears as a subscript (for example, in Nu and Lu ) it is intended as a label to indicate that this instance pertains to the u case Importantly note that we allow the coefficient in front of the |u] to be a set comprised of elements (L, t) for all times t ∈ N . This allows us to maintain information about the contribution to the phase coming from each part of the path. We also generalize and allow paths for arbitrary N . The set, V , of paths we consider may be arrived at by various means in this more general setting. For the moment we simply note that we do not require that every entry have the same N .   Next we will look at how the path integral can be obtained from LV . First we must provide a more general notion than [u|v] applying more generally to situations when u and v may pertain to different N . We need this because the path u may be contained in the path v (where Nu ⊂ Nv ): If u ⊆ v then u contained in v

(10)

To clarify, u ⊆ v means that u = vNu where v|Nu = {(y, t) ∈ v : ∀t ∈ Nu }

(11)

[u v] = δ(u − v|Nu )

(12)

Now we define

Note that this is asymmetric (and we have indicated this using which is supposed to be indicative of the example where a longer path gets shortened). If u is not contained in v then [u v] = 0. We also define [u Av (Nv )|v] = Au (Nu )δ(u − v|Nu )

(13)

This is non-zero only when u is contained in v, and in this case we keep only the elements of Av (Nv ) pertaining to elements in Nu . We will say W V

⇐⇒ ∀ u ∈ W ∃ v ∈ V such that u ⊆ v

(14)

If W V then each path in W is contained in some   path  in V .  In the case where V [ti , tf ]V , we can use LV to calculate S[yi ,ti ,yf ,tf ] as follows. First we project  %   V [ti ,tf ] = L

v∈V [ti ,tf ]

 %  Dv |v] [v (LV )

(15)

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Then we integrate and exponentiate   % % i  V [t ,t ]  V [ti ,tf ] = (exp int)op L i f S h¯

(16)

We use the notation (exp hi¯ int)op to indicate that we have an operator acting linearly  %  V [t ,t ] on each term in L i f that integrates over elements in the set, Lu (Nu ), and then  %  (t ,t ) exponentiates inserting the factor hi¯ . Once we have S i f we can use (5) to get  %  V [x ,t ,x ,t ] the amplitude. Note that we could simply have projected down to L i i f f directly in (15) and then we only require that V [xi , ti , xf , tf ]V . We choose the slightly more complicated route here for pedagogical reasons as we will set up a similar approach for Quantum Gravity in the next section. Finally, we are not restricted to having the Lagrangian in the coefficient. More generally, we can consider extended A-states  |A ] =

Du cu Au (Nu ) |u]

(17)

Au (Nu ) = {(A, t) : t ∈ Nu }

(18)

where

for the quantity A calculable at each time t. Also we can, in general, let the coefficient cu depend on u. One reason for allowing such a dependence is that some paths may be blocked or be partially absorbed. We have dropped the superscript, V , in |A ] as now we can regard this as integration over all u with cu equal to zero for u not in the given set. It is worth noting that the object  |] =

Du cu |u]

(19)

actually contains all the information we need. We can set up a linear operator that acts on each |u] and brings out Au (Nu ) as a coefficient in front of this term. We will call the object in (19) the extended state.

5 From the Path Integral to Extended States in Quantum Gravity We can set up a similar path integral for Quantum Gravity. First we define u = {(, p) : ∀p ∈ Mu }

(20)

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where  is a list of all the fields—the matter fields plus the metric field (so  is playing an analogous role to y of Sect. 4). Further, Mu is a subset of manifold points (analogous to Nu of Sect. 4 which was the union of disjoint time intervals). Typically we would like Mu to be well behaved. We will assume it is union of disjoint manifolds with corners [32, 34]. Roughly speaking, a manifold with corners is one that can everywhere be locally covered by the points in [0, ∞)k RN−k —this allows it to have boundaries and corners (in the vicinity of these, the integer, k, is greater than zero). See [23] for more motivation for using such manifolds with corners. When we refer to a “manifold” in what follows we are referring to these well behaved subsets of manifolds (the union of disjoint manifolds with corners) which are, in any case, a generalisation of the usual notion of manifolds. Consider a boundary, b, at which we can impose boundary conditions a. We need not go into detail for the time being except to note that physically meaningful boundaries and boundary conditions should be invariant under diffeomorphisms (and, more generally, under the quantum version of such diffeomorphisms). A treatment of boundary conditions invariant under diffeomorphisms appropriate to this situation is given in [23]. In Sect. 6 we treat quantum diffeomorphisms and in Sect. 7 we say what the beables are. Boundary conditions must be beables in this sense. We can now calculate the amplitude associated with the given boundary conditions as  Du eiS/ h (21) u∈V [a]

Here V [a] is the set of all u consistent with boundary conditions, a, are analogous to (yi , ti , yf , tf ) from Sect. 4 and the boundary, b, is analogous to (ti , tf ). As before, we can consider the more general object  %   V [b] = S

u∈V [b]

Du

1 iS/h¯ |u] e Z

(22)

where  V [b]% is the set of all u consistent with any boundary condition at b. We can  use SV [b] to calculate the amplitude in (21) using  u∈V [a]

Du eiS/h¯ =

 v∈V [a]

  Dv [v S[{a}]

(23)

This object contains integrals (in the form of S) and also commits to a certain boundary condition. We can consider a more general object still—this is the extended L-state for Quantum Gravity:  %   V L =

u∈V

Du

1 Lu (Mu ) |u] Z

(24)

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where Lu (Mu ) = {(L, p) : ∀p ∈ Mu }

(25)

Note that we allow V in (24) to be more general than V [b]. The manifolds, Mu , pertaining to u ∈ V do not have to pertain to the region defined by some boundary conditions b. They can be any (reasonably well behaved) manifolds.   As in Sect. 4, we can set up notions that allow us to project LV . First we say that u is contained in v iff u ⊆ v. This is equivalent to saying that u = v|Mu where v|Mu = {(, p) ∈ v : ∀p ∈ Mu }

(26)

[u v] = δ(u − v|Mu )

(27)

[u Av (Mv )|v] = Au (Mu )δ(u − v|Mu )

(28)

We define

and

Finally, we say W V iff, for each u ∈ W , there exists a path v ∈ V that contains u. If V (b)V then we can project (just as we did in (15)) obtaining  %   V [b] = L

v∈V [b]

 %  Du |v] [v (LV )

(29)

 %  Now we can obtain SV [b] by applying the integration and exponentiation operator   % % i  V [b]  = (exp int)op LV [b] S h¯

(30)

From this we can use (23) to calculate the amplitude associated with a particular boundary condition a via the path integral.   As before we can consider other objects. A more general object than LV is the extended A-state:  |A ] = Du cu Au (Mu ) |u] (31) where A can be any tensor field (the Lagrangian is then a special case). We have dropped the V superscript as this is superseded by the cu dependence. Also, we can start with the extended state  |] = Du cu |u] (32)

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Acting on this with linear operators can be made to bring out the information we need to calculate the other objects.

6 Quantum Diffeomorphisms A classical diffeomorphism, ϕ, is a smooth invertible map on the manifold taking p to ϕ(p). This induces transformations on tensor fields that live on the manifold which we denote by saying that A(p) goes to ϕ ∗ A(ϕ(p)). If we act on u with a diffeomorphism then we have   ϕ ∗ u = (ϕ ∗ A, ϕ(p)) : ∀ϕ(p) ∈ ϕ(M)   = (ϕ ∗ A, p) : ∀p ∈ ϕ(M) where the second line follows as ϕ(p) is a dummy variable in the first line. We will first define a restricted notion of a quantum diffeomorphism which we then use to motivate the general definition. The basic action of a restricted quantum diffeomorphism is to perform a classical diffeomorphism on each of the different |u] terms. A restricted quantum diffeomorphism is a linear operator, ϕ, defined by a set of classical diffeomorphisms, {ϕu : ∀u} that acts on |A ] as follows  ϕ |A ] = ϕ

Du cu Au (Mu ) |u]

 =

Du cu eiθ(u,ϕu

∗ u)

  ϕu ∗ A(ϕu (Mu )) ϕu ∗ u

(33) (34)

where   ϕu ∗ A(ϕ(Mu )) = (ϕu ∗ A, p) : ∀p ∈ ϕ(M)

(35)

θ (u, v) = −θ (v, u)

(36)

and

This phase function is some given function prescribed by the theory (its form may, though, depend on what type of tensor field A is). It need only be defined for arguments u and v which can be transformed into one another by a diffeomorphism. We require it to be anti-symmetric so that if we find a quantum diffeomorphism that undoes the action of a previous quantum diffeomorphism, then |A ] returns to its original version without accumulating any phase factors. One possible choice is θ (u, v) = 0

(37)

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as this is antisymmetric. At least in the case where A is a scalar field we are motivated to choose θ (u, v) = 0 since then we want the extended L-state, |L ], to have real coefficients before and after a change of quantum coordinates so that we get the correct path integral. In general, however, there may be physical reasons for some other choice. In particular, there is a non-trivial phase associated with the quantum frame of reference change in the work of Giacomini et al. It may be a challenge to find physically meaningful functions of u and v. These restricted quantum diffeomorphisms are not necessarily invertible. This is because we could have ϕu ∗ u = ϕv ∗ v (where u and v are different). This does not matter as, by virtue of the way we will set the theory up, no physical information is lost under quantum diffeomorphisms. We will now give the general definition of a quantum diffeomorphism. These also are not invertible. However, as we will see, the action of a quantum diffeomorphism on any given |A ] can be reversed by an appropriate choice of quantum diffeomorphism. A quantum diffeomorphism, ϕ, is given by a set 



3 aα ϕuα

: ∀u

(38)

α

where aα is real for all alpha and



2 α aα

= 1. The sum over the label α could instead

be an integral. The action of ϕ on |A ] is as follows  ϕ |A ] =

Du cu



α ∗ u)

aα eiθ(u,ϕu

  ϕuα ∗ A(ϕu (Mu )) ϕu ∗ u

(39)

α

It is clear now that, although quantum diffeomorphisms are not invertible, their action on a given |A ] can be inverted. Since quantum diffeomorphisms are not invertible, they do not form a group. They do, however, form a monoid (that is a semigroup with an identity element). Just as classical General Relativity is invariant under diffeomorphisms, Quantum Gravity is, we propose, invariant under quantum diffeomorphisms. This has particular significance for the ontologically real quantities in the theory.

7 Beables The term beables was coined by John Bell to refer to the ontologically real quantities in a theory. In General Relativity, the beables are those functions of solutions that are invariant under diffeomorphisms (see discussion in [23]). Thus, if the world is described by u then it is equally well described by ϕ ∗ u for any ϕ and beables are

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functions, B(u), having the property that B(u) = B(ϕ ∗ u)

∀ϕ

(40)

We cannot talk about the beables in a particular patch of the manifold as a diffeomorphism will replace the fields living there with different fields. Thus, beables, in General Relativity are in some sense nonlocal. What are the beables in Quantum Gravity? Here we propose that they are quantities that are invariant under quantum diffeomorphisms. Thus, B(|]) is a beable if and only if B(|]) = B(ϕ |])

∀ϕ

(41)

This means that ϕ |] contains the same physical information as |]. A natural question that arises here is whether this way of formulating beables will offer any resolution to the interpretational problems of quantum theory. In particular, will it resolve the measurement problem and will it offer some way of understanding Bell-type nonlocality? A good way to develop an intuition with regard to these questions would be to look at some examples. For the time being we can note that transformations between different quantum coordinate systems can remove superpositions and entanglement (this is true in the scheme of Giacomini et al. [13] and will also hold here). This offers some hope that macroscopic superpositions and entanglement based nonlocality are gauge artifacts and will not truly be present amongst the beables. We can use beables to select on extended states. For example, we may have seen a certain outcome, β, in some experiment, E, and the extended state must be consistent with this. Then we have beable value BE (|]) = β

(42)

This constrains the extended state associated with this outcome. One way to do this is using operational space as outlined in [23]. Operational space is given by nominating an ordered set of scalars, S = (S1 , S2 , S3 , . . . , SK ). If we plot u into operational space then we will obtain a surface that is N dimensional at most (where N is the dimension of spacetime). We can select on u that plot into certain regions, A, of operational space. Then we require that the beables satisfy Bop space (|]) ∈ A

(43)

This means that we can only have terms, |u], in the expansion of |] that plot into A. The discussion of operational space in [23] is applicable here.

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8 Quantum Coordinate Systems In General Relativity, coordinate systems are introduced to cover a manifold. In our treatment of Quantum Gravity, the situation is complicated by the fact that we have a set of manifolds, {Mu }, rather than a single manifold. Nevertheless, we can set up a map that identifies points on different manifolds in {Mu } in some region. We can then lay down coordinates. We will call these quantum coordinate systems. This will be appropriate as we will be able to transform between them by means of a quantum transformation corresponding to a quantum diffeomorphism. Indeed, a diffeomorphism is the abstract version of a change of coordinate system. Likewise, we will be able to think of a quantum diffeomorphism as the abstract version of a change of quantum coordinate system. First, we choose a set of smooth invertible maps ϕu→v (p)

(44)

p ∈ Ou ⊆ Mu

(45)

ϕu→v (p) ∈ Ov ⊆ Mv

(46)

ϕu→w (p) = ϕv→w ◦ ϕu→v (p) ∀ p ∈ Ou

(47)

for all u and v mapping

to

such that

so points are mapped consistently between {Ou }. We will call the maps ϕu→v identification maps. Note, the notation ϕu→v may be a little misleading. These maps do not map u to v, but rather they map a subset of points (those in Ou ) in the manifold associated with u to a subset of points (those in Ov ) in the manifold associated with v. We can always do this by choosing Ou that are coverable by points in Vu ⊆ RN .1 We can now cover these points, so identified, with coordinates, x = (x μ : μ = 1, 2, . . . N). To do this we can set up a bijection, xu (p) from the points p ∈ Ou to the points in Vu ⊆ RN for some particular u then use the map ϕu→v to generate a coordinate system for every other element of {Ou }. xv (p) = xu (ϕv→u (p))

1A

(48)

subtlety arises if some of the Ou include points in the boundary of Mu . Then the maps ϕu→v are only invertible for points in Ou . In particular, it may then be that there are points in Ov that do not map to points in Ou . In this case we can simply map from some reference set, O , to each of the Ov . For the most part, we will not concern ourselves with this subtlety.

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In this way we have laid down a coordinate system that covers part of each element of {Mu } identifying points with the same coordinates, x. This quantum coordinate system only covers part of each manifold, Mu . This is analogous to the way a (classical) coordinate system (some times called a chart) only covers part of a manifold. In the classical case, a set of charts that cover the manifold is called an atlas. We could consider setting up a quantum atlas consisting of enough quantum coordinate systems (we could also call them quantum charts) to cover every part of every Mu . In Sect. 12.4 we will discuss the tentative idea of quantum manifolds which provides an implementation of a quantum atlas.

9 Quantum Coordinate Transformations We can perform a classical coordinate transformation simply by transforming x μ as in General Relativity. 

x μ → x μ = f ({x μ })

(49)

This changes the name of the coordinate at each point. However, it does not break the identification map between the elements of {Mu }. A more radical transformation is a quantum coordinate transformation which does break this identification map. To do this we act on the maps ϕu→v with the elements of {ϕu } (that can be used to define a restricted quantum diffeomorphism) to obtain a new identification map  ϕu→v → ϕu→v = ϕv ◦ ϕu→v ◦ ϕu−1

(50)

between the elements in the set {Ou } (where Ou = ϕu (Ou )). Each element of this set is covered by coordinates xu (p) = xu (ϕu (p)). It is worth noting that the quantum aspect of the quantum coordinate system is in the identification between manifold points since this is something that can be changed by a quantum coordinate transformation (but not by a classical coordinate transformation). A quantum coordinate system is simply attached to a set of manifolds (this is passive in that it does not change the extended state, |A ]). A transformation between different quantum coordinate systems is associated with a set of diffeomorphisms on each of these manifolds. This set of diffeomorphisms can be used to implement a quantum diffeomorphism on an extended state, |A ]. What is the relationship between this quantum diffeomorphism with its action on an extended state and the quantum coordinate system? To investigate this first note that we can take an active (rather than passive) approach. Thus, when we want to identify points in Ou and Ov we can perform an active transformation so that these identified points actually coincide so that the identification maps become equal to the identity. To do this, we choose some particular u = u˜ (it does not matter which one) with an associated Ou˜ . Then we map all other Ou to overlap with Ou˜ this by

204

L. Hardy

using the transformation maps ϕu→u˜ . The transformation of |A ] is  %    ˜ |A ] −→  Du cu eθ(u,ϕu→u˜ ∗u) ϕu→u˜ ∗ ϕu→u˜ ∗ u A =

(51)

This is a quantum diffeomorphism which transforms the points in Ou in each |u] term so that they coincide with the points they are identified with in Ou˜ . We can associate coordinates with Ou˜ by means of the map xu˜ (p)

(52)

Since the Ou sets have been transformed so that the point p in the transformed set coincides with the point it is identified with in Ou˜ , this coordinate map associates these coordinates for all terms in the extended state (for the part of the manifold we are interested in). Had we chosen u˜  instead of u˜ we would get a different extended state. We can map between these cases by applying the same diffeomorphism, ϕu→ ˜ u˜  to each term of the expression on the left in (51). We can write this as  %  % ˜ ˜ A = ϕu→ ˜ u˜  A

(53)

Note this can be regarded as a classical diffeomorphism since every |u] term is subject to the same diffeomorphism. % ˜ Once we have a  A in this form (so the identified points in the Ou ’s coincide) then we can apply a (restricted) quantum diffeomorphism to it  %  %  % ˜ ˜ ˜ = ϕ  A A −→  A

(54)

obtaining a new extended state now having the points for the sets Ou = ϕu (Ou ) coinciding for different u (here {ϕu } are the maps associated with the quantum diffeomorphism). We choose the diffeomorphism, ϕ such that it the sets Ou are coincidence with the set Ou˜ (to be able to do this we need to make certain assumptions about the “shape” of these sets so they can continue to coincide). We choose an identification map (for the quantum coordinate system) that is now the identity map with respect to these new transformed sets Ou . This will change the identification between points in the manifold associated with different u thus constituting a quantum coordinate transformation. The new coordinates are given by xu˜ (ϕu˜ (p))

(55)

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Thus, if we take an active point of view then we see that quantum diffeomorphisms (of this type) on the extended state are associated with quantum coordinate transformations. One point that is worth making is the following. A point, x, in a quantum coordinate system is associated with a set of points, one in each manifold, Mu . If we perform a classical diffeomorphism then x is transformed to x  . Under this classical transformation we do not break the identification map and so we might say that x keeps its identity (and is merely relabeled by x  ). However, if we perform a quantum diffeomorphism, then x may lose its identity in the sense that it is no longer associated with any x  in the new quantum coordinate system. This is because quantum diffeomorphisms break the identification between points in the different manifolds. It is instructive to consider an example in which a quantum coordinate transformation can be used to transform from a coordinate system in which one quantity is definite to one in which another is. We can nominate a set of scalars, S = (S1 , S2 , . . . , SK ) (in [23] such sets were used to set up an “operational space”). Each scalar, Sk (p), in this can be built out of the tensors in  and the action of the covariant derivative, constructed so that all indices are summed over. We can set up a quantum coordinate system in which points are identified between different Mu that have the same S. We could consider this as going into the quantum frame of reference “co-moving” with S (note, however, this frame of reference will not be unique if, for some u, more than one point, p ∈ Mu , is mapped to the same S). We  ). We can, instead, might consider a different set of scalars, S = (S1 , S2 , . . . , SK  consider a quantum coordinate system in which these points are identified. When we are in the first quantum coordinate system we expect S to be indefinite (so points identified with the same x have different values for S ). Similarly, when we are in the second quantum coordinate system, we expect S to be indefinite. An appropriate quantum coordinate transformation will take us between these two cases.

10 Implementing the QEP The Quantum Equivalence Principle (QEP) says that, in the vicinity of any point, we can always find a quantum coordinate system such that we have definite causal structure. By “point” we mean any point that can be defined with respect to some quantum coordinate system. Such a point is given by specifying some particular point, p1 ∈ Mu˜ , for some particular u = u˜ along with a set of identification maps {ϕu→u } which identifies p1 with a point ϕu→u (p1 ) for every other Mu . Consider ˜ ˜ the extended g-state    g = Du cu gu (Mu ) |u] (56)

206

L. Hardy

where g is the metric tensor (considered as an abstract tensor). To implement the QEP we simply need to find a quantum diffeomorphism such that the metrics gu are equal in the vicinity of the point in question with regard to the  quantum  coordinate system. We will describe one way to do this. We can act on g with a quantum diffeomorphism to bring the points identified with p1 (for the other u) into coincidence (as discussed in Sect. 9)  %   ˜ (57) g = ϕ g where ϕ is associated with the set {ϕu→v } of diffeomorphisms that bring the points corresponding to p1 into alignment along with a set of points in the vicinity (in associated Ou sets). At this stage the metric living on the different Mu can be very different in the vicinity of p1 . Now we perform a second Now we perform a second 

quantum diffeomorphism, ϕ , which leaves p1 g-state is  % %   ˜ ˜g  g = ϕ 

(58)

where this quantum diffeomorphism is associated with a set, {ϕu }, of diffeomorphisms chosen such that now the metrics gu are equal to first order in the vicinity of p1 . Thus, if we write  %  ˜ ˜ u ) |u] g = Du c˜u g˜ u (M

(59)

then we require g˜ u (x1 + δx) = g˜ v (x1 + δx) + O(δx 2 ) ∀ u and v s.t. cu = 0 and cv = 0 

(60)

We can think of the second quantum diffeomorphism (ϕ ) as rotating the light cones associated with these different metrics (for the different u) so that they align and also stretching the manifolds so that the conformal factors associated with the metrics agree. If there is a time direction (past to future) at each p then we can rotate the light cones so that they agree on future and past. We can always satisfy (60) because we can always transform the metric in the vicinity of any point to the Minkowski metric. The transformation does a little more than implied by the quantum equivalence principle. In addition to aligning causal structure (light cones) it also matches the conformal factors. We could relax the latter imposition so that we have a statistical mixture over different conformal factors (corresponding to different clock rates at x1 for the different manifolds. However, we have enough freedom to actually align these conformal factors so we will assume that we do this in what follows (we might call the fact that we can do this the strong quantum equivalence principle).

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11 General Relativity 11.1 The Problem of Relativistic Gravity After Special Relativity had been discovered by Einstein in 1905 along with the space-time picture of Minkowski from 1907 it was understood that physical theories should be formulated as Special Relativistic Field Theories. Maxwell’s theory of electromagnetism, already published some years earlier in 1861, was one such theory. Fluid dynamics and other theories would subsequently be given formulations as Special Relativistic Field Theories. However, in 1907, Newton’s theory of Gravity resisted efforts to formulate it in these terms. Thus, the stage was set for consideration of the following problem. The problem of Relativistic Gravity is to find a physical theory that reduces in appropriate limits to the theory of Newtonian Gravity on the one hand and to Special Relativistic Field Theories (SRFT) on the other. Newtonian Gravity ←− Relativistic Gravity −→ SRFT

11.2 How Einstein Solved the Problem of Relativistic Gravity Einstein solved the problem of Relativistic Gravity in the form of General Relativity. His starting point was a realisation he had in 1907—namely the equivalence principle. He called this the “happiest thought in my life” [41]. The equivalence principle acts as a bridge between the old physics and the new physics. It is the essential clue in working out how to reverse the arrows so we have Newtonian Gravity −→ Relativistic Gravity ←− SRFT While the equivalence principle formed the right starting point, there was still much work to be done to go from this realisation to the full theory of General Relativity (and it took him until 1915 to complete this task). It is worth outlining in some detail how Einstein did this. First, let us be clear about what General Relativity is. The theory is captured by three elements[24] A prescription for converting the field equations in a Special Relativistic Field Theory into general relativistic field equations. This prescription (sometimes called minimal substitution)works as follows. • The coordinates, x μ¯ , of the global inertial reference frame are replaced by general coordinates, x μ . This is done by replacing all indices μ, ¯ ν¯ , . . . by μ, ν, . . .

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L. Hardy

• The Minkowski metric, ημν is replaced by the general metric, gμν . • All partial derivatives, ∂μ are replaced by covariant derivatives, ∇μ . The field equations obtained in this way are called the matter field equations. An addendum. The general metric gμν introduces an additional ten real parameters into the theory (for four-dimensional spacetime). Thus, to have a complete set of field equations we need (it would seem) an additional ten field equations. Einstein provided just such a set in the form of the Einstein field equations Gμν = 8πT μν These equations satisfy a form of energy momentum conservation (namely ∇μ T μν = 0) because of the mathematical identity ∇μ Gμν = 0. However, this means that Einstein’s field equations actually only furnishes us with six independent field equations (since the mathematical identity shows that four of them are related). It would seem that we do not have a complete set of field equations after all. In fact we do not need extra equations because of the next element. An Interpretation. The beables (physically real quantities) are those quantities that are invariant under diffeomorphisms. To elaborate, we can regard a solution, u, to the field equations (both the matter and Einstein field equations) as a set u = {(, p) : ∀p ∈ M} that specifies the values of all the tensor fields (denoted by ) at each point, p, in some manifold, M. The need for the second element follows from the first. The need for the third element follows from the missing equations in the second and is, in any case, necessitated by the fact that all the equations are invariant under general coordinate transformations. Thus, acting on a solution with a general coordinate transformation produces another valid solution. To see how this works, consider the following. When we represent the points, p ∈ M with coordinates, x μ , a diffeomorphism corresponds to a coordinate transformation represented by four equations: x μ →  x μ ({x μ }). These four equations correspond to the missing equations of the second element. These three elements are motivated in a bigger environment employing principles, mathematical structures, and using the old theories of Newtonian Gravity and Special Relativistic Field Theories. This is captured in Fig. 1. The black arrows represent lines of influence. For example, the equivalence principle is motivated by physics already in Newton’s theory of gravity (of course, the principle actually goes back to Galileo’s observation that different masses fall at the same rate). The equivalence principle motivates the move to general coordinates which is the starting point in setting up the mathematical structure used in General Relativity. The equivalence principle finds a particular role in the prescription (by replacing derivatives with covariant derivatives). The principle of general covariance states

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Fig. 1 The elements of General Relativity are motivated by principles, mathematical structures, and the old physical theories. The black arrows indicate influences. The red arrows indicate how the old physical theories can be obtained as a limiting case

that the laws of physics should be written in a way that they take the same form in any coordinate system. This motivates adopting the use of tensor fields in expressing physical laws. Einstein was motivated by the Poison equation formulation of Newtonian Gravity to look for an equation that is second order in derivatives of the metric (since the metric is playing the role of Newton’s gravitational potential). Figure 1 is not complete. Additional principles could be included. Mach’s principle influenced Einstein’s thinking but it is not clear that it plays a direct role in dictating the mathematical form of the elements of General Relativity. The equivalence principle is closely related to the principle of local flatness (that we can always find a reference frame with respect to which the metric is Minkowski in the vicinity of any given point). These two principles become equivalent under the assumption of metric compatibility (that ∇μ g μν = 0) [23]. The fact that we adopt a torsion free covariant derivative might warrant some sort of motivating principle (it follows from requiring that covariant derivatives, like regular partial derivatives, commute when acting on scalar fields). There is much additional mathematical structure not explicitly mentioned. There are particular tensor fields that play an important role such as the metric tensor (gμν ), the energy-momentum tensor (T μν ), and the

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Einstein tensor (Gμν ). There are the notions of coordinate transformations and, relatedly, diffeomorphisms. And there are the Bianchi identities. All these objects might be represented in an enriched diagram. One might argue for additional lines of influence. For example, the mathematical notion of tensor fields is important for every element of General Relativity. Also, we can argue that the principle of energy-momentum conservation is motivated by Newtonian Gravity as well as Special Relativistic Field Theories. However, it is the version from SRFTs (that ∂μ T μν = 0) that finds particular application. In any case, the diagram is already busy enough so less pertinent lines of influence have been omitted. The red arrows in Fig. 1 indicate that General Relativity has appropriate limits to Newtonian Gravity and to Special Relativistic Field Theories—thus solving the problem of Relativistic Gravity. It is interesting that General Relativity modifies both the old theories in order to do this—there is, perhaps, a lesson for Quantum Gravity in this.

12 Quantum Gravity 12.1 The Problem of Quantum Gravity Today we face a problem that is analogous to the problem of Relativistic Gravity. We have two physical theories that are each successful in their own realms (General Relativity and Quantum Theory) but they do not fit together. The stage is set for consideration of the following. The problem of Quantum Gravity is to find a theory that reduces in appropriate limits to General Relativity on the one hand and to Special Relativistic Quantum Field Theory (SRQFT) on the other. General Relativity ←− Quantum Gravity −→ SRQFT

Special Relativistic Quantum Field Theory seems like the appropriate version of Quantum Theory for the limiting case here. Of course, there remain technical, structural, and conceptual questions about formulating Quantum Field Theories and we might even hope that we gain some insight into these by solving the problem of Quantum Gravity (for example, Ding Jia has suggested that indefinite causal structure may offer a route to regularising Quantum Field Theory [29]).

12.2 A Proposed Path to a Theory of Quantum Gravity Can we take a path that is analogous to Einstein’s path to solving the problem of Quantum Gravity? One way to think about this is to appropriate Fig. 1. This idea

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Fig. 2 We can consider attempting to construct a theory of Quantum Gravity by following a schema that is analogous to that used to obtain General Relativity. There is much work to be done in elucidating what the different parts of this diagram might mean in this case. The present paper makes concrete proposals for some of the elements

is illustrated in Fig. 2 which has been converted from Fig. 1 by changing the “old theories” to Special Relativistic Quantum Field Theories and to General Relativity and inserting the word “quantum” where appropriate. Proposals for some elements of Fig. 2 have been provided already in this paper. We have a quantum equivalence principle. We have a notion of quantum coordinates. We have a notion of quantum diffeomorphisms which clarifies what the interpretation means. The principle of general quantum covariance can be extrapolated from the classical principle: The principle of general quantum covariance says that the laws of physics can (and should) be written in such a way that they take the same form in any quantum coordinate system

A similar idea was suggested by Giacomini et al. [13]. They set up a notion of quantum reference frames at a given time (so they are in space but not in space-time) and posited that physical laws should be covariant with respect to transformations between such frames. In fact they go a bit further and prove that the Schroedinger

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equation is covariant. We have discussed the interpretation element of Quantum Gravity in Sect. 7. The notion of an extended A-state does not seem to quite do the job of tensors in General Relativity so some work is necessary there (we will make some proposals for this in Sect. 12.5). We do not have a suitable notion of a quantum covariant derivative but there is reason to be hopeful that this can be built given the notions we do have.

12.3 How Can We Use the QEP? Now that we have the quantum equivalence principle, how should we use it to construct a theory of Quantum Gravity? We can get some suggestions from Fig. 2. There are two lines of influence coming out of the quantum equivalence principle in Fig. 2: (1) one goes to general quantum coordinates; (2) the other goes to the prescription. We have already shown how we can set up a notion of general quantum coordinates motivated by the QEP. What about the prescription? In General Relativity the equivalence principle is used to convert special relativistic field equations to general relativistic ones in such a way that we maintain the property that there exists an inertial coordinate system in the local vicinity of any given point. To do this requires a notion of covariant derivative. Thus, in the quantum context, we wish to find a way to convert special relativistic quantum field equations for the matter degrees of freedom into quantum gravity field equations in such a way that we maintain the property that there exists a causally definite quantum coordinate system in the vicinity of a point. Aligning the metric for the different |u] terms by applying a quantum diffeomorphism will not, in general, also align the matter fields. However, this will allow us to implement causality conditions locally at any given point. This seems to be the most promising avenue for research. It is worth noting that causality is not listed among the principles of Fig. 1. This is because Einstein did not make explicit use of causality in setting up the theory of General Relativity. Nevertheless, General Relativity is causal. Local disturbances cannot propagate outside the light cone determined by the metric. This is particularly interesting in the context of General Relativity because we can use disturbances in the metric itself (the very object that determines what we mean by causality) to send signals. It is, then, striking that causality comes out of General Relativity given that it was not put in. It would be interesting to pursue a different route to General Relativity wherein causality was explicitly used as a principle in the derivation of the theory. If we could do this in the classical context, then it might shed light on how to use causality as a principle in obtaining Quantum Gravity exploiting the quantum equivalence principle along the way.

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12.4 Quantum Manifolds I will now outline some tentative ideas in which we make sense of a notion of quantum manifolds with quantum tensor fields on them. First note that, a (classical) tensor field is specified by giving its components at each point, p, in a manifold. The point, p, has no intrinsic physical properties of its own (physical properties only emerge through relational properties between the various fields defined at p). Let us recall a few definitions we have used so far. An extended A-state is written  |A ] = Du cu Au |u] (61) The set, Au , is defined as Au = {(A(p), p) : ∀p ∈ Mu }

(62)

u = {((p), p) : ∀p ∈ Mu }

(63)

Here u is

Thus, u, contains full information about the given configuration of classical fields. The idea I want to propose here is to disregard this aspect of u and regard it as indexical—that is regard it as being of a similar status to a point p in a manifold, M. This is a big step as u is already a set of fields defined on a manifold. However, we can think of u as being represented as a point in its own space, S. We can take the following steps to implement this. First we disallow any operations that act on |u] to bring out tensor fields from . Second, we can imagine “scrambling” or “forgetting” the physical information in u so we are just left with the topological information. We could do this by applying some arbitrary diffeomorphism. We do, however, need to keep three pieces of information. First we need to keep Mu . Second, we need to keep the information as to whether or not two elements, u and v in S can be transformed into one another by a diffeomorphism (which we denote u 6 v). This enables us to apply quantum diffeomorphisms. Third, we need to keep the partial order u ⊆ v so we know how to evaluate [u v]. We can use S and Mu to build a space which we will call a quantum manifold. A quantum manifold is an element, Q, in a quantum fibre bundle which will be defined below. First we define the simpler notion of a basic quantum fibre bundle (this does not include the u 6 v and u ⊆ v structure) which is modeled on the definitions of a fibre bundle and a manifold. We have a base space, S, and at each point, u ∈ S is attached a manifold, Mu . We impose what we call the locallocal triviality condition intended to ensure that we can smoothly sew the quantum manifold together.

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A basic quantum fibre bundle is defined by (Q, S , π, M) such that: • for each q ∈ Q, π(q) ∈ S ; • for each u ∈ S , M(u) = Mu where Mu is a manifold (recall that this can have corners according to our usage of the term). • there exists a bijection, h, between points q ∈ Q and (u, p) where u = π(q) and p ∈ Mu . h(q) = (u, p) Further we require that the following local-local triviality conditions hold: • Q is covered by open sets, Ri , so that 

Ri = Q

i

• there exists a bijection, gi , such that gi (Ri ) = π(Ri ) × V and where V ⊆ RN ; • the sets, Oui ⊆ Mu , defined by p ∈ Oui iff h−1 (u, p) ∈ Ri are open and form a cover for Mu so that 

Oui = Mu

i

• the maps ωui (p) from p ∈ Oui to x ∈ V defined by ωui (p) = proj2 g(h−1 (u, p)) satisfies the usual axioms for such a cover for a manifold. These are: (1) for points −1 (x)) is smooth; and (2) ω(Ouj ∩ Oui ) is open. Note p ∈ Ouj ∩ Oui we require ωuj (ωui that proj2 projects onto the second factor in the cartesian product π(Ri ) × V returning an x ∈ V .

The manifolds, Mu , may be topologically distinct from one u to another (whereas the fibres in a standard fibre bundle are all topologically equivalent). However, the local spaces, Oui , are topologically equivalent to V (for all u). Hence we choose to build a something like fibre bundle with fibres V while ensuring that the Oui sets can be sewed together to form a manifold at each u. More attention is required to what types of maps h and gi are. Above they are described as bijections. However, we may want to impose continuity and, even, smoothness on them also. Certainly they need to be smooth enough that the Oui sets satisfy the smoothness condition given. However, given that the manifolds, Mu , at different u may be

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topologically different, these maps may also have to have discontinuities. Assuming we can count the distinct topologies, we may correspondingly be able to impose the constraint that h and g have a countable set of discontinuities and are otherwise smooth and still have a useful definition for quantum manifolds. One of the two local’s in “local-local triviality condition” comes from the local sewing together of the Oui ’s and the other from the usual local triviality condition for fibre bundles. We leave for future work to prove whether or not Q is actually a manifold in the usual technical sense. However, it deserves the name “quantum manifold” because it is the quantum analogue of a manifold as used in (classical) General Relativity. The functions gi (·) and h(·, ·) can be used to implement a quantum coordinate system that covers those u ∈ π(Ri ). The quantum coordinate associated with q ∈ Ri is xi (q) := proj2 gi (q)

(64)

(note that the i subscript indicates that this coordinate charts Ri ). We can use gi and h to construct an identification map from the points in Oui to the points in Ovi for the quantum coordinate system as follows: ϕu→v = proj2 h(gi−1 (v, xi (h−1 (u, p))))

(65)

This maps a point p ∈ Oui to a point q ∈ Q using h−1 . Then it uses xi to map this to some coordinate x. Then, using gi−1 , it maps this coordinate to a point q ∈ Q for which π(q) = v. Using h we map to a point (v, p) where p ∈ Ovi . Then finally, we project on to the second entry using proj2 and obtain p ∈ Ovi . Since we have a set of quantum coordinate systems (or quantum charts) that cover the whole of the quantum manifold (as we vary over i) we have a quantum atlas in the sense discussed at the end of Sect. 8. We call the above a basic quantum fibre bundle since, as stated earlier, we actually need to add extra structure corresponding to the fact that elements u and v of S have two relationships between them in the physics we have considered that we would like to keep at an abstract level (that is, without explicit reference to the physical fields, , defined on the manifolds Mu and Mv ). First, two such elements may be diffeomorphism equivalent (either can be transformed into the other by a diffeomorphism). We denote this by u 6 v. This is a transitive relationship (if u 6 v and v 6 w then u 6 w). Second, one such element may contain the other which we denote by u ⊆ v. This is a partial order. This gives us the full definition we seek A quantum fibre bundle is defined by (Q, S , π, M, 6, ⊆) such that it is a basic quantum fibre bundle in the elements (Q, S , π, M) and further: • 6 is a transitive relationship that can hold on pairs of elements u, v ∈ S such that u6v

⇒

Mv = ϕ(Mu ) for some ϕ

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• ⊆ is a partial order on elements in S such that u ⊆ v ⇒ Mu ⊆ Mv • the following completeness condition holds if ∃ u, v, w ∈ S s.t. u ⊆ v 6 w then ∃ y ∈ S s.t. u 6 y ⊆ w

Now we have enriched these quantum manifolds with the additional abstract structure required to be able to do those manipulations on |u] we discussed earlier— evaluating [u v] and applying quantum diffeomorphisms. The conditions given on manifolds for when u 6 v or u ⊆ v are necessary but not sufficient conditions (as indicated by the ⇒) because, in the actual physical examples, the fields defined on these manifolds also have to bear a certain relationship for 6 or ⊆ to hold. The completeness condition expresses the idea that if u is contained in v and v is transformed to w by ϕ then there should exist some element y that is obtained by acting on u with ϕ.

12.5 Quantum Tensor Fields We can specify a quantum tensor field by providing a tensor A(q) at each point q ∈ Q where this tensor has components defined with respect to the tangent space of Mπ(q) (thus the indices, μ, from 1 to dim(Mu )). We can lay down a quantum coordinate system, x, in some region R ⊆ Q as explained at the end of Sect. 12.4. Then we can represent the quantum tensor field explicitly by giving components δ... Aγμν... (x, u)

where u = π(q). We can sum over indices in the usual way. For example, γ Cγμ Dν = Bμν

Here we perform the summation at each point (x, u). We can use a quantum tensor field to define an extended state  |] =

  Dπ(q) (A(q), p) : ∀p ∈ Mπ(q) |π(q)]

(66)

Thus, we can go from quantum tensor fields to the path integral and recover quantum predictions in the usual way.

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13 Questions, Comments and Conclusions In this paper we have found a way to implement the quantum equivalence principle I originally proposed in [24]. The main idea is that a quantum coordinate system sets up an identification between the different manifold points corresponding to different terms in a quantum superposition and then associated coordinates, x, with the points. A quantum diffeomorphism can break this identification and so corresponds to a bigger class of symmetries than simple classical diffeomorphisms. We showed how we can use a quantum diffeomorphism to implement the quantum equivalence principle. We then discussed the conceptual structure of General Relativity (shown in Fig. 1) and conjectured that a similar structure (shown in Fig. 2) may be possible for Quantum Gravity using the quantum equivalence principle. There are a number of questions that emerge from this project. First, it is not clear that this is the only (or indeed the best way) to implement the idea of quantum coordinate systems for implementing the quantum equivalence principle. In particular, it is not clear that the ideas here can be used to account for the (albeit discrete) situation considered by Guerin and Brukner [15] and by Oreshkov [38]. It would be interesting to explore the connections between the approach to quantum coordinates considered here and quantum reference frames as considered by Giacomini, Castro-Ruiz, and Brukner (GCB). Here are a few features of the GCB approach that are not features of the approach here. First, the GCB approach uses a 3 + 1 split into space and time. Second, the approach of GCB places the reference frame on a particular quantum system. Third, the GCB approach eliminates the state associated with the quantum system the quantum reference frame is associated with (though it seems that they could keep this part of the state without affecting the main points they make). A major issue we have not resolved is fixing the phase function, θ (u, v), used in defining quantum diffeomorphisms (see Sect. 6). One possibility is that we simply set it equal to zero. However, there may be physical motivations for some other choice. In particular, this phase function may be fixed by considering the relationship between the approach to quantum coordinates presented here and the GCB approach to quantum reference frames (in which a phase is acquired on performing a quantum reference frame transformation). We have not developed a theory of measurement for extended A-states beyond saying that they can be used to calculate amplitudes via the path integral. We may be able to do much more. First, we can select on extended A-states by demanding that they have certain beable properties (these beable properties corresponding to the outcomes) as discussed in Sect. 7. We could use operational space corresponding to coincidences in the values of some set of scalars (as defined in [23]) for this purpose. Second, we may be able to take ratios of functions formed on extended Astates to determine whether probability ratios for different such outcomes are well defined (independent of choices elsewhere) and, if they are, what these probabilities are equal to (this is the formalism locality approach outlined in [20]). This kind of approach may take us beyond the path integral way of calculating probabilities.

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To push forward this project we need to say what object is analogous to the covariant derivative. In General Relativity, the covariant derivative replaces the partial derivative in field equations (this happens in the prescription). We need to find an analogous procedure for quantum field equations. Acknowledgements I am deeply grateful to Flaminia Giacomini, Esteban Castro-Ruiz, and ˇ Philippe Guérin explaining to me their work as well as broader discussions. I am grateful to Caslav Brukner for email correspondence. I am also very grateful to Ding Jia for explaining his superposed spacetimes approach and for comments on an earlier version of this paper. I am grateful to Joy Christian for discussions on fibre bundles and the role non-trivial geometries may play in Quantum Gravity. I am grateful to Zivy Hardy for support and the people at Aroma Cafe where most of this work was done. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. This project was made possible in part through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation. I am grateful also to FQXi for support through the grant FQXi-RFP-1824 entitled “Operationalism, Agency, and Quantum Gravity”.

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The D-CTC Condition in Quantum Field Theory Rainer Verch

Abstract A condition proposed by David Deutsch to describe analogues of processes in the presence of closed timelike curves (D-CTC condition) in bipartite quantum systems is investigated within the framework of local relativistic quantum field theory. The main result is that in relativistic quantum field theory on spacetimes where closed timelike curves are absent, the D-CTC condition can nevertheless be fulfilled to arbitrary precision, under very general, model-independent conditions. Therefore, the D-CTC condition should not be taken as characteristic for quantum processes in the presence of closed timelike curves in the sense of general relativity. This report is a very condensed summary of the publication (Tolksdorf and Verch, Commun Math Phys 357:319–351, 2018). A new result showing that the D-CTC condition can be approximately fulfilled by entangled states is also presented.

1 The D-CTC Condition: Bipartite Quantum Systems The D-CTC condition was introduced in 1991 in a seminal article by David Deutsch [9] as a condition for processes involving an analog of closed timelike curves (CTCs) in quantum circuits. Such a process in a quantum circuit is symbolically depicted in Fig. 1. In more concrete terms, the simplest form of a quantum circuit is a bipartite quantum system with subsystem Hilbert spaces HA and HB and H = HA ⊗ HB as the total Hilbert space of the system, together with a unitary operator U : H → H describing the dynamics (“one-time-step time evolution”) of the system, coupling the two subsystems. −T symbolizes a “time-step backward in time”, meaning that the partial state of the full system on the system part B after applying U is the same as before applying U on system part B. More formally, Deutsch’s condition on quantum circuits with processes that provide “analogues of closed timelike curves”, that means, involving (dynamical)

R. Verch () Institut für Theoretische Physik, Universität Leipzig, Leipzig, Germany e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 F. Finster et al. (eds.), Progress and Visions in Quantum Theory in View of Gravity, https://doi.org/10.1007/978-3-030-38941-3_9

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Fig. 1 A process in a quantum circuit is represented a unitary operator U describing the dynamical coupling of two system parts (denoted by A and B). A “step backward in time” is symbolized by −T ; the B-part of the result of the process is again fed into the process as initial state of the B-part

U

T

“backward time steps”—referred to as D-CTC condition—is verbally described as follows: Given a unitary U on H and a partial state (density matrix) %A on system part A, a state (density matrix) % of the full system is said to fulfill the D-CTC condition if the restriction of % to system part A coincides with %A and if U %U ∗ and % agree when restricted to system part B. At the level of equations, this is expressed in the following way: • Given: U unitary on H, %A density matrix on HA A density matrix % on H fulfills the D-CTC condition if • TrB % = %A ⇔ Tr(%(a ⊗ 1)) = TrHA (%A a) and • TrA U %U ∗ = TrA % ⇔ Tr(%(1 ⊗ b)) = Tr(U %U ∗ (1 ⊗ b)) for all a ∈ B(HA ) and b ∈ B(HB ). Deutsch has shown that there is always a solution to the problem of finding a density matrix state % fulfilling the D-CTC condition if the Hilbert spaces HA and HB are finite-dimensional: Given U and %A = density matrix on HA , there is %B = density matrix on HB such that % = %A ⊗ %B fulfills the D-CTC condition. According to Deutsch, the proof is based on a fixed point argument: The map S : %B → TrA (U (%A ⊗ %B )U ∗ ) possesses fixed points. The deeper reason why the result holds—and why a fixed point argument can be used—is that for quantum mechanical systems with finite dimensional Hilbert spaces, the state space is convex and closed.

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2 The D-CTC Condition and Dynamics on CTC Spacetimes Spacetimes with CTCs are possible in General Relativity. Their physical realization as solutions to Einstein’s equations of gravity and matter is doubtful. The common understanding is that unusual matter properties (e.g. negative energy densities) are required in order for spacetimes with CTCs to occur in general relativity, i.e. as solutions to Einstein’s equations, and that this is not feasible for macroscopic matter [16]. Deutsch [9] has proposed that quantum circuits with backward time-steps can be seen as analogues of dynamical (quantum) systems on spacetimes with CTCs. The paradigm example to this end is the 2D (Deutsch-) Politzer spacetime which arises by cutting out two finite spacelike slits separated by a finite time segment in two-dimensional Minkowski spacetime and by identifying the “inner rims” and “outer rims” of the cuts as depicted in Fig. 2. For further discussion, see the refs. [7, 14, 25]. One can consider (possibly multi-component) wave functions ψ(t, x) on Politzer spacetime fulfilling a hyperbolic wave equation which may be brought into the form ∂t ψ(t, x) = Dx ψ(t, x) where Dx is a suitable (possibly matrix-formatted) differential operator acting with respect to thex coordinate. (E.g. for a Klein-Gordon field φ(t, x), one would choose  φ(t,x) ψ(t, x) = ∂t φ(t,x) .) The identification of the cuts in the Politzer spacetime imposes boundary conditions like lim ψ(−τ + ", x) − ψ(τ − ", x) = 0

"→0+

along the cut line, and one may regard them as analogous to Deutsch’s condition TrA (U %U ∗ ) = TrA (%) on viewing U as performing the analogue to the “time-evolution” ψ(−τ, . ) → ψ(τ, . ) . Fig. 2 The Politzer spacetime arises by cutting two spacelike line segments out of 1 + 1 dimensional Minkowski spacetime and by suitably identifying the cuts

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It is basically this analogy that serves as motivation to interpret the D-CTC condition for bipartite systems stated above as representing processes in the presence of closed timelike curves. This proposal by Deutsch has stimulated several investigations of the D-CTC condition including experimental tests; the references [1, 3, 5, 24, 27] are only a few examples. However, authors of more recent publications went beyond Deutsch’s original interpretation of the D-CTC condition which was mainly at the level of an analogy. E.g. in [27] it is claimed that bipartite quantum systems in which the D-CTC condition is fulfilled have experimentally been constructed, and that “. . . quantum mechanics therefore allows for causality violation without paradoxes whilst remaining consistent with relativity”. Taken literally, this amounts to a much stronger interpretation of the D-CTC condition, basically stating that it is characteristic of quantum processes in the presence of closed timelike curves in the sense of general relativity. From our perspective, the question now arises if this stronger interpretation is justified, and how one could attain an answer to that question—bearing in mind that, apart from the analogy with fields on the Politzer spacetime, the D-CTC condition in its original formulation refers to general bipartite quantum systems without an a priori given relation to spacetime structure. The connection between quantum physics and special or general relativity is provided by relativistic quantum field theory, brought to the fore at its best in the local, operator algebraic approach [15]. In fact, in [33] it is shown that the quantized massless scalar field can be constructed on the Politzer spacetime in compliance with the principles of isotony of locality, upon making suitable adaptations due to the occurrence of closed timelike curves, like the F-locality concept due to Kay [19]. We refer to the references for full discussion, on which we touch briefly in the following section.

3 Relativistic Quantum Field Theory Relativistic quantum field theory, especially in the local, operator algebraic framework, addresses the localization of observables and processes in the sense of special relativity, or in the sense of general relativity when suitably extending the framework as quantum field theory in curved spacetimes supplied with local covariance [4, 13, 15]. Basic elements of the such extended framework, describing a quantum field theory on a d + 1 dimensional Lorentzian spacetime M with metric g and taken to be orientable and time-orientable, are: A quantum field theory on M is given by a collection A(O) of operator algebras indexed by subsets O of M. Hermitean elements in A(O) represent observables which can be measured within O, and therefore, the collection of operator algebras is subject to the following conditions, • Isotony: O1 ⊂ O2 ⇒ A(O1 ) ⊂ A(O2 ) • Locality: O1 and O2 causally separated ⇒ A(O1 ) and A(O2 ) commute

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There are examples where the subsets O of M are a priori “extended” [23, 26]. If the A(O) are “large” for arbitrarily “small” neighbourhoods of points in M, the QFT is called arbitrarily localizable (this is the standard case if the A(O) derive from a Wightman quantum field). If the spacetime M with metric g is globally hyperbolic, implying the absence of CTCs, then this works fine; and there are many examples. In fact there is reason to assume that every physically realistic QFT on a globally hyperbolic spacetime complies with this basic structure. (See [13] for discussion.) If a spacetime contains CTCs, this is far from clear. • In general, constraints stemming from CTCs could make the A(O) small • Isotony and locality become highly non-trivial in presence of CTC constraints— arbitrary localizability won’t hold in general In other words, the most basic conditions or principles generally assumed for observable quantities of quantum field theories on globally hyperbolic spacetimes (which are safe against the occurrence CTCs) won’t be applicable for QFTs on spacetimes containing CTCs. Therefore, it is not even clear what structural properties a QFT on a spacetime containing CTCs should have, and given the lack of a generally accepted guideline to that effect, one could take quite diverse positions. A position one could take may aim to make the local algebras of observables a priori as large as possible and to view consistency conditions arising from the presence of CTCs as a means to classify various possibilities for CTCs to arise. A contrasting position may consist in trying to generalize the principles of QFT on globally hyperbolic spacetimes in such a way to quantum field theory on more general spacetimes that the consistency conditions arising from the presence of CTCs render any quantum field theory on such spacetimes inconsistent (inexistent). The latter view has essentially been adopted in the publications [16, 19, 20] to which we refer for considerable further discussion. See also [10, 21, 22] for complementary views. However, the task we have set ourselves—to investigate the D-CTC condition within a framework that refers to the principles of special or general relativity—is without the mentioned difficulties as long as we stay within the established approach of quantum field theories on globally hyperbolic spacetimes.

4 The D-CTC Condition in QFT on Globally Hyperbolic Spacetimes Now we assume that we are given a globally hyperbolic spacetime M with metric g together with an arbitrarily localizable QFT given by a collection of operator algebras A(O) complying with isotony and causality where the O range over all open, relatively compact subsets of M.

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Fig. 3 Sketch of two spacetime regions (in two-dimensional spacetime) which are causally separated, i.e. there is no causal curve connecting the closures of the spacetime regions OA and OB . These spacetime regions represent the causal completions of the times and locations where two causally separated observers carry out measurements on their respective parts of the system

Bipartite systems are represented by pairs operator algebras A(OA ) and A(OB ) for causally separated spacetime regions OA and OB as illustrated in Fig. 3 (see [17, 34]). Furthermore, we add the (not needed but) technically convenient assumption that the A(O) are von Neumann algebras on a Hilbert space H. This amounts to the choice of a Hilbert space representation of the QFT. In QFT on Minkowski spacetime, this would typically be the vacuum representation and a similar choice could be adopted if the spacetime is stationary. For a QFT on a general curved spacetime, the Hilbert space representations induced (by means of the GNS representation) by states fulfilling the microlocal spectrum condition are regarded as describing physically relevant system configurations [12, 13, 18]. With these assumptions in place, we will now investigate the extent to which the D-CTC condition can be fulfilled. We put this in the following form: D-CTC Problem Given a unitary U in H and a density matrix state ωA (a) = Tr(%A a)

(a ∈ A(OA ))

on A(OA ), is there a density matrix state ω(c) = Tr(%c) on B(H) whose partial state on A(OA ) agrees with ωA and which is U -invariant when restricted to A(OB ), i.e. ω(a) = ωA (a) ω(U ∗ bU ) = ω(b)

(a ∈ A(OA )) (b ∈ A(OB )) ?

Our first result displays conditions under which the D-CTC problem has no positive answer. To this end, we make the assumption that the spacetime is ultrastatic, i.e. it takes the form M = R × & with g = dt 2 − h(&) where & is a d-dimensional

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manifold with a complete Riemannian metric h(&) . Thus, there is a global time coordinate t ranging over a “time axis” R, and the spacetime metric has a very rigid time-symmetry with respect to this time-coordinate. We furthermore suppose that • the QFT admits a time-shift symmetry implemented by a continuous unitary group Vt = eit H with generating Hamilton operator H ≥ 0, such that Vt A(O)Vt∗ = A(Ot ), where Ot = {(t  + t, σ  ) : (t  , σ  ) ∈ O ⊂ R × &} • the QFT admits a vacuum vector, i.e. a unit vector ∈ H so that Vt = • the QFT fulfills additivity: if a family {Oν } of open, relatively compact subsets of M covers O, then the von Neumann algebra generated by all the A(Oν ) contains A(O). • the QFT fulfills the timelike tube property: for any open (non-void) O, the von Neumann algebra generated by all the Vt A(O)Vt∗ , t ∈ R, coincides with B(H). The timelike tube theorem can be seen as a variant of the Reeh-Schlieder property (see below) which holds under very general conditions given the other assumptions [2, 30]. The above conditions are fulfilled e.g. for any QFT on Minkowski spacetime that derives from a Wightman-type quantum field. A density matrix state ω(c) = Tr(%c) has finite energy if %=

1 E %E ˜ , Tr(E %E) ˜

where E denotes a spectral projector of H corresponding to any finite spectral interval. A density matrix state ω is an analytic state for the energy if it is a limit of finite energy states preserving the analyticity properties of finite energy states with respect to Vt . Under the stated assumptions,1 any analytic state has the Reeh-Schlieder property:

ω(c∗ c) = 0 ⇒ c = 0

(c ∈ A(O)) .

For more on the Reeh-Schlieder property, see [15, 28, 29, 31]. We can now state our first result. Theorem 1 Assume that the spacetime is ultrastatic and that the QFT admits a ground state and fulfills additivity and the timelike tube property as described above. Then for given unitary U in H and density matrix state ωA on A(OA ), there can be no density matrix state ω with both of the following properties: (i) ω is a solution to the D-CTC problem (ii) Both ω and ω(U ∗ . U ) are analytic for the energy For the proof of this statement, see [33]. Analytic states show long-range entanglement as a consequence of the Reeh-Schlieder property [8, 34]. If entanglement is taken as a distinguishing feature of a quantum theory (as opposed to a classical 1 For the way the Reeh-Schlieder property is formulated here, it must be assumed that the closure of the spacetime region O has a non-void open causal complement.

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physical theory, without a priori uncertainty relations), then the result might be taken as indirectly supporting the view that in a quantum theory, the D-CTC condition is a characteristic feature for the occurrence of CTCs (since in a QFT on a spacetime without CTCs, entanglement and D-CTC condition are not coexistent). However, in the light of the following results, this position appears to us as a stretched interpretation of the counterfactual. The next result we are going to present shows that under very general conditions, the D-CTC condition can always be fulfilled approximately, to any required precision, in arbitrarily localizable QFTs on any globally hyperbolic spacetime. Thus, we now assume that the manifold M with Lorentzian metric g is a generic globally hyperbolic spacetime—no symmetry assumptions are being imposed. We consider an arbitrarily localizable QFT on this spacetime, again described by a family of von Neumann algebras A(O) on a Hilbert space H, with O ranging over the open, relatively compact subsets of M. Isotony and locality are imposed as before. A further assumption that we add is the split property. The QFT fulfills the split property if, • given a pair of strictly causally separated spacetime regions OA and OB , and • given a pair of density matrix states ωA (a) = Tr(%A a)

(a ∈ A(OA )) ,

ωB (b) = Tr(%B b)

(b ∈ A(OB )) ,

there is a density matrix state ω(c) = Tr(%c) on B(H) which extends the two partial states as a product state (uncorrelated state), i.e. ω(ab) = ωA (a) · ωB (b)

(a ∈ A(OA ) , b ∈ A(OB )) .

We have used here the term “strictly causally separated” in order to emphasize that the closures of the spacetime regions OA and OB need to be causally separated which implies that the closures of the regions cannot touch (however this agrees with our previous definition of causally separated spacetime regions in this paper). The split property means that the subsystems of the QFT experimentally accessible within the spacetime regions OA and OB can be made statistically independent, or isolated from one another, by preparing suitable physical states once the spacetime regions are strictly causally separated. The split property has been proved to hold for free QFTs and generalized free fields with a sufficiently regular mass spectrum; it has also been proved for interacting QFTs in 1 + 1 dim. It is expected to hold generally for physically interpretable QFTs. For considerable further discussion, see the publications [6, 11, 17, 32] and references cited therein. We can now state our second result. Theorem 2 Assume that the QFT is on a general globally hyperbolic spacetime and fulfills the split property. Then, given any unitary U in H and any density matrix

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state ωA (a) = Tr(%A a) on A(OA ), there is an approximate solution to the D-CTC problem in the following sense: Given arbitrary ε > 0 (small) and R > 0 (large), there is a density matrix state ω = ωR,ε on B(H) such that • ω(a) = ωA (a) (a ∈ A(OA )) • |ω(U ∗ bU ) − ω(b)| < ε (b ∈ A(OB ) , ||b|| < R) For the proof, see again [33]. It follows from the proof that the state ω which provides an approximate solution to the D-CTC problem is highly non-unique; moreover, it is classically correlated (i.e. not entangled) across the bipartite system defined by the local algebras of observables A(OA ) and A(OB ). This is considerably different from our first result, where states with a high degree of entanglement have been shown not to provide exact solutions to the D-CTC problem. However, if an approximate solution to the D-CTC problem is allowed to be approximate on the A(OA ) part of the QFT as well, then there are also states which are analytic in the energy as solutions to the D-CTC problem. This is the statement of our next observation, which appears here for the first time. We again invoke the assumptions made for Theorem 1 , that is, the spacetime M with Lorentzian metric g is assumed to be ultrastatic, and the QFT is assumed to possess an associated time symmetry implemented by a continuous unitary group Vt (t ∈ R) on the Hilbert space H with a generating Hamilton operator H whose spectrum is non-negative. Furthermore, it will be assumed that there is a vacuum vector , that the system of local observable algebras of the QFT fulfills additivity, and that the timelike tube property holds. In addition, we assume that the split property holds. Theorem 3 Let U be a unitary operator on H and ωA a density matrix state. Furthermore, let ε > 0 and R > 0 be given. Then there is a density matrix state ωan on B(H) which is analytic in the energy and has the property that     ωA (a) − ωan (a) + ωan (U ∗ bU ) − ωan (b) < ε holds for all a ∈ A(OA ) and b ∈ A(OB ) which fulfill the bound ||a|| + ||b|| < R. Proof Let En be the spectral projectors of the Hamilton operator H corresponding to the spectral intervals [0, n](n ∈ N). Furthermore, if % is any density matrix on H, define the sequence of density matrices %n =

1 En %En . Tr(En %En )

(1)

Clearly, each %n induces a density matrix state which is analytic in the energy. It is plain to check that, given ε > 0 and R  > 0, there is n ∈ N so that sup |Tr(%c) − Tr(%n c)| < ε

||c||