TIME AND SCIENCE: Physical Sciences and Cosmology [3, 1 ed.]

Table of contents :
CONTENTS
Time and Science: Foreword
Introduction to Volume 3
Chapter 1 The Physics of “Now”
1 Introduction
2 A Model IGUS
3 The Present is not a Moment in Time
3.1 Some features of Minkowski space
3.2 The past, present, and future of the robot
3.3 The common present
4 Why Don’t We Remember the Future?
5 Alternatives to Past, Present, and Future
5.1 Different organizations of temporal information
5.2 Different laws, different scales
6 Conclusion
Acknowledgments
Appendix A: The Cosmological Origin of Time’s Arrows
References
Chapter 2 Discovering Physical Time within Human Time
1 Introduction
1.1 Two times problem
1.2 Introduction of the IGUS view to solve the problem
1.3 The many IGUS views
2 A Dualistic IGUS View of Manifest Time
2.1 Definitions
3 Dualistic Components of Manifest Time
3.1 “Present” (unique [moving] and no unique present)
3.2 Persistence (enduring self) and impermanence (ephemeral self)
3.3 Change (dynamic and completed)
3.4 Motion (dynamic movement) and completed movement
3.5 Temporality and temporal order
3.6 Speed of time and duration judgments
3.7 Other components of Manifest Time
3.8 Flow whoosh and dynamism
4 Summary of the Dualistic Mind Theory
5 Dualistic IGUS Claims and Predictions
5.1 Evolution of IGUSes
5.2 Consequences of a loss or dysfunction of the illusory system
5.3 Testing the dualistic mind view
6 Conclusions
Acknowledgments
References
Chapter 3 New (and Old) Work on the Fundamentality of Time
1 Introduction
2 Doing Away with Time
3 Platonia, Superspace, and Timelessness
4 Neo-Heracliteanism: Time Reinstated
5 Allocentric Versus Egocentric Representation
6 A Place for Will and Creation?
References
Chapter 4 The Layers That Build Up the Notion of Time
1 Introduction
References
Chapter 5 Senses in Which Time Does and Does Not Exist
1 Introduction
2 Newtonian Dynamics Expressed Relationally
3 A Relational Universe
4 Building the House
5 Subsystems and the Arrow of Time
6 Newtonian Big Bangs
Acknowledgments
References
Chapter 6 The Complex Timeless Emergence of Time in Quantum Gravity
1 Preambles
1.1 Our focus: physical time as deduced from our best theories
1.2 Emergence and its many kinds
1.3 Making things concrete: an example of emergent behavior
2 What is Time, in General Relativity?
2.1 Manifold points, diffeomorphisms, and background independence
2.2 Relational time (and space)
3 What is Time in Quantum Gravity if it is Quantized GR?
3.1 Quantum GR, quantum fields, quantum (relational) time
3.2 The timeless emergence of time from quantum time
4 What is Time in Quantum Gravity if it is not Quantized GR?
4.1 A timeless ontological emergence of time
4.2 The timeless emergence of time via a continuum approximation
4.3 What kind of emergence is this?
5 Things Get Worse, for Time: Quantum Gravity Phase Transitions and Geometrogenesis
5.1 Inequivalent continuum phases and phase transitions
5.2 The fundamental ontology is even more timeless
5.3 Geometrogenesis
5.4 Geometrogenesis as a physical process: a temporal characterization?
6 Concluding Remarks
References
Chapter 7 Time and Durations in Relativistic Physics
1 Datations and Durations: Time and Proper Times
1.1 The basis: events, space-time, histories
1.2 Measuring time?
1.3 Proper times
2 Time From Chroincidence
3 No Chroincidence in the Real World
4 Relativistic Delays
5 A Twin Story
6 Delay as a Geometrical Effect
7 A Crucial Experiment
8 The Shapiro Delay
9 Blue- and Redshifts
10 The Kinematical Shift
11 The Einstein Effect
11.1 The black hole case
12 Spectral Redshift and Energy
13 Observations of Einstein effect
13.1 Einstein effect on Earth
13.2 Redshift from the Sun
13.3 From Sirius
13.4 Clocks
13.5 The GPS system
14 Linking Delays and Shifts
15 What Remains?
References
Chapter 8 Problem of Time: Lie Theory Suffices to Resolve It
1 Introduction
2 Time and Background in Each Paradigm of Physics
2.1 Space and time in Newtonian Physics
2.2 Relational critique
2.3 Special Relativity (SR)
2.4 GR from the spacetime (S) perspective
2.5 Killing vectors
2.6 Dynamical and Canonical formulation of GR (C perspective)
3 Relationalism as Implemented by Lie Derivatives
3.1 State Space Relationalism
3.2 Temporal Relationalism
3.3 Combining Temporal and Configurational Relationalism
4 Closure, as Implemented by Lie Brackets
4.1 Lie (and Poisson) algebraic structures
4.2 Generalized Lie Algorithm (GLA)
5 Assignment of Observables
6 Constructability, as Implemented by Deformations if Rigidity Holds
7 Wheelerian 2-way Route Between Dynamical and Spacetime Lie Claws
7.1 Spacetime Constructability from Space
7.2 Refoliation Invariance
8 Conclusion
8.1 Recovery of previous literature’s individual Problem of Time facets
8.2 Facet interferences explained
8.3 End summary
8.4 Further outline of global and quantum versions
Acknowledgments
References
Chapter 9 Views, Variety, and Celestial Spheres
1 Introduction
2 Summary of the Causal Theory of Views
2.1 A view may be expressed as a quantum state in a celestial sphere
3 Relativistic Measures of Differences of Views
3.1 Best matching and local gauge invariance
3.2 The view as an unordered set of incoming energy-momentum
3.3 The view as a state in a boundary Chern–Simons theory
3.4 The ladder of dimensions
3.5 Quantum gravity and spin networks
3.6 The Chern–Simons boundary action
3.7 Some remarks
4 The Dynamics
4.1 The variety as potential energy
4.2 The dynamics of difference: the half-integral
4.3 Semiclassical limit: the emergence of Minkowski space-time
5 Conclusions
Acknowledgments
References
Chapter 10 Scientific Cosmogony, the Time in Quantum Relativistic Physics
1 Introduction
2 The New Standard Model of Cosmology
2.1 A short historical survey
2.2 The assets of ΛCDM
2.3 Summary of the outcome of ref. [7]
3 The Problem of Time in Cosmology
3.1 The “extra-mundane time” of de Sitter
3.2 A beginning of the world before the beginning of space and time
4 As a Conclusion
References
Chapter 11 PT Symmetry
1 The Condition of PT Symmetry
2 Complex Number System
3 A Fundamental Symmetry of Complex Numbers
4 Complex Numbers and PT Symmetry
5 Extension of PT Symmetry to CPT Symmetry
6 What is Symmetry and Why is it Useful?
7 An Elementary PT-symmetric Classical Physical System
8 An Elementary PT-symmetric Quantum-mechanical Physical System
9 First Experiments Involving PT-symmetric Optical Systems
10 Constructing Model Non-Hermitian PT-symmetric Hamiltonians
11 Some Other PT-symmetric Hamiltonians
12 Conservation of Probability
13 Laboratory Studies and Some Practical Applications of PT Symmetry
14 Future Possible Theoretical Applications of PT Symmetry
Acknowledgments
References
Chapter 12 Experimental Evidence for Time Reversal Violation
1 Introduction
2 Experimental Studies of T Reversal Symmetry
2.1 Detailed balance
2.2 Neutrino oscillations
2.3 Permanent electric dipole moments
2.3.1 Neutral meson systems: tests of CP, T, and CPT invariance
2.3.2 T violation with neutral K mesons
3 Experiments with Entangled Neutral Mesons
3.1 The BABAR experiment with entangled B mesons
4 Conclusions
References
Chapter 13 Free Will and the Arrow of Time
1 Introduction: The Biocosmology Revolution
2 A Simple Story
3 Einstein’s Message
4 Back and Forward in Space-time
5 Obstinate Theorists
6 My Traitor’s Heart
7 The Initial Conditions Problem
7.1 The past hypothesis
7.2 Boltzman’s equation
7.3 The multiverse
7.4 Experimental data and the 3.7 billion-year-old scientific method
8 The Block Universe
9 Time Asymmetry — My “Traitor’s’’ Heart
10 The Non-organicist Neurosciences on Quantum Mechanics in the Brain
10.1 Time scales of the brain
10.2 What gives the brain its advantage in the world?
10.3 What is a “detection” in physics?
11 Novelty
12 At Sea
13 My opinion: The Future and Beyond
Acknowledgments
References

Citation preview

TIME AND SCIENCE Volume 3

Physical Sciences and Cosmology

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TIME AND SCIENCE Volume 3

Physical Sciences and Cosmology

Edited by

Rémy Lestienne Centre National de la Recherche Scientifique, France

Paul A Harris Loyola Marymount University, USA

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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Published by World Scientific Publishing Europe Ltd. 57 Shelton Street, Covent Garden, London WC2H 9HE Head office: 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

TIME AND SCIENCE In 3 Volumes Volume 1: The Metaphysics of Time and Its Evolution Volume 2: Life Sciences Volume 3: Physical Sciences and Cosmology Copyright © 2023 by World Scientific Publishing Europe Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-1-80061-997-5 (set_hardcover) ISBN 978-1-80061-998-2 (set_ebook for institutions) ISBN 978-1-80061-999-9 (set_ebook for individuals) ISBN 978-1-80061-372-0 (vol. 1_hardcover) ISBN 978-1-80061-373-7 (vol. 1_ebook for institutions) ISBN 978-1-80061-384-3 (vol. 1_ebook for individuals) ISBN 978-1-80061-374-4 (vol. 2_hardcover) ISBN 978-1-80061-375-1 (vol. 2_ebook for institutions) ISBN 978-1-80061-385-0 (vol. 2_ebook for individuals) ISBN 978-1-80061-376-8 (vol. 3_hardcover) ISBN 978-1-80061-377-5 (vol. 3_ebook for institutions) ISBN 978-1-80061-386-7 (vol. 3_ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/Q0405#t=suppl Desk Editor: Rhaimie B Wahap Typeset by Stallion Press Email: [email protected] Printed in Singapore

TIME A ND S C I EN CE: F O R EWO R D Carlo Rovelli

Time is never an emotionally neutral topic. In The Direction of Time, one of the most lucid books on the nature of time, Hans Reichenbach suggests that it was in order to escape from the anxiety that time causes us that Parmenides wanted to deny its existence; Plato imagined a world of ideas that exist outside of it, and Hegel speaks of the moment in which the Spirit transcends temporality and knows itself in its plenitude. It is maybe in order to escape this anxiety that we have imagined the existence of ‘eternity’: a strange world outside of time that we would like to be inhabited by gods, by a God, or by immortal souls. The opposite emotional attitude, the veneration of time — that of Heraclitus or Bergson — has given rise to just as many philosophies, perhaps without getting us any nearer to giving us a clear shared understanding of what time really is. Our deeply emotional attitude towards time may have contributed more to the construction of cathedrals of philosophy than has logic or reason. In the course of the last centuries, the advance of scientific knowledge has definitely revealed that a number of “natural” intuitions about the nature of time are wrong: as soon as we step out from the limited domains of our daily routine, time behaves surprisingly differently from what we intuitively expect. In particular, the unexpected temporal structure of the world revealed by relativistic physics and the invariance under time (better: CPT) reversal of microphysics have forced us to reconsider old assumptions and adjourn old debates. The closer we look to the nature of time, the richer and complex are the problems that appear. The extraordinary richness and variety of the perspectives presented in this collection testify to this depth. v

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But something about the nature of time keeps remaining slippery and perhaps mysterious. I think that it is exactly this complexity that makes the problem of time so enchanting: we have to constantly modify the conceptual structure we use to think about temporality, when we consider different orders of phenomena: from the still uncertain theories of quantum (space-) time to the violent temporal distortions of general relativity, up from the lack of directionality of microphysics to the time arrow of macrophysics, and up still to the peculiar time extension, via memory and anticipation, of the products of the biological evolution, and finally to the ample control of temporality that the specific mental evolution of our species and its recent civilization and culture have produced. Our thinking and our emotions, our very nature, can only exist in — are byproducts of — the oriented time of macrophysics. There isn’t a single time; there are layers and layers of structures that make up our sense of time. We can only ponder about time from within this peculiar macroscopic temporal structure that makes us. And perhaps the uppermost of the layers, the one nourished by the emotion of time, by the emotion of impermanence, is what time really is for us. Will we be able to disentangle the rich puzzle of time more clearly in the future? Maybe yes. But what we have learned so far about the nature of time is already astonishing. The fact that we still disagree on so many subtle points, as the pages that follow testify, does not render this path of discovery less valuable. On the contrary, it makes it richer, and more fascinating. My wish is that among the young readers of the confused and apparently contradictory ideas in this text, some will find a way to shed more light on the nature of time.

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IN TRO D UC T I ON TO VO LU M E 3 The present volume of the Time and Science Series is devoted to the physical sciences and cosmology. “Today, more than ever, physicists and cosmologists alike are troubled by the question “what is Time”? Is it an ontological property — a necessary ingredient for a physical description of the world — or a purely epistemological element that is relative to our situation in the world? For many, relativity (and particularly general relativity), as well as the reconciliation with quantum mechanics in the elaboration of a quantum theory of gravitation, points to a negative answer to the first alternative, and leads them to deny the objective reality of time. For others, the answer is nuanced by the evidence of the emergence of the temporal property when one climbs the scales of the complexity of systems and/or the applicability of the statistical laws of thermodynamics. But for some, the illusion of the unreality of time comes from certain confusions that they denounce, and they instead plead for the re-establishment of time at the heart of physical theories. We open the volume with a reflection on the relationship between cognitive time, lived time, and the time of relativistic physics and its spacetime. To this end, we first reproduce the article by the physicist James Hartle, published in 2005 in the American Journal of Physics, titled “The Physics of ‘Now’.” He had the idea to model, a little like Turing and Turing’s machine, the essential mechanisms of the acquisition of temporal information in humans, or to be more general, in any machine of acquisition and use of the ambient information, called IGUS (Information Gathering and Utilizing System). For him, this machine necessarily knows the distinctions between present, past, and future; these categories of information do not contradict the laws of physics and can fit a physical description in terms of spacetime (the cognitive present is not a spacelike surface in spacetime, it is a local and thick present, or specious, as William James would say). The article analyzes the characteristics of each of these vii

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categories with regard to the laws of physics, taking into account in particular the delayed character of electromagnetic waves and the universal law of entropy increase. The following article by Ronald Gruber, Carlo Montemayor, and Rick Block continues in Hartle’s wake to explore the relationship of psychological time and physical time. The authors have indeed seen that the “time flow” aspect of the psychological present requires the use of Hartle’s “IGUS” model to interrogate its relationship with physical time. They favor a dual-time model, for which the conscious experience in humans of the past, present, and future trilogy does not contradict the laws of physics, but the dynamism of time, its sensible flow, would be an illusion, experimentally demonstrable. The authors note that the brain has been endowed with a capacity for truthful experience that is consistent with physics, but that truthful experience also has illusory components adopted by evolution to make the brain more functional. Dean Rickles and Jules Rankin propose a different vision of the problem of the reality of Time. They examine the possibility of abandoning the Platonic representation of configuration space (what Lee Smolin calls the Newtonian approach to physics) by something new; on this occasion, they review the Smolin’s argument, developed in particular in his book Time Reborn (2013), according to which the Newtonian approach can only describe closed systems studied by an external observer equipped with a clock, but is unsuitable for any description of the Universe as a whole. Further on, they briefly describe, and it seems to me, keeping their distance, Smolin’s attempt to propose a new interpretation of quantum mechanics around the “Principle of Precedence.” They insist more on Smolin’s program, which is to re-establish the aspects of lived temporality in the description of reality, such as the distinction between past, present, and future, those of flow and evolution, and even of the qualia of consciousness, and at the same time the possibility of introducing the appearance of genuine novelty. We note the proximity of this program to the one proposed in his time by Alfred North Whithead, in spite of the failure of the new theory that this author developed to replace the theory of Relativity of Einstein.1 Finally, the authors insist on the possible relation between the appearance of novelty and free will, which they see as perhaps the most credible flaw in determinism.  See for example R. Lestienne, Alfred North Whitehead: Philosopher of Time, World Scientific Publishing, 2022. 1

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In the next chapter, Julian Barbour develops his reflection on the theme of existence or not, on a fundamental level, of time in physics, within the framework of a universe composed of a countable number of particles, governed by the Laws of Newtonian Physics. He shows how time can, in this framework, be understood as a construction whose elements are the forms of the universe that are suitably defined. His program begins with a reformulation of Newton’s laws based on what Einstein called “Mach’s principle”, that is, in which the gravitational pull on all the masses of the universe brings out the existence of a set of frames of reference for which Newton’s laws are valid. Although the arguments posed by Barbour remain within the framework of Newtonian physics, their validity for relativistic physics is defended with conviction. The author notes in particular that the Leibnizian principle of the identity of indistinguishables, which inspired Einstein in the development of his theories, is found again in the present presentation. In a relational universe of a countable number of particles of fixed total energy, Jacobi’s principle of the extremalization of action leads to a physical theory for which trajectories are represented by geodesics and time is basically absent. However, time can be constructed from the average motion of all particles. This time, says J. Barbour, is not really a clock time, but rather a time representative of the history of the universe as soon as one extends the reflection on the model to considerations of entropy. Lee Smolin then takes us into the most recent developments of his construction of a causal physical theory: relational in the Leibnizian sense of the term, and purely objective. In constructing his theory, Smolin has rid himself of any a priori prejudice on the nature of the spacetime framework to be used. The fundamental elements of this theory are not only the “events,” but the “views,” i.e., the elements with the web of relations propagated at the speed of light that the events receive from previous events and which determine its nature. In a way, locally, it is the temporal reverse of the vision that we have of the world at a given moment. It is the total transfer of energy and impulse that the event receives, and which respects the law of conservation of these entities that matters. In his theory, the dynamic evolution of the world is determined by the differences between the views that constitute it, through a new entity that the author (in a joint work with Julian Barbour) called the “variety.” The universe is governed by a law that tends to maximize variety. What is remarkable is that in developing this theory, Smolin realizes that it is not only consistent with relativity but also fits perfectly with ix

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non-relativistic quantum theory (in particular to recover the property of entanglement), with prospects for extensions to general relativity and a relativistic quantum theory. Last but not least, the theory retains the fundamental and ontological character of time; for Smolin, time remains the alpha and omega of a dynamic description of the world, but it is a time internal to the latter. The chapter “Time and Durations in Relativistic Physics” by Marc Lachièze-Rey reminds us of the difference between these two concepts. He emphasizes that relativity, whether special or general, is a theory that deals deeply with durations (i.e., time intervals as determined by the causality propagated between localized events) and not with times (i.e., dates). The author reminds us that “the second is a unit of duration, rather than a unit of time,” since any clock only marks the duration between two events, two clicks of the clock’s own time. Of course, it is still possible and useful to introduce a time, for example, the cosmic time, but this one is not strictly ontological; it depends on an approximate model of the universe, such as the Friedman-Lemaître model or its avatars. All the consequences of the theory of relativity concerning the measurement of time are then reviewed, for example, the measurement of the redshift accompanying the elliptical trajectory of the star S2 around the supermassive black hole at the center of the galaxy, or the desynchronization of hyper-precise atomic clocks when one of them is moved a few tens of centimeters in height. Lachièze-Rey recognizes that the causality regulated structure of the spacetime of general relativity introduces the notion of proper time, which is the only properly temporal concept of the theory, but that one can speculate that in its future developments even this last one will appear as superfluous, being simply emergent in relation with the introduction of causality. The following contribution by Daniele Oriti describes the difficulties surrounding the development of the most actual and active part of contemporary theoretical physics: the search for a quantum theory of gravity. He also explains why and in what sense the majority of the various attempts in this direction give up the status of the fundamental variable of time in physics and keep it only as an emergent variable. Moreover, it is only the weak version of the word emergence that is most often implied. At the beginning of his argument, Oriti notes that the spacetime of general relativity and its event-points do not have any physical meaning, that they are not part of the ontology of the world, because the world in this theory is only made of fields, and the physical observables of the x

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theory are in fact the quantitative relations between these fields. Strictly speaking, contrary to all previous physics, there is no fundamental time in this theory. It is only from the dynamics of the fields that one can make the choice of a clock with its domain of applicability. The previous contribution by Lachièze-Rey underlined the causal structure of spacetime in relativity. The situation in the quantum extension of the theory will necessarily be much more complex because the causal relations themselves will have to be subject to situations of superposition and fluctuations. Time itself will have to be a quantum object. We are thus entering a field where the very notion of reality must be questioned. Oriti suggests that the solution probably lies in a new metaphysics where ontology would be a complex notion with multiple levels, and where fundamental entities and emergent entities would share the property of reality. Moreover, the construction of the ‘final’ theory is further complicated by the fact that in the quantum regime, we probably have to deal with various phase transition phenomena, which are in themselves as many cases of emergence. At what level should we place “geometrogenesis”? Insofar as time disappears from the fundamental entities, this one should probably be understood as a kind of “a-chronic emergence”. We can see that for Daniele Oriti the collaboration of mathematicians, physicists, and philosophers is required to clarify a still very foggy landscape. In his article “The Layers That Build up the Notion of Time,” Carlo Rovelli begins by emphasizing, as J.T. Fraser did in his time, the various layers that constitute the notion of time. He also reminds us that relativity refutes the “naive” idea of simultaneity. For him, neither the idea of presentism nor that of eternalism are adequate to represent temporality. In fact, relativity offers a third possibility, in which events appear and disappear without there being a global present. As Marc Lachièze-Rey argued, physics only tries to answer two distinct questions: the relative succession of events (duration), and the Newtonian idea of a measure of the flux of time in itself. In fact, so far, it only answers the first one. Carlo Rovelli’s research revolves around the loop quantum gravity theory. It includes, for example, the possibility that between two events A and B, the time measured by the same clock could be not only quantified in discrete values, but also reveal itself in the interval in superposition of two different values. Rovelli also reflects on the phenomenon of the arrow of time, and like others, he insists that this arrow belongs exclusively to xi

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the macroscopic description of phenomena. This fact leads him, along with many thermodynamic theorists, to the idea that it is a perspective effect linked to the coarse-graining process. He concludes that our lived time, the nostalgia of the past, the anticipation of the future, and our feeling of the flight of time are obviously related to physical time, but are not universal aspects of physical temporality. Edward Anderson’s chapter continues the debate on the “Problem of Time,” i.e., on the impossible ontological nature of time, illustrated in Physics by the contradictions between relativity and quantum mechanics. The author borrows the basic principles of relationalism, laid down by Leibniz: there is no time for a universe considered in its entirety, although the universe by itself constitutes a perfect clock. As with Leibniz, two completely identical systems are only one and the same. Adding Mach’s principle, one sees that time is an abstraction elaborated from the observed changes in the universe. According to Anderson, special relativity and general relativity have addressed and solved the debate between relationalism and absolutism. From there, Einstein built theories of relativity. The difficulty in solving the Time Problem is to reconcile these theories with quantum mechanics, but keeping an integral relationalism, i.e. independence from the background. This dilemma is particularly subject to the problem of the nine facets of Time, stated by Wheeler, De Witt, and Dirac, then by Kuchaï and Isham, intervenes. Anderson proposes that Lie theory (Lie hooks, Lie algebras, and Lie algebroids), with the relationalism it supports, allows us to mathematically solve the Time Problem and its facets while realizing independence from the background. Symmetry considerations play a fundamental role in physics. A profound symmetry in particle physics is the CPT theorem, which states that if time and space are reflected (that is, t -> –t and x -> –x) and all particles are replaced by their corresponding antiparticles, the measured properties of the physical system will remain invariant. For the past two decades, Carl Bender has been studying physical systems that possess a simple but related symmetry called PT symmetry. These are systems described by Hamiltonians that remain unchanged if time and space are reflected. In his present contribution, Bender shows that Hermiticity, the symmetry that a Hamiltonian has traditionally been believed to possess since the very beginning of quantum mechanics, can be replaced by PT symmetry, which is a weaker but more physical (and less mathematical) condition than Hermiticity. xii

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Physical systems that are not Hermitian but are PT symmetric are perfectly valid and acceptable in quantum theories — for example, they have unitary (probability-preserving) time evolution. Moreover, such systems can exhibit remarkable and unexpected properties, such as new kinds of phase transitions, which conventional Hermitian systems do not have. For the last decade, many examples of PT-invariant systems have been studied in laboratories. Such systems have new kinds of optical, acoustic, mechanical, and electronic properties. One can even conceive of PT-symmetric gravitational systems in which two massive particles repel instead of attracting one another. This raises an interesting possibility, namely that the observed acceleration in the expansion of the universe is governed by a slightly modified theory of gravity that has a PT-symmetric component. As we know, applying the time-reversal transformation T consists in changing in the equations describing the evolution of a system each appearance of the symbol t into –t; this amounts in short to watching the film of its evolution in reverse, and to wondering if such an evolution is physically possible. Of course, like a diver coming out of the water and landing on the diving board, the practical impossibility of the reverse evolution of such macroscopic systems is obvious, but this is due to entropy considerations, which are not our subject. In his contribution “The Experimental Evidence for Time Reversal Violation,” David Hitlin talks about microscopic systems at the level of elementary particle interactions, and reviews the experimental situation of different tests of T-symmetry invariance or violation in particle physics. Since weak interactions weakly but clearly violate CP symmetry, they must necessarily also violate T symmetry, as far as the CPT theorem mentioned above has the universal applicability it claims. Indeed, the detailed study of ϒ(4S) meson decays into pairs of intricate B0–B0 particles produced at the e+–e– PEP-II collider at Stanford University (built especially for this study), among other tests, has clearly confirmed such a violation. On the other hand, there is today no evidence of a violation of the T symmetry in strong interactions (although there is the possibility of extremely low violation rates in the framework of the standard model of elementary particles); the corresponding tests have given, in recent years, only a maximum limit for its possible violation rate. This is, in particular, the case of the search for dipole electric moment in elementary particles, obviously forbidden by a T symmetry. The violation of T-symmetry in strong xiii

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interactions would have spectacular consequences for the development of physical theory; it could notably allow us to understand why the universe we know seems to be made only of ordinary matter and not equally of islands of matter and others of antimatter. The last paper in this volume, by Marina Cortês, is devoted to a survey of the recent history and current status of the Time Problem under the author’s careful and knowledgeable eye. As is well known, she is the first author of a paper written with Lee Smolin, “The Universe as a Process of Unique Events” (2014). This article, among many others she cites, helped focus the attention of theoretical physicists and philosophers of science on the connection between temporal irreversibility and the nondetermination of the future, and even with free will. The personal survey she presents, while not exhaustive, leads her to state her conviction that the future progress of knowledge, notably through the engaged dialogue between philosophers, neuroscientists, and physicists, will entail a further move away from the discouraging vision of a block universe for which everything would be written in advance. Let me add that in such a prospect, the concept of Time, while losing some of its apparent simplicity, would certainly regain much of its metaphysical value. Rémy Lestienne

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C O NT ENTS Time and Science: Forewordv Introduction to Volume 3vii Chapter 1

The Physics of “Now”

1

James B. Hartle Chapter 2

Discovering Physical Time within Human Time

27

Ronald P. Gruber, Carlos Montemayor and Richard A. Block Chapter 3

 ew (and Old) Work on the Fundamentality N of Time

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Dean Rickles and Jules Rankin Chapter 4

The Layers That Build Up the Notion of Time

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Carlo Rovelli Chapter 5

Senses in Which Time Does and Does Not Exist

105

Julian Barbour Chapter 6

 he Complex Timeless Emergence of Time in T Quantum Gravity

137

Daniele Oriti Chapter 7

Time and Durations in Relativistic Physics Marc Lachièze-Rey xv

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Chapter 8

Problem of Time: Lie Theory Suffices to Resolve It

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Edward Anderson Chapter 9

Views, Variety, and Celestial Spheres

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Lee Smolin Chapter 10 S  cientific Cosmogony, the Time in Quantum Relativistic Physics

269

Gilles Cohen-Tannoudji and Jean-Pierre Gazeau Chapter 11 PT Symmetry

285

Carl M. Bender Chapter 12 Experimental Evidence for Time Reversal Violation

311

David G. Hitlin Chapter 13 Free Will and the Arrow of Time Marina Cortês

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Chapter 1

THE PHY S IC S O F “NOW” James B. Hartle* University of California, USA [email protected]

The world is four-dimensional according to fundamental physics, governed by basic laws that operate in a space-time that has no unique division into space and time. Yet, our subjective experience of this world is divided into present, past, and future. This paper discusses the origin of this division in terms of simple models of information gathering and utilizing systems (IGUSs). Past, present, and future are not properties of four-dimensional space-time, but notions describing how individual IGUSes process information. Their origin is to be found in how these IGUSes evolved or were constructed. The past, present, and future of an IGUS are consistent with the four-dimensional laws of physics and can be described in four-dimensional terms. The present, for instance, is not a moment of time in the sense of a spacelike surface in space-time. Rather, there is a localized notion of the present at each point along an IGUS worldline. The common present of many localized IGUSes is an approximate notion appropriate when they are sufficiently close to each other and have relative velocities much less than that of light. But modes of organization that are different from the present, past, and future can be imagined that are also consistent with the physical laws. * Reproduced from Hartle, J.B., “The Physics of ‘Now’”, American Journal of Physics, 73, 101–109, 2005, with the permission of the American Association of Physics Teachers.

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We speculate why the present, past, and future organization might be favored by evolution and be therefore a cognitive universal.

1 Introduction A lesson from the physics of the last century is that on length scales much greater than the Planck length characterizing quantum gravity, the world is four-dimensional with a classical space-time geometry. There is neither a unique notion of space nor a unique notion of time. Rather, at each point in spacetime, there are a family of timelike directions and three times as many spacelike directions. Yet, in this four-dimensional world, we divide our subjective experiences into past, present, and future. These seem very different. We experience the present, remember the past, and predict the future. How is our experience organized in this way? Why is it so organized? What is the four-dimensional description of our past, present, and future? Is the division into past, present, and future the only way that experience can be organized? This chapter is concerned with such questions. The general laws of physics by themselves provide no answers. Past, present, and future are not properties of four-dimensional spacetime. Rather, they are properties of a specific class of subsystems of the universe that can usefully be called information gathering and utilizing systems (IGUSes) [1]. The term is broad enough to include both single representatives of biological species that have evolved naturally and mechanical robots that have been constructed artificially. It includes human beings, both individually and collectively, as members of groups, cultures, and civilizations. To understand the past, present, and future, it is necessary to understand how an IGUS employs such notions in the processing of information. To understand why an IGUS might be organized in this way, it is necessary to understand how it is constructed and ultimately how it evolved. Questions about past, present, and future, therefore, are most naturally in the domains of psychology, artificial intelligence, evolutionary biology, and philosophy.1 However, questions concerning past, present, and future cannot be completely divorced from physics. For instance, the notions must be describable in four-dimensional terms to be consistent with the  See, e.g., [2] for a collection of philosophical papers on time. This chapter does not aim to discuss or resolve any of the philosophical debates on the nature of time. 1

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fundamental picture of spacetime. Furthermore, as we review, the distinctions between the past, present, and future of an IGUS depend on some of the arrows of time that our universe exhibits, such as that summarized by the second law of thermodynamics. In the tradition of theoretical physics, we illustrate these connections with simple models of an IGUS — achieving clarity at the risk of irrelevance. Our considerations are entirely based on classical physics.2 One such model IGUS, a robot, is described in Section 2. It is simple enough to be easily analyzed, but complex enough to suggest how realistic IGUSs distinguish between past, present, and future. The four-dimensional description of this robot is discussed in Section 3. There we will see that the robot’s present is not a moment in spacetime.Rather, there is a present at each intant along the robot’s worldline, consisting of its most recently acquired data about its external environment. The approximate common notion of “now” that could be utilized by a collection of nearby robots moving slowly with respect to one another is also described. Section 4 describes the connection of present, past, and future with the thermodynamic arrow of time and the radiation arrow of time. It addresses the question, “Could we we construct a robot that will remember the future?” Section 5 describes alternative organizations of a robot’s experience that are different from past, present, and future, but equally consistent with the four-dimensional laws of physics. The possibility of these alternative organizations shows that past, present, and future are not the consequences of these laws. We speculate, however, that, as a consequence of the locality of the laws of physics, the past, present, and future organization may offer an evolutionary advantage over the other modes of organization for sufficiently localized IGUSs. A past, present, and future organization may then be a cognitive universal [5].

2  A Model IGUS Imagine constructing a robot that gathers and utilizes information in the following manner (Fig. 1.1):  The arrows of time in the context of quantum mechanics, as well as the quantum mechanical arrow of time, are discussed in [3] in the framework of a time-neutral generalized formulation of quantum theory. The author knows of no obstacle of principle to extending the present classical discussion to quantum mechanics in that framework. For the special features of history in quantum mechanics, see, e.g., [4]. 2

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Figure 1.1   Information flow in the robot described in the text is represented schematically in this diagram. The internal workings of the robot are within the dotted box; its external environment is without. At every proper time interval τ *, the robot captures an image of its external environment. In the example illustrated, this is of a stack of cards labeled a, b, c, etc., whose top member changes from time to time. The captured image is stored in register P0, which constitutes the robot’s present. Just before the next capture, the image in P3 is erased and images in P0, P1, and P3 are shifted to the right, making room for the new image in P0. The registers P0, P1, P2, and P3 therefore constitute the robot’s memory of the past. At each capture, the robot forgets the image in register P3. The robot uses the images in P0, P1, P2, and P3 in two processes of computation: C (“conscious”) and U (“unconscious”). The process U uses the data in all registers to update a simplified model or schema of the external environment. That is used by C, together with the most recently acquired data in P0, to make predictions about its environment to the future of the data in P0, make decisions, and direct behavior accordingly. The robot may therefore be said to experience (through C) the present in P0, predict the future, and remember the past in P1, P2, and P3.

Information Gathering: The robot has n + 1 memory locations, P0, P1, … Pn, which we call “registers” for short. These contain a time series of images of its external environment assembled as follows: At times separated by intervals τ *, the image in register Pn is erased and replaced by the image in Pn–1. Then, the image in Pn–1 is erased and replaced by the image in Pn–2, and so on. For the last step, the robot captures a new image of its external environment and stores it in register P0. Thus, at any one time, the robot possesses a coarse-grained image history of its environment extending over time (n + 1) τ *. The most recent image is in P0, the oldest is in Pn. Information Utilization: The robot employs the information in the registers P0, P1, … Pn to compute predictions about its environment at times to the future of the data in P0 and direct its behavior based on these predictions. It does this in two steps in two processes of computation:

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(1) Schema: The robot’s memory stores a simplified model of its environment containing not all the information in P0, P1, … Pn but only those parts important for the robot’s functioning. This model is called a schema [1]. At each time interval τ *, the robot updates its schema making use of the new information in P0 and the old information in P1 … Pn through a computation process we denote by U.   The schema might contain the locations and trajectories of food, predators, obstacles to locomotion, fellow robots, etc. It might contain hard-wired rules for success (e.g., get food — yes, be food — no), and perhaps even crude approximations to the rules of geometry and the laws of physics e.g., objects generally fall down. It might contain summaries of regularities of the environment abstracted from the information gathered long before the period covered by registers P0, … Pn or explained to it by other robots, etc.3 (2) Decisions and Behavior: At each time interval τ *, the robot uses its schema and the fresh image in P0 to assess its situation, predict the future, and make decisions on behavior to exhibit next through a computation process denoted by C. This process is distinct from U. The important point for this chapter is that the robot directly employs only the most recently acquired image in register P0 in this computation C. The information in P1, … Pn is employed only through the schema. It seems possible that such a robot could be constructed. As a model of sophisticated IGUSs, such as ourselves, it is grossly oversimplified. Yet, it possesses a number of features similar to those in sophisticated IGUSs that are relevant for understanding past, present, and future: the robot has a coarse-grained memory of its external environment contained in registers P0, P1, … Pn . The robot has two processes of computation, C and U. Without entering into the treacherous issue of whether the robot is conscious, the two processes have a number of similarities with our own processes of conscious and unconscious computation:  A history book is a familiar part of the schema of the collective IGUS linked by human culture. It is a summary and analysis of records gathered at diverse times and places. It is true whether the history is of human actions or the scientific history of the universe. The schema resulting from the reconstruction of present records simplifies the process of future prediction. (For more on the utility of history, see [4]). 3

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(1) U computation provides input to decision-making C computation. (2) There is direct input to C computation only from the most recently acquired image in register P0. The images in P1, … Pn affect C only through the schema computed in U. (3) Information flows into and out register P0 directly used by C. Equally evident are some significant differences between the robot and ourselves. Our information about the external environment is not exclusively visual; neither is it stored in a linear array of registers, nor is it transferred from one to the other in the simple manner described. Input and records are not separated by sharp time divisions. We can consciously access memories of other than the most recent data, although often imperfectly and after modification by unconscious computation. This list of differences can easily be extended, but that should not obscure the similarities discussed above. The analogies between the robot and ourselves can be emphasized by employing everyday subjective terminology to describe the robot. For example, we will call C and U computation “conscious” and “unconscious” secure in the confidence that such terms can be eliminated in favor of the mechanical description we have employed up to this point if necessary for clarity. In this way, we can say that the robot “observes” its environment. The register P0 contains a record of the “present”, and the registers P1, … Pn are records of the “past.”4 When the register Pn is erased, the robot has “forgotten” its contents. The present extends5 over a finite interval6 τ *. The robot has a conscious focus on the present, but only access to the past through the records that are inputs to the unconscious computation of its schema. The robot can thus be said to “experience” the present and “remember” the past. The “flow of time” is the movement of information into the register of conscious focus and out again. Prediction requires computation — either conscious or unconscious — from memories of the present and past acquired by observation and is thus distinct from remembering.  For the moment we take the records P1, … Pn to define the past. Section 4 will connect this notion of past with other physical notions of “past”, for instance that defined by the time direction toward the big bang. 5  As James [6] put it more eloquently: “... the practically cognized present is no knife-edge, but a saddleback, with a certain breadth of its own, on which we sit perched, and from which we look in two directions in time.” 6  For human IGUSes the time τ * can be taken to be of order the .1s, separation time needed to discriminate between two visual signals [7]. 4

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The subjective past, present, and future, the flow of time, and the distinction between predicting and remembering are represented concretely and physically in the structure and function of the model robot. We now proceed to describe this structure and function in four-dimensional terms.

3  The Present is not a Moment in Time In the following, we describe the robot of Sec. 2 in terms of special relativity.7 For simplicity, we consider the flat spacetime of special relativity (Minkowski space). But with little change, it could be a curved spacetime of general relativity. 3.1  Some features of Minkowski space We begin by Recalling a few important features of four-dimensional spacetime, which are illustrated in Fig. 1.2.8 Events occur at points in spacetime. At each point Q there is a light cone consisting of two parts. The future light cone is the three-dimensional surface generated by the light rays emerging from Q. The past light cone is similarly defined by the light rays converging on Q. (The labels “future” and “past” are conventions at this point in the discussion. In Section 4 we will define them in the cosmological context.) Points inside the light cone of Q are timelike separated from it; points outside the light cone are spacelike separated. Points inside the future light cone of Q are in its future; points inside the past light cone are in its past. Points outside the light cone are neither. The center of mass of a localized IGUS, such as the robot of Sec. 2, describes a timelike world line in space-time. At each point along the world line, any tangent to it lies inside the light cone so that the IGUS is moving at less than the speed of light in any inertial frame. A moment in time is a three-dimensional spacelike surface in spacetime — one in which any two nearby points are spacelike separated. Each spacelike surface divides spacetime into two regions — one to its future and one to its past. A family of spacelike surfaces such that each point in space-time lies on one and only member of the family specifies a division

 An abbreviated version of this discussion was given in [4].  For a classic text on special relativity from a spacetime point of view, see [8].

7 8

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Figure 1.2  Space-time concepts. This space-time diagram represents a three-dimensional slice of a four-dimensional flat space-time defined by three axes, ct, x, and y, of the four specifying an inertial (Lorentz) frame. The ideas of event, light cone, world line, and spacelike surface are illustrated. An event is a point in space-time, like Q. Each point has a future and past light cone. A spacelike surface like the one illustrated defines an instant in time. Each such surface divides space-time into two regions, conventionally called the future of this surface and its past. There is an infinity of such families and thus infinitely many different ways of defining instants in time and their futures and pasts. In the context of cosmology, the past of a spacelike surface is defined to be the region closest to the big bang, and the future is the region furthest away.

of spacetime into space and time. Surfaces defined by constant values of the time of a particular inertial frame are an example (e.g., the surfaces of constant ct in Figs. 1.2–1.4). Different families of spacelike surfaces define different notions of space and different notions of time, none of which is preferred over the other. 3.2  The past, present, and future of the robot Figure 1.3 shows the world line of the robot introduced in Section 1.2 with the world line of an object in its environment that appears in the robot’s stored images. The robot illustrated has a short memory with only four registers P0, P1, P2, P3 whose contents in each interval τ * are indicated by the content of the boxes to the right of the world line. These contents change at proper time intervals τ * as described in Section 2. The contents of the register P0 defining the robot’s present do not define a spacelike surface representing a moment in time. They do not even define an instant along the robot’s world line because the contents 8

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Figure 1.3   A space-time description of the present and past of the robot whose information processing is illustrated in Fig. 1.1. In addition to the world line of the robot, the figure shows the world line of an external object that is the source of its images such as the stack of cards in Fig. 1.1. This source changes its shape at discrete instants of time demarcated by ticks, running through configurations … b, c, d, e, f, g, … The configuration in each time interval of the object’s world line is labeled to its left. At discrete instants separated by proper time τ * the robot captures an image of the object. The light rays conveying the image from object to robot are indicated by dotted lines. The images are stored in the registers P0, P1, P2, and P3 described in Fig. 1.1. The contents of these registers between image captures are displayed in the boxes with P0 on the left and P3 on the right. The history of the contents of the register P0 constitute the four-dimensional notion of present or “now” for this robot (the heavily outlined boxes in the figure). The present is not one instant along the robot’s world line, much less a spacelike surface in spacetime. Rather, there is a “present” for each instant along the robot’s world line extending over the proper time τ *. In this way, the evolution of the contents P0 can be described four-dimensionally and is fully consistent with special relativity. In a similar way, the contents of P1, P2, and P3 constitute a four-dimensional notion of past.

of P0 are constant over proper time intervals τ *. Rather, there is content defining the present for every instant along the world line. For each point along the world line, the most recently acquired image defines the present. This is the four-dimensional description of the present. In a similar way, the data acquired earlier and stored in registers P1, P2, P3 define the robot’s past for each point along the world line. Thus, there is no conflict between the four-dimensional reality of physics and the subjective past, present, and future of an IGUS. Indeed, as 9

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defined above, the subjective past, present, and future are four-dimensional notions. They are not properties of spacetime, but of the history of a particular IGUS. In Section 5 we will see that IGUSes constructed differently from our robot could have different notions of past, present, and future. All of these are fully consistent with a four-dimensional physical reality. However, there is a conflict between ordinary language and the fourdimensional, IGUS-specific notions of the past, present, and future. To speak of the “present moment” of an IGUS, for instance, risks confusion because it could be construed as referring to a spacelike surface in spacetime stretching over the whole universe. No such surface is defined by physics.9 In fact, the “moment” in the context of this section refers to the most recently acquired data of an IGUS. This is not a notion restricted to one point on the IGUS’ world line which somehow moves along it. Rather, it is a notion present at every point along the world line. 3.3  The common present The previous discussion has concerned the present, past, and future of individual localized IGUSs. We now turn to the notion of a common present that may be held by collections of IGUSs separated in space. When someone asked Yogi Berra what time it was, he is reported to have replied, “Do you mean now?”10 The laughter usually evoked by this anecdote shows how strongly we hold a common notion of the present. More precisely, different IGUSs agree on “what is happening now.” This section is concerned with the limitations of the accuracy of this agreement that arise both from the construction of the IGUSs and the limitations of defining simultaneity in special relativity. We continue to use robot model IGUSs to make the discussion concrete. Figure 1.4 shows the world lines of two robots in spacetime together with the intervals on their world lines that define their individual notions of now. There are at least two reasons why there is no unambiguous notion of a common “now” that can be shared by the two robots. The first

 There is no evidence for preferred frames in spacetime and modern versions of the Michelson-Morley experiment and other tests of special relativity set stringent limits on their existence. The fractional accuracy of these experiments range down to 10-21 making Lorentz invariance at accessible energy scales one of the most accurately tested principles in physics. See e.g. [9]. 10  Lawrence E. Berra, a catcher for the New York Yankees baseball team in the 1950s. See [10]. 9

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Figure 1.4  The ambiguous common present. This space-time diagram shows the world lines of two similarly constructed robots A and B. The intervals of proper time of length τ * over which the contents of the registers defining their individual presents are constant are demarcated by ticks. A common present would be defined by an identification of each interval on one world line with that on the other “at the same time” (to the accuracy τ *). But special relativity allows many different such identifications. Using the constant t surfaces of the inertial frame illustrated is one way to define a common present, but any other spacelike surface such as the one shown would do equally well. The range of ambiguity for intervals on B that could be said to be “at the same time” as one interval on A is shown as a shaded region. The figure shows two robots separated by a distance (e.g., as defined in the rest frame of one) over which the light travel time is longer than τ *. In this situation the ambiguity in the definition of a common present is much larger than τ *. However, if the distance between the robots is much smaller than c τ *, and if their relative velocity is much less than c so this continues to be the case, then the ambiguity is much smaller than τ *. An approximate common present can then be defined.

is the elementary observation11 that the present for each individual robot is not defined to an accuracy better than τ *. The second reason arises from special relativity. A precise common now would specify a correspondence between events on the two world lines. Such a correspondence would specify a notion of simultaneity between events on the world lines. But there is no unique notion of simultaneity provided by special relativity. Rather, there  The ambiguity in the common present arising from the finite time of IGUS operation has been discussed in [11], which includes a review of current neurophysiological data bearing on this question. 11

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are many different notions corresponding to the different possible spacelike surfaces that can intersect the two world lines. Figure 1.4 illustrates the range of ambiguity. To illustrate the ambiguity of the present more dramatically, imagine you are a newscaster on the capital planet Trantor of a galaxy-wide empire some hundreds of thousands of years in the future. News of events all over the galaxy pour in constantly via electromagnetic signals. You want to broadcast a program called The Galaxy Today, reviewing important events in the last 24 hours (galactic standard). But what time do you assign to the latest news from the planet Terminus at the edge of the galaxy, 60,000 light-years away? There is an inertial frame12 in which those events happened within the last 24 hours. But in the approximate rest frame of the galaxy, they happened 60,000 years ago. There is thus no unambiguous notion of “present” for the collective IGUS consisting of the citizens of the galactic empire because the ambiguities in defining simultaneity are large compared to the time scales on which human events happen. You could, of course, fix on the time of the galactic rest frame as a standard for simultaneity. But in that case the only comprehensive program you could broadcast would be The Galaxy 60,000 Years Ago. One imagines the audience for this on Terminus would not be large, since the program would be seen 120 000 years after the events there happened. The satellites comprising the Global Positioning System are an example of a collective IGUS closer to home that faces a similar problem. The special relativistic ambiguity in defining simultaneity for two satellites is of order the light travel time between them in the approximate inertial frame in which the Earth is at rest. This is much larger than the light travel time across the few meters accuracy to which the system aims to locate events. Precise agreement on a definition of simultaneity is therefore needed. Each satellite clock is corrected so that it broadcasts the time of a clock on the Earth’s geoid (approximately the ocean surface) [12]. No such agreement on a definition of simultaneity appears to be a prerequisite for the everyday notion of now employed by human IGUSs. Rather, we seem to be employing an approximate imprecise notion of the  For example, there is the inertial frame moving with speed V with respect to the galactic rest frame such that 12

∆t′ = γ [∆t – V (∆x/c)] where γ = (l – V2) –1/2, ∆t′ = 24 h, ∆t = 6 × 104 yr, and ∆x = 6 × 104 light-years. The required velocity is within a few parts in 107 of the velocity of light.

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common present as appropriate in everyday situations and characterized by the following contingencies: (1) The time scale of perception τ * is short compared to the time scales on which interesting features of the environment vary. (2) Individual IGUSs are moving relative one another at velocities small compared to c. (3) The light travel time between IGUSes in an inertial frame in which they are nearly at rest is small compared to the time scales τ *. Contingency (1) means that the ambiguity in the now of each IGUS is negligible in the construction of a common present. Contingency (3), based on (2), means that the ambiguity arising from the definition of simultaneity is negligible.13 Collections of robots satisfying contingencies 1–3 can agree on what is happening now. Consider just two robots — Alice and Bob. Alice can send Bob a description of the essential features of the image currently in her register P0. Bob can check whether these essential features are the same as those of the image in his register P0. Bob can check whether these essential features are the same as those of the image in his register P0 at the time of receipt. He can then signal back agreement or disagreement. As long as the light travel time is much shorter than τ * (contingency 3), and as long as the essential features vary on much longer time scales (contingency 1), Alice and Bob will agree. Contingency (2) ensures that this agreement will persist over an interesting time scale. Thus, Alice and Bob can construct a common present, but it is a present that is local, inherently approximate, and contingent on their relation to each other and their environment. This approximate common now is not a surface in space-time. No modification of the laws of physics is needed to understand the common now of a group of IGUSs, as has sometimes been suggested [14]. The common nows of IGUSes meeting the above contingencies will coincide approximately with constant time surfaces in any inertial frame in which they are approximately at rest. But these frames are not singled out by the laws of physics. Indeed, the experimental evidence against preferred frames is that special relativity is extraordinarily good [9]. Rather,  These kinds of contingencies and the synchronization protocols necessary when they are violated have been discussed in [13]. 13

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the frames are singled out by the particular situations of the IGUSes themselves.

4  Why Don’t We Remember the Future? The fundamental dynamical laws of physics are invariant under time reversal to an accuracy adequate for organizing everyday experience.14 They are time-neutral. The Einstein equation of general relativity and Maxwell’s equations for electrodynamics are examples. But the boundary conditions specifying solutions to these equations describing our universe are not time symmetric. The universe has a smooth (near homogeneous and isotropic) big bang at one end of time and a very different condition at the other end. This other end might be the unending expansion driven by the cosmological constant of the simplest cosmological models favored by observation [15]. Or with different assumptions, it could be a highly irregular big crunch. In any event, one end is different from the other.15 By convention, this paper refers throughout to the times closer to the big bang as the “past” and times further away as the “future.” Asymmetry between past and future boundary conditions is the origin of the various time asymmetries — arrows of time — exhibited by our universe. The arrow of time associated with the second law of thermodynamics is an example.16 The operation of Section 2’s robot is not time neutral in at least two respects. First, the robot receives information about external events in its past (closer to the big bang) and not the future.17 Second, its processing of  The effective theory of the weak interactions applicable well below the Planck scale is not time-reversal invariant. This is important, for instance, for the synthesis of baryons in the early universe but negligible for the functioning of our robot. See, e.g. [3] for further discussion. 15  We thus exclude, mainly for simplicity. the kind of cosmological model where initial and final conditions are related by time symmetry that have sometimes been discussed (e.g. [3. 19. 20.21]). 16  For reviews of the physics of time asymmetry from various perspectives, see [3, 16, 17, 18, 19]. There is also some discussion in the Appendix. These are only a few of the references where these issues are treated. 17  More precisely, registered signals originate from events within the past light cone of their reception event. We use “future” and “past” in the present discussion understanding that in each case these are defined by an appropriate light cone, as described in Section 3.1. 14

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the received information is not time-neutral. The flow of information from recording to erasure defines a direction in time. As mentioned in Section 2, that direction gives a concrete model for the subjective feeling of inexorable forward progression in time commonly called the “psychological arrow of time.” In natural IGUSs, such as ourselves, information flows from the past to the future. More specifically, the records in registers P1, … Pn are of external events to the past of those in P0 in decreasing order of time from the big bang. This is the reason that Sec. 2’s robot could be said to be experiencing the present, remembering the past, and predicting the future.18 Could a robot be constructed that receives information from the future? Could one be constructed whose psychological arrow of time runs from future to past with the consequence that it would remember the future? Both of these possibilities are consistent with time neutral dynamical laws. But two familiar time asymmetries of our universe prohibit such constructions as a practical matter. These are the radiation arrow of time and the arrow of time associated with the second law of thermodynamics. Records of the future are possible as in a table of future lunar eclipses.19 Indeed, records of the future are the outcome of any useful process of prediction. But our robot’s records of the future are obtained by computation, whereas its records of the past are created by simple, automatic, sensory mechanisms. These are very different processes, both physically and from the point of view of information processing by the robot.20 By a robot that remembers the future, we mean one constructed21 as in Section 2 with the records in registers P1, … Pn of events to the future of P0 .  In Section 1.2 we defined the robot’s past to be the records in the registers P1, … Pn . If information flows from past to future in the robot, that notion coincides with the physical past defined as the direction in time towards the big bang. From now on, we assume this congruence except where discussing its possible violation, as in this section. 19  We use the term “record” in a time-neutral sense of an alternative at one time correlated with high probability with an alternative at another time — future or past. Thus, there can be records of the future. 20  Realistic IGUSes, such as human beings, also create records of the past by computation as in differing interpretations of past experience, and as in the collective construction of human history, and the history of the universe. However, we did not explicitly endow our robot with these functions. 21  In Section 5 we will describe a different construction of a robot which could be said to remember the future. 18

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First, consider the question of whether information from the future could be recorded by the robot. In our universe, electromagnetic radiation is retarded — propagating to the future of its emission event. That time asymmetry is the radiation arrow of time. The electromagnetic signals recorded by the robot propagated to it along the past light cone of the reception event. The images received by the robot, whether of the cosmic background radiation, distant stars and galaxies, or the happenings in its immediate environment, are therefore all from past events. As far as we know, all other carriers, neutrinos for example [22], are similarly retarded. One reason the robot doesn’t remember the future is that it receives no information about it. Irrespective of the time its input originates, could a robot like that in Section 2 be constructed whose psychological arrow of time is reversed, so that internally information flows from future to past? In such a robot, the events recorded in registers P1, … Pn would lie to the future of that in P0 — further from the big bang. The robot would thus remember the future. Such a construction would run counter to the arrow of time specified by the second law of thermodynamics, as we now review.22 All isolated subsystems of the universe evolve toward equilibrium. That is statistics. But the preponderance of isolated systems in our universe are evolving toward equilibrium from past to future, defining an arrow of time. That is the second law of thermodynamics that is expressed quantitatively by the inexorable increase of an appropriately defined total entropy. If the robot processes information irreversibly, then its psychological arrow of time must generally be congruent with the thermodynamic arrow of time. The formation of records is a crucial step. An increase in total entropy accompanies the formation of many realistic records. An impact crater in the moon, an ancient fission track in mica, a darkened photographic grain, or the absorption of a photon by the retina are all examples. But an increase in total entropy is not a necessary consequence of forming a record. Entropy increase is necessary only on the erasure of a record [25]. For the model robot discussed in Section 2, the only part of  Many other authors have connected the “psychological” arrow of time with that of the second law. See. e.g. [17. 23. 24]. 22

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its operation in which entropy must necessarily increase is in the erasure of the contents of the register Pn at each step.23 However, that is enough. To see that, imagine the process of erasure running backward from future to past. It would be like bits of smashed shell reassembling to form an egg. To construct a robot with a reversed psychological arrow of time, it would be necessary to reverse the thermodynamic arrow not only of the robot but also of the local environment it is observing. That is possible in principle. However, because we have a system of matter coupled to electromagnetic radiation, it would be necessary to deal with every molecule and photon within a radius of 2 × 1010 km to reverse the system for a day. More advanced civilizations may find this amusing. We can have the same fun more cheaply by running the film through the projector in reverse. The origin of both the thermodynamic and radiation arrows of time are the time-asymmetric boundary conditions that single out our universe from the many allowed by time-reversible dynamical laws. These boundary conditions connect the two arrows. A brief sketch of the relevant physics is given in the Appendix, although it is not necessary to understand the main argument of this chapter. But it is interesting to think that our subjective distinction between future and past can ultimately be traced to the cosmological boundary conditions that distinguish the future and past of the universe.

5  Alternatives to Past, Present, and Future The preceding discussion suggests that the laws of physics do not define unambiguous notions of the past, present, and future by themselves. Rather, these are features of how specific IGUSs gather and utilize information. What then is the origin of the past, present, and future organization of information in familiar, naturally occurring IGUSs? Is it the only organization compatible with the laws of physics? If not, does it arise uniquely from evolutionary imperatives, or is it a frozen accident that  Possibly an isolated robot could be constructed on the principles of reversible computation [26] that would not have an erasure step. However, it seems unlikely that the whole system of robot plus a realistic observed environment could be reversible. 23

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took place in the course of three billion years of biological evolution? This section discusses such questions. Certainly, some features of the laws of physics are essential prerequisites to the functioning of the robot in Section 2. There would be no past, present, or future at all if spacetime did not have timelike directions. The fact that IGUSs move on timelike rather than spacelike world lines is the main part of the reason they can have an approximate common now rather than an approximate common “here.” An IGUS functioning in a spacetime where it moved along a closed timelike curve could not maintain a consistent notion of the past and future. Likewise, a local distinction between past and future would be difficult to maintain in the absence of the arrows of time discussed in Section 2. But the features of the physical laws of dynamics and the initial condition of the universe that are necessary for a past, present, and future organization of temporal information are consistent with other organizations of this information, as we now show. 5.1  Different organizations of temporal information Perhaps the easiest way of convincing oneself that the notions of past, present, and future do not follow from the laws of physics is to imagine constructing robots that process information differently from the one described in Section 2. We consider just three examples: (1) The Split Screen (SS) Robot. This robot has input to C computation from both the most recently acquired data in P0 and from that in a different register Pj that was acquired a proper time τ s ≡ Jτ * earlier along its world line. Thus, there is input to conscious computation from two times. (2) The Always Behind (AB) Robot. This robot has input to C computation only from a particular register PK, K > 0 and the schema. That input is thus always a proper time τ d ≡ Kτ * behind the most recently acquired data. (3) The No Schema (NS) Robot. This robot has input to C computation from all the registers P0, … , Pn equally. It employs no unconscious computation and constructs no schema, but rather takes decisions by conscious computation from all the data it has.

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There seems to be no obstacle to constructing robots wired up in these ways, and they process information differently from the present, past, and future organization with which we are familiar with.24 An SS robot would have a tripartite division of recorded information. Its present experience, its now, would consist of two times (P0, Pj ) equally vivid and immediate. It would “remember” the intermediate times (P1, … , PJ–1), and the past (PJ+1, … , Pn) through the U process of computation and its influence on the schema. The AB robot would also have a tripartite division of recorded information. Its present experience would be the content of the register PK. It would remember the past stored in registers PK+1, … , Pn . But also, it would remember its future stored P0, … , PK–1 a time τ d ahead of its present experience.25 (Or perhaps we should say that it would have premonitions of the future.) What would discussions with an AB robot be like, assuming that our information processing is similar to the robot discussed in Section 2? Assume for simplicity that we and the AB robot are both nearly at rest in one inertial frame and that contingencies (1) – (3) of Section 3.3 are satisfied. The AB robot would seem a little slow — responding in a time τ d or longer to questions. Its answers to queries about “What’s happening now?” would seem out of date. It would always be behind. The NS robot would just have one category of recorded information. Conversations with an NS robot would be impressive, because it would recall every detail it has recorded about the past as immediately and vividly as the present.26 The laws of physics supply no obstacle of principle to the construction of robots with exotic organizations of information processing, such as the SS, AB, and NS robots. But are such organizations a likely outcome of  Some idea of what the notion of “present” would be like for some of these robots could be had by serving time in a virtual reality suit in which the data displayed was delayed as in the AB case, or in which there was an actual split screen as in the SS case. An alternative realization of the SS robot’s experience might be produced by electrical stimulation of the cortex that evokes memories of the past which are comparably immediate to the present [27]. 25  That is not in conflict with the discussion in Section 4 because each record is still of events in the past to the proper time it was recorded. 26  Perhaps not unlike conversations with Ireneo Funes in the Borges story Funes the Memorious [28]. 24

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biological evolution? Can we expect to find such IGUSs in nature on this or other planets? We speculate that we will not. It is adaptive for an IGUS of everyday size to focus mainly on the most recently acquired data as input for making decisions. The effective low-energy laws of physics in our universe are local in spacetime, and the nearest data in space and time is usually the most relevant for what happens next. A frog predicting the future position of a fly needs the present position and velocity of the fly, not its location 10 s ago. An AB frog would be at a great competitive disadvantage by not focusing on the most current information. An SS frog would be wasting precious conscious focus on data from the past that is less relevant for immediate prediction than current data. An NS robot would make inefficient use of computing resources in giving equal focus to present data and data from the past whose details may not effect relevant future predictions. Employing a schema to process the data is plausibly adaptive because it is a more efficient and faster way of processing data with limited computing resources. The collective IGUS linked by human culture certainly evolved to make use of schema rather than focus on the individual records that went into them. For instance, prediction of the future of the universe is much simpler from a Friedman–Robertson–Walker model characterized by a few cosmological parameters than directly from the records of the measurements that determined them [4]. 5.2  Different laws, different scales Something like the SS organization of temporal information might be favored by evolution if the laws of physics were not local in time. Suppose, for example, that the position of objects to the future of a time t depended,27 not just on the force acting and their position and velocity at that time, but rather on their position at time t and on earlier times t – h and t – 2h for some fundamental fixed time interval h. Then an  It is not difficult to write down dynamical difference equations with this property, for instance in a one-dimensional model we could take 27

F (t ) = m

[x (t ) − 2 x (t − h ) + x (t − 2 h )] 2h ²

where x(t) is the body’s position, m its mass, and F(t) the force. However. such equations are not consistent with special relativity and the author is not suggesting a serious investigation of alternatives to Newtonian mechanics.

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organization such as the SS robot with conscious focus on both the most recently acquired data and that acquired at times h and 2h ago might be favored by evolution. Similarly, different organizations might evolve if the IGUS is not smaller than the scale over which light travels on the characteristic times of relevant change in its environment. The present, past, and future organization is unlikely to serve such an IGUS well because these notions are not well defined in these situations, as discussed in Section 3.3. As mentioned there, the galactic empires beloved of science fiction would be examples of such IGUSs. Faster-than-light travel inconsistent with special relativity is often posited by authors whose stories feature these empires to make their narratives accessible to IGUSs like ourselves that do employ — however, approximately — a present, past, and future organization of information.

6 Conclusion A subjective past, present, and future are not the only conceivable way an IGUS can organize temporal data in a four-dimensional physical world consistently with the known laws of physics. But it is a way that may be adaptive for localized IGUSes governed by local physical laws. We can conjecture that a subjective past, present, and future is a cognitive universal [5] of such localized IGUSes. That is a statement accessible to observational tests, at least in principle.

Acknowledgments The author thanks Murray Gell-Mann for discussions of complex adaptive systems over a long period of time. He is grateful to Terry Sejnowski and Roger Shepard for information about the literature in psychology that bears on the subject of this paper. Special thanks are due to Roger Shepard for an extended correspondence on these issues. Communications with Jeremy Butterfield, Craig Callender, and Matt Davidson have been similarly helpful in terms of philosophical literature. However, the responsibility rests with the author for any deficiencies in providing relevant references to the vast literature of these subjects, about which he is largely ignorant. Research for this work was supported in part by the National Science Foundation under grant PHY02-44764.

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Appendix A: The Cosmological Origin of Time’s Arrows The origin of our universe’s time asymmetries is not to be found in the fundamental dynamical laws, which are essentially time-reversible. Rather, both the radiation and the thermodynamic arrows of time arise from the special properties of the initial condition of our universe.28 This appendix gives a simplified discussion of these special features starting, not at the very beginning, but at the time the hot initial plasma had become cool enough to be transparent to electromagnetic radiation. This is the time of “decoupling” in cosmological parlance — about 400 000 years after the big bang or a little over 13 billion years ago. As Boltzmann put it over a century ago, “The second law of thermodynamics can be proved from the [time-reversible] mechanical theory if one assumes that the present state of the universe ... started to evolve from an improbable [special] state” [30]. The entropy of matter and radiation usually defined in physics and chemistry is about 1080 in the region visible from today at the time of decoupling (in units of Boltzmann’s constant). This seems high, but it is in fact vastly smaller than the maximal value of about 10120 if all that matter was dumped into a black hole [17]. The entropy of matter early in the universe is high because most constituents are in approximate thermal equilibrium. However, the gravitational contribution of the smooth early universe to the entropy is near minimal, and entropy can grow by the clumping of the matter arising from gravitational attraction leading to the galaxies, stars, and other inhomogeneities in the universe we see today. Amplifying on Boltzmann’s statement above, the explanation of why the entropies of isolated subsystems are mostly increasing in the same direction of time is this: the progenitors of these isolated systems were all further out of equilibrium at times closer to the big bang (the past) than they are today. Earlier, the total entropy was low compared to what it could have been. Therefore, it has tended to increase since. The radiation arrow of time can also be understood as arising from time-asymmetric cosmological boundary conditions applied to timereversible dynamical laws. These are Maxwell’s equations for the electromagnetic field in the presence of charged sources. Their time-reversal invariance implies that any solution for specified sources at a moment of 28

 For example, Hawking’s “no boundary” wave function of the universe [29].

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time can be written in either of two ways: (R) a sum of a free field (no sources) coming from the past plus retarded fields whose sources are charges in the past, or (A) a sum of a free field coming from the future plus advanced fields whose sources are charges in the future. More quantitatively, the four-vector potential Aμ(x) at a point x in space-time can be expressed in the presence of four-current sources jμ(x) in Lorentz gauge as either A∝ ( x ) = A ∝in ( x ) + ∫ d 4 x J Dret ( x − x J ) j∝ ( x J )

( R)

A∝ ( x ) = A ∝out ( x ) + ∫ d 4 x J Dadv ( x − x J ) j∝ ( x J )

( A).

or

Here, Dret and Dadv are the retarded and advanced Green’s functions out for the wave equation and Aµ in ( x ) and Aµ ( x ) are free fields defined by these decompositions. Suppose there were no free electromagnetic fields in the distant past so that Aµ in ( x ) ≈ 0. Using the R description above, this time-asymmetric boundary condition would imply that present fields can be entirely ascribed to sources in the past. Fields are thus retarded and that is the electromagnetic arrow of time. This explanation needs to be refined for our universe because, at least if we start at decoupling, there is a significant amount of free electromagnetic radiation in the early universe constituting the cosmic background radiation (CMB). Indeed, at the time of decoupling, the energy density in this radiation was approximately equal to that of matter. Even today, approximately 13 billion years later, after being cooled and diluted by the expansion of the universe, the CMB is still the largest contributor to the electromagnetic energy density in the universe by far. The CMB’s spectrum is very well fit by a black body law [31]. This strongly suggests that the radiation is disordered with maximal entropy for its energy density. There is no evidence for the kind of correlations (sometimes called “conspiracies”) that would tend to cancel Aµ out ( x ) in the far future and give rise to advanced rather than retarded effects.29 29

 For an experiment that checked on advanced effects, see [32].

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The expansion of the universe has red-shifted the peak luminosity of the CMB at decoupling to microwave wavelengths today. There is thus a negligible amount of energy left over from the big bang in the wavelengths we use for vision, for instance. The radiation used by realistic IGUSs is therefore retarded. A contemporary robot functioning at wavelengths where the CMB is absent will therefore receive information about charges in the past. This selection of wavelengths is plausibly not accidental, but adaptive. A contemporary robot seeking to function with input from microwave wavelengths would find little emission of interest, and what there was would be overwhelmed by the all-pervasive CMB, nearly equally bright in all directions, and carrying no information.

References   [1] M. Gell-Mann, The Quark and the Jaguar, W. Freeman, San Francisco (1994).   [2] J. Butterfield, The Arguments of Time, British Academy and Oxford University Press, Oxford (1993).   [3] M. Gell-Mann and J. B. Hartle, Time Symmetry and Asymmetry in Quantum Mechanics and Quantum Cosmology, in Proceedings of the NATO Workshop on the Physical Origins of Time Asymmetry, Mazagón, Spain, September 30– October 4, 1991, ed. J. Halliwell, J. Pérez-Mercader, and W. Zurek, Cambridge University Press, Cambridge (1993), gr-qc/9309012.   [4] J. B. Hartle, Quantum pasts and the utility of history, Physica Scripta, T76, 67–77 (1998), gr-qc/9712001.   [5] R. N. Shepard, Perceptual-cognitive universals as reflections of the world, Psychonomic Bulletin & Review, 1, 2 (1994); How a cognitive scientist came to seek universal laws, ibid., 11, 1–23 (2004).   [6] W. James, Principles of Psychology, Henry Holt, New York (1890).   [7] See, e.g., G. Westheimer, Discrimination of short time intervals by the human observer, Experimental Brain Research, 129, 121–126 (2000), and G. Westheimer, Visual signals used in time interval discrimination, Visual Neuroscience, 17, 551–556 (2000).   [8] E. F. Taylor and J. A. Wheeler, Spacetime Physics, W. Freeman, San Francisco (1963).   [9] For a review see M. P. Haugan and C. M. Will, Modern tests of special relativity, Physics Today, 40, 69–76 (1987), or C. M. Will, The confrontation between general relativity and experiment, Living Reviews in Relativity, 4, (2001), (http://www.livingreviews.org/lrr-2001-4). For a modern version of the Michelson-Morley experiment, see, in particular, A. Brillet and J. L. Hall,

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Improved laser test of the isotropy of space, Physical Review Letters, 42, 549 (1979). [10] Y. Berra, The Yogi Book, Workman Publishing, New York (1998). [11] C. Callender, The Subjectivity of the Present, Centre for Time, Department of Philosophy, University of Sydney Sydney, unpublished conference (2006). [12] N. Ashby, Relativity and the global positioning system, Physics Today, 55, 41–47 (2002). [13] J. Butterfield, Seeing the present, Mind, 93, 161–176 (1984). [Reprinted in Questions of Time and Tense, ed. R. Le Poidevin, Oxford University Press, Oxford (1998).] [14] See, e.g., the preface to F. Hoyle and G. Hoyle, The Fifth Planet, Harper and Row, New York (1963). [15] See, e.g., D. N. Spergel, et al., First year Wilkinson microwave anisotropy probe (WMAP) observations: determination of cosmological parameters, The Astrophysical Journal Supplement Series, 148, 175 (2003); J. L. Tonry, et al. Cosmological Results from High-z Supernovae, The Astrophysical Journal, 594, 1–24 (2003). [16] P. C. W. Davies, The Physics of Time Asymmetry, University of California Press, Berkeley (1976). [17] R. Penrose, Singularities and time asymmetry, in General Relativity: An Einstein Centenary Survey, ed. S. W. Hawking and W. Israel, Cambridge University Press, Cambridge (1979). [18] H. D. Zeh, The Physical Basis of the Direction of Time, Springer, Berlin (1989). [19] H. Price, Time’s Arrow and Archimedes’ Point, Oxford University Press, Oxford (1996). [20] R. Laflamme, Time-symmetric quantum cosmology and our universe, Classical and Quantum Gravity, 10, L79 (1993), gr-qc/9301005. [21] D. Craig, Observation of the final condition: extragalactic background radiation and the time symmetry of the universe, Annals of Physics (NY), 251, 384 (1995), gr-qc/9704031. [22] See, e.g., K. Hirata et al., Observations of a neutrino burst from the supernova SN1987A, Physical Review Letters, 58, 1490–1493 (1987). [23] H. Reichenbach, The Direction of Time, University of California Press, Berkeley (1956). [24] S. W. Hawking, The Direction of Time, New Scientist, 115, 46 (1987). [25] R. Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Development, 5, 183–191 (1961). [26] See, e.g., C. Bennett, Logical reversibility of computation, IBM Journal of Research and Development, 17, 525–532 (1973); E. Fredkin and T. Toffoli, Conservative logic, International Journal of Theoretical Physics, 21, 219–253 (1982).

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[27] W. Penfield and L. Roberts, Speech and Brain Mechanisms, Princeton University Press, Princeton (1959). [28] J. L. Borges, Ficciones, Grove Press, New York (1962). [29] S. W. Hawking, The quantum state of the universe, Nuclear Physics B, 239, 257 (1984). [30] L. Boltzmann, Zu Hrn, Zermelo’s Abhandlung Uber die mechanische Erklärung irreversibler Vorgange, Annalen der Physik, 60, 392, (1897), as translated in S. G. Brush, Kinetic Theory, Pergamon Press, New York (1965). [31] D. J. Fixsen et al., The cosmic microwave background spectrum from the full COBE FIRAS data set, The Astrophysical Journal, 473, 576 (1996). [32] R. B. Partridge, Absorber theory of radiation and the future of the universe, Nature, 244, 263–265 (1973).

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Chapter 2

DIS C OVE RI NG PHY SI CA L TI M E W I T HI N HU MA N TI M E* Ronald P. Gruber,† Carlos Montemayor‡ and Richard A. Block§ Stanford University, USA [email protected] ‡ San Francisco State University, USA [email protected] § Montana State University, USA [email protected] †

There is a gap between the time of physics and that of the human brain. Bridging it can be treacherous, and some refer to the endeavor as the “two times problem.” In order to justify the existence of human time, many attempts are made by physicists to reify temporal experiences, in particular the experiential “flow” of time. An alternative approach has been for brain scientists to castigate static space-time views derived from Einstein’s Block Universe. The theory proposed here opts for a compromise dualistic view. The claim is made that within the cranium, there exists a complete and independent veridical system of time, one that is compatible with modern space-time cosmological views. However, the brain, through a process of evolution, developed a complementary illusory system that provides a more satisfying experience of physical time and better adaptive behavior. The Dualistic Mind view is based upon Hartle’s * This article has been modified and extended from Gruber, R.P., Block, R.A., and Montemayor, C. (2022). Physical Time within Human Time, Frontiers in Psychology, 13. https://doi.org/10.3389/fpsyg.2022.718505 (CC BY 4.0).

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information gathering and utilizing system (IGUS), which solves a big part of the “two times problem.” It provides a framework to demonstrate that the experiential past, present, and future is not a property of fourdimensional space-time but one of many notions describing how individual IGUSes process information. The dualistic view proposed here builds upon the IGUS view to provide a complete explanation of major temporal experiences. This view is based in large part upon experimentation involving all of the temporal experiences and then makes predictions accordingly.

1 Introduction 1.1  Two times problem Ancient Greek philosophers began the debate as to whether or not time is an illusion. Currently, most physicists opt for Einstein’s view that the “past/present/future are illusions even if stubborn ones” (Davies, 2002). For some, time does not exist at a fundamental level, but is derived (Barbour, 1994; Rovelli, 2011). Others present a space-time cosmology that provides a mechanism to account for the “flow” that the brain undeniably experiences. Ellis (2014; also, Elitzur 1992) proposes a universe that grows, the edge of which provides the passage (flow) that humans experience. Aert’s (2018) theory of “Refounding Relativity” provides a method to account for the reality of “change.” Some philosophers such as McTaggart (1908) took an extraordinary position that time is logically an illusion. More recently, Price (2011) made the case that the components of the flow of time (FOT), including motion and the moving present, are subjective. Some academics have the opposite view. Philosophers of physics, such as Broad (1923), and more lately Capek (1991) and Maudlin (2002), have made room in their metaphysics for objective flow. Prominent among physicists who opt for the view that basic human time should be reified are Smolin and Unger (see Smolin, 2013, 2015; also, Unger and Smolin, 2010). They provide a cosmological natural selection (CNS) theory that is based upon Temporal Naturalism, one thesis of which is that time, as a succession of moments, is “real.” This view also implies that the dynamic experiences of time as experienced by humans, e.g., change and temporality, are “real,” including motion (personal communication, 2016). Lastly, the “present,” according to Temporal Naturalism, is 28

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considered unique. To their credit, the CNS theory is falsifiable. For example, another thesis of the CNS theory is that there is an evolution of laws with respect to that time. To be clear, although varying constants may be part of the Smolin theory, fundamental constants might also vary in the context of different views of time, a topic beyond the scope of this chapter. That notion is not new, as Dirac (1979) proposed, for entirely different reasons, that the constants such as G and particle masses change with time. Others have concurred with this possibility (Gruber and Brahm, 1993). However, providing testable explanations, as the CNS theory does, makes it noteworthy despite being in the minority. A number of philosophers recognize that “philosophical speculations need to be disciplined in the face of hard facts of neuroscience and experimental psychology” (Montemayor, 2013). Most recently, Callender (2017) examined the flow of time carefully, noting that it is part of what he calls “manifest time,” which includes the experiences related to the “now” and the past/future asymmetry. Then, he suggests a parallel between the “two times” problem and Eddington’s “two tables problem,” which he views as two interpretations. One is a manifest table that looks and feels solid. The other is a scientific table composed of molecules in between, which is much space. Recently, Buonomano and Rovelli (2021) summarized the current disagreements between neuroscience and physics. The most concerning problem remaining for Rovelli is that of the “flow” of time, which he attributes largely to entropy. By contrast, Buonomano questions the validity of the Block Universe. Both understandably want to reify human time. Understandably, it is a terrible feeling to believe that some of your perceptions are illusions. Rickles (2014) discussed the problem in terms of a distinction between the “world in the head” and the “world outside of the head. Perhaps the best and most succinct summation of the “two times problem” was provided when Gleick (2011) reported on the physicist Feynman’s view on the illusion of time: It seemed to Feynman that a robust conception of “now” ought not to depend on murky notions of mentalism. The minds of humans are manifestations of physical law, too, he pointed out. Whatever hidden brain machinery created (one’s) coming into being must have to do with a correlation between events in two regions of space — the one inside the cranium and the other elsewhere “on the spacetime diagram.” On Richard Feynman, 1963 29

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This is a concise passage that contrasts and relates human time to physical time. Feynman was asking for a physical explanation for the human experience of time. He felt that there must be a fundamental connection between the two times and that it would ultimately involve physics. 1.2  Introduction of the IGUS view to solve the problem In response to Feynman, the problem is approached here with a unified theory of manifest time (combined human temporal experiences). First, it involves explication of the IGUS “information gathering and utilizing system” view proposed by Hartle (2005, 2014). It mathematically demonstrates that the experiential past, present, and future are not properties of four-dimensional space-time, but notions describing how individual IGUSes process information. For that reason, the conflict between physics and psychology for this particular aspect of temporal experiences should not, it says, exist. The potential problems with using a robot model or system as an experimental platform for the study of human behavior are reviewed by Datteri (2021). Hartle’s IGUS view is widely accepted, among physicists at least, to bridge the gap between physics and psychology for issues relating to past/present/future and the “now.” He provides a means to reconcile the physical “now” with the experiential “now.” He starts with the proposition that the world is four-dimensional according to fundamental physics, governed by basic laws that operate in a space-time that has no unique division into space and time. He discusses the origin of this division (into present, past, and future) in terms of simple models of information gathering and utilizing systems (IGUSes). Past, present, and future are not properties of four-dimensional space-time, but notions describing how individual IGUSes (robots) manipulate information. Their origin is to be found in how these robots were constructed. There is a localized notion of “present” at each point along an IGUS worldline. But modes of organization that are different from present, past, and future can be imagined that are consistent with the physical laws. Loosely speaking, it is being suggested that the past/present/future is outside of physics, as is the case for music. With help from the psychologist Shepard (1994, 2004) he proposes a falsification test. He suggests that it would be possible to construct IGUS robots that process information differently and therefore experience different “presents.” For example, a robot with a split visual 30

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Figure 2.1   Schema for the model information gathering and utilization system (IGUS). At every proper time interval t*, the robot captures an image of its external environment. In this case, the robot experiences a stack of cards labeled a, b, c, d, e, f, etc., whose top member changes from time to time. The IGUS robot chooses how to route and utilize its information. Figure modified from Hartle (2005).

system (SS robot) could experience the present with one half screen and events from the immediate past with the other half screen. Doing so would confirm that there is no unique “present.” A simple schematic of the human (model) IGUS is seen in Fig. 2.1. At every proper time interval t*, the robot acquires an image of its outside environment. In this case, the robot experiences a pile of cards labeled a, b, c, d, e, f, etc., whose top card changes periodically. The IGUS robot decides how to route and utilize that information. The robot utilizes the images in registers P0, P1, P2, and P3 in two computational processes: C (conscious) and U (unconscious). Process U uses the data in every register to update an uncomplicated model or schema of the outside environment. A schema of the outside environment is utilized by C along with the most recently received data in P0 to predict events of its environment to the future of the data in P0, formulate decisions, and execute behavior. In sum, the Hartle view allows one to appreciate the subjectivity of past/present/future that is said to be compatible with physics. He helps resolve the issue raised by Lestienne (1988, p. 219) that the “past, present and future are only categories of our mind and are not given a meaning except in them.” Regarding the actual “flow” of time, Hartle attributes that to the movement of information in and out of the C (consciousness) register. In other words, the experiential flow component of the FOT is attributed to the utilizing system of the robot and not to the time of physics. 31

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1.3  The many IGUS views The IGUS concept to deal with the “two times problem” was picked up by Jenann Ismael (2015; 2017). She augmented the IGUS by including “flow” and higher-level temporal beliefs — which, figuratively speaking, are “gadgets” to the robot. Dorato and Wittmann (2019), Huggett (2018), and Hertzel (2016) all share the view of the importance of IGUS to explain passage (flow). Recently, Callender (2017, 227) provided the much needed expansion of the model IGUS and relied upon it for the “beginnings of a theory of time flow.” He augmented the IGUS robot with even more “gadgets” to account for many other phenomena, including objective temporal experiences but also others such as “motion qualia,” all of which come under the heading of “manifest time.” Recently, we introduced yet another IGUS version, but one involving a “dualistic model” (Gruber et al., 2020b). After acknowledging that the “present” component of the FOT requires an IGUS model for its explanation, it was postulated that each of the other components of the FOT have both an illusory and a non-illusory (“real”) aspect. That model is derived in part from 10 chosen space-time cosmologies (see Table I in Gruber et al., 2020b). Many of the space-time cosmological claims take Einstein’s Block Universe as a basis. Notable is the view that objects in the universe are, in reality, events that “happen” (Rovelli, 2011, 2018). In other words, the Block is not “frozen” as originally interpreted. Some space-time views modify the Block to allow it to grow and thereby provide an objective basis for temporal “flow” (Ellis, 2014). All the experiences of the FOT are then contrasted with what these space-time cosmologies have to say about those very same phenomena. Then, it is possible to construct a dualistic mind model based upon Hartle’s IGUS robot. Without a doubt, there will be controversy here simply because the 10 space-time views are not unanimous and it is necessary to decide which ones will most likely be correct and sustained.

2  A Dualistic IGUS View of Manifest Time The Dualistic View proposed here suggests that upon careful inspection, one will discover a complete and independent veridical system of temporal experiences. That system is completed or augmented by an illusory system. In this review the major temporal experiences will be explicated. An analysis of more minor temporal experiences is provided by Callender 32

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(2017). By analyzing known veridical and illusory components of passage, a dualistic classification is derived, and in turn more veridical components of flow (passage) are found, as will be described below. Furthermore, it is argued that the veridical system begets a corresponding illusory system of temporal experiences. That dualistic construction is then applied in the form of add-on “gadgets” to Hartle’s “model IGUS” (representing the human), resulting in a dualistic IGUS that represents manifest time. It is argued that the illusory system’s sole purpose is to enhance the human experience of time. It is said to be the product of natural selection. The net result for the two times problem is that there may be less of a need to reify the “flow” of the FOT. Knowing that there is a complete and independent veridical system might lessen the need to deny aspects of space-time theories that stand on solid ground. Finally, knowing that the illusory system is, if necessary, dispensable should soften the perception that our brains have been left out of the physics of time. 2.1 Definitions Before explication, it is necessary to avoid confusion by defining terms that have often been conflated during the past debates involving the “two times problem.” The term dualistic means nothing more than that there are two types of experiences (or cognitions) for each component of manifest time, including the flow of time (FOT). The term dualistic should also not be confused with dualism — a view in the philosophy of mind that mental phenomena are, in some respects, non-physical, or that the mind and body are distinct and separable. Dualistic is also not to be confused with the important discovery of phylogenetically and dual temporal cognitive systems (Hoerl and McCormack, 2019). It is also not the dual model in the philosophy of time that distinguishes conscious from unconscious time perception based on agency, which is used to clarify the metric from the subjective requirements of time cognition (Montemayor, 2017a; 2019). The term illusion refers to a perception that has no basis in reality, which in turn defers the problem to what the currently accepted laws of physics suggest. It has long been an ambiguous term that needs defining (Buonomano 2017). Consider the perceptual completion that the brain provides to fill the retinal blind spot. On the one hand, “filling in” is illusory. On the other hand, the brain guesses correctly. One would say that the “perceptual completion” of that perception is not an illusion; it provides no false information the way a mirage does. It provides helpful 33

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information and is more accurately referred to as a “perceptual add-on,” one that is veridical (not contradicted by accepted physics). When only cognition is involved, such as a myth or belief, it can be referred to as a cognitive add-on. However, the need to make a distinction between the terms cognitive and perceptual is not critical, as Mroczko-Wasowicz (2016) questions the close relationship between the two. The word illusion is retained here because it is engrained in the literature and less cumbersome than a “perceptual add-on.” We are not using the completely different definition of illusion by Hoffman (2012). For him, “perception has not evolved to report truth, but instead to guide adaptive behavior.” Therefore, he defines an illusion as any perceptual phenomenon that does not guide that behavior. Lastly, there is the new term, “manifest time,” coined by Callender (2017). It is the sum of human temporal experiences that are readily perceived and recognized by the mind. It includes experiences associated with “subjective time” and the flow (passage) of time. It is meant to be a complete collection of what is perceived. All major dualistic components of manifest time will be elaborated upon here. The three commonly associated with flow (passage) are: (1) a unique (moving) present, (2) dynamism of change/motion, and (3) directionality (temporality).

3  Dualistic Components of Manifest Time 3.1  “Present” (unique [moving] and no unique present) Of the common components of flow, one of the most hotly debated is the alleged unique “present.” It is first from a list of several components of manifest time (Fig. 2.2). To experimentally confirm that the present is not unique, Gruber and Smith (2019) chose to test Hartle’s IGUS hypothesis that a new “present” can be fabricated, suggesting that the current one we humans possess is not unique. The experiment involved the construction of a split screen (SS) robot using a VR headset screen containing two “presents” of slightly different local time intervals. However, upon construction, it was immediately apparent that the observer did not experience two simultaneous “presents,” because there was no sense of immersion in the environment. To create this experience of immersion, a similar robot was created in which the observer is permitted to alternate between “past” and “present” screens ad libitum — the intermittently behind (IB) robot. By being able to switch between equally realistic time periods, the 34

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Figure 2.2  Major components of manifest time. Veridical components beget illusory components.

observer experienced what was intended in the split screen (SS) robot, except in an alternating instead of a simultaneous manner. The participant was also allowed to go back and forth between “past” and “present” ad libitum by pressing a button. Unsolicited participant comments that it clearly felt like “being in the past” were received. A few participants even indicated that they sometimes “got lost” between what was “past” and what was “present.” A worldline description of the “present” for an IB robot is given in Fig. 2.3. It shows the worldline of an external object that is the source of its images, such as the stack of cards in the prior figure. This source changes its shape at discrete instants of time delineated by ticks, passing through configurations c, d, e, f, and g. As an example, the object E is recorded as e in two adjacent registers — thus, e,e. The number of registers for e is simply proportional to the duration of the observation. The image in each register is then experienced as Ce. These e’s are experienced again as Ce when each e moves to another register that is further away (along the worldline). In short, the robot is permitted to utilize information as it chooses. In this case, the present from register e is experienced 35

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Figure 2.3   A worldline description of the intermittently behind (IB) robot.

at two different points along the worldline, i.e., the IB robot experiences the same present twice. It should be noted that some of Hartle’s IGUS robots do exist in humans to a limited extent (Gruber, 2008). There is an always behind (AB) robot. This robot has input to C computation only from a certain register PK > 0, and also the schema. That input is, therefore, always a proper time behind the most received acquired data. The AB robot would also possess a tripartite division of information it records. Its present experience would be the substance of the PK register. It would remember the past that was stored in the PK+1, …, Pn registers. However, it would remember its future, which has been stored in P0,…PK–1, at a time that is beyond its present experience. Its answers, in other words, to questions such as “What’s happening now?” would be out of date by a little bit. On the other hand, it would have premonitions for future events. Another IGUS example is a savant, the no schema (NS) robot, that receives input to C computation from all the registers P0 – Pn (to the same degree). The NS robot employs no unconscious computation; it also constructs no schema, but instead makes decisions by conscious calculations from all the data it possesses. Savants process almost every bit of incoming information. And they recall 36

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it just as fast, and almost as vividly as information of the present. However, an NS robot would not make efficient use of computing resources by giving equal attention to the “present” data and data from the past, the details of which may not have any effect on important future predictions. Verifying Hartle’s prediction suggests that the brain has, as one of its fundamental experiences of manifest time, an experience of a potentially variable past/present/future and an illusory unique present. To be clear, although it was possible to construct a robot other than the “model robot,” the IB robot does not prove that the “moving present” is an illusion. It only establishes that there is a notion of a “present” at each point along the worldline. The actual “moving present” is a dynamistic illusory experience that is more related if not identical to the experience of “moving” — in other words “motion,” which is described as an illusion below. In agreement, Romero (2015) indicates that for physics “there is no ‘moving present’ only an ordered system of events.” 3.2  Persistence (enduring self) and impermanence (ephemeral self) For years, the FOT debate revolved largely around two FOT levels: the past/present/future illusion, particularly the “moving present” (the upper level) and the dynamic temporal experiences such as motion (the lower level) (Gruber et al., 2015). However, another related phenomenon is made more evident by Price (2011) and Ismael (2011), who discovered what they would refer to as a double illusion. It turns out that the subjective or illusory phenomenon of the FOT rests upon another illusion that in some sense, they say, is more important. It is the phenomenon of persistence, specifically the “persisting self” (Ismael 2007, 2011). Under the name of “enduring self,” Paul (2016) reviews this matter thoroughly. A human needs to feel that she persists and is not simply a conglomerate of impermanent events, as space-time cosmologies suggest. The observer, in a unique (moving) present, wants to believe that she is a single individual and not multiple momentary individuals extending backwards in time. In the Block Universe, persistence of that sort has no place. When consoling the wife of his best friend Besso, who had just died, Einstein said, in effect, that Besso was still there (Elitzur, 1992; Davies, 2002). The implication was clear: there is a Besso who may be dead, but another who is alive is in the past. Clearly, this time illusion is difficult to accept for 37

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most. However, it is much easier to acknowledge impermanence as veridical if one happens to hold the space-time view that the universe is composed of events, and that the observer, too, is basically a series of complex events (Rovelli 2018; Romero, 2015). Balashov (2007) reviewed this issue of persistence as an illusory phenomenon in the physics of Minkowski space-time. The two opposite views regarding persistence are known as endurantism (or three-dimensionalism) and “perdurantism” (four-dimensionalism). According to the former, objects are extended in three spatial dimensions and persist through time by being wholly present at any moment at which they exist. On the latter, opposing account, objects are extended both in space and time, and persist by having “temporal parts,” no part being present at more than one time. Only perdurantism is compatible with space-time physics and suggests that object persistence is not veridical. With the hypothesis that experiential persistence is an illusory experience, a pilot experiment was performed to demonstrate that it can be precluded (Gruber et al., 2020b). Observers (“human IB robots”) wearing the backwards-in-time VR apparatus were allowed to watch a remotecontrolled toy dog roaming about as they went “back and forth in time.” They lost the experience of persistence. Going back and forth into the past (e.g., 30 seconds back), an observer would note a moving toy dog. When she was in the past, she might see the dog to her right, even though it was actually located to her left. When she returned suddenly (in a fraction of a second) to the present, she would see the dog to her left where it actually is. The experience was that the dog did not appear to be the same dog, because it could not have traveled several feet that quickly. The explanation is based upon the principle of spatiotemporal priority, which occurs for the well-studied phenomenon of “object persistence” (Scholl 2007). When deciding whether an object is the same persisting object from an earlier time, factors relating to how and where that object has moved will almost always trump factors relating to what the object looks like. For example, when a car goes through a tunnel, it is considered to be the “same” car if there are no major changes to it and if the time of exit is appropriate for the entering velocity. If it exits almost the same time as it entered, it is “not same.” Not all share the view that persistence is illusory, and that impermanence (ephemerality) is veridical (reviewed by Haslanger and Kurtz, 2006). However, it would seem that any argument on behalf of persistence likely stems from a desire not to be ephemeral, i.e., not to be a fleeting 38

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individual. In fact, it is more likely that the whole adaptive purpose of persistence illusion is to provide a singular “self” that can move along with the “present.” Other philosophers do support the physics view. Paul (2010) takes the position that persistence, just like apparent motion, is an illusion, and refers to it as “apparent persistence.” The illusory cognitive add-on of a persisting or enduring “self” allows the observer to either view herself as stable with events and the “present” moving by relative to her, or vice versa, that she is moving relative to events of the environment. Callender (2017) considers the “self” important enough to add it as a gadget to the IGUS. The entire phenomenon of self and its illusory aspects are beyond the scope of this chapter but are reviewed by Hood (2012), Klein (2004), Blackmore (2012), and Velleman (2016). It is worth noting that a case can easily be made that veridical impermanence is a viable alternative to the enduring self. There is a large group of people who opt for the view that the self is not persistent, or at least should choose not to be persistent. Outside of Western civilization, there are those who subscribe to that view. It is a belief among those adhering to Buddhism (Struhl, 2020). It is a belief that the individual is really — or at least should consider herself to be — “ephemeral” (impermanent). Recognition of the dualistic nature of this particular temporal phenomenon may help legitimize the self-illusion (enduring self) and foster reconciliation with the Eastern view of self. 3.3  Change (dynamic and completed) It was not that many years ago that “change” was discovered to have dualistic experiences (Rensick, 2002). It was definitely not realized until recently how important that understanding of the change phenomenon would be for the two times problem (Gruber et al., 2020b). There is “dynamic change” — the experience of seeing an illusory change occur, such as one color or shape to another. There is also “completed change” — the non-illusory experience that the change “must have occurred.” In a flicker paradigm, Hollingworth (2008) demonstrated it with an initial object (such as a cup) and a different object (another cup) for 250 ms duration each and a blank interstimulus interval (ISI) varying from 200–5000 ms. Participants reported a strong impression of “seeing the change occur” at 200 ms, a weaker impression at 1000 ms, and no impression (completed change) at 5000 ms. After hypothesizing that the experience of “happening” was part of the 39

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“flow” in the FOT phenomenon, Gruber and Block (2013, 2017; see also Gruber et al., 2020a) studied the change of more prolonged, featural processes such as bread toasting and noted the ISIs at which dynamic and completed change occurred. In several sensory modalities (auditory and tactile), the phenomenon was expressed by the participants as experiencing the scene (events) to actually “happen” (dynamic change) vs. knowing it or “not seeing it actually happen” (completed change). Dynamic change is one of many elements of dynamic perceptual completion (DPC), all of which provide the all important experience and phenomenon of temporal continuity for discrete or interrupted perception. Dynamic change and all DPCs are perceptual add-ons. They do not necessarily provide significant information for the observer other than to indicate (loosely speaking) that there are multiple events of unspecified type between two temporally adjacent stimuli. Were there no experiences of DPC, however, it is likely that the brain would cognitively deduce it (i.e., continuity) anyway. Physicists have tried for years to reconcile the phenomenon of “change” in the “frozen” Block Universe. Both Rovelli (2011 97; 2018) and Aerts (2014, 2018) have successfully “unfrozen” it. The “change” in their cosmological theories is said to be real, and most physicists today have adopted that view. When the dualistic view was first introduced (Gruber et al., 2020b), the hypothesis then was that a dualistic temporal experience of “change” should be expected, in particular, a veridical one for physics. The claim then was, and still is, that completed change represents the “change” in physics. A dynamic change simply augments that experience. The fact that the human brain also evokes an illusory form of change does not conflict with its “real” aspect. When analyzing change, it eventually becomes necessary to delve into the phenomenon of “becoming” because of the intimate relationship between the two. Rickles (2014) provides an extensive review of the FOT, noting that there are those who believe there is genuine flow or becoming in the world and those who believe there is just a block of events. Philosophers and physicists have debated the specifics of becoming and its relation to change for some years now (Mellor, 1998). Moreover, Savitt (2002) argues that special relativity has within it two concepts of time: (1) coordinate time and (2) proper time, “the latter being a kind of time perfectly apt for ‘becoming.’” From a physics viewpoint, Aaerts (2018) points out that the apparent contradiction between a process view on reality (where there is a being and a becoming), and a geometrical view (where 40

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there is only a being and no becoming) need not exist; therefore, change is said not an illusion. In sum, the phenomenon of becoming can and should be recognized; but for purposes here, it need not be treated as a separate component of manifest time because the phenomenon of change is dealt with in depth. 3.4  Motion (dynamic movement) and completed movement When the dualistic model was introduced, one of the hypotheses included the notion that it should be possible to discover a corresponding veridical experience for every illusory one. That would not only include change, as just demonstrated, but it would include its most important aspect — motion. Motion is divided into “real” and apparent types. Briefly, apparent motion involves both beta (movement of an apparent object between successive stimuli) and phi (objectless movement) (Steinman et al., 2000). However, the experience of “real motion,” itself, is considered to be an illusory percept for the following reason. Visual perception is generally agreed to be discrete at a rate of 10–13 Hz, with the continuous wagon wheel illusion. As the speed of the wheel increases, a point is reached when it starts to reverse itself — an illusion due to the phenomenon of aliasing (VanRullen et al., 2010; VanRullen and Koch, 2003). It must be acknowledged that a couple of studies are skeptical that discrete perception has been proven by the wagon wheel experiments (Kline and Eagleman, 2008; Holcombe, 2014). Assuming that perception is discrete, Koch (2004, p. 274) suggests that motion is “painted on” each frame. At this time, it is uncertain if that “paint” is phi, beta, or both (VanRullen, personal communication, 2018). Fortunately, the world is quite navigable at a perceptual rate of 13 Hz (although not always comfortably) without the DPC of motion. For example, a bird in flight that flashes on and off to the hunter’s eyes at 10–13 Hz has a staccato type of appearance. However, the bird is visible enough to the hunter, so that an accurate shot is possible. The superimposed motion experience can be thought of as a perceptual add-on, one that augments the perception of movement. Specifically, beta may be filling the gaps with images of the bird and, in that sense, is veridical because the brain guessed correctly. On the other hand, it implies continuity and persistence, which is not veridical. There is also one caveat, in that motion would seem to be essential when considering the debilitating life of akinetopsia patients. However, these patients are missing more than one 41

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type of motion (such as beta or phi). Their perception is erratic, with many frozen, prolonged intervals (Rizzo et al. 1995). By contrast, a patient with cinematographic vision (a flickering series of stills) from a seizure disorder or migraine (Sacks 1999) may get all the essential information regarding spatiotemporal changes. Similar to the situation described above with change, motion (also termed dynamic movement) is the illusory counterpart of a (veridical) completed movement. An experimental example of this in the form of positional change was given by Nakashima and Yokosawa (2012, p. 269) in a flicker change detection task. A 250 ms duration image of a bed oriented to the right was alternated with another oriented to the left, with black (blank) ISIs between them. As the ISI increased, that impression (of bed rotation) weakened and was lost after 1000 ms. Then, participants experienced what can be considered to be completed movement, i.e., “not seeing the change occur.” The similarity to the completed change is noteworthy. From the viewpoint of physics, motion is denied in the Block Universe, “End of Time,” and “Order of Time” space-time cosmological views. Physical continuity is not in the cosmological scheme. Instead, what is expected of them is that events, including cerebral events, be discrete. It is no coincidence, therefore, that the temporal experience of “completed movement” is a discrete process, as is completed change. These two veridical experiences are good examples of how much physical time resides within the human cranium. Meanwhile, the illusory, dynamistic aspect of movement (i.e., motion) satisfies the desire of space-time theorists, such as Dowker (2014), with her space-time atoms view to “breathe life” into the Block Universe. However, it comes from the illusory system within the cranium, not necessarily the edge of an expanding universe. Therefore, we add it as one more “gadget” to the dualistic IGUS (Fig. 2.2). 3.5  Temporality and temporal order The initial dualistic model also hypothesized that temporality should have veridical and illusory aspects (Gruber et al., 2020b). Of note, temporality is defined in many ways. Physics and the psychological sciences often define it as the totality of time experiences within their disciplines. Here, the term is restricted to the before/after human experience. Poppel and Bao (2014) and Ruhnau (1997) provide a basic analysis of that experience in the first several seconds. Experienced time is segmented into a hierarchy of domains or zones. For intervals of 5–30 ms, there is no 42

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experience of before/after. It is the “atemporal” zone. When put together, those ultra-brief zones provide a longer 2–3-second zone that compromises the human experience of the “present” (or “nowness”). Each 2–3 second island of “nowness” is then linked with the previous one, but the continuity of that entire experience is said to be an illusion (Poppel 1997; Ruhnau 1997). Further analysis of that hierarchy of temporality is provided by Montemayor and Wittmann (2014), who emphasize that the continuity of experience (requiring working memory) involves multiple seconds to generate a platform for the narrative self. It is important to note that these intervals comprising the temporality experience are part of the “succession of experiences” (Arstila, 2016). Moreover, there is an additional overlaying “experience of succession” (also known as the “feeling of succession (Hoerl, 2013) for at least the first three zones. The illusory aspect of the temporality experience is brought to the fore by Arstila (2018), who insists that “the succession of experiences and the experience of succession are two different things.” Historically, James (1890, pp. 628–9; see also Block and Patterson, 1994; Block, 1994) expressed this view similarly: “A succession of feelings, in and of itself, is not a feeling of succession. And since, to our successive feelings, a feeling of their own succession is added, that must be treated as an additional fact requiring its own special elucidation.” Plainly put, temporality comprises both the experience of succession and the succession of experiences. To argue that there is an illusory aspect to temporality, consider the thought experiment of C. D. Broad (1923; see also Montemayor, 2009, 2012; Arstila, 2016). When looking at the big hand of a clock, there is an experience of succession and also a succession of experiences as it moves along passing the numbers on the dial. By contrast, when viewing the little (hour) hand, there is no experience of succession because the movement is imperceptible. However, there is still an experience of a succession of experiences when it eventually passes a number. Thus, one is left with temporal order. It is helpful to contrast the experience of succession with that of motion. Arstila (2016, 2018) notes that both experiences are said not to extend time and are part of his “snapshot theory.” He refers to the illusory experience associated with “pure motion” and then notes that the same can be said of succession, referring to it as “pure succession.” Both serve the same purpose for human adaptation by providing a continuity for events and an assumption of “sameness” or persistence, which is contrary to the physics view. There is a caveat when analyzing succession within scenes involving motion, such as the clock hand thought experiment. Is the experience of 43

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dynamism due to pure succession, or is it due to dynamic movement? To avoid that ambiguity, consider the following heartbeat experiment. When listening to the normal first and second heartbeats (“lub-dub”), there is a strong before/after (i.e., succession) feeling. The normal heartbeat is experienced as lub-dub-pause/lub-dub-pause/lub-dub-pause. The lub and dub are of slightly different frequencies and are due to the closing of the atrioventricular valves. The experience is that dub follows lub, not the other way around. Credit for this observation belongs in part to Poppel (1997), who noted that the brain has a tendency to experientially draw a series of two or more chronologically separated stimuli together. Lub does, of course, follow dub and can be experienced as such in terms of temporal order (or sequentiality), but not succession, at least under these circumstances. In short, part of the normal heartbeat experience contains an experience of succession and part does not. When, however, the heart rate is increased, the ejection and fill times approach each other, and the lub-dub and dub-lub intervals approach each other also. Then, the experience becomes one of a continuous succession of heartbeats: lub-dub-lub-dublub-dub.1 Thus, the heartbeat experiment further supports the notion that an experience of succession is separate from a succession of experiences. 3.6  Speed of time and duration judgments Totally unrelated to temporal order but also important is the veridical phenomenon of duration judgment and its corresponding illusory experience — the “speed of time,” which can be thought of as the speed of duration judgments. The phenomenon of duration judgment has been studied thoroughly, with its prospective and retrospective types (Block and Zakay, 1997; Wittmann, 2016; Montemayor, 2017b). Prospective duration judgment involves some sort of timing mechanism. The original “internal clock” has not been discovered (reviewed by Block, 2003), but “population clocks” are a likely answer (Buonomano, 2017). These judgments are also strongly influenced by attention and cognitive load (Block et al., 2010). Retrospective duration judgment involves memory, and in particular the memory and contextual cues of those events (Block et al., 2018), all of which provide objective duration measurements, even if not necessarily accurate. The corresponding illusory experience, the speed of time, is judged when asking “how fast time went,” and it can be thought of as how  A demonstration can be found at https://www.youtube.com/watch?v=ZIbVh6dFLnc.

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quickly the events went by. Droit-Volet and Wearden (2016) studied this phenomenon, referring to it as the “passage of time judgments.” This temporal experience was assessed by asking the participants to indicate how quickly time seemed to pass during a task, an experience that is different than its duration. Their results showed that although an interval can be retrospectively underestimated (such as sitting in a waiting room), time can be judged as passing slowly during that interval. 3.7  Other components of Manifest Time There are other temporal order phenomena that are dualistic. A complete analysis is provided elsewhere (Gruber et al., 2022). One of them is the veridical experience of neural temporal order and its counterpart, the illusory causational temporal order. The brain is constantly rearranging the timing of events. Recall, for example, that the neural arrival times for the auditory and visual components of snapping fingers differ. The temporal order differential needs to be adjusted closer to but not necessarily zero if they are to be experienced simultaneously. Proper cause and effect are required. There is also an illusion that we live in the immediate present. The “postdiction” process modifies the veridical “neural temporal order” to provide what can be called an illusory “causational temporal order” that enables the observer to believe she is in charge. Another component is the specious present and its counterpart, discrete (snapshot) perception. Ordinarily, we don’t think of these temporal experiences in the contest of manifest time. However, upon reflection, it should be apparent that the experiential experience of the “present” is extended (on average 3 s) by contrast to the physical moment. However, perception at its core is generally believed to be discrete and in agreement with any spacetime theory suggesting a universe of discrete events. Lastly, it is quite likely that some animals do not have a specious present, although it is difficult to imagine how that would be tested. With respect to the view from physics, the notion of an extended present is, understandably, not congruent with the “present” of Hartle’s robots and Minkowski space-time. 3.8  Flow whoosh and dynamism Until now, little has been said about (1) “flow” in the “flow of time,” (2) the dynamic aspect of temporal experiences often referred to as “whoosh,” and elsewhere (3) the “feel” for Paul (2010). The exact mechanism behind this 45

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dynamic experience is debatable. Callender (2017, p. 255) submits that “to make our IGUS believe that time whooshes by we need it to represent itself in its model of the world as an enduring self.” By contrast, flow is the dynamic experience of time that seems to mostly concern Rovelli (Buonomano and Rovelli, 2021), who attributes it to entropy. This is his principal issue for bridging the gap between physics and neuroscience. In so doing, he is attempting to reify that particular human experience. It is conjectured here that “flow” represents the dynamism of a few temporal experiences from the illusory system, e.g., motion (dynamic movement) , dynamic change, and the “feeling of succession (“pure succession) of temporality. In the final analysis, these experiences are ones of continuity and implied “sameness,” all of which find no home in most space-time cosmologies.

4  Summary of the Dualistic Mind Theory Now that many “gadgets” have been added to the model IGUS, the dualistic robot with all the significant parameters of temporal experiences is largely complete. There are a few others (Callender, 2017), but space does not permit reviewing them. Moreover, until now, nothing has been mentioned as to how these temporal processes might be implemented. A general discussion of such a temporal process is reviewed by Marchetti (2014) but is beyond the scope of this paper. However, in keeping with the goal of helping to solve the two times problem, the IGUS model for the flow of time is expanded to a dualistic view of manifest time. The “gadgets” figuratively represent the many components of the flow of time and other temporal properties of manifest time. Moreover, the components fall into a veridical system that is compatible with modern space-time cosmology and a corresponding illusory system that augments veridical experiences. The dualistic mind approach avoids implicating many human experiences of time as illusory and removes the compulsion to insist that the experiential “flow” is real or veridical. Both systems of veridical and illusory experiences exist, and are not only not in conflict, but they guide adaptive behavior. The two times problem is thereby offered as a possible solution. To be transparent, there are certain weaknesses with the dualistic mind view. For one, it ignores auditory and somatosensory perception by the dualistic IGUS. The dualistic view does not explore, in depth, the phenomenon of precognition/premonition or déjà vu, temporal experiences that are almost universal other than the one method mentioned in the 46

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always behind (AB) robot. Moreover, by almost exclusively covering results from visual perception but not motor movement, somatosensory, and auditory perception, etc., the model is admittedly biased toward vision. If one wanted to refer to Hartle’s IGUS robot as a visual robot, no good argument could be given other than the fact that special relativity and Minkowski’s flat space-time, of necessity, require photons of light for the IGUS robot to communicate.

5  Dualistic IGUS Claims and Predictions 5.1  Evolution of IGUSes Resolving the unique/no unique “present” issue itself is not enough for Hartle (2005). He also suggests that the small variety of IGUS robots are involved in evolution (in a very general sense), with the most successful IGUS being the “model IGUS” representing most humans. A less successful robot is the NS (“no schema”) robot. As noted above, it is a great conversationalist but wastes too much energy. Another is the AB (“always behind”) robot, which is slow to respond to its environment but an evolutionary possibility. Although the model IGUS is best, it has a meager structure of temporal properties. If one is to construct an actual evolutionary story for the model IGUS robot, it would have to have had objective sensors at the outset in order to engage with the external world and acquire the necessary information for utilization so that it could direct all behavior. Those sensors of objective parameters would include timing, movement, and a detector for temporal order — all the parameters that the acceptable space-time cosmologies expect. For example, a means of detecting duration judgment would be necessary, something that is functionally similar to the hypothetical internal clock. Moreover, movement detectors, functionally similar to Reichardt detectors for visual motion (movement), would be necessary even if more primitive, as is the case for animals in a lower part of the phylogenetic tree. In short, the IGUS would be expected to acquire the ability to evoke all of the major temporal experiences in the veridical system described above and a few more that Callender (2017, p. 261) suggests. The benefits of adding components (“gadgets”) to provide illusory aspects of manifest time are substantial. For example, whereas Rovelli’s “Order of Time” (2018) expects a monitor for order (physical order), the early model IGUS would enhance this by begetting the illusory aspect of temporality (“pure succession”) as a part of the evolutionary changes to 47

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provide continuity. In the case of movement, the gadget of motion also provides continuity, and with it, the experience of “sameness” or object persistence. More importantly, the evolution of persistence (the enduring self), the deepest illusion, provides an indispensable foundation for human personality. Without an enduring self, one cannot engage in mental time travel. It is the capacity to mentally reconstruct personal events from the past (using episodic memory) as well as to imagine possible scenarios in the future (episodic foresight / episodic future thinking). For reviews, see Suddendorf and Corballis (2007), Corballis (2013), and Klein and Nichols (2012). Most animals lack that ability and are unaable to foresee and plan the future. Most, but not all, do not exhibit self-awareness of an enduring self. Although a crude test, many fail the mirror test, in which a spot is placed on the back of their head, and they are allowed to look at a mirror to see if they exhibit self-awareness behavior (Hongshang 2005). These animals assume that the one in the mirror is conspecific (of the same species but not the “sam”), causing them sometimes to react violently. Admittedly, some of these self-recognition tests may be difficult for humans to interpret in taxonomically divergent animals, especially those that lack the dexterity (or limbs) required to touch a mark. 5.2  Consequences of a loss or dysfunction of the illusory system In addition to the positive contributions by an illusory system, there is a potential negative effect due to its absence or dysfunction. This should be expected, because it is seen following the loss of other illusory experiences that the human brain evokes for non-temporal experiences. For example, the patient with amusia from a stroke, particularly if she is a musician, can be devastating (Sacks, 2008), in contrast to the individual who has congenital amusia and never experienced the aesthetics of music to begin with. Consider now the possible harm from a reversion from the “enduring self” to the “ephemeral self” (the impermanence of physics), as occurs in schizophrenia. In general, dysfunction of manifest time is a particular problem for some of these patients (Mishara et al., 2014). Those disturbances of the self are related to alterations in time processing, which include temporal order and temporal continuity (Giersch et al. 2013, 2016). The “self” is normally experienced as being continuous in time. If not, it becomes a “self-disorder.” Martin et al. (2014) proposed that disorganization over time might impact patients’ ability to experience themselves as a continuous self, i.e., the patient may not feel that she is the same person at all times. For more, see 48

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Montemayor and Wittmann (2014). The following excerpts are from an interview with a schizophrenic patient with a self-disorder (Fuchs, 2013, p. 84). Time is also running strangely. It falls apart and no longer progresses. There arise only innumerable separate now, now, now — quite crazy and without rules or order. It is the same with myself. From moment to moment, various “selves” arise and disappear entirely at random. There is no connection between my present ego and the one before.

A few humans with brain damage (particularly to the hippocampus) have been described as no longer being able to time-travel. As a result of not traveling into the future, the person is devoid of aspirations and wishes (Klein and Loftus, 2002; Andelman et al., 2009). In one of their patient’s words: “I take each day one at a time . . . I don’t see past today and tomorrow . . . and can’t picture myself in anything beyond the immediate present.” Patients with this type of hippocampal damage cannot imagine anything they are likely to do on a subsequent occasion. They seem to be living in a “permanent present” (Tulving, 1985). 5.3  Testing the dualistic mind view A theory should not only explain a major phenomenon but also make predictions. Just providing a mental model and expecting it to work would be insufficient. The dualistic view begins by invoking a number of hypotheses that were subsequently tested (Gruber et al., 2020a). Most important is the hypothesis that a complete and independent veridical system contains all the important temporal experiences to sustain the human for adaptation. That includes veridical change, as turned out to be the case with “completed change” and veridical movement, which was discovered in the form of “completed movement.” The Hartle hypothesis that a past/present/future notion is, in and of itself, consistent with Minkowski space-time physics was verified. Regarding predictions, more evidence is expected, indicating that there is a complete and independent physical system for temporal experiences independent of the illusory system. For example, although it is clear that completed movement exists, it would be helpful to demonstrate that under some as yet to be described circumstances a human can function with all visual perception exhibiting only completed movement without dynamic movement (motion). 49

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The dualistic view predicts the existence of a discrete (snapshot) perception in the absence of the specious present. A possible experiment to consider is as follows. When a series of visual images, e.g., a walking scene, with a large interstimulus stimulus interval (ISI) e.g., much greater than 3 s, is presented, a snapshot-like experience is expected, devoid of not only motion and “happening (which has been demonstrated by Gruber and Block (2017)) but also the specious present. Perhaps the most important prediction is that temporal order in the absence of the experience of succession (“pure succession”) can be sustained and is sufficient for adaptation. Currently, the thinking from the clock thought experiment is that somewhere between seconds and an hour of constant observation, “pure succession” drops out and temporal order remains. An experiment similar to others that tests for the upper and lower limits of temporal order could be conducted. By varying the interstimulus interval and/or other parameters, a point is expected to be reached when only temporal order of temporality exists. Lastly, one other prediction relates to the pilot experiment involving VR to demonstrate that the experience of persistence is illusory. It requires full experimental verification.

6 Conclusions Reconciling the time of physics with the time of human experience, known as the “two times problem,” may be possible after all. The IGUS model of Hartle suggests that the experiential past, present, and future are not properties of four-dimensional spacetime, but notions describing how individual IGUSes, including humans, process information. That IGUS concept is used to explain the temporal experiences of impermanence and (illusory) persistence. That, in turn, explains the origination of the “self.” The IGUS model is then upgraded to explain other temporal experiences of manifest time which include change, motion and temporality. Schematically, “gadgets” representing components of manifest time are added to the IGUS; each of these “gadgets” is dualistic. For every veridical experience of time there is a corresponding illusory (“outside of physics”) experience that the brain possesses. Moreover, all the components of veridical time are possessed by the brain with corresponding non-veridical aspects. For example, the veridical (completed) movement experience is accompanied by an illusory experience of motion (dynamic movement) which contains a superimposed “flow” experience of time. 50

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The Dualistic Mind view also claims that because of natural selection the veridical system begot the illusory system for a much better temporal system overall in order for the human to be more functional. Thus, Feynman is correct in that physics successfully crossed the bridge into the cranium. Then the brain embellished it with an illusory system for better adaptation. Partial loss or dysfunction of either system is problematic for adaptation if not survival. The dualistic view as a proposed theory then makes a number of predictions for future investigation. With the dualistic view in mind, the compulsion to minimize the illusory nature of some temporal experiences should be reduced. There should be no need to reify the “flow” or “whoosh” experience of time. They are wonderful experiences but subjective, nonetheless. A better view is argued here that there is one time from physics; its parameters reside within the brain; and are all there with their “illusory” counterparts.

Acknowledgments We would like to thank James Hartle, Julian Barbour, James Kalamas, and Ryan P. Smith for helpful discussions of this complex topic. Also, many thanks to David Karp and Stephen Waddell for their technical expertise, which made the VR experiment possible.

References Aerts, D. (2014). Quantum theory and human perception of the macro-world. Frontiers in Psychology 5:554. Aerts, D. (2018). Relativity theory refounded. Foundations of Science 23:511–547. Andelman, F., Hoofien, D., Goldberg, I., Aizenstein, O. and Neufeld, M. Y. (2009). Bilateral hippocampal lesion and a selective impairment of the ability for mental time travel. Neurocase 16:426–435. Arstila, V. (2018). Temporal experiences without the specious present. Australasian Journal of Philosophy 96:287–302. Arstila, V. (2016). The time of experience and the experience of time, in Philosophy and Psychology of Time, Springer, Cham, pp. 163–186. Balashov, Y. (2007). About stage universalism. The Philosophical Quarterly 57:21–39. Barbour, J. (1994). The emergence of time and its arrow from timelessness in Physical Origins of Time Asymmetry, eds. J. Halliwell et al., Cambridge University Press, Cambridge.

51

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Block, R. A. (2003). Psychological timing without a timer: the roles of attention and memory, in Time and Mind II: Information Processing Perspectives, ed. H. Helfrich, Hogrefe and Huber, Cambridge, pp. 41–59. Block, R. A., Grondin, S. and Zakay, D. (2018). Prospective and retrospective timing processes: theories, methods, and findings, in Timing and Time Perception: Procedures, Measures, and Applications, eds. A. Vatakis, F. Balcı, M. D. Luca, and A. Correa, Brill, Boston, pp. 32–51. Block, R. A., Hancock, P. and Zakay, D. (2010). How cognitive load affects duration judgments: a meta-analytic review. Acta Psychologica 134:330–343. Block, R. A. and Patterson, R. (1994). Simultaneity, successiveness, and temporalorder judgments, in Encyclopedia of Time, ed. S. L. Macey, Garland, New York, pp. 555–557. Blackmore, S. (2012). She won’t be me. Journal of Consciousness Studies 19:16–19. Block, R. A. (1994). James, William (as psychologist) (1842–1910), in Encyclopedia of Time, ed. S. L. Macey, Garland, New York, pp. 320–321. Block, R. A. and Zakay, D. (1997). Prospective and retrospective duration judgments: a meta-analytic review. Psychonomic Bulletin Review 4:184–197. Broad, C. D. (1923). Scientific Thought, Routledge and Kegan Paul, London. Buonomano, D. V. (2017). Your Brain is a Time Machine, W. W. Norton, London. Buonomano, D. and Rovelli, C. (2021). Bridging the neuroscience and physics of time. arXiv preprint arXiv:2110.01976. Callender, C. (2017). What Makes Time Special? Oxford University Press, Oxford. Capek, M. (1991). The New Aspects of Time: Its Continuity and Novelties. Boston Studies in the Philsophy of Science, Kluwek Academic Publishers, Dordrecht. Corballis, M. C. (2013). Mental time travel: a case for evolutionary continuity. Trends in Cognitive Science 17:5–6. Datteri, E. (2021). Robots and bionic systems as experimental platforms for the study of animal and human behaviour, in Italian Philosophy of Technology, Springer, Cham, pp. 181–197. Davies, P. (2002). That mysterious flow. Scientific American 287(3), 40–47. Dorato, M. and Wittmann, M. (2019). The phenomenology and cognitive neuroscience of experienced temporality. Phenomenology and the Cognitive Sciences, 1326:18–25. Dowker, F. (2014). The birth of spacetime atoms as the passage of time. Annals of the New York Academy of Sciences 1326:18–25. Droit-Volet, S. and Wearden, J. H. (2016). Passage of time judgments are not duration judgments: evidence from a study using experience sampling methodology. Frontiers in Psychology 7:176. Elitzur, A. C. (1996). Time and Consciousness: The Uneasy Bearing of Relativity Theory on the Mind-Body Problem, The Weizmann Institute of Science, pp. 543–550. Ellis, G. (2014). The evolving block universe and the meshing together of times. Annals of the New York Academy of Sciences 1326:26–41. 52

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Fuchs, T. (2013). Temporality and psychopathology. Phenomenology and the Cognitive Sciences 12:75–104. Giersch, A., Lalanne, L. van Assche, M. and Elliott, M. A. (2013). On disturbed time continuity in schizophrenia: an elementary impairment in visual perception? Frontiers in Psychology 4:1–10. Giersch, A., Laianne, L. and Isope, P. (2016). Implicit timing as the missing link between neurobiological and self disorders in Schizophrenia? Frontiers in Human Neuroscience, 10:1–13. Gleick, J. (2011). Genius: The Life and Science of Richard Feynman, Open Road Media, New York. Gruber, R. P. and Brahm, D. (1993). Clock asynchrony and mass variation. General Relativity and Gravitation 25:361–364. Gruber, R. P. (2008). Neurophysics of the flow of time. Journal of Mind & Behavior 29:239–254. Gruber, R. P. and Block, R. A. (2013). The flow of time as a perceptual illusion. Journal of Mind and Behavior 34:91–100. Gruber, R. P., Bach, M. and Block. R. A. (2015). Perceiving two levels of the flow of time. Journal of Consciousness Studies 22:7–22. Gruber, R. P. and Block, R. A. (2017). Dynamic perceptual completion of scenes (indifferent sensory modalities) as a result of modal completion. American Journal of Psychology 130:23–34. Gruber, R. P., Smith, R. and Block, R. A. (2018). The illusory flow and passage of time within consciousness: a multidisciplinary analysis. Timing and Time Perception 6:125–153. Gruber, R. P. and Smith, R. (2019). An experimental information gathering and utilization systems (IGUS) robot to demonstrate the physics of now. American Journal of Physics 87:301–309. Gruber, R. P., Smith, R. and Block, R. A. (2020a). Dynamic perceptual completion and the dynamic snapshot view to help solve the ‘two times’ problem. Phenomenology and the Cognitive Sciences 19(4):773–790. Gruber, R. P., Montemayor, C. and Block, R. A. (2020b). From physical time to a dualistic model of human time. Foundations of Science 25:927–954. Hartle, J. B. (2005). The physics of now. American Journal of Physics 73:101–109. Hartle, J. B. (2014). Classical and quantum framing of the now. Physics Today 67:8. Haslanger, S. and Kurtz. R. (2006). Persistence, MIT Press, Cambridge. Hertzel, M. (2016). Making sense of temporal passage. Intuitions 13:35–48. Hoerl, C. and McCormack, T. (2019). Thinking in and about time: a dual systems perspective on temporal cognition. Behavioral and Brain Sciences 42:e244. Hoerl, C. (2013). A succession of feelings, in and of itself, is not a feeling of succession. Mind 122(486):373–417. Hoffman, D. (2012). The construction of visual reality, in Hallucinations, eds. Blom, J. and Sommer, I., Springer, New York, NY. 53

Ti m e a n d S c i e n c e

Holcombe, A. (2014). Are there cracks in the façade of continuous visual experience? in Subjective Time: The Philosophy, Psychology and the Neuroscience of Temporality, eds. Arstila, V. and Lloyd, D., Cambridge, Boston Review, pp. 179–198. Hollingworth, A. (2008). Visual memory for natural scenes, in Visual Memory, ed. Luck, S. and Hollingworth, A., Oxford University Press, Oxford, pp. 137–162. Hongshang, Y. (2005). On the uniqueness of self-face recognition. Psychological Science 28:1517–1519. Hood, B. (2012). The Self-Illusion, Oxford University Press, New York. Huggett, N. (2018). Reading the past in the present, in Time’s Arrow and the Probability Structure of the World, eds. Loewer, B., Weslake, B. and Winsberg, E., Harvard University Press, Cambridge. Ismael, J. (2007). The Situated Self, Oxford University Press, New York. Ismael, J. (2015). On whether the atemporal conception of the world is also a modal. Analytic Philosophy 56:142–157. Ismael, J. (2017). Passage, flow, and the logic of temporal perspectives, in Time of Nature and the Nature of Time. Boston Studies in the Philosophy and History of Science, Vol. 326, eds. Bouton, C. and Huneman, P., Springer, Cham, pp. 23–38. Ismael, J. (2011). Temporal experience, in The Oxford Handbook of Philosophy of Time, ed. Callender, C., Oxford University, Oxford, pp. 460–482. James, W. (1890). The Principles of Psychology, Dover, New York. Klein, S. B. (2004). The cognitive neuroscience of knowing one’s self, in The Cognitive Neurosciences, ed. Gazzaniga, M. S., Boston Review, pp. 1077–1089. Klein, S. B. and Lofthus, J. (2002). Memory and temporal experience: the effects of episodic memory loss on an amnesic patient’s ability to remember the past and imagine he future. Social Cognition 20:353–379. Klein, S. and Nichols, S. (2012). Memory and the sense of personal identity. Mind 121:677–702. Kline, K. A. and Eagleman, D. M. (2008). Evidence against the snapshot hypothesis of illusory motion reversal. Journal of Vision 8:1–5. Koch, C. (2004). The Quest for Consciousness, Roberts, Englewood. Lestienne, R. (1988). From physical to biological time. Mechanisms of Ageing and Development 43(3):189–228. Marchetti, G. (2014). Attention and working memory: two basic mechanisms for constructing temporal experiences. Frontiers in Psychology 5:880. Martin, B., Giersch, A., Huron, C. and Wassenhove, V. (2013). Temporal event structure and timing in schizophrenia: preserved binding in a loner ‘now.’ Neuropsychologia 51:358–371. Maudlin, T. (2002). Remarks on the passing of time. Proceedings of the Aristotelian Society 102:237–252. McTaggart, J. (1908). The unreality of time. Mind 17:457–474.

54

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Mishara, A., Lysaker, P. and Schwartz. M. (2014). Self-disturbances in schizophrenia: history, phenomenology, and relevant findings from research on metacognition. Schizophrenia Bulletin 40:5–12. Montemayor, C. (2009). The Psychology of Time and Its Philosophical Implications, Rutgers The State University of New Jersey-New Brunswick. Montemayor, C. (2012). Minding Time: A Philosophical and Theoretical Approach to the Psychology of Time, Vol. 5, Brill. Montemayor, C. (2017a). Time perception and agency: a dual model , in The Routledge Handbook of Philosophy of Temporal Experience, ed. Phillips, I., Routledge, New York, pp. 201–212. Montemayor, C. (2017b). Conscious awareness and time perception. PsyCH Journal 6(3):228–238. Montemayor, C. (2019). Early and late time perception: on the narrow scope of the Whorfian hypothesis. Review of Philosophy and Psychology 10(1):133–154. Montemayor, C. (2013). Minding Time: A Philosophical and Theoretical Approach to the Psychology of Time, Leiden, Brill. Montemayor, C. and Wittmann, M. (2014). The varieties of presence: hierarchical levels of temporal integration. Timing & Time Perception 2:325–338. Mroczko-Wasowicz, A. (2016). Perception-cognition interface and cross-modal experiences: Insights into unified consciousness. Frontiers in Psychol 7:1593. Nakashima R. and Yokosawa. (2012). Sustained attention can create an (illusory) experience of seeing dynamic change. Visual Cognition 20:265–283. Paul, L. (2010). Temporal experience. Journal of Philosophy 107:333–359. Paul, L. A. (2016). The subjectively enduring self, in Routledge Handbook of the Philosophy of Temporal Experience, ed. Phillips, I., Routledge, London. Pöppel, E. (1997). The brain’s way to create ‘nowness’, in Time, Temporality, Now, eds. Atmanspacher, H. and Ruhnau, E., Springer Verlag, Berlin, pp. 107–120. Pöppel, E. and Bao. Y. (2014). Temporal windows as bridge from objective time to subjective time, in Subective Time. The Philosophy, Psychology, and Neuroscience of Temporality, eds. Lloyd, D. and Arstila, V., MIT Press, Cambridge, pp. 241–261. Price, H. (2011). The flow of time, in The Oxford Handbook of the Philosophy of Time, ed. Callender, C., Oxford University Press, Oxford, pp. 276–311. Rensink, R. (2002). Change detection. Annual. Review of Psychology 53:245–277. Rickles, D. and Kon, M. (2014). Interdisciplinary perspectives on the flow of time. Annals of the New York Academy of Sciences 1326(1):1–8. Rizzo, M., Narrow, M. and Zihl, J. (1995). Motion and shape perception in cerebral akinetopsia. Brain 118:1105–1127. Romero, G. (2015). Present time. Foundations of Science 20:135–145. Rovelli, C. (2011). Forget time. Foundations of Physics 41:1475. Rovelli, C. (2018). The Order of Time, Riverhead Books, New York.

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Ruhnau, E. (1997). The deconstruction of time and the emergence of temporality, in Time, Temporality, Now, eds. Atmanspacher, H. and Ruhnau, E., SpringerVerlag, Berlin, pp. 53–70. Sacks, O. (2008). Musicophilia, Random House, New York. Sacks, O. (1999). Migraine: Revised and Expanded, Random House Inc., New York. Savitt, S. (2002). On absolute becoming and the myth of passage. Royal Institute of Philosophy Supplement 50:153–167. Scholl, B. (2007). Object persistence in philosophy and psychology. Mind and Language 22:563-591. Shepard, R. (2004). How a cognitive scientist came to seek universal laws. Psychonomic Bulletin & Review 11:1–23. Shepard, R. N. (1994). Perceptual-cognitive universals as reflections of the world, Psychonomic Bulletin & Review 1:2–28. Smolin, L. (2013). Time Reborn, Mifflin & Harcourt, Boston; Houglton. Smolin, L. (2015). Temporal naturalism. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 52:86–102. Steinman, R., Pizlo, Z. and Pizlo, F. (2000). Phi is not beta. Vision Research 40:2257–2264. Struhl, K. J. (2020). What kind of an illusion is the illusion of self? Comparative Philosophy 11(2):8. Suddendorf, T. and Corballis. M. (2007). The evolution of foresight: what is mental time travel, and is it unique to humans? Behavioral and Brain Sciences 30:299–313. Tulving, E. (1985). Memory and consciousness. Canadian Psychologist 26:1–12. VanRullen, R. and Koch. C. (2003). Is perception discrete or continuous? Trends in Cognitive Sciences 7:207–213. VanRullen, R., Reddy, L. and Koch, C. (2010). A motion involved illusion reveals the discrete nature of visual awareness , in Space and Time in Perception and Action, eds. Nihawan, R. and Khurana, B., Cambridge University Press, New York. Velleman D. 2016. So it goes. The Winnower 16:1–16. Wittmann, M. (2016). Felt Time: The Psychology of How We Perceive Time, MIT Press, Cambridge.

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Chapter 3

N E W (A ND O LD ) WOR K O N THE FUN D AM E NTA LI T Y OF TI M E Dean Rickles* and Jules Rankin** University of Sydney, Australia *[email protected] **[email protected]

We review some recent strands in the philosophy and physics of time. We explain the new orthodoxy denying time any fundamental status, and then turn to several proposals that deny this. There is a standard procedure to square the opposing views by looking at the way agents are embedded in the universe. This has also been questioned by some recent work. We focus on Lee Smolin’s recent work in this vein and on his view that novelty and freedom pose problems for anti-fundamentalist views. We ultimately defend this component.

1 Introduction [T]he growth of our knowledge has led to a slow disintegration of our notion of time. Carlo Rovelli, The Order of Time Time, of course, is real: ask any woman who has just seen the first wrinkle on her face in the mirror. The fault must lie, therefore, with the

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analyses of time; and it is the task of philosophers to cure themselves of their homegrown paradoxes and perplexities. Richard Gale, The Philosophy of Time: A Collection of Essays

As Lee Smolin described in a 2014 interview, his primary field of research, quantum gravity, is one “where it’s assumed that time is unreal, that time is an illusion” (Smolin, 2014). This formed something of a working hypothesis for quantum gravity researchers, and still does to a large extent, as the above quotation from Carlo Rovelli (another quantum gravity researcher) indicates.1 It was Smolin’s parallel work on what he calls “cosmological natural selection” that dislodged this basic hypothesis for him, since it involves the notion (inspired by John Wheeler) that laws of physics must themselves evolve in time (rather than sitting imperiously outside the concrete world, where they direct events from eternity), implying that time is perhaps the most fundamental thing of all.2 However, this work is hardly mainstream, so very few have followed in his footsteps. But there are other recent attempts to work time into physics at a fundamental level: Dolev (2005), Dowker (2020), Ellis (2012), Fuchs (2014), Gisin (2017), Maudlin (2010), Norton (2010), and others. What characterizes many of them is a tendency to promote creation, novelty, and even choice to play a central role in the constitution of the universe. If one wants novelty and freedom, the story goes, then one had better eliminate the block universe, for what place is there for uncertainty and openness in a closed block containing all happenings? Other work, e.g., of James Hartle (2004) and Jennan Ismael (2017), has attempted to offer a kind of reconciliation of the two opposing viewpoints, showing them to be simply different, dual ways of looking at one and the same thing, much like the proverbial blind men and the elephant. This chapter will offer a review and assessment of this recent work, and connect it with some older work from the turn of the last century. Ultimately, though this is not by any means a sustained defense, we come down on the side of those who wish to see time in a more fundamental role for reasons of providing a home for real possibility and novelty, especially given  We explain where this orthodoxy stems from below.  In this, Smolin does not go “full Wheeler,” since the latter believed that there must be an additional substratum (pregeometry) underlying space-time, from which the chronogeometrical structure of the world emerged, so that time is not considered fundamental. 1 2

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quantum mechanical theorems of the Bell–Conway–Kochen–Specker “incompleteness” variety.

2  Doing Away with Time In Mazagon, Spain, at a 1991 NATO Advanced Research Workshop on time asymmetry in physics, Julian Barbour took a straw-poll in which he posed the following question to the participants: Do you believe time is a truly basic concept that must appear in the foundations of any theory of the world, or is it an effective concept that can be derived from more primitive notions in the same way that a notion of temperature can be derived in statistical mechanics? (Barbour, 1991, p. 405)

The idea that time is not fundamental won by majority vote, but, somewhat surprisingly perhaps, certainly not unanimously: of 42 participants, 20 denied a fundamental time, 12 were unsure, and 10 believed time was fundamental. It must be said at the outset that the timeless idea is extremely wellmotivated in quantum gravity, as it is in theories of classical relativity, too. We begin by briefly presenting the key ideas leading to this idea before exploring the recent work that aims to put time back into the center of physics and physical theories. But we must be careful with talk of the reality of time in such contexts. Among the physicists at least, only Julian Barbour truly argues that time itself does not exist in the physical world.3 Most other quantum gravity researchers view time as perfectly real, but emergent in some sense or other. There appear to be two key approaches to time: (1) to look for answers from physics; (2) to look for answers from the study of the mind and brain, and from our experience. Often, the latter investigations motivate a denial of physical time (or at least certain key features, such as flow), while the former investigations motivate a search for mind-based features that can explain what is seen as the “illusion of time.” Yet, it is also clear that our experience of time is a key feature in the physical  And even this must be taken cum grano salis, since he allows that instants of time (“Nows”) exist, only not in a continuum but ordered in a relative configuration space that he names “Platonia.” One could certainly argue that Barbour has simply provided a new account of time rather than eliminating it. We say more about this in Section 3. 3

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models we construct, which are to some extent there to save the phenomena in the sense of our direct experience. Precisely this experiential foundation can be found at the root of conflicts over the reality of time: some take our experience of flow to point to a world that flows, while others doubt the veracity of this experience and try to eradicate it from physics, as one would with any other human projections. Perhaps the most famous argument against time in philosophy is John McTaggart’s finding that the intrinsic A-series (flow determinations: past, present, and future) leads to contradictions and yet also seemingly demanded by the concept of time. The B-series, with its eternal ordering of all events into earlier-than, later-than, and simultaneous-with, is, then, not sufficient to account for time. Recent work, however, has shown how they can in fact be made compatible by viewing something like an A-series as emergent from the interplay of an embedded agent in a B-theoretic block. We return to this idea below. In physics, there appear to be three chief arguments against the reality of becoming, having to do with the nature of laws, the principles of special relativity, and the treatment of time in general relativity, where it appears as an artifact of a coordinate representation: 1. Time-Symmetric Fundamental Laws: The laws of general relativity and quantum theory are time-symmetric: performing the mapping t → –t generates another solution of the equations of motion. This fact is supposed to tell against the reality of flow, since surely flow demands a one-way direction in which such flow occurs. Yet physical laws seem to involve no such direction of time, which appears only as a statistical feature of the world. 2. The Block of Special Relativity: Most philosophers who believe in the unreality of objective becoming are motivated by special relativity. Here, the problem stems from the relativity of simultaneity, which appears to conflict with the necessity (for flow) of a privileged present providing the pulse of the universe.4  Einstein himself is, of course, usually taken to have believed the block picture. But, as Rudolf Carnap noted: 4

Einstein said the problem of the Now worried him seriously. He explained that the experience of the Now means something special for man, something essentially different from the past and the future, but that this important difference does not and cannot occur within physics. That this experience cannot be grasped by science seemed to him a

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3. Time as a Gauge Parameter: Going from the special to the general theory of relativity (GR) introduces what is often referred to as the Problem of Time in modern physics. There are broadly two facets that both arise from general relativity. The first is from classical GR, while the second is a consequence of initial work in canonical quantum gravity — though it also arises in many other approaches to quantum gravity that demand diffeomorphism-invariant observables.   The first is in large part due to how general covariance is implemented in GR. Specifically, the diffiomorphism invariance of the theory results in the fact that the transformation of time evolution becomes a gauge transformation. If we take the orthodox view of gauge transformations, that they represent non-physical transformations (i.e., motion along a gauge orbit), then we arrive at the seemingly paradoxical conclusion that temporal labeling is physically meaningless. This can also be viewed from the perspective of observables, which, in general relativity, must be diffeomorphism-invariant. Clearly, if time is an example of a diffeomorphism, then observables must not change over time, leading to a frozen world.   Secondly, the fact that time does not appear in the fundamental variables of quantum gravity does not thereby imply that time is a subjective notion. Rather, it is something that emerges from the basic degrees of freedom: a relative quantity (for relative quantities are immune to the changes generated by diffeomorphisms). We see from this snapshot that time is in trouble across the major theories of physics. We can find similar trouble lying in the brain sciences. In this case, we find the isolation of mechanisms in the brain responsible for the phenomenological quality of flow,5 as well as evidence for the nonlinear nature of the construction of one’s present temporal snapshot of the matter of painful but inevitable resignation. So he concluded “that there is something essential about the Now which is just outside the realm of science.”   Hence, it is possible that he saw in physics an incompleteness in addition to his more wellknown qualms with quantum mechanics and its apparent denial of the objective world itself. 5  For example, lesions in the area of the visual cortex known as V5 can lead to a condition known as akinetopsia, in which the flowing quality of experience is compromised. See Vaina (1995) for a philosophical presentation.

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world.6 And yet, to paraphrase Galileo, it flows! The world of experience, to some extent, remains a buzzing, blooming confusion. How can it be that the kinds of worldly features that seem so crucial to us as humans (the fixity of the past and the ability to have an effect on the world in the future) are illusory? Perhaps they are not ...

3  Platonia, Superspace, and Timelessness There is no such thing as spacetime, quantum mechanics tells us. Spacetime is a purely classical concept. (Wheeler, 1978, p. 84)

Before considering the recent work that aims to reinsert time in the world, let us first use Julian Barbour’s infamous denial of time to get a better sense of the physics that is supposed to lead us to timeless positions. This will prove useful since it relates to many of these alternative positions, including Smolin’s. We take Julian Barbour’s controversial denial of time to amount to realism about “superspace” (i.e., the configuration space of general relativity, in which no temporal dimension appears [cf. (Barbour, 1991, p. 409)], but there are instants). Hence, this basic ontology is very different from the usual space-time models, which have instants arranged along some preferred direction (or family of directions). The configuration space in this case is rather complicated, being a quotient space of the space of 3-metrics by the diffeomorphism group (i.e., in which points related by diffeomorphisms are identified, thus generating singularities).7 This space of configurations is then supposed to be the basis for the illusion of time. From this space, indeed, everything comes, including time asymmetries. If you want to recover something, then you must find an appropriate universal wave-function on this space in which it

 We have in mind here the recent “illusory rabbit” experiments by Shinsuke Shimojo’s group at UC Berkeley (Stiles et al., 2018), in which the brain inserts “past experience” (a visually experienced flash, between two flashes, that does not exist outside the brain) based on an incoming signal (two flashes but three audible beeps) over a stretch of time. 7  More recently, with the advent of his shape dynamics program, the space is a conformal superspace in which conformal transformations are quotiented out along with diffeomorphisms. 6

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“resides.” It is, ultimately, Everettian, deriving all things from patterns or structure of the universal wave-function.8 To call it “timeless” is a topological statement in this case: it amounts simply to a denial of a fundamental one-dimensional ordering of configurations. Rather, fundamentally, a different patterning orders them, while the familiar one-dimensional ordering is our doing. Indeed, the basic idea of a timeless configuration space for general relativity can be found in Wheeler’s early descriptions of superspace (Wheeler, 1967): Superspace, rather than any one universe, is the arena in which the dynamics proceeds. Normally this circumstance does not come to our attention. We think of spacetime and the time-ordering of events as all quite clean cut and deterministic. However, this option disappears in the final stages of gravitational collapse. There the dimensions of the dense star core, or the dimensions of the recontracting universe as the case may be, shrink to values comparable to 10−33cm. Then the familiar every day large-scale view of what dynamics is all about becomes completely inapplicable. Then we have to abandon the concept of space-time. Then it makes absolutely no sense to ask what happens “next” in collapse process. Still less does it make sense to speak of an “oscillating” universe. “Time” has lost all meaning. Without time, repetition in time is also meaningless. From the point of view of the mathematical formalism the phase of complete gravitational collapse marks region in superspace where the probability amplitude for many alternate histories of the universe are strongly coupled together. In the dynamics of collapse, the alternative histories that come into the analysis are alternative histories of the universe — or, to use sharper words, alternative universes. Should you think of “us” as present in those alternative geometry, tracing out those alternative histories and experiencing alternative sensations?

 Abner Shimony was not impressed with the claims of timelessness here: “He smuggles in time. Everybody who tries to get rid of time and then explains the derivation smuggles it in. He does it by that model of cards, putting data on cards, and then he arranges the cards. Tell me about arrangement without time” (Myrvold and Christian, eds., 2009, p. 457). The best review of Barbour ’s (early) program to eliminate time is by Butterfield (2002). 8

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Even to raise such questions is to open the door to all kinds of mysticism, and misunderstandings. We have to turn away such questions as being altogether lacking in any well-defined meaning. What we can say within the context of the mathematical formalism, I believe, is this: These are regions of superspace where histories of the universe of quite diverse character come into strong coupling, one with another. Gravitational collapse is the physical acts that ties one such geometric dynamic or history to another.

Hence, the idea of a space of 3-geometries (that is, in which each “point” is a space of 3-dimensions with symmetries eliminated: an intrinsic representation of space at an instant) is Wheeler’s.9 Classically, one can envisage general relativity as a dynamical theory of these spaces. Quantum theory, however, has no way of generating determinate histories (or orbits through this space), just as there are no trajectories for particles. In this case, space-time is not a well-defined quantum concept. Rather, we are pushed, in Wheeler’s mind, to “spacetime foam” in which topology and metric fluctuate. The disappearance of time from physics was, in any case, for Wheeler, already a foregone conclusion as soon as gravitational collapse to singularities was properly understood: the big bang, the big crunch, and black holes are “gates of time.” This is essentially the conceptual basis for the well-known Wheeler– DeWitt equation. Superspace provides the domain for a wave equation describing the universe as a whole, and the wave-function was initially seen as a dynamic equation. However, one quickly bumps into the frozen formalism problem, which does not allow a space-time to be constructed from the elements of the domain. Yet Barbour’s space of configurations, just like Wheeler’s superspace, is a block of sorts (cf. Blodwell, 1985), even if it is infinite-dimensional rather than 4-dimensional (see Fig. 3.1). Space-time is not subject to dynamic evolution, and neither is superspace. Yet, the latter is supposed to be the proper representational

 Note that Barbour states that he came upon the superspace idea quite independently through an analysis of Mach’s principle (see his letter to Wheeler, 9 Febuary 1974 [John Archibald Wheeler Papers, 1880–2008: Mss.B.W564]). His approach is more general than Wheeler’s in applying to a broader range of theories, of which general relativity is just one. 9

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Figure 3.1   The structure of superspace and its relation to the space-time block, according to John Wheeler (Wheeler, 1968, p. 264). Here, A–E are 3-geometries of space (a small sample of a space that contains every conceivable shape), in this case undergoing expansion and contraction. Space-time is then the structure whose orthogonal cross-sections (leaves of space at a time) give the A–E, but the resulting space-time is also one of many other possible spacetimes (leaves of history) that can be charted through superspace — it is the job of law (the Wheeler–DeWitt equation) that determines which space-time emerges.

vehicle for thinking about generally relativistic theories. Indeed, Barbour’s treatment of the illusion of time is similar to the “allocentric” versus “egocentric” representations we will see below: there’s no time in the block; rather, it is a feature of conscious beings embedded in the block. To save the apparent phenomenon of change in Barbour’s configuration space, we must extract some way of parameterizing the instants in a way that does not depend on an external time parameter. Wheeler himself adopted the view that observer-participators must function to “select” some spatial instants in order to actualize them (with probabilities given by the Wheeler–DeWitt equation) and generate, as classical approximations, the ordinary concepts of “time,” “space,” and “space-time,” “before,” and “after,” and so on. In this sense, it is clear that Wheeler viewed his superspace in a more modal

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sense, as a space of possible 3-spaces: the Wheeler–DeWitt equation then assigns probabilities to entire histories.10 This brief review gives a general flavor of the general relativityinspired anti-fundamentalists. Others follow a similar line. For example, Rovelli’s thermal time hypothesis is an attempt to build a linear ordering by defining a thermal clock to be precisely one whose pointer reading moves linearly with respect to Hamiltonian flow. This, as with Wheeler’s, is a statistical concept, so Rovelli too denies a fundamental status to time, though he provides a proposal for its emergence: time is that variable that in some sense simplifies the description of the macro-system. His relational approach also has a similar flavor, in which time is extracted from the physical degrees of freedom of the theory of loop quantum gravity. Again, the source of this viewpoint lies in the interpretation of space and time coordinates as gauge degrees of freedom (Rovelli calls them “partial observables”) that need to be fixed by physical degrees of freedom, in this case relative quantities (“complete observables”) involving relations between partial observables (e.g., one can define an instant of time using a coincidence of the values of a pair of physical fields that while individually are not gauge-invariant, but when related like this are gauge-invariant). Hence, regardless of the radical nature of the denial of time, a necessary feature of all such proposals is the recovery of our experience of time; the phenomena must be saved, regardless of whether the noumena (i.e., the world outside the subjective perspective) match up.

4  Neo-Heracliteanism: Time Reinstated Time would continue to pass for the smoldering ruins were we and all sentient beings in the universe suddenly to be snuffed out. John Norton

According to Norton (2010), the only reason to believe that time’s passing is an illusion (that the noumena and phenomena diverge, as it were) is that it does not appear in our physical theories (for the kinds of the reason  In this context, Wheeler quotes approvingly from William James: “Actualities seem to float in a wider of possibilities from out of which they were chosen; and somewhere, indeterminism says, such possibilities exist, and form part of the truth” (Wheeler, 2014, p. 273). 10

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sketched above). It is, then, he thinks, sheer “vanity” to make the leap from this to the denial of what appears to Norton an inescapable and obvious fact about reality: time exists. Of course, Norton is right that if we dismiss time, then we are spared the task of explaining how it could exist in the world at a fundamental level. It simply becomes a matter for neuroscientists or becomes part of some emergentist account in which a simple mapping between mind and world can be found. Specifically, it is a matter of locating which aspects of time can be found in fundamental physics and which can be recovered from our particular psychological embedding (which might simply have to do with the energy scale we occupy). The more radical claim that some make is that since we cannot identify features of our psychological experience with features of fundamental physics, we ought to introduce those features directly into formalism. This is a very nontrivial leap to make and has been rightly met with skepticism. Let us briefly consider a pair of these approaches. Maudlin’s Temporalizing Space: Instead of representing time topologically (composed of an open set), Maudlin (2010) proposes a Directed Linear Structure as the basis for constructing a space-time geometry. Maudlin’s view is less radical than those found below since it is consistent with the orthodox Block universe picture. It is motivated by questioning whether topological geometry might be the best tool for describing more fundamental quantum space-time. His goal is to introduce a fundamental directedness into our formal representation of time, which is seemingly not present in the usual picture. However, as has been previously noted by Price (2011), temporal directedness is not sufficient to fully introduce notions of flow or passage that the authors below advocate. Dowker’s Space-Time Atoms: Causal set theory is seen as a pathway to introducing passage and becoming. Dowker explores a specific model of causal sets referred to as classical sequential growth (CSG), which is a stochastic model of granular space-time. One of the primary benefits of the CSQ model is that the “birth” of the individual nodes of the causal set provides an “objective correlate of our subjective perception of ‘time passing’” (Sorkin, 2007). However, this is rather hard to swallow both on account of this being a Planck scale phenomenon, and because the apparent passage of time in our subjective perception seems not to be of the same kind as this “birthing process,” but rather is linked to more general 67

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changes in the environment. Making the link between theory and experience here certainly requires a more sturdy bridge.11 Hence, we are not convinced by either approach. Nor are we so convinced by the several approaches based on models of quantum collapse. The idea with these is to reestablish a direct mind-world link (similar to the causal-set proposal) by isolating real transitions (the wave-function collapses) in the physics that will reflect our experience as of flow. The problem with these is that the mechanism for collapse is not itself on stable ground, and there are issues with making such collapse consistent with relativistic principles — however, we find such collapse proposals rather plausible, as we explain in the final section. Lee Smolin has developed a very different kind of neo-Heraclitean account, which we turn to now. However, like Maudlin and Dowker, this account, at least one component of it, also focuses on the nature of the mathematical structures employed in physics, tracing the timelessness in physics also to the timelessness in the abstract representational structures. There are, in fact, several threads making up Smolin’s project of “Temporal Relationalism/ Temporal Naturalism” that will require some untangling — only one of these threads we will take up in the final section. Curiously, Lee Smolin’s temporal realism invokes Julian Barbour’s shape dynamics (SD), which contains a candidate for a privileged present that is consistent with the results of classical general relativity. This is the case because shape dynamics was shown to be equivalent and, in fact, a dual representation of classical general relativity (Gomes and Koslowski (2012); Gomes et al. (2011)). What shape dynamics tells us, as Gryb and Thébault (2016) put it, is that our classical description of gravity is Janus-faced; there is an underdetermination of the symmetries of classical gravity, such that it admits these different descriptions with different accompanying temporal ontologies. One with a privileged, global present (and a distinguished slicing of space-time into 3-spaces), and one without (with many-fingered time resulting from foliation invariance). How does shape dynamics provide a privileged present? Starting from classical general relativity, we can employ a procedure of symmetry  An analysis by Hudetz (2015) examining the application of Maudlin’s framework to casual sets suggests that it is just as fruitful as standard topology in describing the theory of causal sets. However, it may not be necessary, as “the theory of linear structures is not more expressive than standard topology.” 11

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trading such that we trade the foliation invariance of Einsteinian general relativity for the invariance under spatial conformal transformations. Physically, this means that the relativity of simultaneity is traded for the relativity of size, resulting in an equivalent but ontologically very distinct theory. The issue Smolin faces, as we will see below, is that shape dynamics does not meet all of his ambitions for a new cosmological framework. One of those issues is that since Smolin aims to rid physics of any timeless, Platonic mathematical representations, which immediately poses a problem when adopting shape dynamics as a candidate for capturing the privileged present. Namely, it is precisely because of the underdetermination of formalism that makes either representation as good as the other. We have no positive reason, all things being equal, to choose one representation over the other. Smolin’s move here is to invoke further motivations to justify such a choice. The theme of dual representation is one that has a rich history in physics and is one we wish to build on with our approach to understanding the complete nature of time. As previously stated, the purported tension between competing images of time arises from ultimately compatible dual representations. In Smolin’s overarching project of temporal naturalism, there are three main threads: (1) opposing the timeless role that our mathematical representation of fundamental laws take, (2) how this comes to produce genuine novelty, and (3) the role and explanation of conscious experience/ qualia. Each of these threads, Smolin admits, is speculative in their implementation and is more on par with guiding principles than fleshed out theories. Let’s start with (1). The lack of any referent of the flow or passage of time in fundamental physics is taken to be an inadequacy in formalism. A radical approach is to abandon the Platonic representation of configuration space in favor of something new. Smolin describes the problem as follows: The argument that time is not a fundamental aspect of the world goes like this. In classical mechanics one begins with a space of configurations of a system S. Usually the system S is assumed to be a subsystem of the universe. In this case there is a clock outside the system, which is carried by some inertial observer. This clock is used to label the trajectory of the system in the configuration space C. The classical trajectories are then extrema of some action principle, δ I = 0. 69

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Were it not for the external clock, one could already say that time has disappeared, as each trajectory exists all at once as a curve γ on C. Once the trajectory is chosen, the whole history of the system is determined. In this sense there is nothing in the description that corresponds to what we are used to thinking of as a flow or progression of time. Indeed, just as the whole of any one trajectory exists when any point and velocity are specified, the whole set of trajectories may be said to exist as well, as a timeless set of possibilities. Time is in fact represented in the description, but it is not in any sense a time that is associated with the system itself. Instead, the t in ordinary classical mechanics refers to a clock carried by an inertial observer, which is not part of the dynamical system being modeled. This external clock is represented in the configuration space description as a special parameterization of each trajectory, according to which the equations of motion are satisfied. Thus, it may be said that there is no sense in which time as something physical is represented in classical mechanics, instead the problem is postponed, as what is represented is time as marked by a clock that exists outside of the physical system which is modeled by the trajectories in the configuration space C. Kauffman and Smolin, 1997, p. 118

Hence, Smolin views his project not simply as putting time back into physics; he also wants a new view of mathematics that is also naturalist. The two are thus deeply entangled, with mathematical naturalism and temporal naturalism understood as codependents. One of the ways in which Smolin attempts to abandon the use of timeless mathematics is his suggestion that cosmological dynamical laws themselves are not immune to the flow of time. For this to be the case, the laws themselves must evolve somehow. In other words, Smolin’s goal is to escape the aforementioned configuration space realism. This is motivated by the Principle of Sufficient Reason targeted toward the explanation of fundamental laws (as Smolin puts it, our cosmological theories should give an answer to the question “why these laws?”). Several well-known cyclic cosmological models provide candidates for this kind of evolution. As we will see below, an analogy with biological evolution is drawn to provide an explanatory framework for understanding how this may reproduce the physics we observe. An immediate objection one has toward this approach of removing the timeless cosmological laws by allowing the laws themselves to evolve 70

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is to ask, “By what law do the cosmological laws evolve?” As previously mentioned, there are cosmological models that allow for the evolution of the constants/laws. However, the form those models take is precisely the form that Smolin wants to avoid, timeless representations. This would seem to reintroduce timeless, higher-order laws that undermine the approach of evolving laws. Smolin does not provide a definitive answer to the problem of how higher-order cosmological laws may escape the problems of timelessness. However, he gestures at the idea of interpreting the metaphysical character of mathematics, not as Platonic, timeless objects, but rather as objects that are evoked in some moment of time. This may initially sound like an epistemological claim about our access to the mathematical nature of the world, but Smolin wants to make this into a metaphysical one. He says that when we evoke mathematical structures, we bring something into existence that wasn’t there before. Not only that, but “Nature has this capacity as well and uses it on a range of scales from the emergence of novel phenomena which are described by novel laws to the emergence of novel biological species which play novel games to dominate novel niches.” However, it is not clear what is meant by “Nature” in this case. The goal of this approach, as with Smolin’s entire project, is to center the fundamentality of the flow of time on how we understand the world. From this interpretive foundation, how does Smolin proceed? Smolin first claims that our aforementioned standard way of doing cosmology falls into a glaring fallacy. This standard way of doing physics, which he refers to as the “Newtonian paradigm,” is summarized as doing “physics in a box”.12 Smolin’s intuition here is that since our standard formalism for describing physical systems implicitly assumes an external environment/clock/observer, and the whole universe is precisely that on which we cannot have an external perspective, then it follows our description of it using the Newtonian paradigm is fallacious. Hence, the interpretation of the Newtonian way of doing physics (namely, trajectories in configuration spaces) is taken to be merely summarizing records of past observations of subsystems of the universe. They are not to be in “isomorphism [with] the natural world”. Smolin suggests that, when doing this new cosmology, the usual approach forces us into a timeless conception of the laws. His proposed solution is that the distinction between physical states and governing  Counterintuitively “Newtonian” is this case not meant to refer to classical physics, since Smolin wants to include standard quantum mechanics under this label. 12

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laws breaks down such that the distinction becomes only an approximation when describing subsystems. He provides a toy matrix model as a proof of concept for such an approach (Smolin, 2015b). However, in the very same article, he says that to realize this model one must “embed the configuration space of states, C and the landscape parameterizing laws, L into a single meta-configuration space, M.” Which brings us back to a timeless configuration space, seemingly defeating the purpose of the model in the first place. Expanding on this idea, Smolin suggests that the higher-order “metalaw” may be part of a large equivalence class, each of which performs the function of a metalaw out of a kind of necessity. He conjectures that “any metalaw that could serve as such is equivalent to any other.” (Smolin, 2015b) This, along with some natural assumptions, Smolin conjectures, is what would determine the evolution on M in the previous paragraph. This is intended to be analogous to biological evolution, where the fitness function that defines evolution is itself subject to evolution. The next primary motivation for Smolin’s project is to introduce genuine novelty into fundamental physics. This is seen to be the means by which we can escape the Platonic representation of mathematics and recover notions of flow and genuine temporal becoming. This motivation is more aligned with the results of quantum mechanics than general relativity and informs the larger project of formulating quantum gravity. Smolin proposes an extension to a real ensemble interpretation of quantum mechanics (Smolin, 2012a), which he refers to as the “Principle of Precedence.” The ensemble interpretation ascribes a certain frequentist flavor to the probabilities of quantum measurements. Namely, the outcomes of quantum measurements are sampled from the ensemble of all previously identically prepared states. The Principle of Precedence goes one step further and claims that if a quantum state has no precedent in the history of our universe, its outcome is not determined by any prior law (Smolin and Unger, 2015a; Smolin, 2012b). Intermediate states, with only a few precedents, have their outcomes “maximally free” in the sense defined by Conway and Kochen (2009) such that the outcomes of experiments can’t be predicted by knowledge of the past. This turns out to be a highly nonlocal approach, since the state needs to “know” if there has been a precedent state, requiring some access to all prior states of the universe at once. This leaves room for genuine novelty of outcomes to arise for states without precedent while respecting existing deterministic laws for states with established precedent. This interpretation invites a kind of 72

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possibilism, or “growing block” picture, as opposed to a standard presentist picture, due to the asymmetry in Smolin’s characterization of the past and the future in a way that is dissimilar to a standard presentist. The last key motivation for Smolin’s project is dissatisfaction with the fact that an explanatory story for consciousness is seemingly missing from the orthodox picture of the world provided by physics. Notably, he raises objections towards the block universe picture provided by relativity. Specifically, his characterization of the physicist’s conception of the block, as he puts it, “assumes that the universe’s history is the result of deterministic and immutable laws acting on initial conditions, and this is what I will refer to when I speak of the block universe” (Smolin, 2015c). While Smolin is more ambivalent towards what he calls the “philosopher’s block,” which is characterized as the set of events that are the case if we let the entire history of the universe play out. He sees this picture as at least compatible with his project, if irrelevant. His main objection stems from his claim that consciousness/qualia “allow the presently present moment to be distinguished intrinsically without regard to relational addressing.” From this starting point, he presents an interpretation of quantum mechanics to address this problem. Trying to provide a solution to the hard problem of consciousness and qualia that takes advantage of quantum mechanics is a well-worn strategy. The intuition is that quantum mechanics is mysterious, consciousness is mysterious: maybe we can kill two birds with one stone. Smolin has an ambitious proposal that is tied to the Principle of Precedence introduced above. Smolin suggests that it may be precisely those states without precedence that produce, or are correlated with, qualia. It is plausible that each physical state a given brain finds itself in has not been reproduced at any previous instant in our universe and hence is without precedent. This approach would be most closely mapped onto a panpsychist approach to consciousness, and would only ascribe qualia to sufficiently complex states. However, there are several places where Smolin’s panpsychism is idiosyncratic. The first is how he describes qualia as “intrinsic” in a way very different from how other panpsychist philosophers would use the term. “Intrinsic” in the philosophical sense is used to describe aspects of the world that are not captured by the dynamical description,13 but Smolin lumps energy and momentum together as “intrinsic” to matter. 13

 Which is precisely why panpsychism is attractive to the naturalistically inclined.

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What Smolin has in mind when talking of intrinsic properties is in contrast to relational properties. He is ambiguous about the dynamical nature of qualia, just that the word intrinsic should encompass both dynamical and internal aspects of matter. It is also the case that the Principle of Precedence as an explanatory framework for consciousness inherits the usual objections to panpsychism, such as the combination problem. If the precedence of states are defined with respect to their wave-function, then it’s not clear how the decohered wave-functions of macroscopic systems (like our brains) are carved up into one unified conscious perspective. Smolin himself touches only briefly on the connection with pansychism (Smolin and Unger, 2015a), leaving much room for further analysis. Smolin’s project is an ambitious one, with many disparate ideas brought together with the aim of solving many disparate puzzles. This makes an effective summary difficult. However, the driving motivation for the project is clear. The conceptual basis of time in fundamental physics requires reimagining. This is in part due to the larger project of constructing a coherent theory of quantum gravity and also due to foundational concerns in explaining consciousness within our physical theories (Myrvold and Christian, eds., 2009, p. 466). One obvious response to Smolin’s demand for finding a place for time at a fundamental level is that it comes from experience, and yet that experience is severely limited. We cannot boost ourselves into frames near the speed of light, so our experiences cannot encompass such regimes. This is why we are led into physics in the first place to supplement our limited range of experience. Moreover, it is clear that human time involves a large constructive component. This underpins much of our experience of time, and indeed, albeit in a small way, our experience can be modified somewhat.14 What does this fact tell us about reality? If our experience of time is constructed in various ways, is it legitimate to use this to inform a conception of time in the universe, as Smolin and others suggest? After all, what Smolin means by “temporal qualia” is simply temporal phenomenology. Until we have established the extent of the brain’s role in this, we might  We have in mind the examples mentioned in footnotes 5 and 6. It seems clear that much of the construction going on here — and in the case of Benjamin Libet’s earlier experiments apparently showing that decisions are made before we become consciously aware of making them — is a matter of solving the integration problem of bringing together strands of input from receptors across different times in such a way as to generate a unified experience. 14

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be wise to withhold making too quick inferences from such temporal phenomenology to the reality of temporal passage.

5  Allocentric Versus Egocentric Representation There is an obvious sense that one can deal with the world in two disjoint ways. One can either consider entire, evolved, complete histories as objects, or one can take elements or cross-sections of such histories as objects. It is, of course, rather natural to think of the block as decomposed into sections, like slices through a loaf of brad, but one can also go in the other direction and think of the cross-sections as building the block. Of course, if there is such a block (and this is a big if15), then we would have a duality of descriptions. One can either view things externally, with time contained within the block, which is itself timeless — we’ll call this the “allocentric” viewpoint — or one can view things internally, with each point or cross-section picking out a different set of past and future events — we’ll call this the “egocentric” viewpoint, since it is tied to the here and now of an observer or agent. Of course, in the latter view, an embedded entity would not be able to see beyond its location to the other times, other than via memory and anticipation. Hence, the embedded perspective appears to be associated with time as we experience it. Indeed, with the insertion of the so-called “past hypothesis,” giving us a low-entropy initial condition (i.e., at the big bang), then we can avail ourselves of an entropy gradient within the block that for an embedded agent would generate the appearance of a temporal arrow, with memory records and a knowledge asymmetry masking future events duly supervening atop of the gradient. This section will explicate current attempts at how these different representations relate to one another and are consistent with one another. It seems to have become near-orthodoxy that this duality resolves the apparent conflict between timeless physics and our time-laden experience. Indeed, this procedure fits physics’ orthodox aim to provide a frameindependent description of the world (that is, allocentric representations), while our experience is precisely relativized to a particular frame  One might think that special and general relativity have already clinched this, showing incontrovertibly that the block must exist if Lorentz invariance is given an orthodox spatiotemporal interpretation. However, this completely ignores the spanner that quantum mechanics can throw into the works, as discussed in the next section. 15

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(involving egocentric representations) alongside our “cognitive engines, and cultural influences” (Callender, 2016). Hence, our common sense experience of time seems to be missing in the general physical description because it only becomes manifest in the frame-dependent description. As we saw above, Lee Smolin links this to the distinction between Platonic and Causal (Myrvold and Christian, eds., 2009, p. 466), where logical (eternal) relations oppose dynamical ones. But there has, along the lines of the duality, been work on finding our dynamic experience of time within the seemingly contradictory, timeless fundamental physics. One of the most famous of these is the model agent suggested by Hartle (2004), called an Information Gathering and Utilizing System (IGUS). In this model, Hartle explores how the interaction between internal information processing and the external environment reproduces certain aspects of our subjective experience — namely, the asymmetry of our temporal experience of the past and future as well as our common present. This would situate the phenomenon of flow, not as a fundamental aspect of physics, but as an emergent aspect of our psychology. In short, the allocentric description has everything to do with how agents are embedded into the four-dimensional manifold, which, within that embedded perspective, gives rise to our temporal experience. This model would be an example of Russell’s “normal observer” (Russell, 1946), an idealized agent that we construct to model our own perceptual relationship with the physical world in order to justify those very perceptual experiences. However, analogous to idealized economic agents, this approach is fraught with difficulties since we run the risk of prescriptivizing our experience to fit our model instead of the other way around. A more elaborate model that locates flow in an agent (and in a way that does not conflict with physics) is the so-called “dual process theory,” which finds both kinds of representation at work in the brain. The dual process theory of Chris Hoerl and Teresa McCormack (2019) can be viewed as a psychological interpretation of the embedded agent solution of the apparent conflict between block and flow pictures. By focusing on the way the brain processes temporal phenomena, Hoerl and McCormack provide a way of accounting for flow in a way that can be consistent with a block. The conflict arises at the level of a pair of systems that each deal with time in very distinct ways. On the one hand, there is something like a block, dealing with past, present, and future events in an even-handed way. On the other, there is a system that only deals with a present window (the specious present) of experience (which is set by 76

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universal perceptual thresholds). These can be taken as corresponding to the allocentric and egocentric representations, respectively; the former is necessarily indirect and cognitive (calling on episodic memory anticipation and such like), while the latter is largely perceptual and, more or less, direct.16 We might view these as corresponding to the thinking slow and thinking fast systems of Daniel Kahneman: the slow system is a rational one, used for planning and decision, while the fast system is for reacting and interacting with the world in the here and now. Of course, taken at face value, this could be taken to provide an eliminativist view of time, since it locates flow squarely in the brain, and in Darwinian terms, not considering whether there in fact is an isomorphic mapping onto the world itself. However, there are some, including Smolin, who are neither convinced by this analysis nor convinced that the project of recovering our experience of time from an overarching allocentric representation is possible, even in principle. This concern is one motivator for Smolin in reimagining the paradigm through which we study physics. This disposition has already been met with some pushback, with Price (2013), for example, calling it “unnecessary,” since based on what he sees as a false implication between the block universe picture and the kind of determinism Smolin chafes at. We turn to this in the next and final section.

6  A Place for Will and Creation? Let us begin this final section with a few passages from kindred thinkers to motivate the position we are about to lay out and defend. [People] do not sufficiently realize that their future is in their own hands. Theirs is the task of determining first of all whether they want to go on living or not. Theirs the responsibility, then, for deciding if they want merely to live, or intend to make just the extra effort required for fulfilling, even on their refractory planet, the essential function of the universe, which is a machine for the making of gods. (Bergson, 1932, p. 317) The scientific world-picture vouchsafes a very complete understanding of all that happens — it makes it just a little too understandable. It al lows you to imagine the total display as that of a mechanical clockwork, which  In fact, musicologists have made similar suggestions for some time. See, e.g., London (2012). 16

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for all that science knows could go on just the same as it does, without there being consciousness, will, endeavour, pain and delight, and responsibility connected with it — though they actually are. (Schrödinger, 1951 — This book has been refered to in the reference list.) “Law without law”: it is difficult to see what else than that can be the “plan” of physics. It is preposterous to think of the laws of physics as installed by a Swiss watchmaker to endure from everlasting to everlasting when we know that the universe began with a big bang. The laws must have come into being. (Wheeler, 1978, p. 11)

And the most important early voice, for our purposes: Indeterminism ... says that the parts have a certain amount of loose play on one another, so that the laying down of one of them does not necessarily determine what the others shall be. It admits that possibilities may be in excess of actualities, and that things not yet revealed to our knowledge may really in themselves be ambiguous. Of two alternative futures which we conceive, both may now be really possible; and the one becomes impossible only at the very moment when the other excludes it by becoming real itself. Indeterminism thus denies the world to be one unbending unit of fact. It says there is a certain ultimate pluralism in it; and, so saying, it corroborates our ordinary unsophisticated view of things. To that view, actualities seem to float in a wider sea of possibilities from out of which they are chosen; and, somewhere, indeterminism says, such possibilities exist, and form a part of truth. Determinism, on the contrary, says they exist nowhere, and that necessity on the one hand and impossibility on the other are the sole categories of the real. Possibilities that fail to get realized are, for determinism, pure illusions: they never were possibilities at all. (James, 1884, p. 151)

In De Divinatione, Marcus Tullius Cicero describes various oracular methods for predicting the future, comparing those with such power to the gods (the “divi” in “divination” refers, of course, to gods: the divine). Science is essentially the modern incarnation of this old idea of divination — called the “mantic” arts in Greek. No longer are bird patterns (augury) or the inspection of the entrails of sacrificed animals (haruspicy) used, namely signs of future events (omens), but instead a principle of causality that involves the future being necessitated by past conditions through laws of nature operating on such conditions. No longer 78

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considered divine but the very opposite. In fact, very often the older predictions were made on the basis of reasonable correlations: herons avoiding a storm; oxen having “premonitions” of rain and so on.17 Cicero himself believed such methods worked, and were not so surprising, since the future “already” exists, only distant in time (an early block universe picture). Signs of the future are “stored” in present causes, much as a seed stores its future expression (Cicero, 2006, p. 87). One can easily discern an early statement of causal determinism here. In his oft-cited work A Philosophical Essay on Probabilities,18 Pierre Simon, Marquis de Laplace, gives us what we now think of as the classic statement of causal determinism (largely drawn from the modeling of planetary systems in terms of differential equations), involving a being with perfect predictive and computational power: We ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it — an intelligence sufficiently vast to submit these data to analysis — would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes. (Laplace, 1902,...)

Laplace’s demon was able to compute all quantities at all times — “demon” here refers, of course, to the Greek word for a deity or genius. This is often expressed as “knowing the future,” yet it works in both directions and amounts to the idea that knowing the state of the world at one time (along with the world’s laws) would allow you to know the state of the world at all times: to know the world simpliciter. Again, Cicero spoke similarly that all things exist, only more or less distantly in time, with “signs” for all such things stored in the present condition of the world. This Laplacian version is as much of a block as modern  Nicholas Denyer discusses the protoscientific nature of divination in Denyer (1985). For a more detailed account, see Harrison (2011). 18  Laplace was responding to the 1756 Berlin Academy competition on “Whether the truth of the principles of statics and mechanics are necessary or contingent?” One can assume that Laplace knew Cicero’s work, since Cicero wrote almost identically about an intelligence who, in knowing the pattern of all causes, would know all things that are going to be. 17

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conceptions in the following way. Just as Everettian interpretations require epistemic probabilities,19 so any block view, in which all events are given, also requires epistemic probabilities to make sense of contingency. How could it be otherwise if everything is already there? But given this, where is the role for creation? For novelty? For action with a purpose? This seems to have been largely the flavor of the old debate between Henri Bergson and Einstein. While not exactly happy with the situation, Einstein was resigned to the fact that the world is really just a machine, with us as mere cogs — he was, of course, famously disgusted with Bohr’s version of the world that inserted subjectivity into the foundations of physics. We find the same battle played out in Huw Price’s recent review of Lee Smolin’s book (Price, 2013, p. 960). Price writes that “[f]or us, “now” is like “here” — it marks where we ourselves happen to stand, but has no significance at all, from the universe’s point of view.” This is what the fundamentalists deny, and we also deny it. Price attempts to deflate Smolin’s wheeling out of the “block = determinism” idea: But although many people assume that the block picture is necessarily deterministic, this is simply a mistake (“albeit a persistent one,” as Einstein might have put it). There is no more reason why the block view should require that the contents of one time be predictable from a knowledge of another, than it is somehow a constraint on a map of the world that the topography at the Tropics be predictable from that at the Equator. I can think that there’s a fact about what I had for lunch last Tuesday — or will have for lunch next Tuesday — without thinking that there’s any decisive present evidence about these matters, such that Laplace’s demon could figure it out, based on what he could know now.

However, as John Earman (2006, p. 1389) has rightly pointed out, this conflates two senses of determinism: worldly possibility (a matter for ontologists) and predictability (a matter for epistemologists). The former  At least, according to Everett himself. Any uncertainties are, like the embedding response to the apparent conflict between experience and reality concerning temporal flow, relative to an agent who assigns some pre-measurement uncertainty (an internal wave-function). But it is a matter of “incomplete information,” with the “outside wave-function” having full information. The experience of temporal flow, likewise, is thought to be a matter of incomplete knowledge, with information update via new records being generated, but still with an unknown future due to knowledge asymmetry. (There is some debate, however, about the nature of probabilities in the Everett interpretation: see Wilson (2013) for a review.) 19

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might well be true; the world might be a causal deterministic system, with each effect necessarily coming from a cause. But that does not necessarily imply that we can know the future from the past. We do not get to help ourselves to worldly indeterminism from unpredictability, which can simply be from incomplete information. We’re concerned with ontological determinism, and I suspect so are fellow fundamentalists, such as Smolin. Thus, Price writes: There is no more reason why the block view should require that the contents of one time be predictable from a knowledge of another, than it is somehow a constraint on a map of the world that the topography at the Tropics be predictable from that at the Equator. I can think that there’s a fact about what I had for lunch last Tuesday — or will have for lunch next Tuesday — without thinking that there’s any decisive present evidence about these matters, such that Laplace’s demon could figure it out, based on what he could know now. Earman (2006, p. 1389)

But the crux, at least for us, is not whether the contents of time are predictable from one moment to the next, but whether the universe “knows” what it will do at other times — that is, regardless of what we decision-making creatures have decided to do. To a certain extent, it will know what will happen next, under constraints of locality and basic laws, but there is a degree of malleability inserted through our free choices. This is what concerns Smolin, as we read him, and it concerns us too: do our free choices play any kind of role in the creation of the state of the universe at the next step? It is about possibility, not knowledge. This is what William James referred to as pluralism, with its object “the Pluriverse”: a universe always under construction (cf. James, 1909, p. 226), with some “loose play.” Yet the previous section might seem to suggest fairly conclusively that there is no reason to believe that anything the advocate of real flow allows cannot be encompassed in the viewpoint. However, we think there might be something to the fundamentalist’s intuition that genuine novelty and free choice is simply not possible in a block view.20 Such an inequivalence in what can be repre We barely dip our toes into this business of free will, and the various links or not to determinism and so on. But we note that there does seem to be an almost dismissive attitude to the notion of freedom, creation, novelty, and such notions, as if they were relics of a bygone era. However, pace Russell, like the monarchy, they are still with us. Let us borrow from 20

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sented in the two pictures would be enough to at least pour some water on the anti-fundamentalist’s flames. Of course, the response of the advocate is simply to point out that future events exist, just as past one’s do, but they don’t exist now: place yourself at the relevant location in the block and voila they exist. This does not satisfy the demand for creation. Genuine novelty, as we have seen, is part and parcel of Smolin’s temporal naturalism. He writes: In Temporal Naturalism, Lee Smolin explains how the activity of time is a process by which novel events are generated out of a presently existing, thick set of present events (Smolin, Time and Science, Vol. 1, p. 42).

We don’t wish to get involved in the definition of “the present,” and we have already had some reason to doubt the relevance of shape dynamics here. Certainly, George Ellis appears to adopt the view that there must be a single advancing sheet of reality, following at least the contours of Smolin’s naturalism. However, the singular nature of this advancing slice does not accord with our view of choice and freedom, which is more haphazard. Ellis’s view is in many ways a simple objective temporal realism. Ditto the accretion model drawn from causal set theory. To a large extent, we can choose which location of space we might occupy. By varying our speed, we get to choose how quickly we occupy a new location. Not so with time. We cannot get in a car to get to a different moment faster. Or so it seems from our ordinary conception. Of course, relativity theory appears to show that we could, in principle, do exactly that. Of course, relativity also points to the fact that presentness is localized or frame-relative. The notion of a frozen block has a profound impact on how we are to view our perceptions of the world, for the world as experienced (as unfolding) is unreal — or at least can be transformed William James (1884) the basic outline of determinism as follows: “[determinism] professes that those parts of the universe already laid down absolutely appoint and decree what the other parts shall be. The future has no ambiguous possibilities bidden in its womb; the part we call the present is compatible with only one totality. Any future complement other than the one fixed from eternity is impossible. The whole is in each and every part, and welds it with the rest into an absolute unity, an iron block, in which there can be no equivocation or shadow of turning”. Yet, we seem to subject ourselves, as physicists and/or philosophers, to a severe cognitive dissonance here. While we sometimes act as if driven by an external force, it is plain that we have choices, and we act on them, determining the way the universe develops around us.

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away by a simple change of representation. This is the substance of Einstein’s much-quoted remark about the distinction between past, present, and future being a stubbornly persistent illusion, of course. But the world’s not being a pre-fabricated entity does not immediately imply the need for a single, objective present. It is also possible that the world is built much as William James thought of our experience of it, namely in “buds or drops”.21 Ultimately,22 we indeed espouse a Jamesian view of the link between possibility and determinism (and free will = local creation). Any notion of a block in which there are laid out all events that have happened and will ever happen has eo ipso eliminated a certain type of possibility and freedom: the idea that things might go otherwise in the future and that we have some influence over this in a present moment. A modal realist might point out that other worlds ground such possibilities, but this takes away the agency of that particular agent at that particular time, since again what we are led to is a meta-block (now modal, along the lines of a configuration space) in which all possibilities are laid out once again. In this sense, switching to another view of quantum mechanics, such as Everettian approaches in which there is simply a universal wave-function on a 3N-dimensional configuration space, does nothing to pull us out of this neo-Megarian23 dilemma in which all possibilities are actual, and so not really possibilities at all. Neither does it help to point out that there is a causal web holding the elements of the block together, revealing dependences, for this makes it appear once again as thoroughly deterministic in the sense of the absence of real possibilities. To conclude: There are two tasks ahead, as we see it, the first is more general: to discern what is constructed (and so largely projected onto

 And since James also believed that the world was, in a sense, nothing but pure experience, then it follows that he too viewed the world itself in the same way. John Wheeler espoused a similar view: “the law of higgledy-piggledy” (named after the Victorian scientist John Herschel, Senior Wrangler of Cambridge in 1913). 22  Though a sustained presentation and defense will occupy a future paper (but only if we decide to make this future happen!). 23  Recall that the Megarians were logical determinists of a stripe that involve the identity of truth (or actuality) and necessity. As Diodorus Cronus is said to have stated, “only what is actual is possible.” The locus classicus is Nicolai Hartmann, “Der Megarische und der Aristotelische Möglichkeitsbegriff: Ein Beitrag zur Geschichte des ontologischen Modalitätsproblems” (Aus den Sitzungsberichten der Preussischen Akademie der Wissenschaften Phil.-hist. Klasse, 1937, X.). 21

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reality), from what is in genuine isomorphism with reality, to discern map from territory. The second task is to more closely assess the relationship between the block picture and the internal picture (what we call allocentric and egocentric pictures). In particular, it remains to be seen whether it is always a live option for the defenders of the block to help themselves to a completed version of the apparently evolving internal view. We are not convinced that it is, especially in quantum mechanical scenarios involving, e.g., Kochen–Specker or delayed-choice phenomena, which can be interpreted as forbidding the view of the world as a wholly given entity.24

References Barbour, J. (1991) The emergence of time and its arrow from timelessness. In J. J. Halliwell, Pérez-Maercader, and W. Zurek, eds., Physical Origins of Time Asymmetry (pp. 405–414). Cambridge University Press. Blodwell, J. F. (1985) Whither space-time? Quarterly Journal of the Royal Astronomical Society 26(3): 262–272. Butterfield, J. (2002) Reviewed Work: The end of time: the next revolution in our understanding of the universe by Julian Barbour. The British Journal for the Philosophy of Science 53(2): 289–330. Callender, C. (2016) Time lost, time regained. In A. I. Goldman and B. P. McLaughlin (eds.), Metaphysics and Cognitive Science. Oxford University Press. Conway, J. H. and S. Kochen (2009) The Strong Free Will Theorem. Notices of the AMS 56(2): 226–232. Denyer, N. (1985) The case against divination: an examination of cicero’s “de divinatione”. Proceedings of the Cambridge Philological Society 31: 1–10. Dolev, Y. (2016) Relativity, global tense and phenomenology. In Y. Dolev and M. Roubach (eds.) Cosmological and Psychological Time (pp. 21–40). Springer. Dowker, F. (2014) The birth of spacetime atoms as the passage of time. Annals of the New York Academy of Sciences 1326(1): 18–25.  In this sense, we readily sympathize with Chris Fuchs’s struggles with the block universe (https: //arxiv.org/abs/1405.2390), and believe that this assumption of a duality between internal and external representations is at least partly to blame. It is an assumption that can certainly be questioned. Indeed, Fuchs presents, as I’m sure he would be the first to agree, a very Jamesian physics with will, experience, and chance weaving the fabric of reality. Ditto Smolin, who is perhaps even more concerned with the moral implications of the block and the Jamesian-possibilist viewpoint, in which the future of the universe is shaped by our choices. 24

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Dowker, F. (2020) Being and becoming on the road to quantum gravity; or, the birth of a baby is not a baby. In N. Huggett, K. Matsubara, C. Wüthrich (eds.), Beyond Spacetime: The Foundations of Quantum Gravity (pp. 133–142). Cambridge University Press. Earman, J. (2006) Aspects of determinism in modern physics. In J. Earman and J. Butterfield (eds.), Handbook of the Philosophy of Science. Volume 2: Philosophy of Physics (pp. 1369–1434). Elsevier. Elitzur, A. C., and S. Dolev (2005) Becoming as a bridge between quantum mechanics and relativity. In R. Buccheri and A. Elitzur (eds.), Endophysics, Time, Quantum and the Subjective (pp. 589–606). World Scientific. Ellis, G. F., and R. Goswami (2012) Spacetime and the passage of time. In A. Ashtekar and V. Petkov (eds.), Springer Handbook of Spacetime (pp. 243– 264). Springer. Fuchs, C. A., and R. Schack (2013) Quantum-Bayesian coherence. Reviews of Modern Physics 85(4): 1693. Fuchs, C. A., M. Schlosshauer, and B. Stacey (2014) My struggles with the block universe. arXiv:1405.2390. Gisin, N. (2017) Time really passes, science can’t deny that. In R. Renner, S. Stupar (eds.), Time in Physics (pp. 1–15). Springer. Gomes, H., S. Gryb, and T. Koslowski (2011) Einstein gravity as a 3D conformally invariant theory. Classical and Quantum Gravity 28(4): 045005. Gomes, H., and T. Koslowski (2012) The link between general relativity and shape dynamics. Classical and Quantum Gravity 29(7): 075009. Gryb, S., and K. Thébault (2016) Time remains. British Journal for the Philosophy of Science 67(3): 663–705. Harrison, P. (2011) Wrestling with Nature: From Omens to Science. University of Chicago Press. Hartle, J., and S. Hawking (1983) Wavefunction of the universe. Physical Review D 28: 2960. Hartle, J. B. (2005) The physics of now. American Journal of Physics 73(2): 101– 109. Hoerl, C., and T. McCormack (2019) Thinking in and about time: a dual systems perspective on temporal cognition. Behavioral and Brain Sciences 42(e244): 1–69. Hudetz, L. (2015) Linear structures, causal sets and topology. Studies in History and Philosophy of Modern Physics 52: 294–308. Ismael, J. (2015) From physical time to human time. In Y. Dolev and M. Roubach (eds.) Cosmological and Psychological Time (pp. 107–124). Berlin: Springer. Ismael, J. (2017) Passage, flow, and the logic of temporal perspectives. In C. Bouton and P. Huneman (eds.), Time of Nature and the Nature of Time (pp. 23–38). Berlin: Springer. James, W. (1884) The dilemma of determinism. In The Will to Believe and Other Essays in Popular Philosophy (pp. 145–183). Cambridge University Press, 2014. 85

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James, W. (1909) The Meaning of Truth: A Sequel to Pragmatism. New York and London: Longmans, Green. Kauffman, S., and L. Smolin (1997) A possible solution to the problem of time in quantum cosmology. arXiv preprint gr-qc/9703026. London, J. (2012) Hearing in Time: Psychological Aspects of Musical Meter. Oxford University Press. Maudlin, T. (2010) Time, topology and physical geometry. Aristotelian Society Supplementary Volume 84(1): 63–78. Myrvold, W. C., and J. Christian, eds. (2009) Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle. The Western Ontario Series in Philosophy of Science 73, Springer. Norton, J. (2010) Time really passes. Humana Mente 13. Price, H. (2013) Rebirthing pains. Science 341(6149): 960–61. Price, H. (2011) The flow of time. In C. Callender (ed.), Oxford Handbook of Philosophy of Time. Rickles, D. (2011) A philosopher looks at string dualities. Studies in History and Philosophy of Modern Physics 42(1): 54–67. Rovelli, C. (2020) Space and time in loop quantum gravity. In N. Huggett, K. Matsubara, and C. Wüthrich (eds.), Beyond Spacetime: The Foundations of Quantum Gravity (pp. 117–132). Cambridge University Press. Russell, B. (1945) Physics and experience. The Henry Sidgwick Lecture Delivered at Newnham College, Cambridge University Press. Smolin, L. (2012) Real ensemble interpretation of quantum mechanics. Foundations of Physics 42(10): 1239–1261. Smolin, L. (2012) Precedence and freedom in quantum physics. arXiv preprint arXiv:1205.3707. Smolin, L. (2014) The metaphysical baggage of physics. Nautilus, January 30, 2014: http://nautil.us/issue/9/time/the-metaphysical-baggage-of-physics. Smolin, L. (2015) Unification of the state with the dynamical law. Foundations of Physics 45(1): 1–10. Smolin, L. (2015) Temporal naturalism. Studies in History and Philosophy of Modern Physics 52: 86–102. Smolin, L. (2020) Temporal relationalism. In N. Huggett, K. Matsubara, and C. Wüthrich (eds.), Beyond Spacetime: The Foundations of Quantum Gravity (pp. 143–175). Cambridge University Press. Sorkin, R. D. (2007) Relativity theory does not imply that the future already exists: a counterexample. In V. Petkov (ed.), Relativity and the Dimensionality of the World (pp. 153–161). Berlin Heidelberg: Springer. Stiles, N., M. Li, C. A. Levitan, Y. Kamitani, and S. Shimojo (2018) What you saw is what you will hear: two new illusions with audiovisual postdictive effects. PLoS One 13(10): e0204217.

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Unger, R. M., and L. Smolin (2015) The Singular Universe and the Reality of Time. Cambridge University Press. Vaina, L. (1995) Akinetopsia, achromatopsia and blindsight: recent studies on perception without awareness. Synthese 105: 253–271. Wardel, D., ed. (2006) Cicero: On Divination, Book 1. Oxford University Press. Wheeler, J. A. (1968) Our universe: the known and the unknown. The American Scholar 37(2): 248–274. Wheeler, J. A. (1978) Frontiers of time. Revised manuscript of lectures presented August 1–5 and to appear in the volume, Rendiconti della Scuola Internazionale di Fisica “Enrico Fermi,” LXXII Corso, edited by N. Toraldo di Francia and Bas van Fraassen, Problems in the Foundations of Physics, North Holland, Amsterdam. Wilson, A. (2013) Objective probability in everettian quantum mechanics. The British Journal for the Philosophy of Science 64(4): 709–737.

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Chapter 4

T H E L AY ER S T HAT BU I LD U P THE NO T IO N OF TI M E Carlo Rovelli* Aix-Marseille University, Université de Toulon, France Perimeter Institute for Theoretical Physics, Canada The Rotman Institute of Philosophy and Western Ontario University, Canada *[email protected]

1 Introduction Much confusion and disagreement around the notion of time is due to the fact that we commonly fail to recognize that we call “time” is a variety of distinct notions that are only partially related to one another. The time of the researcher in quantum gravity (like myself), the time of the cosmologist, the time of the scientist that studies black holes, the time of the engineering designing a GPS system, the time of biology, the time of the train-station master, the time of the historian, the time of the lover waiting for her love to arrive, the time of the old man thinking about his life, and the time of the kid dreaming her future are obviously somewhat related, but they are all profoundly different. For a particle physicist or a black-hole researcher, for instance, it is obvious that the time elapsed between two events depends on the path taken, while for the train-station master, this is inconceivable, and for the biologist it may be conceivable, but it remains totally irrelevant. The time in Newton’s equations does not distinguish past from future; the time of biology needs to. And so on. 89

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None of these “times” is false. None of them is “illusory” or “not real.” They are all, each of them, appropriate concepts for organizing phenomena at some scale, in some domain. But they are only valid on some scales, for different regimes, and in different contexts. Some connections between these times are obvious. For instance, the subtleties of relativistic time become negligible, irrelevant, when relative velocities are small. Other connections are more complicated; they have required centuries of scientific investigations to be sorted out. Some nodes of the overall story are not fully clear yet. The common mistake, which I find repeated over and over again, is to take any one of these notions of time, good and appropriate in some domain, and mistakenly think that this particular one is universal, necessary, it is “what time really is,” or “what we mean by time.” In other words, the common mistake is not in the fact that we understand time is this or that manner: it is to think that this or that manner are universal features of temporality. To understand time, we have to break it apart. We have to study its various layers (Fraser, 1975): these have different properties, which in turn are sourced in approximations, in particular conditions (Rovelli 2018a). I illustrate below a number of distinct notions of time and their differences; they are all relevant for describing the real world. What is important in what follows is to realize that many “obvious” properties of time are actually the results of different kinds of approximations and idealizations.

2  Let me start with a well-known example. Take two events. For instance, I clap my hands and call this event A. Then, I clap my hands again and call this event B. We can say that between event A and event B some time has lapsed, and this elapsed time is a quantity tAB that can be characterized as the reading of a (good) clock. This is a definition of time that can make many physicists, sport arbiters, and engineers happy. How universal is it? In our everyday experience, it largely suffices. But any good relativist knows well that this does not work in general. The reason is that if they are sufficiently precise, two identical clocks measure different values tAB if

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they move differently between the two events A and B. Any clock that moves (fast enough) away from A and then comes back (fast enough) to B measures a shorter time than a stationary clock. So the time lapse between A and B is in fact arbitrarily short if the clock measuring it is moved fast enough. This is a fact. We may try to correct the previous definition by stating that the “true” time lapse between event A and event B is the time measured by a stationary clock: a clock that remains in my hand and does not move. This, we may think, is the longest possible time span between A and B. But this would be wrong again. The clock that measures the longest time span between A (the first clap of my hands) and B (the second clap of my hands) is actually not the clock that remains near my hands. It is a clock that I throw upward at the moment of A and catch back when it falls down precisely at B. (The reason is that this clock is in free fall during the flight, hence follows a geodesic, which can be shown to maximize the proper time between the extremes of the flight). We could define time in this way, but it follows that if we do so, we cannot synchronize clocks over a region. If I clap three times, the total time is not the sum of the two intermediate times. There is no common clock time; each clock measures its own “proper” time along its path, and the proper times do not match on arrival. The time of a relativist is definitely quite different from the universal common clock time of our daily experience. This is just how nature happens to work. There is nothing wrong with the idea of a common universal time, but this idea only works within an approximation where we disregard relativistic effects. General relativistic time is not unique and not universal.

3  The same is true with regard to cosmological time. Cosmologists use the notion of “cosmological time,” which is roughly defined as the time that a clock sitting inside a galaxy would have measured it had started at the Big Bang. But this notion of time is an approximation, because when two galaxies merge (as our Milky Way and Andromeda are headed to do in the future), they have in general different proper times from the Big Bang. Which of the two has the “right” time? The question is obviously

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meaningless. Cosmological time is defined only under the (false) assumption of exact homogeneity. Beyond this approximation, cosmological time does not correspond to clock time. Even within the homogeneity idealization, the simultaneity surfaces defined by cosmological time are not simultaneity surfaces in the sense of special relativity, as can be easily shown. Cosmological time is a fully conventional arbitrary labeling of space-time events. Neither our common non-relativistic time, nor cosmological time, nor full relativistic times, are “true times.” They are related but distinct temporal notions, with different properties. The unicity of our commonsense notion of time is the result of an approximation, like the apparent flatness of the surface of the Earth.

4  A good example of the misunderstanding caused by the failure to appreciate that the notion of time is complex and not monolithic is the erroneous argument that Einstein’s relativity implies a static four-dimensional “block” universe (Putnam 1967). The argument goes as follows. Given two events A and B, with B in the past of A, the theory of relativity states that it is possible to find a third event C that is “simultaneous” to A with respect to some observer but also “simultaneous” to B with respect to some other observer. Since two simultaneous events are said to be both “real now” in common language, it follows that all the events of the universe are all “real now.” Hence, the universe is a four-dimensional “block universe,” which is “without becoming.” The argument is wrong because it is based on the mistaken assumption that the only manner time can function is as in our naïve nonrelativistic intuition, where simultaneity can be defined in a transitive and observer-independent manner. Einstein’s definition of simultaneity is relational and not transitive. This misunderstanding leads to a false alternative between Presentism and Eternalism (Rovelli 2019). Presentism is the picture of reality based on the idea that there is a common objective “now” all over the universe. This idea is based on the commonsense impression that what we see around us is the common and objective “now.” The discoveries of the physics of the last century have conclusively shown that this view is indefensible, as is the idea that the 92

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Earth is the center of the universe. The commonsense impression that what we see is a shared “now” is based on the fact that our senses are imprecise and we do not resolve the light traveling time. It is much like the fact that, around us, we see that the Earth is flat. Eternalism is the idea that there is no temporal becoming and that our sense of temporal becoming is illusory. What the universe truly is, according to eternalism, is a static four-dimensional block, where future events are “real now” as present events are. This too is an absurd position, because the four-dimensional space-time of relativistic theories is a manifold of events, namely of “happenings,” and there is nothing “static” about them. There is nothing in relativity that motivates us to say that a future event is “real now,” in any reasonable sense. General relativity is about happenings, not about anything static. The source of the confusion is the mistaken idea that either time is exactly as it is in our naïve intuition, or it does not exist: it is illusory. This leads to the mistaken idea that Presentism and Eternalism are the sole alternatives: they are not. Relativity offers a third possibility, where things happen, but there is no global present. A physicist working with relativity easily develops the proper intuition to deal with a local becoming that is not organized in a single global shift from a universal now to the next (Rovelli 2019). It is a notion of time that lacks a global character.

5  Before Newton, in the long intellectual tradition that goes from Aristotle to Descartes, as well as in common parlance, “time” was mostly understood as a counting of events: day, night, day, night, and so on. It is Newton who introduced the idea that time passes “by itself,” irrespective of the events happening. (In his famous defense of the relational notions of space, against Newtonian’s absolute space, Leibniz was largely articulating (also with new arguments) notions of space and time that were traditional, not novel.) A few centuries later, Newtonian time looks “natural” to us. Detaching the idea of an autonomously flowing time from the actual events of the world was instrumental for Newton to build the conceptual structure of mechanics, and it proved a very effective move. This implies that there are two distinct notions of times that are often at play (Newton clearly distinguishes the two in his Principia (1687)). The 93

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first is the old notion of time as the simple relative succession of events. The second is the Newtonian idea of a metric (measurable) time flowing by itself. The difference is substantial. For instance, if truly nothing happens, Newtonian time continues to pass, while it is meaningless to say that the counting of happenings continues if nothing happens. The traditional notion is relational: time is a relation between events. No events happening, no time. The novel notion of time introduced by Newton refers to a (peculiar) entity, which exists by itself. Once again, neither of the two is right or wrong: relational time and Newtonian time are just two notions playing a role in the organization of the phenomena. Newtonian time definitely turned out to be far more effective than relational time in science for a few centuries. But the Einstein revolution changed the game in an unexpected manner, which in a sense led to a compromise between the two previous notions of time. Newtonian time was first brought together with Newtonian space (another peculiar entity introduced by Newton) to form spacetime. Then, this spacetime itself was identified by Einstein as a physical field: the gravitational field. As such, it is definitely an entity, but not a peculiar entity anymore. It is a rather mundane entity, a field like the other fields that modern physics uses to describe the world. Notice that even the causal structure of spacetime is just a property of a field, the gravitational field. Thus, Newtonian time is reduced to an aspect of a field, an ensemble of events itself, after all. The traditional relational notion of time, on the other hand, remains useful, of course. We can still count events in a general relativistic context and call “time” their relations, which are well accounted for by the mathematics of general relativity. We can still count day, night, day, night ... and call this “time”, also in a general relativistic solar system. Again, we have a multiplicity of notions of time, and we understand their relations: how some emerge within approximations from others in special regimes.

6  There is no consensus on a quantum theory of gravity yet, and no direct empirical support for any of the current tentative theories. But there are 94

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tentative theories of quantum gravity that coherently and consistently merge quantum theory and general relativity. Among the best is loop quantum gravity, with which I work (Rovelli, 2004). Loop quantum gravity describes the quantum behavior of the gravitational field, namely of (relativistic) space-time. It describes, for instance, the possibility that between two events A and B even the time measured by the same clock could be in quantum superposition of two different values. Recently, the possibility of measuring this effect (predicted by all current quantum gravity tentative theories) in the laboratory has been considered by some laboratories (Bose, 2017; Marletto, Vedral, 2017). In loop quantum gravity, the relational notion of time, namely the possibility of counting series of events, remains in place, but Newtonian time, like the proper times of general relativity, is replaced by quantum variables. Like all quantum variables, these can be discrete, have a probabilistic dynamics, and be in superposition (Rovelli, 2018b). Hence, the basic equations of quantum gravity are very likely not to include a time variable (De Witt, 1967). The basic equations of loop quantum gravity, indeed, don’t have a time variable (Rovelli, 2004). This, however, is no big deal, because classical general relativity can be equally formulated without using a time variable, for instance in its Hamilton-Jacobi formulation. There is nothing mysterious in the absence of a time variable, in spite of the discussion that this fact has generated and unfortunately still generates. The reason for the absence of a time variable is simply that in general relativistic physics, as mentioned above, there is no preferred time variable, no preferred interval between A and B. Hence, it is rather obvious that the theory can be formulated without a preferred time variable. The theory gives the relative evolution of physical variables with respect to one another. For instance, it gives the evolution of local variables with respect to local clock times and the relative evolution of clock times with respect to one another. Consider, for instance, the example described above, where I keep one clock in my hand, throw the second upward and catch it when it falls back. It is a fact that the time between the launch and catch measured by the two clocks is different. Which of the two measures the true elapsed time? Is the theory describing how the flying clock evolves in the true time defined by the stationary clock, or vice versa? The question is meaningless: the theory tells us how the clocks change one with respect to the other. 95

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The conceptual structure of the theory is clear and well defined without any need to single out a preferred variable and call it real time (Rovelli, 2011). Loop quantum gravity allows us to (tentatively) compute what happened around the Big Bang, in the interior of black holes or in other situations where space-time is not classical, in a comprehensible and predictive (although not yet empirically supported) way, without any need of having a specific time variable, or a preferred notion of time, besides the ancient relational idea that we can call time any counting of events.

7  A striking feature of the entire structure of elementary physics (classical mechanics, relativistic field theory, quantum mechanics (see DiBiagio, 2020), quantum field theory, the Standard Model, general relativity, etc.) is the fact that the basic equations do not distinguish the past from the future (provided we also swap parity and charge, namely swap the names “left” with “right” and “positive” with “negative” charge) (Price, 1997). This is in striking contrast with the manifest irreversibility of most phenomena around us, which makes the past direction of local time dramatically different from its future direction. All observable distinctions between past and future can be traced to a fact: the tendency to equilibrate towards the future, but not towards the past (Myrvold, 2021). Contrary to what the study of the statistical mechanics of gases suggests, in the real world, equilibration is generally very slow (the universe is still very far from equilibrium). Many realistic thermalization times are huge (in fact, cosmological): the sun is hot, Earth cool, after billions of years of slow equilibration. While equilibration towards the future is intuitively comprehensible, the lack of equilibration towards the past is puzzling. This is usually denoted as the puzzle of the past low entropy, or the statement of the “past hypothesis.” In the following paragraph, I mention some current ideas on the possible ground of this macroscopic asymmetry; here I discuss what it implies about time. The key to understanding what this macroscopic asymmetry implies about time is to realize that the difference between the past and the future 96

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pertains to the macroscopic description of phenomena only. There is no way of even naming it in terms of microphysics alone. While some macroscopic histories (breaking eggs) are more commonly observed than their time reversals (eggs recombining), every single micro-history is equally peculiar and equally unlikely to be found in the universe, both in its ahead and in its time reversed version. (The reason is that no single micro-history is ever commonly observed.) It is only when we patch micro-histories in groups (when we coarse grain) that we begin to see irreversibility. In the past, the actual micro-history of the world belongs to a macro state with an intact egg; in the future, it belongs to a much wider macro state of broken eggs. Irreversibility is a property of the macrovariables, not the micro-history. It is important to stress that the time orientation of all irreversible phenomena is grounded in this basic arrow of time. Irreversible phenomena are all macroscopic and statistical in nature. That is, they all exist thanks to a microphysics with many degrees of freedom where energy dissipates. Among these are the fact that we have traces of the past (and not corresponding traces of the future) (Rovelli, 2020a), agency (Rovelli, 2020b), and time travel (Lewis, 1993). The fact that all arrows of time depend on the entropic one is important and is often misunderstood. An important consequence of this fact is that since phenomena like thinking, being conscious, and similar, are all irreversible, they are necessarily macroscopic. They cannot be elementary. The key point here is that time orientation is not a necessary property of some “elementary” basic notion of time. It is only a feature of the approximate notion of time we use in coarse-grained “approximate” accounts of the natural phenomena. The orientation of time is ubiquitous, but is not fundamental. The time orientation of our thinking, of our living, of biology, of evolution, of the notion of causation that we employ, the abundance of traces of the past that has no equivalent in the future... all these phenomena are grounded in the macroscopic approximation. They are all (in a wide acceptance of the term) of entropic origin. The time of microphysics has no orientation. There is no contradiction between this fact and the fact that we have no reason to believe that everything we observe supervenes on microphysics. This is strongly counterintuitive, but it is an inescapable consequence of what we have learned so far about the natural world. The very specific

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“temporal” features of the time variable have to be studied with statistics, thermodynamics (Rovelli, 1993), and perhaps the lack of full information implied by quantum theory (Connes and Rovelli, 1994), not in any a priori intuition about time flow. Our intuitions have developed for our macro world, so it is of no surprise that they are misleading.

8  The existence of the entropy gradient in the macroscopic account of our universe remains puzzling. It can be taken as a contingent fact or as a law (“the past hypothesis”) (Albert, 2000). It might also be a perspectival phenomenon. The possibility that the arrow of time is perspectival (Rovelli, 2016) is a speculative but intriguing possibility. It is based on the fact that any macroscopic account of phenomena is determined by a choice of macroscopic variables. This choice is not arbitrary; it is dictated by the available physical interactions with the system. Hence, any macroscopic account is fundamentally relational; it is relative to a second physical system that interacts with the first via an interaction that involves only a (relatively) small number of variables: these variables are the macroscopic variables. This observation opens up the intriguing possibility that the asymmetry of the arrow of time is not due to the fact that the past microstate is by itself non-generic. Rather, it is due to the coarse graining defining the macrophysics. This is determined by the physical way in which we interact with the rest of the universe. Hence, it may be that what is special is not the micro-history of the world, but our being part of a subset of the world that interacts with rest via relatively few macroscopic variables, which define the peculiar coarse graining with respect to which there is an entropy gradient. If this is correct, the distinction between past and future could be a majestic phenomenon, but a perspectival one, like the apparent rotation of the universe around us. The rotation of the sky we observe around us is a perspectival phenomenon due to the fact that we live on a spinning planet. It is a real but perspectival phenomenon. To understand time, and to understand all the phenomena we witness, we have to take into account the perspective from which we witness them (Ismael, 2007).

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9  Experiential time is a far more complex phenomenon than physical time. I believe that a lot of confusion about time is based on the fact that we have a rich temporal experience, and we erroneously pretend to project it all over elementary physics, where it does not belong (Husserl, 1928). In this, we are as primitive as the ancient Greeks when they tried to understand the dynamics of the material world in terms of the “basic forces” of “love” and “hate.” Experiential time is generated by the way our brain works. A specific activity of the brain is to store information about past events. This is possible precisely because the brain is a macroscopic object functioning in an environment far away from equilibrium, with a constant income of free energy (we live near an infernal furnace burning at 6000 degrees, the sun). Hence, the functioning of the brain does not make up a time arrow; it gets it from the macroscopic physics that defines it and the entropic time arrow of the environment. The brain stores information in the form of memories and constantly retrieves and utilizes some of these memories. This has the effect that, in a sense, our experiential present is impregnated by aspects of the past. Furthermore, recent neuroscience suggests that one of the main activities of the brain (if not the main activity) is to constantly compute possible futures, trying to anticipate them (Buonomanno, 2017). Hence, we equally live with vivid awareness of some future (possible) events. The result of all this is that our experience includes a sort of temporal window formed by the set of events included in memories and anticipations. This is the window we actually call “time.” (See the notion of Lichtung in Heidegger, 1950). We presumably have this basic structure in common with mammals, at least. Our species, on the other hand, is probably characterized by an increased span of this window, thanks to collective memory (and science) that, compared to other mammals, gives us access to larger chunks of the past repetition and a greatly increased capacity of future planning that thus gives us a wider, and perhaps richer (or poorer?) sense of the future. In any case, we literally do not experience just the present but also the past and the future. This experience of time depends on the fact that we have a brain. The beautiful clock in my living room keeps time much better than me, but it has no comparable experience of the past and the 99

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future in any remotely connected sense. It does not remember yesterday and is not waiting for tomorrow. Our experiential time is grounded in our complexity. The sense of the speed at which times flows, the nostalgia of the past, the anticipation of the future, all the rich phenomenology of our experience of time obviously are grounded in physics, but are not universal aspects of physical temporality. Yet, it is hard to make an abstraction from them. It is hard to renounce the intuition that the Big Bang, 14 billion years ago, is “very far,” while what happened 3 seconds ago is “just now,” and to focus on the fact that the sense of closeness or distance is an effect of the specific capacity of our memory. The perceived speed at which time passes obviously has far more to do with the working time of our neurons than with anything pertaining to mere physics. To understand time, we have to learn to separate our intuition about time, which is grounded in this rich experiential time, from the concepts of time that turn out to be needed to describe the events of nature. I think that too often we fail on this.

10  Table 4.1. Here is a brief table listing some notions of time with their properties: Time notion

Some typical properties

Experiential time

Feels like having a flowing speed.

Brain structure

Irreversible time

Oriented. Past differs from future.

Disregard microscopic degrees of freedom

Newtonian time

Unicity. Single now in the universe.

Disregard relativistic phenomena

Special relativistic time

Independent from actual events happening.

Disregard relativistic gravity

Cosmological time

Global.

Disregard inhomogeneity

Proper time in general Measured by local clocks. relativity 100

Ground

Disregard quantum phenomena

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Time notion

Some typical properties

Ground

Independent evolution Arbitrary. variable in quantum gravity

Not needed to describe happenings

Relational time

Arbitrary counting of happenings

Always available.

11  Experiential time is not only heavily infiltrating and (mis-)guiding our intuition when we try to understand the temporal structure of reality, but is also heavily emotionally charged. Two of the most influential investigators of the nature of time, who are perhaps at the farthest extremes of the cultural spectrum, are Hans Reichenbach (1958) and Martin Heidegger (2010). Quite surprisingly, both of them point out that much of philosophy — and other human enterprises — can be read as efforts to escape the anxiety that time causes. I suspect that this anxiety is not just an emotional disturbance that fogs our efforts to get intellectual clarity about the notion of time. It is actually the other way around: time for us is largely precisely this anxiety, the emotional underpinning that drives the constant process of utilizing memories to build our future or protect us from it. Separating these entrenched intuitive aspects of time for us from the temporal notions we use to organize and understand the physical world does not mean that we have to choose one of the two times as the true one, as opposite intellectual traditions unfortunately do. Indeed, I do not think that it makes much sense to say that experiential time is authentic, as some contemporary schools pretend, while natural time is constructed. Nor do I think it makes much sense to say that physical time (which one?) is real, and what we experience is illusory. To get clarity about the complex structure of the natural world, of which we are part, we have to distinguish the multiple layers that build up the complex phenomenology of time. We have to get some clarity about the multiplicity of structures that we negligently and generically call “time.”

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References Albert, D. (2000). Time and Chance. Cambridge, Harvard University Press. Aristotle. Physics, IV, 219 b 2; see also 232 b 22-3, http://classics.mit.cdu/ Aristotle/physics.html. Bose, S., Mazumdar, A., Morley, G. W., Ulbricht, H., Toros, M., Paternostro, M., Geraci, A., Barker, P., Kim, M. S., and Milburn, G. (2017). A spin entanglement witness for quantum gravity. Phys Rev Lett, 119, 240401. Buonomano, D. (2017). Your Brain is a Time Machine: The Neuroscience and Physics of Time, New York, Norton. Connes, A., and Rovelli, C. (1994). Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories’. Classical and Quantum Gravity, 11, 2899–2918. De Witt, B. S. (1967). Quantum theory of gravity. I. The canonical theory. Physical Review, 160, 1112–1148. Di Biagio, A., Donà, P., and Rovelli, C. (2020). Quantum information and the arrow of time. arXiv: 2010.05734. Fraser, J. T. (1975). Of Time, Passion, and Knowledge, New York, Braziller. Heidegger, M. (1950). Holzwege’, in Gesamtausgabe, Vol. V, Klostermann, Frankfurt. Off the Beaten Tracks, Cambridge University Press, 2002. Heidegger, M. (2010). Being and Time. Translated by Joan Stambaugh, revised by Dennis J. Schmidt. Albany, New York, SUNY Press. Husserl, E. (1928). Vorlesungen zur Phänomenologie des inneren Zeitbewusstseins, Niemayer, Halle a. d. Saale. The phenomenology of internal time-consciousness, J. S. Churchill tr., Indiana University Press, 2019. Ismael, J. T. (2007). The Situated Self. New York, Oxford University Press. Lewis, D. (1993). `The paradoxes of time travel’. Am Philos Q, 13, 1976, 145–152, reprinted in R. Le Poidevin and M. MacBeath (eds.), The Philosophy of Time. Oxford, Oxford University Press. Marletto, C., and Vedral, V. (2017). Witness gravity’s quantum side in the lab. Nature, 547(7662), 156–158, https://doi.org/10.1038/547156a. Myrvold, W. (2021). Beyond Chance and Credence. Oxford UP. Newton I, Philosophiae Naturalis Principia Mathematica, Book I, def. VIII, Scholium. Price, H. (1997). Time’s Arrow and Archimedes’ Point: New Directions for the Physics of Time. Oxford University Press. Putnam, H (1967). Time and physical geometry. Journal of Philosophy, 64, 240–247. Reichenbach, H. (1958). The Philosophy of Space and Time. New York, Dover. Rovelli, C. (1993). Statistical mechanics of gravity and the thermodynamical origin of time’. Classical and Quantum Gravity, 10, 1549–1566. Rovelli, C. (2004). Quantum Gravity. Cambridge University Press. Rovelli, C. (2011). Forget time. Foundations of Physics, 41, 1475–1490, https://arxiv. org/abs/ 0903.3832. 102

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Rovelli, C. (2016). Is time’s arrow perspectival? [History of physics; General relativity and quantum cosmology; High energy physics — theory], in K. Chamcham, J. Silk, J. Barrow, and S. Saunders (eds.), The Philosophy of Cosmology (pp. 285–296). Cambridge University Press. Rovelli, C. (2018a). The Order of Time. London, Penguin. Rovelli, C. (2018b). Space and time in loop quantum gravity, in N. Huggett, K. Matsubara, and C. Wuthrich (eds.), Beyond Spacetime: The Foundations of Quantum Gravity (pp. 117–132). Cambridge University Press 2020, arXiv:1802.02382. Rovelli, C. (2019). Neither presentism nor eternalism. Foundations of Physics, 49(12), 1325–1335. Rovelli, C. (2020a). Memory and Entropy, 24(8), 1022. http://arxivorg/abs/2003. 06687. Rovelli, C. (2020b). Agency in Physics, in “Experience, abstraction and the scientific image of the world. Festschrift for Vincenzo Fano”, Franco Angeli editore, 2021, http://arxiv.org/abs/2007.05300.

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Chapter 5

S EN S E S IN WHI CH TI M E DO ES AN D D OE S NO T EX I ST Julian Barbour* University of Oxford, UK [email protected]

In this chapter, I argue that time itself does not exist but, as argued by Ernst Mach, is an abstraction derived from change. To fully vindicate Mach’s intuition, it is necessary to consider the manner in which the possible shapes of a dynamically isolated universe evolve. At the classical level, a complete theory of time can be extracted from the Newtonian theory of the gravitational interaction of point particles.

1 Introduction Ernst Mach said [1], “It is utterly beyond our power to measure the changes of things by time. Quite the contrary, time is an abstraction at which we arrive by means of the changes of things.” I believe the aphorism is true if by time we mean duration.1 Ontological status does not pertain to time, but it does to things from which we can derive duration as an abstraction. The abstraction is a construct. Duration exists in the construct, not in the things from which it is made. Those are the two “senses” of my title. Rather than duration, I will mostly use time. * Visiting Professor in Physics at the University of Oxford 1  In the Principia, Newton said time “by another name is called duration.” It is what clocks measure.

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If the construct deserves to be called time, it, like a house, needs a secure foundation. I seek it in geometry. Galileo said, “He that attempts natural philosophy without geometry is lost.” Geometry, for its part, is founded on experience. Greek for earth is ge¯; geometry derives from the measurement of land. It is possible that Egyptian surveyors discovered Pythagoras’s theorem while measuring fields in the Nile Delta using bamboo canes. Congruence establishes proof in geometry. Solids realize congruence to great accuracy. Suppose two canes laid next to each other in Alexandria are found to have the same length. Take one to Luxor and back. If they are now not as accurately congruent, it is almost always easy to find out why — somebody cut a bit off one. Reliability like this, given to us by nature as a gift, is what makes all experiments possible. Euclidean geometry, the foundation that I use to exemplify a possible “construction of time,” is itself an abstraction that experience suggests can help us understand what happens in the world. There are facts that reveal the remarkable structure of the world. Measure the N (N − 1)/2 distances between N points on houses, rocks, and trees. If N ≥ 5, it is a fact that the distances satisfy, to the accuracy of measurement, certain algebraic relations. The N (N − 1)/2 distances are not all independent; they can be derived by simple computation from 3N Cartesian coordinates ascribed to the positions. These, however, are not uniquely determined; without the distances being changed, the coordinates can be acted on by Euclidean translations and rotations. Only 3N − 6 of the coordinates are independent; they define the configuration of the points. In fact, since the unit of measurement — defined nominally by two marks on a “master ruler” — is arbitrary, only 3N − 7 numbers are independent. They define the shape of the N points. They are pure numbers and characterize the shape as it is — intrinsically. Anything else we add is extraneous.2

 In themselves, individual points have no attributes. They acquire them through their distance relations to the rest of the “universe” of points. The identity of each point is defined by the entire universe as follows: Suppose given, relative to a nominal scale, all the separations rij between the points and calculate the root-mean-square length lrms = Σ i< j rij2 . Define the relative separations as rij = rij /lrms . Then the N – 1 primary attributes of particle i are the rij , j ≠ i. To these can be added secondary “branch” attributes: the primary attributes of each of the points j ≠ i. This leads to ternary attributes and so forth until all branching possibilities have been exhausted. Except in degenerate cases due to symmetry within the shape, for example, an equilateral triangle, any two particles a, b will be identified as a pair by rab. Three particles a, b, c will be identified individually by rab , rac , rbc. This process of identification through attributes is manifestly holistic in that it requires knowledge of the complete shape. 2

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The passage from the N (N − 1)/2 distances to the 3N − 7 numbers is impressive data compression: from ≈ N 2 to ≈ N, essentially the square root. Geometry is a distillation of facts. In despair at not being able to comprehend existence, Goethe’s Faust cries out, “What it is that, in its innermost core, holds the world together.”3 For Galileo it is geometry: “He that attempts natural philosophy without geometry is lost.” The manner in which geometry binds things together is reflected in the fact that the N (N − 1)/2 numbers, by virtue of their values, proclaim to those in the know what they are — distances between N points in space. Note that, in contrast, the data-compressed numbers by themselves tell you nothing; they could have been pulled at random out of a hat. In physics, time as the independent variable, denoted by t, is like that. Where does it come from? In effect, Newton said time, like space, just is absolute, ontologically prior to things and change. He introduced his absolute concepts, which in modern treatments of dynamics are implicitly retained in the notion of an inertial frame of reference, to counter what he saw as a fatal flaw in Descartes’s theory of motion.4 Newton accepted that the closed, essentially Aristotelian cosmos enclosed by a sphere of fixed stars that defined rest and position for Copernicus and Kepler should be replaced, as in the Cartesian mechanical philosophy, by a universe in infinite Euclidean space filled with material bodies all moving relative to each other. Descartes, however, asserted, on the one hand, that bodies move solely relative to each other and, on the other, that any undisturbed body has an innate tendency to move uniformly in a straight line. Newton, with the aspiration to create a theory of motion as mathematically rigorous as Euclidean geometry, recognized the potential of what became his first law of motion but also the impossibility, in the Cartesian mêlée of countless bodies all moving relative to each other, of confirming that any one of them moves between collisions in a straight line. He therefore defined the motion of bodies relative to absolute space and time and succeeded brilliantly. But he never made good on his promise at the end of the Scholium  “Was ist es, was die Welt im Innersten zusammenhält?”  I discuss Newton’s reaction to Descartes in [2], chapter 11. Critical for understanding Newton’s position, which he justified at best obscurely and without mention of Descartes in the famous Scholium at the start of the Principia, is his earlier unpublished paper known by its incipit De gravitatione, which first came to light in the 20th century. Much confusion in the debate about whether motion is absolute or relative has arisen from the bizarrely contrived manner in which Descartes, unbeknown to Newton, sought to avoid Galileo’s condemnation by the Inquisition while still retaining the essence of the Copernican revolution. 3 4

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to show how the absolute motions can be determined from the observed relative motions. There are increasingly sophisticated levels at which one can show how time emerges as a construct from changes of things: • Accept as correct Newton’s laws for a finite number N of gravitating point particles and show how the resulting observable relative motions are related to the absolute theory from which they are derived. This is what Newton failed to do. I will show that when it is done, some justification does emerge for both of Newton’s absolutes but in a form which suggests the need for an alternative foundation of dynamics. • Reformulate Newton’s laws for N particles in a form that implements Mach’s idea, dubbed Mach’s principle by Einstein, that the masses of the universe give rise to local inertial frames of reference in which Newton’s laws of motion hold. This eliminates absolute time entirely and yields time as the promised construct. This, the main part of my paper, will also show that a residual absolute element related to spatial scale survives if a classically expanding universe is to be described. I conjecture that only a quantum theory of the universe can eliminate this last residuum of Newton’s absolutes. • Reformulate Einstein’s general theory of relativity and establish the conditions under which similar conclusions can be drawn. This goes beyond the scope of the present paper, so I merely refer the reader to [3], chapter 8, and its notes, in which references to original papers are given. This shows that the essential structure of general relativity, when formulated as a dynamical theory, as opposed to a spacetime theory, matches the Machian reformulation of Newtonian theory. I also make occasional comments in footnotes that will allow readers familiar with the concepts and issues in the Hamiltonian representation of general relativity [4, 5, 6] and the attempts at canonical quantization of gravity [7, 8] based on it to see how the concepts and issues that arise in Newtonian theory reappear in a manner that is essentially the same, except that the group transformations involved are no longer ones that depend on a finite number of parameters but ones that depend on groups. The main conclusion is that when the changes of things are described with appropriate geometrical concepts, Mach’s aphorism is to a large 108

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degree vindicated in a modified form of Newtonian theory and is likely to be so in Einstein’s theory. What precisely a quantum theory of gravity will reveal remains an open issue, but I think it is highly likely that change, defined geometrically, is primary and that time derives from it.

2  Newtonian Dynamics Expressed Relationally Paradigmatic in dynamics, and therefore adequate for my purposes, is the N-body problem: a finite number N of point particles interacting through Newtonian gravity. It can be formulated in a reduced representation, in which only directly observable — physical — degrees of freedom (dofs) appear, or in an extended representation with additional dofs that admit some freedom of choice. Comparison reveals the role that Newton’s absolutes play. The extended representation uses the most fundamental concept in Newtonian dynamics: configuration space N [9].5 For N particles of masses mi, i = 1, … , N, in three-dimensional space it has 3N dimensions. A corresponding number of Cartesian coordinates xi fix the particle positions; associated with them is the Newtonian kinetic metric

dsN =

N

∑ m dx i

i

i

⋅ dx i , (1)

which is the mass-weighted distance in N with dxi the change in position of each particle i from one instant of time to a later one. If the N particles are assumed to model a universe — an “island universe” in Euclidean space — no imagined external scales can weigh it, so the total mass M = ∑ iN mi is nominal and only N − 1 mass ratios are physical. The first (trivial) step in the reduced representation is to set M = ∑ iN mi = 1. Much more significant are two decisive steps that follow directly from the basic facts of Euclidean geometry. As noted above, the 3N Cartesian coordinates are not uniquely fixed by measurement with rulers. First, they change under the transformations of the Euclidean group: translations and rotations. However, these do not, as noted, change the measured distances rij = |xi − xj| between the particles. The rij are invariants of the Euclidean group. It is helpful here to recall the principles for  The corresponding concept in general relativity in the case of spatially closed universes is Riem, the space of all possible Riemannian metrics on a closed manifold. 5

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which Leibniz argued during the final two years his life (1715–1716) in the well-known correspondence he had with Clarke [12], who defended Newton’s claim that positions in absolute space are the foundation on which dynamics must be created even if they cannot be directly observed. The debate never really got to grips with the real problem, which results from the fact that the distances between bodies in the universe undoubtedly change. However, for unchanging separations, Leibniz introduced a conception that became established about a century and a half later with the creation of group theory. He pointed out that, at any given instant, only the separations between bodies are observable. If some collection of bodies constituted the entire universe, nothing observable could be discerned were they supposed “moved” in absolute space by a translation in some direction or rotated as a whole. These, of course, are typical examples of the not-yet-formulated notion of group transformations, which are now interpreted as merely changing the representation of something without changing its essential, or physical, properties. Leibniz clearly anticipated this interpretation of group transformations in the formulation of his principle of the identity of indiscernibles (PII): two things identical in all their aspects must in reality be one and the same thing [11, 12]. This principle allows one to pass from the Newtonian configuration space N to the 3N − 6-dimensional relative configuration space (RCS) as defined in [10] by quotienting with respect to the Euclidean group.6 Leibniz could have gone one step further by including a change of scale among the transformations that alter nothing observable. This follows from the fact that since the universe is everything that exists, there cannot be a ruler outside it to measure its size. Size is intrinsic, not extrinsic. For example, the separation between one pair of particles can be “measured” intrinsically as a ratio of the separation between another pair. Such ratios are invariants of the similarity group. In the undertaking of this paper, in which time is to be created as a construct in which Euclidean geometry is the foundation, the similarity group is supreme in establishing what we mean by identity when invoking the PII to define quantities that may be called physical. We acknowledge this by quotienting the Newtonian configuration space N by the similarity group of Euclidean  In general relativity, again for spatially closed universes, one passes analogously from Riem to superspace by quotienting with respect to three-dimensional diffeomorphisms, which, considered as group transformations, are infinite-dimensional generalizations of Euclidean transformations. 6

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translations, rotations, and dilatations. The resulting space, with 3N − 7 dimensions, is shape space S.7 Each point in S is defined by the N − 1 mass ratios of the points and the shape they form. These shapes are what, in this idealized exercise, I identify as Mach’s “things.”8 In the fourth of his letters to Clarke, Leibniz introduced a second principle, referring to “those great principles of a sufficient reason and of the identity of indiscernibles.” In fact, in his various writings, Leibniz introduced several other key principles besides these two, the first of which I will abbreviate to PSR. There is an excellent discussion of them and their interconnections in [11]. It does not help that Leibniz coupled the PSR with a different “great” principle in his Monadology, in which he states that “Our reasonings are grounded upon two great principles, that of contradiction, in virtue of which we judge false that which involves a contradiction, and true that which is opposed or contradictory to the false” (paragraph 31); “And that of sufficient reason, in virtue of which we hold that there can be no fact real or existing, no statement true, unless there be a sufficient reason, why it should be so and not otherwise, although these reasons usually cannot be known by us” (paragraph 32). One gets the impression that besides the logical requirement, critical in mathematics, of non-contradiction, he does see the PSR as his crowning principle and seeks to derive the others, including the PII, from it. It is, however, noted in [11] that there are grounds for seeing the PII as a “stand-alone” principle. This will be relevant below, in which the statement of the PSR quoted above from Monadology will be my guide. Turning now to application of the principles, Spekkens [13] has drawn attention to the creative use that Einstein, whether or not he knew of Leibniz’s ideas, made in effect of the PII, first in the creation of special relativity and then general relativity. As I will argue below, the strength of the PII is most readily recognized if applied to the whole universe, as Leibniz did in his correspondence with Clarke [12]. Einstein did not in fact apply it in such an all-encompassing context. At the very start of his 1905 paper that created the special theory of relativity, he pointed out that  The corresponding space in general relativity is conformal superspace, which is obtained from Riem by quotienting by both diffeomorphisms and conformal transformations [8]. 8  Newton and Leibniz referred to bodies but in effect treated them as point particles, which I think is permissible in a conceptual elucidation of the nature of time. I should also mention that in quantum mechanics there are many situations in which individual particles cannot be regarded as distinguishable. This fact can be taken into account without having to jettison a Machian representation of quantum particles but I won’t go into that in this paper. 7

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the then existing theory of electromagnetic effects described observationally identical phenomena (dependent solely on the relative motion of a magnet and a conductor) in ontologically different ways (depending on whether the magnet or the conductor were taken to be moving). Soon after that, he made a similar observation about gravitational fields, noting that uniform acceleration of a lift could not be distinguished within it from a gravitational field acting on the lift when at rest. Spekkens points out that Einstein was in effect using the PII as a criterion to judge the quality of a theory: if epistemologically identical phenomena are represented in a theory as ontologically distinct, this is to be regarded as a defect of the theory and a sign that it needs to be replaced by one that is not in conflict with the PII. This was undoubtedly a powerful heuristic; Spekkens argues that the PII is more important than the PSR, disagreeing in his paper with me in an assertion I had made that Einstein had been guided by the PSR. When I read [13], I was persuaded by Spekkens’s arguments, but recent discussion with Pooya Farokhi has made me come to think the PSR, which is certainly a metaphysical principle in that it calls for a theory to be as perfect as possible, should rightly be regarded as occupying the pinnacle in Leibniz’s thought. I would put it thus: if in a theory observationally identical causes lead to different effects, then the theory is defective and needs to be replaced by one that is better. As we will see, it is the introduction of causality, expressed through the nature and number of Cauchy data needed in a dynamical theory, as a criterion that gives the edge to the PSR while retaining an essential supporting role for the PII. This may seem obscure at the moment, but a critical example that is to follow should, I believe, make things clear. A further ideal in theory construction to be used below is closure. It is achieved in the relational setting through the assumption that the universe consists of a finite though arbitrarily large number of particles. Einstein commented that in a “self-contained” conceptual world as conceived by Mach, “the series of causes of mechanical phenomena [is] closed” [14]. Closure, surely only attainable, if at all, in the theory of the whole universe, is a theoretician’s prime desideratum. In the present context, the need to choose some arbitrary number N of particles in the N-body model may seem a defect. As explained in [3], it disappears in the context of an Einsteinian universe spatially closed up on itself. Before we consider the application of these principles, I want to alter Mach’s “changes of things” not just to “changes of shapes” but to 112

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“differences of shapes.” The discovery and development of thermodynamics and statistical mechanics in the 1850s brought to light the mystery of the “arrow of time” and is what leads me to prefer “difference of shape.” Newton’s laws do not discern a direction of time. Any given solution of them with time assumed to flow in one direction can be run “backwards” by, at some instant, exactly reversing all the particle velocities and commencing the solution of the equations with the positions achieved at that instant. The system, evolving in accordance with just the same Newtonian laws of motion, simply retraces the path it has taken. This property of the laws is called time-reversal symmetry. The mystery is the manifest disagreement between the symmetry and the pervasive unidirectionality — the irreversibility — of all the phenomena we observe. For example, we and all the stars get older in the same direction, and the proverbial broken egg never “unbreaks.” It is repeated experience of things like this that gives us the impression that time always moves forward and associates with the word “change” the sense of “going forward,” whereas Newton’s laws allow us equally well, at least conceptually, to “go backwards.” But in whichever way we do go, forwards or backwards from the present, what we find in either direction is different from the present we left. That is why I prefer “difference of shape” to “change of shape,” though I will often use the word change. This comment also relates to the “construction,” from timeless things, of all aspects normally associated with time; the direction of time is, after all, the one of which we are most powerfully aware. I will, in fact, argue against the nearly universal view, born of the apparently universal validity of the laws of thermodynamics (especially the second), that unidirectionality — irreversibility — is never found in any individual solution (a microhistory) of the laws of dynamics. The claim is that an arrow of time must have a statistical origin and results from bundling together a Gibbs ensemble of microhistories, which is then shown to exhibit entropic behavior. This is true in systems that are confined — technically have a phase space of bounded Liouville measure — but when that is not so, counterexamples exist.9 We will find them by “projecting” N-body

 The stimulus to the discovery of thermodynamics in 1850 was Carnot’s 1824 theory of steam engines. They can only operate with steam confined to a cylinder. Matching this, all thermodynamic experiments were made on systems in a physical container, while statisticalmechanical studies presupposed confinement in a conceptual box. There is no box in the N-body universe. As we will see, the standard story is inverted. 9

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solutions to S. This changes the perspective, from laws formulated locally in a Newtonian spatiotemporal framework to a law of the universe formulated in the arena of shape space S. Projection to S means the removal from N-body solutions obtained in N of all the information except the system’s successive shapes. They become points on a smooth curve in S. All that remains of the Newtonian spatiotemporal structure are the ratios of the separations rij that define the shapes.10 However, residues do “show up” in S. To identify them, the first step is to list the “absolutes” that Newton introduced when he created dynamics. There are five: time (in its sense as duration), the direction of time, position and direction in space, and spatial scale. By virtue of Galilean invariance, position in space disappears without trace in S; directions of time emerge in each solution curve in a manner I will discuss after the other three absolutes: duration, direction in space, and scale. They are manifested through the effect of five pure numbers: • By the velocity decomposition theorem [15], the kinetic energy T decomposes at any instant into three orthogonal components Ts, Tr, Td, respectively the kinetic energies in change of shape, rotation, and overall expansion (d for dilatation). These three dimensionful quantities combine into the two dimensionless ratios Tr/Ts and Td/Ts. • Unless the angular momentum is zero, two numbers define its direction relative to the current shape of the system. • The ratio of the kinetic and potential energies, T/VNew, is a fifth dimensionless number. These numbers are critically important since they bear on what is perhaps the greatest of Newton’s achievements: the creation of dynamical laws that permit prediction of the future state of a physical system, the solar system for example, given the form of the laws that govern the system and certain data, called Cauchy data, that specify the state of the system at a given initial instant. In the case of the solar system, the laws are Newton’s three laws of motion, together with his law of universal  If all the particles have the same mass, nontrivial dynamics still exists without the Newtonian concept of mass. However, forces remain. Critically, their effect can be expressed through pure numbers derived in the framework of Euclidean geometry using the full set of similarity-group transformations. At the particle level, such a treatment implements Einstein’s dream of reducing physics to geometry. 10

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gravitation, according to which two bodies of mass m1 and m2 attract each other with a force proportional to mimj/rij, where rij is the distance between the particles at the considered instant. The Cauchy data are the positions and velocities (or momenta) of the particles at the considered instants. The five numbers above are encoded in 6N standard N-body Cauchy initial data for any Newtonian dynamical system formulated in absolute space and time (or, in modern terms, an inertial frame of reference), but in S are not natural shape degrees of freedom (dofs). They, and with them the Cauchy data one would expect in S, are only 6N −15 in number and are the 3N −7 angles that define a shape and the 3N −8 that define a direction at it in S. In the simplest case of the three-body problem, two angles α and β define the shape of the triangle that has the three particles at its vertices, while dα /d β defines the direction in which the shape is changing. The simplest and simultaneously most predictive evolution law of the universe one can propose is a geodesic law. For one, an initial point and initial direction — in the three-body case α , β , and dα /d β — uniquely determine a solution. Any such geodesic determines a mere succession of shapes — there is no criterion that distinguishes one direction along it compared with the opposite direction. In geodesic theory, the additional Newtonian kinematic structure associated with absolute time and space is eliminated entirely. The decisive fact about general Newtonian theory when represented in S is that it fails to meet the ideal of a geodesic theory by five data. Unlike standard Cauchy data in N, they cannot be deduced from an infinitesimal section of a solution curve in S. In that arena, they become additional variables that are non-standard from a shape-dynamic relational point of view.11 As opposed to the shape dofs required in geodesic theory, one must have access to a finite segment of the evolution curve in S in order to determine their values. By definition, the universe is everything. Within it, only ratios are determinable. The mismatch between the natural expectation of what variables should govern its evolution and the behavior of a Newtonian universe is a puzzle. In fact, all but one of the non-geodesic — non-intrinsic — variables can be eliminated without  In [12], Leibniz did employ the PII very effectively to identify the epistemologically problematic aspect of Newton’s absolute space and time and thereby, as noted above, anticipate the basis of group theory, but he did not identify the resulting defect in Newtonian dynamics — its failure to satisfy the PSR when its solutions are represented in truly observable terms — shapes of the universe. It was only Poincaré, in a perceptive analysis that I highlight in [16], who made the first significant step in that direction in his [17]. 11

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undermining our ability to create what I believe is a satisfactory model of the universe.

3  A Relational Universe In opposing Newton’s ideas, Leibniz was, as previously discussed, an early advocate for a relational universe. In this section, I will use his two principles, the PSR and the PII, to justify my claim that all but one of the five non-geodesic dofs can be eliminated. I will highlight their power by showing how the problematic nature of Newtonian absolutes shows up if we use them to describe an “island universe” in Euclidean space. I begin with the last of the five numbers identified in Section 5.2, the one related through energy directly to time. It is readily eliminated. In standard Newtonian dynamics, time is introduced as an independent variable along with the configuration space N. It is used in the N-body problem to convert mass-weighted displacements miδ xi of all the particles into momenta pi by dividing them one and all by the same time difference δ t and going to the limit in which the numerators (the miδ xi) and the denominator δ t simultaneously tend to zero.12 This mathematical procedure reflects Newton’s claim that absolute time flows uniformly “without relation to anything external.” This means that for one and the same δ t the external change in the universe can be different; namely, there can be a one-parameter family of sets of changes miCδ xi, 0 ≤ C ≤ ∞. Thus, for each value of C, the universe changes by a different amount in the same δ t. As a result, we obtain a one-parameter family of possible expressions for the total kinetic energy T:

T = C2 ∑ i

pi ⋅ pi . (2) 2

Moreover, if provisionally we grant that absolute space does exist and that position and displacements in it are observable but that time itself is not — surely nobody will claim that it is — we have an uncomfortable state of affairs that arises from the fact that Newtonian trajectories become ever harder to distinguish from inertial motion the shorter the interval of them is known. Thus, a planetary orbit is ever better approximated by its tangent as the fraction of the orbit is reduced: given only the 12

 I here use δ to denote finite quantities as opposed to the infinitesimal d in (1).

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tangent, it is impossible to predict the motion. In the solar system, in which the rotation of the earth defines sidereal time, a small, essentially rectilinear and hence tangential section of the orbit is enough to determine the planet’s energy to a good accuracy (the earth rotates many times during a small fraction of a planet’s orbit), but without the clock the planet’s speed cannot be determined and the motion cannot be predicted. Given only the position and the tangent, many different motions are possible. Moreover, in one and the same universe, there could be widely separated identical planetary systems in which, at any given instant, all the planets are in the same positions and with the same tangents to their motions. Completely different motions could then unfold in the different systems. As defined by the positions and tangents, all the systems are indiscernible yet quite different — discernible — motions result. This violates the PSR: no sufficient reason can be given for the different outcomes. As Einstein concluded in the example Spekkens cites, this is a reason to look for an alternative theory. There are two ways to do that, one much more satisfactory than the other. Before we consider them, we need to identify the overarching reason, within standard Newtonian theory, for the different outcomes. It resides in the different amounts of information needed to determine the kinetic energy T and the potential energy V of the system, the sum of which is the total energy E = T + V. Let us suppose “snapshots” taken of the configuration of a considered system at given instants. Two such snapshots separated by a short known interval of time are needed to determine the kinetic energy to reasonable accuracy. In contrast (for known masses), a single snapshot determines the potential energy. At the same time, the evolution is determined jointly by T and V. This is why the PII, in conjunction with the PSR, tells us there is an unfortunate failure of predictablity in Newtonian theory if applied to the universe. The problem does not arise in a subsystem like the solar system, which is not identical to other such systems in the universe and in which rotation of the earth provides a clock to determine the different speeds of the planets. The less satisfactory resolution of the problem is to retain Newton’s notion of time and accept that the universe at large could in principle have any value of its total energy E. One can then invoke the PSR to ask if there is any particular value of E that is distinguished. I think the reader will agree that one sticks out: E = 0. That zero is distinguished is an undoubted fact within the theory of numbers, quite independently of physics applications. But E = 0 is also distinguished within the practice of 117

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physics, in which definite units (seconds, minutes, etc., for time; meters, inches, etc., for distances) are necessarily employed. Now the choice of units is entirely arbitrary; different choices of them lead to different values of E unless E = T + V = 0 ⇒ T/V = −1, which is a scale-invariant requirement since T/V is dimensionless. Thus, one way to satisfy the PSR will be to modify Newton’s law by adding the requirement that T/V = −1 and justify it by saying that the law that governs the universe, as opposed to subsystems of it, must not depend on the choice of arbitrary units. This will be the case if the law of the universe is expressed solely in terms of shape. If nature has plenitude in mind and seeks a rational universe in which more than one shape exists, absolute time cannot be employed to create it. It must be done some other way. It can be in a relational universe, in which everything is done with ratios. We can use the kinetic metric (2) in a simple way to define the ratios of the amount by which individual particles are displaced. For this, we simply divide a nominal individual displacement of each particle by a corresponding nominal quantity, which reflects what they all do:

mi dxˆ i =

mi dx i

∑ i mi dx i ⋅ dx i

. (3)

These quantities express the essence of relational dynamics and reflect our most direct conscious experience of spatial phenomena, which I suspect is due to the fact that the rods and cones in the retina mediate what we directly see as angles. When we look at two stars in the sky at night, in the dark, they excite rods separated by a fraction of the total extension of the retina. The brain translates that into what we see: a separation that is a fraction of the 2π of a great circle around the sky. This reflects the way we define angles in a scale-independent way by dividing a circular disk into segments by radii from its center to the perimeter. The angles are defined as ratios of the distance around the circumference, these ratios being the same however large the circle is made. Solid angles are defined similarly relative to a sphere as seen from its center, just as we see the bowl of the sky. In fact, as I will explain, it is not possible in a description of the universe in classical dynamics to achieve full-scale invariance through exclusive use of ratios like (3). In order to see what can be done and highlight what cannot quite be achieved, let me point out that Newtonian N-particle 118

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solutions that have one fixed energy E can be represented as geodesics in N by means of Jacobi’s principle [9], in which one extremalizes between any two configurations 1 and 2 in N the action

2

i=N

1

i =1

AJac = ∫ dλ (E − V )T (λ ), T (λ ) = ∑

mi dx i dx i ⋅ , (4) 2 dλ dλ

where λ is an arbitrary monotonic parameter, and V = V (xi) is any potential; in the standard account, E is the total energy, but from the point of view of a geodesic law, it is simply some arbitrary constant. Action (4) is reparameterization-invariant: AJac has the same value for all choices of a monotonic λ . Minimization of the action (4) between the two fixed points 1 and 2 in N determines a geodesic whose points correspond to the configurations of a Newtonian solution between them. Jacobi’s principle gives us a timeless theory of two of the most basic properties we associate with time: chronology and duration. Except that it comes without direction, the manner in which chronology arises is trivial. As we walk along a path, one position comes after another. What Jacobi gives us is not successive positions on a path, but successive configurations of the considered system. Except for the fact that we could walk in the other direction, which reverses the order, we cannot require more of chronology. I postpone the issue of direction because it is not directly related to the form of the law, but to the nature of its solutions. To see how duration arises, we need to find the laws of motion that follow from (4). The Euler–Lagrange equations of motion are

d dλ

 E − V  1/2 dx i   T (λ )  1/2 ∂V , (5)  =   dλ   E − V  ∂x i  T (λ ) 

in which λ , which is present not only in the derivatives but also in T as defined in (4), is still freely specifiable, subject to monotonicity. There is nothing that prevents us from choosing it at all stages of the evolution such that T = E − V.

(6)

Let us denote this specially chosen λ by t; then, its increment dt is 119

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dt =

∑ m dx i

i

i

⋅ dx i

2(E − V )

, (7)

and if we use it in the equations of motion (5), they simplify to

d2 x i ∂V (8) = , ∂x i dt 2

which is Newton’s second law. Thus, time emerges in a familiar form from a timeless law through the choice of the arbitrary parameter λ that casts the equations of motion into their simplest form. However, there is much more to the “choice of time” than mere simplicity. Before we come to that, we see that, as defined by (7), time is constructed out of the elements dxi weighted with E − V. In accordance with the logic we have followed, the relation (6) is not what it seems to be, the definition of energy when rewritten as E = T + V, but a special choice made to simplify the equations of motion to the form (8). It is the definition of time, the duration that the universe takes to pass from one configuration to another; we see time emerge as the explicit construct (7).13 You may now be puzzled about the relational status of E in (4). What is the justification for the appearance of such a parameter in a geodesic law? Its presence allows one to determine a one-parameter family of geodesics between two given configurations. However, which one should be chosen? Doesn’t the existence of the family violate the PSR? Leibniz would surely say yes and seek a sufficient reason for one value of E, rather than all the others. The resolution of this difficulty is readily found. As already noted, among all finite real numbers only one is truly distinguished: zero. For a Jacobi action defined this way, the definition of the distinguished curve parameter λ becomes T = −V, which does not imply an imaginary T provided the potential is negative (as the Newton potential is). Then, the expression for the distinguished dt becomes  When astronomers started to doubt whether the rotation of the earth gave a good measure of time, they instead employed the dynamically significant bodies in the solar system to define time and did so essentially in accordance with (7). The corresponding quantity is called ephemeris time [18]. In the case of the solar system, it can be regarded as the time variable which ensures that the equation that expresses energy conservation holds. However, in the absence of a true independent variable, this reduces the definition of either time or energy to a tautology. 13

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dt =

∑ m dx i

i

i

−2V

⋅ dx i

, V = −∑ i< j

mi mj rij

, rij =|x i − x j|, (9)

and is indeed, as Mach required, defined as an abstraction from the changes (differences) of things. This puts the notion of time and the usual manner in which Jacobi’s principle is employed in a different perspective. For centuries, the rotation of the earth, reflected in the sweep of the stars across the heavens, supplied astronomers with their only useful clock; it measured sidereal time. With respect to it, Kepler found that the radius vector from the sun to each planet sweeps out equal areas in the plane of its motion (defined observationally relative to the effectively fixed stars) in equal intervals of sidereal time. When Newton’s laws are used to solve Kepler’s problem, the orbit is first found in effect by Jacobi’s principle in the form r = r(φ ), where r is the distance from the sun to the planet and φ is its azimuthal angle. Energy conservation is then used to find the speed and position in the orbit as a function of time. This device is possible given an independent variable generally called time (for centuries, as I noted, the rotation angle of the earth). However, an independent variable cannot exist for the universe. There is nothing outside it. The only way to describe the universe is through relationships between ratios of quantities of the same kind, for example, angles as fractions of 2π or r defined as the ratio of the earth–sun distance divided by the earth’s diameter. Within the solar system, with the star-studded celestial sphere supplying a background for the definition of angles as fractions of great circles on the sky, Kepler discovered the phenomenon that defines good clocks. Note the plural. It is meaningless to say that a single motion defines time. What justifies the choice of certain motions rather than others? Kepler’s great discoveries included the fact that time defined by the earth’s rotation marched in step with time defined by the area swept out by each planet’s radius vector in its motion around the sun. Let the area swept out by planet i from its perigee be ai and θ be the meridian of some star observed from the earth. Then, Kepler’s second law takes the form dai/dθ = ci, with ci a constant. Including the earth, Kepler knew six planets, and with them, six areas being swept out together with the earth’s rotation angle. From these seven variables, the differential calculus allows us to determine six rates of change, all of which stay mutually constant.

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The seven motions keep good time relative to each other. They “march in step” [18]. About 80 years later, John Flamsteed, the Astronomer Royal, showed with experiments at Greenwich that the recently invented pendulum clocks kept good time not only relative to each other but also relative to those in the heavens. These facts show that a satisfactory theory of time does indeed require much more than choice of an independent variable like λ that simplifies the equations of motion; one also needs a situation in which what appear to be more or less independent motions “march in step.” In fact, comments that Newton made in the Scholium after he had defined absolute time show that he was well aware that different measures of time, essentially the ones I have listed, march in step with each other. About them, he commented, “It may be that there is no such thing as a uniform motion whereby time may be accurately measured. All motions may be accelerated and retarded, but the flowing of absolute time is not liable to any change.” Newton seems to have followed Descartes in believing that the universe consists of infinitely many bodies. Had he assumed only a finite number existed, he could have called the special λ defined by (7) the emergent time of the universe, with respect to which individual motions could indeed be either retarded or accelerated. However, quite contrary to his claim that (absolute) time flows uniformly without reference to anything external, the time defined by (7) is the weighted average of all changes. The recognition that good clocks do exist in nature — since there are motions that march in step — and that their existence is compatible with Newton’s laws (and their modern replacement by relativity theory and quantum mechanics) does not explain how they can have come into existence. We also do not yet have an explanation for the existence of the spatial frame of reference, very well approximated by the stars, within which the earth’s ecliptic and the orbital plane of the other planets are defined. We still have some way to go before we can claim that the “bricks,” Mach’s things, used to build time as a sound construct are possible shapes of the universe that evolve subject to the law that governs it.

4  Building the House The construct just promised is erected in stages. The first involves the process originally called the intrinsic derivative in [10] and then renamed best matching in [19]. I won’t go into all the details, which are given in the 122

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two cited papers and in the recently published [3] in chapter 8. The key thing is that best matching is a procedure that eliminates another Newtonian violation of the PII. Leibniz invoked the principle to point out that translating or rotating a single fixed configuration in absolute space could not change anything observable. More subtle is the problem that arises if one wishes to establish a difference between two configurations that are not identical. It is such a situation that arises in the dynamics of N particles in Euclidean space. A problem arises already before one tries to consider how much time passes as one configuration changes into another that is genuinely different and not just through the translation or rotation that Leibniz considered. To see what the problem is, let us first formulate it in Newtonian terms. In one instant, N particles are assumed to have the positions xi, i = 1, ... , N, and a little later to have the positions x¯ i, i = 1, ... , N . The difference between the two, x¯ i − xi, gives rise to the dxi in (2). In standard Newtonian terms, division by dt yields the initial velocities. We have already seen how that problematic feature can be eliminated by employing a geodesic principle, but now it is Newton’s absolute space rather than time that leads to a conflict with the PII and an associated failure of the PSR. It arises not because there is no observational criterion for determining the position and orientation of either of the two configurations, each considered separately in absolute space, but because of an ambiguity in the difference between their positions and orientations. By Galilean relativity, the absolute position and orientation of the xi configuration have no observable effect, but the mutual orientation of the two configurations is critical since it defines an initial direction of the angular momentum and with it two of the pure numbers listed above. They add two more nonshape variables to the dynamics in shape space and violate the PSR, since in terms of what can actually be observed — the separations between the particles and their difference between the two configurations — all relative orientations of the two configurations are observationally on the same footing. Since angular momentum strongly affects evolution, indiscernible initial data give rise to very different effects. The PSR is violated. To deal with the analogous problem in the case of time, we cut the Gordian knot. We made no attempt to choose between possible time increments, between possible dts, and simply said there is no time at all and passed to a geodesic theory, taking the final step by removing the constant E from Jacobi’s principle. The problem of mutual orientation can 123

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be solved differently, though again, by invoking the PSR. To see this, note first that the kinetic energy (2) is positive definite and depends on the separation between the centers of mass of the two configurations and their mutual orientations. Being positive definite, (2) necessarily has a minimum. It is unique and is realized when the centers of mass coincide and the mutual orientation is such that (2) takes the minimum value. As shown in [10], the angular momentum in this best-matched position tends exactly to zero in the limit that the displacements do so together with dλ in (4). Provided the potential is invariant with respect to Euclidean rotations, which is the case in the gravitational N-body problem, the angular momentum is conserved, including the Machian case in which it is zero. Moreover, energy is conserved, even if it is zero, provided the Hamiltonian or Lagrangian that governs the motion does not depend on time. This leads to a perfect match between a geodesic theory based on best matching in the relative configuration space (RCS), for which the initial data are point and direction in the RCS, and the solutions of the Newtonian N-body problem for which the energy and angular momentum are exactly zero, both being conditions that are conserved. To summarize our progress in the elimination of Newtonian absolutes and, with it, the violation of the PSR, we have arrived at a geodesic principle that we can express in the form 2



A bm Jac = ∫ dλ 1

∑ i< j

mi mj rij

i=N

T (λ ), T (λ ) = ∑ i =1

mi dx ibm dx ibm ⋅ , (10) 2 dλ dλ

where the superscript bm stands for best matched and the Newton gravitational potential has been represented explicitly but without the gravitational constant G, the inclusion of which would merely change the overall value of the action without changing its extremals: the geodesics in which we are interested. In order to highlight a key property of (10), it is helpful to rewrite it in the parameter-free form

A Jac =

2

1 2 ∫1

∑ i< j

mi mj

( ∑ m dx

rij

i

i

bm i

)

⋅ dx ibm , (11)

for which the expression for the time increment dt in (9) becomes

dt =

∑ m dx i

i

bm i

−2V

124

⋅ dx ibm

. (12)

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The variational principle that action (11) defines shows that three of the Newtonian absolutes — time, position, and orientation — are eliminated, but not absolute scale. This is because the Newton potential has dimension [length]−1, which means that although it is invariant under Euclidean translations and rotations, it changes under scaling, as does the action (11) because the term ∑ i mi dx ibm ⋅ dx ibm has dimension [length]2. As a consequence, the action AJac has the dimension [length]1/2 and is not scaleinvariant. It is true that we can use action (11) to construct a geodesic theory in the relative configuration space but not in shape space S. As shown in [20], it is possible to achieve the shape-space ideal if the Newton potential is replaced by a rotationally and translationally invariant potential that has dimension [length]−2, which then balances the [length]2 dimension of the quadratic kinetic term in (11). Such a theory does have attractive and interesting properties but fails completely to model a universe with any resemblance to the one in which we find ourselves. This is because a geodesic theory of this kind in S, when represented in either the Newtonian configuration space N or the RCS, cannot expand — the center-of-mass moment of inertia (half the trace of the inertia tensor), which measures the size of the system, has a fixed (nominal) value. In fact, it is the sole conserved quantity associated with such a theory. As I argue in [3], I think this hints at a deep connection between the question of scale and whatever quantum law it is that governs the universe. I will say something about that at the end of this chapter, but I will here merely show how far we can get in the “construction” of time by eliminating from Newtonian theory all absolutes except scale.

Figure 5.1  Horizontal and vertical stacking of three-body configurations, with the dots showing the positions of the center of mass. For four or more particles, a representation like this requires suppression of one spatial dimension. Figure created by Flavio Mercati.

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Figure 5.1 illustrates the construction by means of “horizontal” and “vertical” stacking [10]. The idea is very simple, especially in the threebody case shown in the figure, because then the motion of the three particles always unfolds in a plane if the angular momentum is zero. This enables us to construct a “card house.” Take any initial configuration of the three particles, and suppose their relative positions marked on a card. Then, all subsequent configurations can be laid successively on top of each other so that best matching is respected with each new addition. One can also suppose that “vertical” distances between successive cards are chosen equal to the best-matched distinguished (12); they become “props” that create the time dimension of Newtonian spacetime, within which the particles move exactly in accordance with Newton’s laws. The resulting cardhouse is a uniquely determined rigid structure that can be supposed to be located anywhere in space. Its overall horizontal and vertical scales, along with its orientation in space, are purely nominal and extrinsic. They have no effect on anything intrinsic, that is, on anything that can be observed within the structure. Moreover, just as is possible with a cardhouse, the whole thing can be dismantled into its constituent cards without loss of any essential information. The cards could be thrown down in a jumbled heap and nevertheless reassembled in perfect order, just as a movie film cut into individual stills can be restored. The construction of the “card house” shows that in a universe of finitely many particles, Newtonian inertia and gravity — as we observe them locally — can be recovered without the dubious notions of absolute space and time. Let there be many particles in the universe and a situation in which a few of them are found, as a subsystem of the universe, separated at a large distance from the other particles. This will model the solar system within the universe. The structure of the “card house,” the frame of the universe, is overwhelmingly the work of the many other particles. They create the local inertial frames of reference in which subsystems of the universe, like the solar system, evolve in accordance with Newton’s laws. It is particularly interesting to see how such subsystems can come to exist in a relational universe.

5  Subsystems and the Arrow of Time It is widely stated that Newton discovered a clockwork universe. Artists like Blake found the image of mechanically revolving cogs repellent — a 126

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universe without history and purpose. I won’t claim to know the purpose of the universe, but I will argue that Newtonian theory has been misunderstood: it is not a theory of clockwork but of history. The false impression arose because dynamics grew, to a large extent, out of the study of the solar system, a subsystem of the universe that we now know had, billions of years before Newton’s discoveries, come into existence with planets that settled down into more or less regular motion. Moreover, astronomers were often called upon, or did it to prove their ability, to predict where the planets would be at specified times, often far in the future. This led to great emphasis being placed on the initial-value problem: given arbitrary initial positions and velocities of a set of dynamically isolated bodies, determine their positions at times far into the future (or the past, for that matter). One can, however, take a broader view of dynamics, idealized for simplicity to point particles: for a given number of particles subject to known laws, establish the qualitative nature of all possible solutions. At least in the case of the Newtonian N-body problem this leads to a strikingly different interpretation of Newton’s achievement: he did not create a theory of clockwork but of history. In 1772, Lagrange won a prestigious prize with a justly famous essay on the three-body problem, an example of which — the motions of the earth, sun, and moon — had given Newton headaches. What is probably the simplest of the calculations in his tour de force was the one that reveals the theory of history at the heart of the N-body problem. If the total energy of the system is subject to the mild condition of being nonnegative — zero or positive — then all the corresponding solutions contain a unique point whose only sensible interpretation is that it is the birth of time and, with it, history. A bold claim, but at least within the N-body problem it will stand and is suggestive. For the mathematical details, I refer the reader to [21, 22]. What Lagrange found, and Jacobi later showed holds for any number of particles, is that the quantity that in Newtonian terms defines the size of the system, the center-of-mass moment of inertia Icm,14

I cm = ∑ mi x icm ⋅ x icm , (13) i

 Equal to half the trace of the moment of inertia tensor; xcm is the center-of-mass position of particle i. 14

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is, in Newtonian terms and except for some very special solutions that I will discuss later, infinitely great in the infinite past, decreases monotonically to a finite minimum value — at what I call the Janus point — and then increases monotonically once more to infinity. By time-reversal symmetry, this qualitative behavior is exactly the same if the direction of time is reversed. Plotted against time, the size is a curve that is concave upward. The unique point of minimum size exists by necessity, a direct consequence of Newton’s laws. Note that in the Leibnizian–Machian ideal form of those laws for a relational universe, for which the energy and angular momentum are both exactly zero, the result still holds. In fact, when we look to describe a relational universe and clear away the accidental redundancies introduced during the creation of dynamics by the inevitable study of subsystems — projectiles, falling apples, and the like — the three architectonic features we find are: vanishing energy, vanishing angular momentum, and the Janus point. Here, I must mention an important thing not yet included in this part of the discussion: not only position but also size is relative. If we want to think about the way the universe behaves, we should not imagine that it has a size defined by an absolute external ruler. What Lagrange had in mind was not the universe, but three bodies that, within it, do have location and size, both necessarily relative to the universe at large. But for the universe as a whole, we can only speak properly about its shape, as I was at pains to emphasize. If a system of particles is taken to model the universe, it is the nature of their distribution that characterizes its shape — to what extent is the distribution of the particles uniform or clustered? Although to my knowledge not noted before [21], significant evidence suggests that there is always some point on the timeline of an N-body “island universe” at which the particle distribution is more uniform than elsewhere. The first evidence comes from the rigorous study [23] of the t → ∞ behavior of N-body solutions. This shows that the system breaks up into clusters that typically decay into single particles and Kepler pairs. By time-reversal symmetry, this must happen in both time directions, so there must be a region between these two limits in which the system is at its most uniform. In fact, it has long been noted that an initially uniform distribution of particles inevitably breaks up into clusters under the influence of gravity. This phenomenon led Penrose [24] to argue that gravitational entropy, unlike normal matter entropy, is least when the matter distribution is uniform. This is the opposite of what is usually observed — the distribution 128

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of matter at maximum entropy (in thermal equilibrium) is invariably uniform. I don’t wish to expand on Penrose’s suggestion here, since my views are adequately presented in [3]. However, it does not seem to have been noted in connection with the behavior just described that, by timereversal symmetry, the evolution when the clustered state has been reached can be reversed, so that the system returns to the original uniform state after which it must necessarily become clustered again, if not immediately, then at least eventually. If not, the claim that an initially uniform state always evolves to a more clustered state is false. This shows that there must be a point of maximal uniformity on the timeline. Numerical calculations in [21] strongly suggest that the point of maximal uniformity will be close to the Janus point, the point of minimal size as measured in the extended representation by the center-of-mass moment of inertia (13). In these calculations, the initial state was found by first specifying an initial distribution with random particle positions and velocities that were then “tweaked” — only small changes were needed — to have vanishing values of the energy, angular momentum, and overall expansion. The results of the calculations are shown in Fig. 5.2; this shows the quantity that my collaborators and I call the complexity in [21] and argue is the mathematically distinguished quantity that characterizes the changing nature of the particle distribution. It is most illuminatingly defined as a pure number that measures the extent to which a finite number N of mass points in Euclidean space is distributed uniformly or non-uniformly. It is the ratio of the two simplest mass-weighted lengths associated with the distribution of particles. For particles with coordinates xi, one is the rootmean-square length rms,15

rms =

1 M

∑m m r i< j

i

, rij =|x i − x j|, M = ∑ mi , (14)

2 j ij

i

and the other is the mean harmonic length mhl:

−1 mhl =

1 M2

∑

mi mj

i< j

rij

. (15)

Thus, the complexity C is 15

C=

rms . (16) mhl

 A simple calculation shows that rms ≡(1 / M 1/2 ) I cm .

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Complexity

Figure 5.2   Growth of the complexity on either side of the Janus point as found numerically by Jerome Barkley for 1000 equal-mass particles and “artist’s impression” of the system’s appearance: very uniform in the region of the Janus point and clustered on either side of it. The figure is reproduced from [21].

Complexity is an invariant of the similarity group: it is unchanged by translation, by rotation, and critically by a change in the overall scale of the distribution of particles. As such an invariant it has pedigree. An examination of Fig. 5.2 shows that it defines bidirectional arrows of time that point in opposite directions away from the Janus point. In them, the complexity increases, from a minimum value in the vicinity of the Janus point, with fluctuations between linearly rising bounds. If we imagine intelligent observers to be localized somewhere in time and at some position in the N-body universe, they must clearly be on one or the other side of the Janus point. They will find a pronounced unidirectionality of the universe’s behavior around them. In one direction of what they call time, the universe becomes more clustered and structured. It will be natural for them, as we do in the universe we inhabit, to say that is the

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direction for the future. In the direction of what it is natural to call the past, the universe “was” more uniform. Note that this behavior is found in every single solution of the kind discussed so far. It does not have a statistical origin; it is not obtained as in conventional statistical mechanics by bundling together a Gibbs ensemble of microhistories and demonstrating the entropic behavior of the ensemble. This counterexample to entropic dogma has existed since 1772. I will here merely mention the argument I have given at length in [3] (see also [22]) for this complete difference between generic N-body solutions and the behavior of entropy in thermodynamics and of Gibbs ensembles in statistical mechanics, its atomistic interpretation. The difference is due to the fact that both the latter were developed in their entirety for confined systems, in a box; in clear contrast, N-body systems are not in a box. This leads me to argue that we should not seek an entropy of the universe; the precondition for its definition — a system with a phase space of bounded Liouville measure — may very well not be met. I need to get back to the promised construct: time, with all its conventional properties from Machian things, which I identified as shapes of the universe. Best matching and the definition of duration by insertion of the “props” dt (7) between shapes has created the cardhouse; we can imagine we hold it in our hands and move it around wherever we like in space. This has no effect on its intrinsic structure, which, in the terminology employed in quantum gravity, is background independent. The cardhouse actually has a structure that resembles an hourglass with the waist the Janus point, at which it pinches into minimal size. The behavior of the complexity now gives us the final property of time: direction in either direction away from the Janus point. Since time in the illustrations in dynamics textbooks is almost invariably taken to flow, not vertically but horizontally, we should now turn the hourglass on its side to complete the picture. However, we cannot follow the other widely employed convention in Newtonian dynamics, which is that time flows from an infinite past on the left to an infinite future on the right. In contrast, we have birectional arrows that point in opposite directions away from the Janus point to infinity, both on the left and on the right.

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Within the picture, we have seen how duration and direction are emergent. Direction we experience more or less directly;16 thanks to the existence of clocks, we can measure duration. They too, along with rods and compasses, emerge within N-body solutions with increasing distance from the Janus point. This is most clearly seen in the three-body problem of vanishing energy and angular momentum. Apart from the special (zero-measure) set of solutions to which I will come, the asymptotic state of such solutions in either direction from the Janus point contains a pair of particles moving in one direction from the three-body center of mass and a single particle moving in the opposite direction. The two particles of the pair settle into evermore perfect Keplerian motion around the pair center of mass, each becoming in the process a rod, clock, and compass all in one: the major axis of each particle is a measure of length and defines a direction, true north if you like, while the clock “ticks” every time an orbit is completed. The two particles of the pair satisfy the condition for being good clocks that I formulated earlier: they “march in step” with each other since they pass through perigee simultaneously. In fact, what we have is even better: continuous analog measurement of time by two clocks in synchrony. Each has just one hand. The two hands on a traditional clock, by which I mean a pre-digital one, were a convenience added to what is the bare necessity — a single hand like the shadow on a sundial. And what is the hand for each clock? It is, of course, the radius vector from the center of mass, which is simultaneously the common focus of the orbits, to each of the two particles of the Kepler pair. The “marks” on the clock face are points strung out at unequal distances along each orbit but which signify equal areas that have been swept out. Equally remarkable is that relative to the Keplerian rod-cum-clock-cumcompass, the single particle moves inertially with ever better accuracy, just as Newton’s first law requires. In fact, one can ask for nothing better, since if a system of many particles is considered, the asymptotic state in  I recently watched a podcast of the Institute of Art and Ideas, in which panelists discussed various aspects of time. Paul Davies, one of them, countered very effectively the idea that time flows by pointing out that it is not time that flows but we who change. To see how apt was Paul’s comment, simply imagine yourself at various stages of your life embedded in the “shapes of the universe,” as arranged in the “hourglass.” You grow and acquire memories, which can surely be encoded in the increasing complexity of the shapes at increasing distance, as measured by integrating the increments dt (12), in either direction from the Januspoint waist of the “hourglass.” 16

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both directions away from the Janus point typically consists of many Kepler pairs, which all march in step with each other, serve as rods with lengths equal to their major axis, and point in directions relative to each other that become ever more firmly fixed. They create a framework in which the laws of dynamics hold with accuracy that steadily approaches the ideal that Newton attributed to the divine artificer. The great difference from the arena in which, according to Newton, the motions of the universe unfold is not invisible absolute space in which each point is indistinguishable from any other but the “hourglass,” within which the position and time of any particle is clearly defined relative to the universe in its totality. I hope the discussion in this section will persuade the reader that Newton’s theory of gravity should not be regarded as one of a clockwork universe, but rather a theory of birth and an unfolding history. That it is such a theory derives from a basic property of the Newton potential — homogeneity, which means that the Jacobi action (11) is merely multiplied by a constant under the scaling of all distances. This leads to a symmetry called dynamical similarity and ultimately explains why the solutions exhibit no interesting structure at the Janus point, but become increasingly structured with increasing distance from it. For a discussion of dynamical similarity, see [25].

6  Newtonian Big Bangs It would be possible to end the “construction of time” at this point. The “hourglass” image shows how all the properties we ascribe to time and clocks emerge from differences of shapes. Of course, I have not considered relativistic effects, but at least to a large degree, they can be obtained, as shown in [26], in a similar manner. However, I would like to include at least a brief mention of a further possibility that I discuss in [27] and that may lead to an even more complete theory of time. The special solutions that I mentioned earlier indicate what the possibility is. This relates to the fact that in the Janus-point solutions the size of the system at the Janus point remains finite. It has, however, been known for over a century that solutions of the N-body problem exist in which the size, measured by (13), becomes zero at what is called a total collision. In one, all the particles collide simultaneously at their common center of mass. Solutions in which total collisions occur are special in that they can only happen if two 133

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conditions are satisfied: the angular momentum of the system must be zero, and its shape must become very special on the approach to the collision, taking the form of what is called a central configuration (see [3, 27]). Specialists who work in the field have not been able to find a way to continue such a solution past a total collision in a unique manner compatible with Newton’s laws; they are therefore assumed to break down at a total collision. The interesting thing about a total collision is that, by the timereversal symmetry of Newton’s laws, one can suppose that such a solution does not end at zero size but begins as a “total explosion,” or Newtonian big bang. What is then fascinating is the shape of the central configuration with which such a solution can only begin. The shape is necessarily an extremum of the complexity (either a minimum or a saddle) and is typically rather uniform. Indeed, it can have the maximum uniformity that is possible, since the complexity always has an absolute minimum, at which the shape is more uniform than any other possible shape. For example, in the three-body problem the minimal-complexity shape is, for any value of the mass ratios, an equilateral triangle, and in the four-body problem it is a regular tetrahedron. For a large number of equal masses, the minimum-complexity shape is extraordinarily uniform, with the particles distributed with an almost constant density within a spherical surface, at which the density abruptly becomes zero. The motions of the particles in the solutions that emanate from such an initial shape begin in a rather special way but then become like those on either side of the waist of the hourglass in the much more general solutions (requiring only that the energy is nonnegative) in which a Janus point exists. At least in the framework of the Newtonian N-body problem these total-explosion solutions offer a remarkably complete theory of time. It is indeed an abstraction from change and in this case is not bidirectional but unidirectional. Time is born in the most uniform state the universe can have and then continues forever as the complexity increases. Apart from their intrinsic interest, one of the reasons for taking totalexplosion solutions seriously is that they may serve as a useful guide to what happens at the big bang in general relativity. As noted earlier, zero is a special number. In a total collision, the size of the N-body island universe becomes zero, which is clearly the reason why such an event can only happen if the shape becomes very special. Since the size of a spatially closed universe tends to zero in general relativity if the evolution is run backwards, it could be that something similar to what happens in a total 134

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collision also happens in general relativity. In fact, in Einstein’s theory there is an effect called asymptotic silence on the approach to the big-bang singularity that is rather similar to the manner in which the shape becomes very special in the N-body problem. The similarity is enhanced in the presence of quiescence, a phenomenon that occurs if a special type of matter is present (a massless scalar field or a stiff fluid) and is the basis of the result obtained in [28]. A more extensive discussion of associated possibilities (which can be found in [3], chapters 16–18, and [27]) would here be out of place, since they are speculative and go beyond the primary aim of this paper, which was to show how time can be understood as a construct built with elements that are shapes of the universe.

Acknowledgments I have benefited from discussions with many people on aspects of this paper and must at least mention, in alphabetical order, Alain Albouy, Alain Chenciner, Tim Koslowski, Flavio Mercati, and David Sloan. Recent discussions with Adam Fountain, Pooya Farokhi, and Kartik Tiwari were a significant help in clarifying the issues related to Leibniz’s philosophy.

References   [1]   [2]   [3]   [4]   [5]   [6]   [7]   [8]   [9] [10] [11] [12]

E. Mach, The Science of Mechanics, Open Court (1960). J. Barbour, The Discovery of Dynamics, OUP (2001). J. Barbour, The Janus Point, Basic Books (2020). R. Arnowitt, S Deser, and C. Misner, “The dynamics of general relativity”, republished in Gen. Rev. Grav. 40, 1997 (2008). P. Dirac, “The theory of gravitation in Hamiltonian form”, Proc. R. Soc. Lond. A 246, 333 (1958). P. Dirac, “Fixation of coordinates in the Hamiltonian theory of gravitation”, Phys. Rev. 114, 924 (1959). K. Kuchaˇr, “Canonical quantum gravity,” arXiv:gr-qc/9304012. ´ Murchadha, “Conformal Superspace: the configuration J. Barbour and N. O space of general relativity,” arXiv:1009.3559 [gr-qc]. C. Lanczos, The Variational Principles of Mechanics, Dover. J. Barbour and B. Bertotti, “Mach’s principle and the structure of dynamical theories,” Proc. R. Soc. (London) A 382, 295 (1982). See the article on Leibniz in the Stanford Encyclopedia of Philosophy, Sec. 3. H. Alexander, ed., The Leibniz–Clarke Correspondence, Manchester University Press (1956). 135

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[13] R. Spekkens, “The ontological identity of empirical indiscernibles: Leibniz’s methodological principle and its significance in the work of Einstein,” arXiv:1909.04628. [14] A. Einstein, The Meaning of Relativity, Methuen (1922). [15] D. Saari, Collisions, Rings, and Other Newtonian N-Body Problems CBMS Regional Conference Series in Mathematics, 104 (2005). [16] J. Barbour, “The definition of Mach’s principle,” Found. Phys. 40, 1263 (2010). [17] H. Poincaré, Science and Hypothesis (1905). [18] J. Barbour, “The nature of time,” arXiv:0903.3489. [19] J. Barbour, “The timelessness of quantum gravity. I,” Class. Quant. Gravity. 11, 2853 (1994). [20] J. Barbour, “Scale-invariant gravity: particle dynamics,” Class. Quant. Gravity. 20, 1543 (2003). [21] J. Barbour, T. Koslowski, and F. Mercati, “Identification of a gravitational arrow of time,” Phys. Rev. Lett. 113, 181101 (2014). [22] J. Barbour, “Entropy and cosmological arrows of time,” arXiv:2108.10074 [gr-qc]. [23] C. Marchal and D. Saari, “On the final evolution of the n-body problem,” J. Diff. Eqs. 20, 150 (1976). [24] R. Penrose, The Emperor’s New Mind, OUP (1989). [25] D. Sloan, “Dynamical similarity,” Phys. Rev. D (2018). [26] F. Mercati, Shape Dynamics. Relativity and Relationism, OUP (2018). [27] J. Barbour, “Complexity as time” online talk at https://www.youtube/ watch?v=vskzeQc9Xpl. [28] T. Koslowski, F. Mercati, and D. Sloan, “Through the big bang,” Phys. Lett. B 339 (2018).

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Chapter 6

THE C OM PL EX T I M ELESS E ME RG EN C E OF TI M E I N Q UA NT U M G R AVI TY Daniele Oriti Ludwig-Maximilians-Universität München, Germany [email protected]

We present the arguments suggesting that time is emergent in quantum gravity and discuss extensively, but without any technical detail, the many aspects that can be involved in such an emergence. We refer to both the physical issues that need to be tackled, by quantum gravity formalisms, to concretely realize this emergent picture of time, and the conceptual challenges that must be addressed in parallel to achieve a proper understanding of it.

1 Preambles The purpose of this contribution is to outline a few key lessons about the nature of time from current physical theories as well as from promising theories under development, and the many ways in which the nature of time should be considered an emergent, non-fundamental notion. We will be as comprehensive as possible, but still offer only a summary of results and arguments that we have discussed in more detail elsewhere [1–4], while we also develop these thoughts further, in some cases. In addition,

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we will have to leave out all the technical/mathematical material on which our understanding of the issues is based, citing useful references when needed. As we will try to convey, there is a lot that can be said about physical time, but it should also be obvious (first of all to readers of this collective book, by a quick look at the table of contents) that “time” is a multifaceted notion, so much that one should maybe speak of the many “times” of our world, from the psychological to the physical, from the physiological to the ecological.1 Thus, the first thing to do is narrow down the scope of our discussion. 1.1  Our focus: physical time as deduced from our best theories We restrict our attention to physical time only, out of necessity, and of limited personal competence. Moreover, we approach the question “what is physical time?” in a rather pragmatic manner (from the perspective of a scientist, at least). By “physical time” we mean “what is expressed in the mathematical models used in physics to account for our experience of the world, i.e., our best current physical theories”. It is important to note that the mathematical models used in physics leave out a number of aspects of the physical world and of our interaction with it. Observations and observers are either not modeled at all or are highly idealized. That is, many aspects of actual, real-world, physical observations are left out altogether (for example, details of the measurement apparatuses we use, or physiological aspects of perception). In addition, any physical theory is in itself a model (or collection of models) of a portion of the physical world, and its “lessons” are thus necessarily partial [6]. This implies that our conclusions about the nature of physical time are going to be, by necessity, the result of some idealization. This is something that we should not see as a problem or a disappointing fact, but something to be embraced, exactly because it leaves much to be explored further, about how to improve the same theories, the relation with other sciences and other aspects of the world, and so on. In any case, we would argue (but not on this occasion) that any understanding is modeling and any model is also the result of some mixture of idealization, abstraction, and approximation. Our main focus, i.e., to illustrate the nature of physical time as deduced from the best current theories and from those under  See [5] and all the resources listed at http://www.studyoftime.org/

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development, is also the result of a basic naturalistic attitude. Any metaphysics (e.g., any statement about the ontology of time) can only be deduced, or at least strongly influenced by, and should be necessarily consistent with, our best scientific theories. We simply state this point here in order to frame our approach better by being transparent about some underlying assumptions, but it is also one that would deserve to be argued for. Luckily, other competent scholars have already done so from different perspectives [7, 8]. 1.2  Emergence and its many kinds The main focus of our discussion of physical time is its emergent nature. We will argue that this is already visible in classical general relativity but it becomes even more radically manifest when we consider the (possible) implications of a theory of quantum gravity. Thus, it is worth clarifying beforehand what we mean by “emergence,” and pointing out the many forms in which emergent behavior and emergent notions can show up. These different manifestations will also enter our discussion about time in quantum gravity (for early work on this issue, see [9]). In order to develop our arguments, we rely on the notion of emergence put forward by Butterfield et al. [10, 11] (see [12] for a broader set of views on the issue of emergence in science): a physical behavior or phenomenon is understood as emergent if it is sufficiently novel and robust with respect to some comparison class, usually associated with the class of behaviors and phenomena it emerges from. Very often (but not as a matter of necessity, so as to be included in the definition) emergent phenomena are associated to some form of limit or to some approximation [13–16] applied to the mathematical model describing them and the physical context they emerge from. This definition is simple and general enough to accommodate all known examples of emergent phenomena in physics. It is also somewhat vague at this stage, but any further refinement would be at risk of restricting too much its scope (leaving out some interesting physical examples) and forcing us into unnecessary complications for our purposes. It agrees with the routine use of the term in physics (which is, in fact, even more vaguely defined). It should be taken as a first step toward a full characterization and understanding of specific emergent phenomena, not the final step, something flexible enough to be adapted and deepened when needed in different contexts.

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One key feature of this definition is that it is not in contradiction or incompatible with reduction (this is basically understood within a mathematical modeling of the relevant phenomena or a logical analysis of the corresponding explanatory links, such as deduction). Indeed, reduction is usually needed to specify the comparison class to identify the novel and robust elements of emergent phenomena. The link between emergence and reduction can be so strong that one could often understand one as the converse, complementary process of the other at the epistemological level. In the context of mathematical modeling of physical systems, one often understands some phenomenon as emergent from another exactly because it has been shown to be deducible (i.e. mathematically derivable, within all sorts of approximations) from the other. It goes without saying, then, that we are speaking here only of what is usually referred to as weak emergence, as opposed to strong emergent behavior [12, 17], with which we will not be concerned. Assuming the same naturalistic attitude we stated above, thus not positing any a priori ontology to which then force our scientific theories, but reading it out (with all inevitable ambiguities) from them, emergent behavior poses an immediate dilemma: which side of the emergence/ reduction relation should be assigned an ontological status? The “fundamental” one only, to which the emergent phenomena can be reduced? Or, do both the emergent and the fundamental sides have a corresponding ontology of equal metaphysical weight? In the following, we do not assume a “fundamentalist” ontology, i.e., we do not assume that the nonsymmetric relation between “fundamental” and “emergent” phenomena is also a primacy relation at the ontological level; we instead adopt an implicit multi-layered ontology.2 In the case of time, our arguments supporting the idea that “time is emergent” do not imply automatically, in our view, that “time does not exist” tout court. In this case, the latter statement should be argued for. Finally, we point out several sub-notions of emergence (see [18–21] for more details). All of them enter our discussion about time as an emergent notion. The first kind of emergence is inter-theoretic emergence, which is basically understood as the converse of inter-theoretic reduction. Here, we speak of a set of physical phenomena as emergent from another, if the  This is not to say that the “fundamentalist” view is not reasonable or correct as a metaphysical position, but just that we do not assume it to be true. 2

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theoretical description of the latter can be reduced to the one of the former (i.e., the theoretical description of the former can be deduced, under some appropriate procedure involving limits, approximations, and more, from the theoretical description of the latter). The second kind is ontological emergence, when we say that a set of entities is in fact emergent from another, through some physical process, and under the assumption that the relation is not symmetric and definitely not one of equivalence. This second kind is distinct from the first, generally speaking, but within our naturalistic assumption, it ends up being so strictly associated with it to be indistinguishable for all practical purposes. A third kind of emergent physical behavior is associated with situations in which one set of phenomena is replaced by a (often radically) different one as one considers different values of some “control” physical quantities or parameters, but without any real asymmetry in the relation between the two sets. One can speak of this kind of emergence when referring to a specific dynamical process, in which the control physical quantities evolve in time from one value to another, producing a change in associated physical phenomena; but one can also speak of this kind of (symmetric) emergence as a change in theoretical description, if referring to the mathematical models used to describe both sets of phenomena and depending on the control parameters. Due to the symmetric nature of the relation, speaking of emergence in this case can be seen as an abuse of language. However, on the one hand, this use of the term is found regularly in the physics literature (and very often in the philosophical literature too); on the other hand, our definition of emergence, given above following Butterfield et al., did not include any asymmetry condition. Thus, we have no reason at this stage to exclude this case from consideration. We point out that no reduction is involved by this third kind of emergence, per se, even if in all of the examples we can think of there is an underlying reduction relation, not between the two sides of this “emergence relation of the third kind,” but between each of them and a third set of phenomena, common to both. Looking at the concrete physical situations in which these three kinds of emergence are identified, they are often labeled as synchronic emergence, for the first two kinds, and diachronic emergence, for the third. This is because the third is often associated with physical processes taking place in time, while the first two do not refer to temporal change at all. As we will discuss, this terminology is problematic, first of all because 141

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at the theoretical level, the third kind of emergence can also be defined in a way that makes no reference to temporal evolution, but also, and most importantly for our present purposes, any implicit or explicit reference to time would obviously mess up the application of these concepts to the case of time itself and for the understanding of its own emergent nature. All these kinds of emergence should be considered in the case of space-time in quantum gravity, and time in particular, as we do in the following. Our discussion should be seen in the context of a growing body of work on space-time emergence in quantum gravity in the philosophy literature [22–28], in addition to the quantum gravity literature. 1.3  Making things concrete: an example of emergent behavior The above discussion is certainly overly sketchy and abstract. Therefore, let us give a concrete illustration of the various notions of emergence, using an example that is as uncontroversial as it can get, at least from the point of view of the usual physics parlance.3 Consider the physical system identified as water molecules.4 They are well described in the mathematical language of non-relativistic quantum mechanics, with a Schrödinger evolution equation for their quantum states, a Hamiltonian encoding the relevant forces between them and, in case, external ones they may be subject to, and their quantum observables: their individual momenta and positions, their angular momenta, etc. This description is in principle valid for any number of them, but when their number is high, collective effects become mostly important, the relevant physical quantities we want to have control of are different, and the quantum description in terms of the individual molecule states is not manageable and, in fact, irrelevant for most practical purposes. We usually switch then to a description in terms of (quantum) statistical mechanics, which is also the relevant mathematical context if we are interested in their behavior at non-zero temperature, which is basically always the case. Other macroscopic and collective quantities, such as the pressure they are collectively subject to or the total occupied volume, become important at this level of description. While the quantum properties of water molecules and their constituent atoms are crucial for their  Still, this example has been extensively analyzed and discussed in the philosophical literature, unraveling a number of interesting conceptual subtleties. 4  In fact, for what follows, any atomic system would work. 3

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microscopic physics, as well as for their higher-level chemical consequences, their macroscopic properties are not much determined by quantum features, and if they are what we are interested in, we can study the system in a classical approximation. In fact, we know that in a given range of temperature and pressure, and when the number of molecules is sufficiently large, the description of the system as a single entity called water, a fluid rather than a collection of molecules, and the theoretical framework of hydrodynamics is the most efficient way of understanding the system. At this level of description, new dynamical quantities, such as the density of the fluid or its collective velocity, and new physical concepts, such as viscosity, vorticity, and so on, are the relevant ones. The situation is more complex than this, though. At the same macroscopic level of description, for large numbers of molecules but different ranges of temperature and pressure, what replaces the microscopic description in terms of molecules is not hydrodynamics, but a description in terms of lattice structures and their deformations. Liquid water turns into ice, a solid, whose physics is very different from that of liquids and requires new concepts and different mathematical tools. At other values of temperature and pressure, without changing the microscopic description of the system, we have vapor instead, with yet another description and different physics. Liquid water, vapor, and ice are different macroscopic phases of the same microscopic system, separated by phase transitions. Where can we speak of emergence in all this? In several places, in fact [29, 30]. The very step from a quantum to a classical description of the water molecules, before one considers their collective behavior, is often taken as an example of emergence — that of a classical world from the quantum one. There is no doubt that classical properties are novel with respect to the typical quantum behavior and robust enough that we can often forget the underlying quantum world. It is then a case of emergence, but one that does not entail any ontological emergence, since the basic entities at both classical and quantum levels are the same molecules, which simply change physical behavior. It is, however, an inter-theoretic emergence, since we do indeed change theoretical framework to describe the same molecules in the two regimes, from quantum to classical mechanics. However, the move from the molecular dynamics to the fluid hydrodynamics is much more radical. It is a case of inter-theoretic emergence as well, but one that entails a switch to a different set of dynamical variables and observables, and marked by the appearance of new concepts 143

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altogether, which are simply not applicable before the switch (what is the viscosity of 10 molecules?). At the same time, we know that all of the above is not incompatible with reduction, since there is a clear sense in which we can reduce hydrodynamics to molecular dynamics and, for example, define hydrodynamic variables in terms of suitable averages of molecular ones. The counterpart of this radical inter-theoretic emergence is also a change in the basic ontology. If we base this on the theoretical description of the system — a collection of molecules, first, each with associated particle-like properties, and a fluid, then, described by continuum fields (density and velocity) — we have ontological emergence, too. This is a consequence of a different type of limit/approximation that the quantum-to-classical one: a continuum and thermodynamic limit, without which we would not have a liquid-like continuum phase. The distinction, conceptual and mathematical, between classical and continuum approximation is also a crucial point in the quantum gravity case. In fact, not only are the two limits very distinct, but the order in which they are taken has, in general, important consequences. If we had considered helium-4 atoms instead of water molecules, approximating them first with classical entities and then considering their continuum limit would have given a very similar hydrodynamic behavior to that of water; on the contrary, taking the continuum limit while retailing their quantum properties would have led us, in the appropriate range of temperature and pressure (plus a few other conditions), to superfluid hydrodynamics first, as a macroscopic consequence of their quantum statistics. One needs a further classical approximation at the continuum level to recover standard hydrodynamics. The many striking properties of superfluids would have remained hidden if we had conflated, conceptually and mathematically, the two types of limits. The result of this limit, however, is not unique. Depending on the value of temperature and pressure, here treated as external control parameters of the theoretical description, from the same molecular system, we could arrive at a solid system or at vapor. The system can organize itself, macroscopically, in different inequivalent continuum phases, which correspond, in fact, to different macroscopic systems with very different properties (and ontology?) and which can all be said to be emergent with respect to the molecular one. These are all examples of intertheoretic and ontological emergence [13, 14, 16]. Notice that the relation between the two descriptions, and the corresponding physical systems is 144

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not symmetric. There is a clear sense in which the fluid comes from the molecules, and not vice versa (not least, because ice and vapor also come from the same molecules). Notice also that there is no dynamical process involved, i.e., nothing that the molecules “do” to become liquid water or ice, no dynamical evolution from molecules to water or ice. It is the theoretical description that changes, not the molecules or the fluid. We can speak of synchronic emergence. However, the existence of different continuum phases allows for phase transitions. These are switches between different theoretical descriptions of the same system (from the microscopic point of view) or between two different physical systems with their associated theoretical descriptions (from the macroscopic point of view). However, they can also be seen as actual physical processes, taking place (in time) when the control quantities (temperature and pressure) change. Phase transitions can be seen, then, as examples of emergence of the third, diachronic kind mentioned above. To conclude, three points are worth mentioning. First, in the case of phase transitions, the emergence relation is symmetric; there is no sense in which one would say that ice is more fundamental than liquid water, or vice versa, either at the ontological or at the theoretical level, and they are on an equal footing with respect to the molecular dynamics. Second, while it is a fact of life that the phase transition can be a temporal process (ice melts, and liquid water boils), its usual description is in terms of equilibrium statistical mechanics where time and dynamics play no role. It is considered an approximation of a more realistic (and much more involved) non-equilibrium description in terms of our standard, non-relativistic temporal evolution with respect to an absolute time, but the equilibrium description is not inconsistent in any way. Third, we find it straightforward to interpret the phase transition as a physical process because the relevant parameters can indeed be controlled and made to change from the outside, by the observer or the experimenter manipulating the collection of molecules in the lab (or at home). The last two points will be especially relevant in the discussion of the emergence of phase transitions in quantum gravity, since in that context, the situation is much more tricky.

2  What is Time, in General Relativity? Let us now turn to time in general relativity, our best theory of time, space, geometry, and, indeed, gravitational phenomena. The reason for 145

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the latter link is that gravitational phenomena, in general relativivity, are equivalent to geometric properties of space-time, i.e., statements about duration of time intervals, distances between objects, relations between reference frames, i.e., the very notions of space and time directions used by different observers, and so on. Gravity is space-time geometry, in general relativity, that is its main lesson. This encapsulates our current best understanding of both gravity and space-time. So we take it seriously and ask what is time in this context. 2.1  Manifold points, diffeomorphisms, and background independence General relativity is a theory of continuum (in fact, smooth) fields defined on a (differentiable) manifold (i.e., a set of points with appropriate regularity properties). Thus, the ontology behind the theory is given by these elements: fields and manifolds. Often, we label the manifold a “spacetime” manifold and identify its points as “space-time events,” i.e., the loci where and when things happen. Among the fields, the metric field enters any determination of the geometric properties of space-time and, in a way, of space-time itself. We routinely speak of spatiotemporal distances between events, space-time curvature at a specific event, time elapsed between two events taking place at different points in space, and causal relations between events. All these quantities are functions of the metric field, leading to the identification of the metric field, thus the gravitational field, with the geometry of space-time itself instantiated by a manifold of events. This is all good, and, as a convenient fiction or an approximate substitute for a more rigorous story, routinely used with success in doing physics. Taken more literally, though, it is unsatisfactory. The reason is that general relativity is invariant under diffeomorphisms of the manifold. Thus, mapping the points of the manifold to one another, and mapping the value taken by all fields defined on it according to their consequent transformation, then the values of fields at different points in the manifold are physically equivalent if they are related by a diffeomorphism transformation.5

 All this can be expressed in terms of coordinates, as is often done, by identifying diffeomorphisms between points as changes of coordinates at the same point; however, this can be very misleading, since physical theories can be written in a coordinate-independent manner, and this does not change in any way their symmetry properties with respect to diffeomorphism transformations. 5

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The debate on the precise implications of this fact is still active [31–33], but we subscribe to the view (rather predominant in the physics community and in a good part of the philosophy community) that the manifold and its points do not really carry any physical meaning. Thus, they are not part of the ontology of the world, if not as providing global (topological) conditions on the fields. The world is made up of fields and fields only. Moreover, diffeomorphism invariance is closely related (in fact, up to some additional subtleties, it can be identified) to background independence [34, 35]. That is, all fields are dynamical entities, subject to their “equations of motion,” constraining their allowed values and mutual relations. Generic solutions of the equations of the theory, i.e., generic allowed configurations of fields, possess no feature that can be used to single out a preferred direction of time or space. This absence of a preferred, non-dynamical, absolute notion of time is often indicated by the statement that “there is no fundamental time” in general relativity [36], contrary to what is the case in all non-general relativistic physics, including standard quantum mechanics and quantum field theory. So, the absence of absolute notion of time in general relativity is a fact. We can get around this uncomfortable fact only when working with specific solutions of the theory, like the flat Minkowski geometry of special relativity or other highly symmetric configurations of the metric;6 in this case, the symmetries of such configurations allow us to identify special directions on the manifold as a preferred temporal (or spatial) direction, and we deal with an absolute (maybe up to some further transformations, e.g., Lorentz) global time. But beyond these special cases, there is no (preferred) time in GR. 2.2  Relational time (and space) If what we are left with are dynamical fields, including the metric (i.e., the gravitational field), where is time in GR? And what do we mean by events and their temporal interval? And how is it that we can routinely use coordinates and manifold points to identify events, and functions of the metric defined at such points to measure time? The general answers to these questions, at least as a matter of conceptual clarity and formal mathematical constructions (the detailed physical constructions can be problematic), are given by a “relational strategy” [37­–40]. If the only things that exist are fields, the only physical observables are relations among (values of) fields.  Or special boundary conditions, corresponding again to highly symmetric geometries.

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In particular, any quantity that we interpret in a spatiotemporal manner, in GR, is a relation between specifically chosen fields, used to define clocks and rods and, by doing so, time and space. All such quantities that correspond to some determination of geometric properties of time and space are then appropriate functions of fields, necessarily including the metric. In this sense, also in any rigorous construction of spatiotemporal quantities, the gravitational field will be a necessary ingredient, justifying its often-stated identification with space-time itself. Let us give a sketchy example. Instead of having two fields each evaluated at point in “time” t identified with the value of some coordinate along a given manifold direction, we use one of the field as a clock and its values to define instants of time, and measure the evolution of the (values of the) other field as a function of (the values of) such a clock (the coordinate time and the direction on the manifold have then entirely disappeared from the physical picture, as they should). The theory itself does not select any physical field as the preferred choice of clock (thus, time). This is the more physical way of characterizing the general covariance of the theory without referring to unphysical coordinates or manifold directions. Most importantly, in general, fields behave like good clocks only approximately, only locally (i.e. for some limited range of their values) or in special (and dynamically chosen) configurations. To have a good notion of time (or, equivalently, a good clock), the physical system used to define it should not interact too strongly with the other fields (including the gravitational field) or with itself; it should not have exceedingly complicated dynamics; and, for the same system to define a globally valid notion of time, further restrictive conditions should be at least approximately true. In particular, in the approximation (or idealized case) in which physical systems do have a vanishing energymomentum, and thus vanishing effect on the geometry of space-time, vanishing self-interaction, and trivial dynamics, then they do behave like simple coordinates labeling manifold points, and can then be used to define time (and space), and the evolution of other fields, forgetting about their own physical nature. In the end, in general relativity, there is no (preferred, external) time, but there are many (infinite) imperfect, approximate physical clocks, each providing a possible definition of physical time with its own limited applicability. In general relativity, then, time is a specific (set of) approximate relation(s) between continuous physical fields, and from them it inherits

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its continuous, ordered (and approximate) nature. A final comment is probably useful. An interesting body of work in the classical and quantum gravity literature concerns the “deparametrization” of general relativity in terms of suitable matter fields [41–44]. This entails rewriting the full theory in relational terms with respect to the reference frame defined by them, in such a way that one does not need to deal with diffeomorphism symmetry anymore, that all resulting relational observables are physical, and that the dynamics takes a more standard form with respect to the time defined by the appropriate component of the matter fields. The drawback of this type of strategy, at the classical level, is that the matter fields that allow for this type of rewriting are always somewhat unphysical. In turn, this is basically inevitable since they have to provide exactly that type of global, perfect clock and rods, with associated globally defined notions of time and space that, as we discussed, we expect to be no more than an idealization or the result of some approximation. The above means that, from the point of view of general relativity, time as an absolute, uniquely identified notion, as in Newtonian mechanics and pre-relativistic physics, is an emergent notion. It appears for special configurations of the gravitational field, in the approximation in which all relativistic effects are suppressed and all physical clocks (that is, other matter fields used as such) behave uniformly, in addition to being assumed to have no impact on other dynamical entities (so that they can be treated as measuring an external parameter). It is then an instance of inter-theoretic (or vertical) emergence. Whether it can also be considered an example of ontological emergence is more dubious. While in the step from relativistic physics to Newtonian mechanics fields are replaced by particles and forces as fundamental entities (this is indeed an example of ontological emergence), those fields whose relations define time in GR would stop being part of the dynamical entities of the world and give rise to an absolute, external, non-dynamical notion of time that can hardly be considered a physical entity on its own.

3  What is Time in Quantum Gravity if it is Quantized GR? We expect, however, that general relativity is not the final story, and that there is more to say about time in physics. We expect this, for example, because of puzzling aspects of astrophysics and cosmology like dark matter and dark energy, which may be explained by modified gravity theories 149

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at the classical level. Most importantly, we expect that general relativity or other classical modified gravity theories should be replaced by a quantum theory of space-time, geometry, and gravity at the more fundamental level. The arguments for this conclusion are many and, even if not fully conclusive, make the conclusion consensual in the community. Therefore, we should consider what happens to time in quantum gravity. The detailed answer will depend on the specifics of the theory, but such a more fundamental theory of quantum gravity has not been established and there are several candidates, which are in fact quite different in their basic mathematical structures and principles, although they also share many ingredients. We have to content ourselves with a less detailed answer based on general aspects shared by several quantum gravity formalisms or simply part of the definition of the quantum gravity problem. 3.1  Quantum GR, quantum fields, quantum (relational) time We would identify as a theory of quantum gravity and quantum spacetime, roughly speaking, any quantum theory that reduces to (some modified version of) general relativity in a classical (and probably macroscopic) approximation. We should also consider the possibility that quantum formalism itself has to be modified to be applied to space-time, but for simplicity let’s assume for now that this is not the case. Then, the simplest possibility, at least conceptually, is that the fundamental quantum theory is the result of “quantizing the classical gravity theory” by one of the many quantization algorithms we have successfully applied to other field theories (canonical, path integral, etc.). Several approaches to quantum gravity can be understood from this perspective [45–47]. Then, the ontology of the resulting quantum theory is the same as that of classical theory. The world remains constituted by continuum fields, among which the gravitational field is the one strictly associated with geometric space-time properties. Physical notions of space and time, as in GR, remain dependent on relational constructions involving several (components of) dynamical fields, except in very special cases where preferred temporal or spatial directions can be identified. However, the same relational constructions should be performed at the quantum level, and the same fields become quantum systems starting from the gravitational field. This step to the quantum domain brings radical changes to our notions of space and time. 150

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The quantum nature of the gravitational field implies the quantum nature of space-time geometry, and the quantum nature of all the other dynamical fields implies the quantum nature of any notion of time constructed using them as relational clocks (more generally, reference frames). Possessing a quantum nature means being subject to uncertainty relations, irreducible quantum fluctuations, some form of contextuality, discreteness of observable values, and, in the case of composite systems, entanglement, in turn challenging our common sense notions of realism, separability, and locality. When these properties are attributed to geometric and spatiotemporal quantities, we clearly enter a new and wild conceptual (and physical) domain. All geometric quantities, such as areas of surfaces, distances, and temporal intervals between events, curvature in a region, are then subject to superposition and quantum fluctuations; they may be forced to have only discrete values; they may be incompatible with one another (like position and momentum of a quantum particle); and they may be restricted to a contextual-only specification. The causal structure of space-time itself (i.e., the list of potential cause–effect relations) will be similarly subject to quantum fluctuations and superpositions. And the list of quantum space-time oddities could go on. The relational strategy for the definition of time will be further affected by the quantum properties of any physical field we choose as our relational frame, in particular our clock [48–50], itself subject to quantum fluctuations, uncertainty relations, etc. So, time will be whatever it was in GR, but quantum. But a quantum time is even farther away from any standard notion of time on which our common sense and our classical physics is based, so that we can definitely say we are in a very different world, or better yet, at a very different, more fundamental level of our understanding of this world. The physical and philosophical literature has barely started to explore these new conceptual depths due to the limitations of our current quantum gravity formalisms, but it should be clear that we will need a profound reconstruction of the basic pillars of both our physics and our philosophy to account for these new aspects. With this quantum step, we must abandon, for example, any value-definiteness of spatiotemporal quantities and possibly any continuous notion of space and time if spatiotemporal observables end up having discrete values [51]. The idea of quantum reference frames and indefinite causal structures or temporal order, moreover, represent new challenges for the foundations of quantum mechanics themselves [49, 50], and together with the notion of a preferred time direction, we are forced 151

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to abandon unitary time evolution as the key dynamical element of quantum mechanics. To make sense of dynamics in the absence of such unitary evolution is sometimes referred to as “the problem of time” in quantum gravity, and it is the direct quantum counterpart of the diffeomorphism invariance of classical general relativity (as such, it affects space in the same manner). As we argued, there is no “solution” to this problem, strictly speaking, as long as the symmetries of classical theory are preserved [52–54]. The absence of a preferred temporal direction is a fact, and the relational strategy is the best way, we maintain, to extract from the theory a physical, if approximate, notion of time and evolution. 3.2  The timeless emergence of time from quantum time The quantum aspects of the story are not minor additions, and they fully justify speaking about the emergence of the general relativistic time from quantum gravity. To start with, they prevent any of the circumvention strategies of the problems given by the absence of a preferred time direction that are available in classical GR, e.g., working with special solutions possessing preferred temporal directions (in quantum theory, there is no state that corresponds to a classical solution of the theory, if not approximately only). This is the sense in which the “problem of time” is worse in quantum GR than in classical GR, even if its origin is the same, i.e., diffeomorphism invariance. The set of approximations and mathematical procedures, choices of special states, and focus on specific observables that lead to classical GR from quantum gravity (realized to various degrees of success in current quantum gravity formalisms, see for example [55–57] in the loop quantum gravity and spin foam context) that go under the collective label of “classical limit” would also lead to recovering the temporal observables are computed as relations between classical fields, from corresponding quantum observables. There will be novelty enough, since the classical limit brings continuous quantities where, probably, there were discrete ones, value-definiteness where there was indefinite temporal order and fluctuating temporal durations, and the classical dynamical aspects of time and its relation to space where we had the limitations of contextuality and non-commutativity of quantum observables. And certainly, the classical world described in its spatiotemporal aspects by general relativity is robust enough, since quantum features of gravity and space-time are so hard to detect (so much so that we have very little guidance from 152

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observations in our search for the more fundamental quantum gravity description). In this respect, we can also see the limitation of the global deparametrization strategy at the quantum level. Deparametrizing the theory at the classical level and then quantizing, e.g., by reduced phase space quantization [42–44] in terms of the canonical decomposition defined by the matter fields introduced as clock and rods, requires neglecting the quantum nature of the clock field itself, thus of an important aspect of its physical nature, and of the effects that its quantum properties may have on other fields. Again, we see that a global notion of time, convenient as it may be and also in this more physical sense of a relational clock, remains an idealization or something that can be valid only in some special approximation of the fundamental theory. This further sense in which time as we know it is emergent, this additional level of emergence is again, first of all, inter-theoretic and synchronic; it corresponds to a change of theoretical framework and associated conceptual one, but it is as such not a physical process in itself and by definition not a temporal one in any case. Thus, it would not qualify as an instance of diachronic emergence. It is not an ontological emergence, either, since we have assumed that the fundamental entities in quantum gravity are the same fields that GR deals with, only turned into quantum entities.7 We lack a complete theory of quantum gravity, though, so we do not know the precise details of the physical circumstances that allow us to pass from the fundamental description of the world, in which all fields including the gravitational one, and thus space-time, are quantum entities, to the one of GR, in which space-time, geometry, and all dynamical fields behave classically. In this situation, it is hard to clearly characterize how this transition to a classical world comes about. We expect it to be the result of a physical process, not only a shift in the most appropriate theoretical description. It should take place whenever the relevant length scales, curvature scales, and related change value from trans-Planckian to

 In our analogy with water molecules and liquid water, we remain at the hydrodynamic level with the system of interest in liquid water and we move to a regime in which the quantum nature of the fluid (not the constituent molecules, whose existence we simply ignore and never enter the picture) can be neglected altogether. The analogy is not as compelling in this case, since liquid water is a fluid whose macroscopic quantum properties can basically always be neglected. A better analogy would be with superfluid, which manifests macroscopic quantum behavior and thus a non-trivial classical limit. 7

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sub-Planckian ones.8 This has happened, for example, in the very early universe, close to the Big Bang singularity predicted by classical theory. In fact, many quantum gravity theorists and cosmologists would expect just that: a transition from a fully quantum epoch in which full quantum gravity is the correct description of the world, close to the Big Bang, to a classical one governed by general relativity (better, semiclassical gravity with quantum matter fields living on classical space-time), when the universe is larger and it has “classicalized” in its spatiotemporal aspects, due also to the interactions between quantum matter fields and quantum geometry. This is all good as far as the general idea of a quantum-to-classical transition being a physical process is concerned. It becomes problematic if we also take our temporal language seriously and interpret the transition as something happening in time. Consider the picture in more detail as we move backward toward the early universe, starting from our present classical one. We select a time variable, first of all, to speak of early and late universe and of cosmological evolution. The specific choice is not so relevant now, but it corresponds to some physical degrees of freedom (a field, a component of it, or a function of it) being used as a clock, with all the other physical quantities expressed as a function of its value. This value should be well-defined to be used as a good reference. In a quantum world, this means that it corresponds to a semiclassical observable, whose mean value we use as reference value and whose quantum fluctuations are negligible compared to it (otherwise, we would not be able to “follow” the evolution at all.9 Then we follow the evolution of the universe and all that it contains in relation to this clock toward earlier epochs. At some point (in clock time), quantum aspects of the world (i.e., of all the physical systems) start becoming relevant. As we move further toward the Big Bang, they become even  Let us stress that this expectation is in line with our effective field theory intuition and is based on established general relativistic and quantum physics, therefore it is very solid but also subject to change in a more fundamental theory. 9  Notice that the possibility that we use instead the eigenvalues of an observable chosen as clock is problematic, although often studied [58], because in this case we would be considering states in which the conjugate observable to the clock time is maximally uncertain, and we should expect these large quantum fluctuations to be physically relevant to, impacting somehow on the evolution of the other physical systems. We would be dealing, in fact, with a highly quantum state of the system; it is not the kind of time variables we use in cosmology. 8

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more relevant, and we enter the full quantum gravity domain. The transition is then understood as a physical and temporal process. Not so fast, though. For this to be the case, the temporal observable defined by our clock should remain (approximately) classical; otherwise, the very notion of relational time evolution would cease to be meaningful. Even within a full theory of quantum gravity, there may well be a physical situation like this, with a specific subsystem, which happens to be the one we have chosen as a clock (maybe exactly for this reason) remains classical, while all the others start manifesting strong quantum fluctuations and other quantum properties. But then there is no emergence of classical, relativistic time from the quantum version of the same we have in the full quantum gravity theory in such transition. It does not correspond to the situation we wanted to interpret as a physical process with respect to time itself. In a situation in which, on the contrary, as we follow the evolution of the universe backwards in relational time, our clock itself starts manifesting more and more its quantum nature, we do indeed face the type of transition from quantum time to classical time (or vice versa, in this case) that we were considering to speak of the emergence of classical time as a dynamical, physical process, but its own temporal characterization stops exactly where (or when) the transition occurs. Beyond that, in the full quantum regime in which we only have quantum time (and space, geometry, and causality), any standard notion of temporal evolution (including the relational one) stops being applicable. No diachronic emergence again. The emergence of time remains timeless. We are going to encounter again this type of situation (an inter-theoretic emergence that is not ontological, that could be in correspondence with a physical process, that can be “met at the end point of some temporal evolution,” but that remains timeless) again in the following section.

4  What is Time in Quantum Gravity if it is not Quantized GR? The story of the progressive disappearance of time in a quantum gravity context, when moving toward the more fundamental level, and of its emergent nature when seen in the opposite direction, takes a more radical turn in quantum gravity formalisms in which the theory is not the straightforward quantum version of classical general relativity (or other gravitational theory for the metric field coupled to matter fields). By 155

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definition, these formalisms are defined in terms of a different ontology. The fundamental entities are not continuum fields (including the metric), be they classical or quantum, and such continuum fields are collective, emergent entities. 4.1  A timeless ontological emergence of time What the new fundamental entities are depends on the specific formalism one considers, and similarly formalism-specific is how different they are from the usual spatiotemporal ontology of fields. The notions of space and time as encoded in field-based observables in general relativity are complex notions, we have explained, involving topological and metric (extension) features, continuity, and directionality, causal, and relational aspects. Therefore, we are in the presence of new entities, whenever we do not deal with continuum fields, and with a more radical absence of the usual notions of space and time, i.e. a non-spatiotemporal level of understanding the physical world, whenever one or more of these features are absent or modified. And the more ingredients of the usual complex notions we drop, the farther away from space and time we end up being. We stress that this is an independent step away from space-time, as we know it with respect to the upgrading of the classical constructions to the quantum level, distinct from the inclusion of the quantum properties of fields into the picture. Examples of candidate fundamental entities of a non-spatiotemporal nature, in the above sense, are the spin networks and spin foams in canonical Loop Quantum Gravity [59], spin foams models [60] and group field theories [61], the abstract simplices and associated piecewise-flat geometries in lattice quantum gravity [62], spin foam models [60], and tensor models and tensorial group field theories [63­–65], the posets of causal set theory [66]. Moreover, in string theory, the same idea of an emergent space-time is central, thanks to non-perturbative results like the string dualities [67], even though there are less clear suggestions for what the fundamental non-spatiotemporal entities could be; in proposals like the IKKT Matrix Theory [68], for example, they are in fact of a similar combinatorial and algebraic nature than in the other approaches. This is a very radical step. The big challenge at the conceptual level, beyond the physical and mathematical ones, is to gain any sort of intuition about the nature of the new fundamental entities, since our physical thinking is grounded solidly in space and time. Understanding is more than just intuition. This change in fundamental ontology requires, by 156

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definition, the development of a metaphysics in which the usual notions of space and time, thus location and ontological distinction based on it, spatial contiguity, temporal change and permanence, and so on, do not feature in the very definition of existence or reality. It is quite a challenge for contemporary and future philosophers, and an important one. A second set of ontological worries concerns space-time in this emergence scenario, and it has to do with how the ontological status of the entities featuring in our physical theories is affected by established emergence relations. If one adopts a “fundamentalist ontology,” according to which only the entities featuring “fundamental” theories are entitled to ontological status, while those appearing in emergent or effective descriptions do not, then the logical conclusion is that space and time cannot be real entities. This metaphysical position is questionable in itself, even before considering quantum gravity. Think of the paradigmatic example of emergence given by molecules and fluids, where this position would imply that fluids do not actually exist in a proper sense. It is clear, however, that a new level of philosophical difficulties arises when it is space and time that are deprived of their status as elements of reality. The way out is some form of “multi-level ontology” in which both fundamental and emergent entities are understood as real. How to articulate in detail and in a solid manner this form of ontology, however, is itself an interesting and non-trivial challenge. 4.2  The timeless emergence of time via a continuum approximation Let’s move back to the physical aspects of space-time emergence in this scenario. Starting from the new candidates for fundamental quantum entities of the world proposed by one or the other of the current quantum gravity formalisms, the emergence of space-time corresponds, then, first of all, to the emergence of fields, including the metric, and (a modified version of) general relativity from their collective quantum dynamics. Once this step is taken, the usual (relational) space-time constructions and notions of relativistic physics can be used, and further down the line, the common sense notions of space and time can be obtained in correspondence with special cases or further approximations. This step is what goes under the generic label of “continuum approximation” in quantum gravity approaches, since in most cases, the new non-spatiotemporal entities are discrete in nature, in one form or another. Understanding how to perform this step is (one of) the main outstanding issue in all quantum gravity 157

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approaches. To do so, these approaches have to first adapt to a background-independent context, and then apply the same methods routinely used in quantum many-body theory to extract macroscopic and collective physics from the fundamental quantum dynamics, e.g., coarse graining techniques and renormalization groups. This is, in fact, a very active research area in fundamental quantum gravity approaches [69–73]. The result would be to pass from the “atomic” description of quantum gravity to the quantum gravity counterpart of an “hydrodynamic” approximation in terms of continuum quantities, within which one could then extract a classical general relativistic dynamics for space-time and geometry. In some approaches to quantum gravity, like tensorial group field theory, this hydrodynamic analogy is in fact quite literally realized in order to extract some effective gravitational (cosmological) physics [4, 74, 75]. The extraction of gravitational dynamics and its approximate rewriting in terms of temporal evolution would then be performed at the effective, hydrodynamic level, adopting once more the relational strategy in terms of the emergent fields (for an example of this procedure, in the same quantum gravity formalism, see [76, 77]). If the starting point is quantum and the result is classical, a “classical approximation” should be involved as well, together with the continuum approximation. However, we stress that this is a distinct step, conceptually, physically, and mathematically. Depending on the details of quantum gravity formalism and of different physical situations, it may take place in conjunction with the continuum approximation, having them somehow intertwined. However, being distinct, they may take place independently. In this case, one should further notice that the order in which they are taken is not irrelevant. Experience with quantum many-body systems teaches us that the quantum properties of the fundamental constituents may be crucial for capturing the correct macroscopic properties of the collective system formed by them. In the analogy with atoms or molecules and fluids, we have quantum atoms whose fundamental description is expressed by a many-body quantum Schrödinger equation or a corresponding quantum field theory, and whose collective physics can be described by the hydrodynamics of a continuum fluid, capturing all the relevant observables and dynamics at a macroscopic scale in which the individual dynamics of the atoms is not of interest and the atoms themselves are not relevant anymore as physical entities.

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But we know very well that there are fluids and superfluids, differing at macroscopic level in a physically (and technologically) very relevant manner, and that specific features of the latter are directly due to the quantum properties of the fundamental atoms, e.g., their bosonic statistics. In order to correctly capture these properties, one cannot take first a classical approximation of the quantum theory of atoms (after which they would be described, say, as classical particles) and then look for a continuum, hydrodynamic approximation. That is, for many systems, it works fine, but for many others, it fails to be physically correct, failing to account for macroscopic quantum effects like superfluidity, superconductivity, quantum phases of matter, etc. Notice that we are not referring only to the quantum properties of the atomic constituents being visible if one looks carefully enough beyond the continuum, hydrodynamic approximation, or at small enough distance scales or at high enough energy. This is not under question. It is a logical necessity that one can go beyond continuum description and be able to distinguish physical effects due to the underlying atomic structure. Otherwise, the atomic description would not qualify as useful as a physical theory. In the quantum gravity case, there is no question that fundamental quantum gravity models should predict modifications of classical GR at high energies or small distances or due to the quantum nature of fields. The same is also true for quantum gravity formalisms that are the straightforward quantization of the classical theory. In a context in which space-time is emergent in this more radical sense, an entirely different type of quantum gravity effect is at least conceivable. Some of them have to do with the very existence of different fundamental entities, and maybe with their discrete nature, but not directly with their quantum properties. Others will be the result of their quantum properties. In both cases, their collective effects can well be macroscopic, i.e., visible at low energy and large distance scales, since they are not captured by the effective field theory intuition based on given space-time structures. An example of a physical phenomenon that has been discussed in this spirit, in a quantum gravity context, is dark energy [78]. What is the case for space-time and gravity? Are some of their features directly due to the quantum properties of the fundamental, non-spatiotemporal quantum gravity constituents? Is our universe akin to a quantum fluid? We do not know yet. We are trying to find out.

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4.3  What kind of emergence is this? In terms of classifications, this more radical type of emergence of spacetime in quantum gravity theories is certainly associated with ontological emergence, as we have stressed. And it is necessarily so in our “naturalistic framework” an inter-theoretic emergence, too. The step from the description of the world in terms of quantum gravity atoms to the one in terms of continuum fields may also be made necessary in correspondence with physical situations, e.g., with cosmological phenomena close to the Big Bang being the natural candidates. It may well be the case that at the high densities found close to the Big Bang, at the very origin of our universe, the “hydrodynamic” description of the universe corresponding to general relativity breaks down not (only) because quantum effects manifesting the quantum nature of fields become important but because the very description in terms of continuum fields, thus continuum space and time, break down. This breakdown could then be understood in terms of the properties and dynamics of the quantum gravity atoms themselves, providing an explanation and a quantitative account for it. The new atomic description offered by a more fundamental theory of quantum gravity would be necessary. Thus, the emergence of space-time from nonspatiotemporal atoms, thus also in this more radical sense, would be as physical as it can get. But would it also be an example of “diachronic” emergence? Would it be a “temporal” process itself? It would be hard to respond in the affirmative in general. Consider again the analogy between atoms and fluids. There are many physical situations in which the atomic structure of fluids is immediately relevant for understanding their properties, and in which, correspondingly, we have to switch from the hydrodynamic description to the quantum atomic one. The example of superfluidity is, again, a good one. But we would not say that something has happened to the atoms that have “produced” the fluid as a result when we look at a Bose condensate manifesting its superfluidity properties more and more (or less and less) in changing physical conditions. We remain in the context of an inter-theoretic relation, with a solid physical motivation and impact. There may be cases in which a “temporal” characterization is, instead, quite appropriate. In the same example of fluids, consider the case in which the fluid becomes less and less dense as a result of its own dynamics. Then, (quantum) statistical fluctuations due

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to its atomic structure could become, in relative terms, more, and more relevant and thus force us to switch to the atomic description of the system. One could then say that the emergence of the fluid from atoms, and vice versa, is associated with a dynamical process as well. Bringing this intuition to the case of time in quantum gravity, however, we face exactly the same type of conceptual and technical difficulties we encountered when discussing the emergence of (classical) time in quantum general relativity from the disappearance of quantum effects from the dynamics of fields. We can hypothetically follow the evolution of the universe toward the would-be Big Bang, only to find out that, sufficiently close to it, the whole continuum description of the universe in terms of spatiotemporal fields needs to be replaced by one in terms of non-spatiotemporal quantum gravity atoms, but there is no way we can speak of temporal evolution once we (have been forced to) adopt such description. This conclusion can only be avoided if some (relational) notion of time is somehow preserved from dissolution into the atomic description, so that it can still be used to describe the dynamics of the “atoms of space” themselves. For example, this would happen in a formulation of the fundamental theory in which a deparametrized dynamics with respect to some internal clock variable can be adopted before quantization, with the consequence that the corresponding relational time is treated as an external parameter and can be used to label the evolution of all the other nonspatiotemporal entities. This procedure is used in some formulations of tensorial group field theory, and in its cosmological applications [79–81], for example. However, as is the case in classical GR, the resulting theory can only be, in general, a special sector of the full theory, whose predictions are at best only approximately valid and only in special circumstances. Specifically, they will only be valid provided one can neglect the quantum and dynamical nature of the degrees of freedom chosen as the relational clock. In a more general situation, though, we cannot expect this to be valid. In other words, by its very definition, the temporal description of the process leading to the disappearance of time ceases to be valid exactly when such disappearance takes place. There was no diachronic emergence across the two descriptions. There can be, at best, a diachronic (i.e., in terms of temporal change) approach toward the disappearance of time, and a diachronic account from the emergence of time onward. This is where the

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analogy with ordinary atoms and fluids that live in space-time fails, and we are left in search of new intuitions.10

5  Things Get Worse, for Time: Quantum Gravity Phase Transitions and Geometrogenesis The story of the emergence of time (and space) is even more complex than this. The same is true, in fact, in the more familiar case of atoms and fluids. 5.1  Inequivalent continuum phases and phase transitions The point is that the continuum limit, encoding the collective behavior of atoms and, more generally, quantum many-body systems is not unique: starting from the same microscopic quantum constituents, their collective dynamics may lead to very different macroscopic phases, that is very

 An important remark should be added. In the context of a quantum gravity formalism that is based on new types of fundamental degrees of freedom, different from continuum fields, one has always the option to take a perspective in which these new structures are purely mathematical artifacts or useful technical tools to arrive at a true physical definition of the theory that is in fact based on the usual continuum fields. Indeed, several practitioners view in this way some of the quantum gravity formalisms we have mentioned. In this case, most of the technical challenges in recovering such a continuum description and a viable gravitational description would remain exactly the same (they are “interpretation-independent”). What changes is, of course, that one would not have to worry about conceptual or physical issues related to the nature of the new fundamental entities (since they would not be “real”) nor about the nature of a new emergence process, since we would be in the same situation as in the previous section. The other difference is that one now has to impose on the theory the requirement that no sign of the structures used as technical tools to recover continuum physics survive the reconstruction procedure. They should entirely disappear from the final result in the computation of any physical quantity. This becomes a key constraint in the definition of the “continuum limit.” If one adopts the opposite viewpoint and regards the new structures as somehow physical or “real,” then the conceptual issues cannot be avoided. Moreover, some signature of their existence should in fact survive the continuum limit; otherwise, it would be entirely vacuous to consider them real. For the same reason, the choice between these two perspectives is not arbitrary, but, as always in physical theories, will have to be decided by observations. Only if such observable consequences of the existence of the new fundamental entities can be theoretically derived, first, and then experimentally confirmed, will the “realist” perspective find strong support. 10

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different macroscopic physical systems.11 The same water molecules that give us liquid water in a certain range of temperature and pressure, as their macroscopic counterpart, can also give us vapor when temperatures are higher, and solid ice when they are lower. And thinking again about superfluids, the same helium-4 atoms that constitute them at very low temperatures will instead produce standard fluids at higher temperatures and gaseous systems at even higher ones. The appearance of different macroscopic phases in the continuum limit of quantum many-body systems is the rule, and a constant source of marvel and challenges, concerned with the investigation of the rich set of new features that is shown in the different phases and of the conditions for the phase transitions leading from one to the other. What should we expect in quantum gravity when studying in detail the collective behavior and continuum limit of the candidate “atoms of space” suggested by any given quantum gravity formalism? Just the same non-uniqueness of the result of the limit, the same richness of emergent macroscopic physics, and the same variety of possible macroscopic phases. Examples of different continuum phases and analyses of the related phase transitions can be found in much of the recent quantum gravity literature, e.g. [82–88]. If any of the proposed quantum gravity formalisms has a chance of being a viable description of the physical world, and of providing a satisfactory explanation for the emergence of space-time and gravitational physics from a more fundamental non-spatiotemporal reality, then at least one of the continuum phases it produces should be properly spatiotemporal and geometric. That is, it should allow reconstruction of effective (and approximate only) gravitational dynamics based on fields, spacetime, and geometry and described by (some possibly modified form of) general relativity. In other words, in at least one such phase, one should find themselves in the situation described in the previous section and, from there, move to the notions of space and time encoded in usual relativistic physics. This may or may not involve, as an intermediate step, a further approximate regime governed by some form of “quantum GR,”  To tell the whole story, the converse is also true. There are usually different microscopic systems that can give rise to the same macroscopic behavior, thus the same macroscopic physical system, with their microscopic differences becoming irrelevant within the limit. This is the phenomenon of universality in statistical and quantum many-body physics. In other words, just like emergence is not a unique relation, reduction is not unique, either, in general. 11

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depending on what is the exact relation between continuum and classical approximations in the quantum gravity formalism being considered. 5.2  The fundamental ontology is even more timeless If one such phase is a physical necessity for the viability of the theory, and the existence of several geometric phases (e.g., describing spatiotemporal and gravitational physics around different backgrounds, or governed by different gravitational theories) is also a possibility, the possibility of nongeometric, non-spatiotemporal phases raises many new issues, both physical and philosophical. The first implication is that the non-spatiotemporal nature of the fundamental entities is more profoundly so that it could have been suggested if there were only geometric phases in the continuum limit of the theory. Indeed, if the continuum limit were to always produce space-time and gravitational physics, one could argue that the fundamental entities, albeit not spatiotemporal or geometric in the usual sense, possessed features that would necessarily lead to the emergence of space-time under collective quantum evolution; they could be seen as possessing some sort of “proto-spatiotemporal” features, some more primitive form of spacetime characterization, whose relation to space-time itself was strong enough to ensure the latter approximation. One could even argue that, in this case, the usual notions of space and time would need to be adapted in the more fundamental theory, but do not cease to be meaningful, and question the very notion of space-time emergence or its relevance. Not so if non-geometric phases can be produced by the same fundamental quantum gravity entities. This very possibility eliminates any necessary link between their features (quantum observables) and space-time: the same quantities that, under some specific conditions, can be used to define and reconstruct time in one continuum phase of the theory, can produce nonspatiotemporal continuum physics under different circumstances (e.g., different values of the parameters entering the definition of the fundamental quantum dynamics). The issue is physical, of course, because it means that the step from quantum gravity to emergent space-time physics is more subtle and more technically challenging than if it were otherwise and that the emergent continuum physics is richer. It also implies that the philosophical challenges to be faced when establishing the new “quantum gravity ontology” of the new non-spatiotemporal entities have to be solved without even the indirect support of spatiotemporal 164

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intuitions that a necessary link between them and space-time would have allowed. In other words, the existence of different continuum phases, including non-spatiotemporal ones, makes the ontological emergence of space-time even more radical, emphasizing the novelty of space and time from the point of view of the fundamental quantum gravity constituents of the universe. 5.3 Geometrogenesis Beyond the ontology of the fundamental entities and the added complexity of the continuum limit that should lead to space-time emergence, the existence of different phases of the quantum gravity dynamics raises interesting new issues at the continuum level related to the physics and philosophy of the phase transitions leading from non-geometric to spatiotemporal phases. Such phase transitions have been dubbed geometrogenesis in the quantum gravity literature [89–92], and (whether interpreted “realistically” or not) are actively investigated in several quantum gravity approaches [82–88]. The first question is how to characterize the existence of the transition. What are the quantum gravity observables that show the relevant discontinuities12 at the transition point? And what are the relevant quantities that can play the role of order parameters, identifying the different phases with their different values? The answers can only be specific to different quantum gravity formalisms, but since it is space-time and geometry that emerge at the phase transition, it must be geometric and spatiotemporal quantities that characterize it by acquiring physically meaningful values only in the geometric phase. Various examples have been suggested in the quantum gravity literature, for example, the metric field components themselves [93] or the universe volume, which would be identically vanishing (in expectation value) in the non-geometric phase, while non-vanishing and with a non-trivial dynamics in the geometric one.13 A second question is, of course, what are the observable features of such phase transitions. What is the physics of geometrogenesis? To answer this, quantum gravity models should identify geometrogenesis  A phase transition is defined by the non-analytic behavior of some observable.  This would be an example of spontaneous symmetry breaking, formally analogous to the Higgs mechanism, by which the Higgs field acquires a non-vanishing expectation value, giving mass to all other fields, in the broken phase. 12 13

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with some precise physical circumstances, and again the most natural suggestion is the physical regime associated, in classical theory, with space-time singularities, like the interior of black holes and the cosmological beginning, the Big Bang. The possible answers to this question are also context specific, with different quantum gravity formalisms offering different proposals. In general, the approach to phase transitions is characterized in terms of strong fluctuations of some key physical quantity (e.g., of the order parameters themselves); thus, this second question can only be tackled together with the first one. However, the physics of phase transitions in quantum many-body systems is very rich, so the range of possibilities is large and should make us optimistic about the potential testability of a geometrogenesis scenario in quantum gravity. It should be clear that the main objective of any phenomenological analysis of the physics of quantum gravity phase transitions should focus on the testable phenomena on the spatiotemporal side of geometrogenesis, i.e., the geometric phase we live in. This is simply because it is hard to imagine how we could directly access the hypothetical non-geometric phase, of whose existence we should then identify instead some interesting indirect testable consequence. For example, in the scenario in which the geometrogenesis phase transition is how quantum gravity replaces the Big Bang singularity in the very early universe, one should look for its possible imprints in the consequent modification of the universe dynamics and in the physics of cosmological perturbations in the very first instants of the history of the universe, in particular in the CMB (cosmic microwave background) spectrum. Indeed, some quantum gravity formalisms are investigating this kind of cosmological scenario, two examples being group field theory condensate cosmology [4, 74, 75] and string gas cosmology [94] (which is strictly speaking an example of the emergence of space only, not time)14, and some more phenomenological proposals for possible signatures of such phase transition have been put forward [95, 96]. 5.4  Geometrogenesis as a physical process: a temporal characterization? In the above discussion, we assumed the perspective in which the geometrogenesis phase transition is a physical phenomenon (not just a  If the pre-transition phase is associated with vanishing, rather than just constant, scale factor. 14

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mathematical artifact), and made some educated guesses about which physical regimes could be associated with it. But is it a temporal process in any sense? Can we say, for example, that the universe underwent a transition to a geometric phase at some point in the past from an earlier non-geometric one? The issue is, of course, with respect to which temporal direction and variable we would make such a statement. The question is very natural, since phase transitions provide prototypical examples of diachronic emergence of novel behavior appearing as a result of the actual temporal evolution of a system. Think, for example, of ice melting as we raise its temperature by pumping heat into the system. Still, it should also be recalled that the typical way in which the phase structure of a physical system and its phase transitions are studied is in the context of equilibrium statistical mechanics and the renormalization group, and temporal evolution does not really play any role in them (by definition, there is no temporal evolution at equilibrium). It is possible to describe the same system out of equilibrium, thus within a formalism that allows us to talk properly of evolution across the phase transition, but it is also much more involved. To prove, using a realistic out-of-equilibrium description, that the phase transitions we normally describe within equilibrium statistical mechanics (thus as associated to different values of [coupling] constants) are in fact processes produced by the (quantum) dynamics of the system is possible in many cases, but it remains an open issue in general. As a result, one normally appeals to the fact that equilibrium statistical mechanics is a very good approximation to the actual physical behavior of dynamical systems whenever their evolution is not too fast, and relies on the possibility of external observers in the lab to adjust the conditions of the physical system and (slowly) tune the values of the parameters/couplings that characterize its description within equilibrium statistical mechanics. Three points need to be noted now to properly appreciate the corresponding situation in quantum gravity. First, although the physics of phase transitions requires the microscopic theory to be properly understood, we are discussing an issue that, in principle, we could investigate at the continuum level, since we are dealing with the interpretation of continuum phases and their associated transitions. Second, a formal description of the phase diagram of quantum gravity models in a context akin to “equilibrium (quantum) statistical mechanics” as opposed to an “out-of-equilibrium” description is inevitable at the fundamental level, simply because at the fundamental level we do not have a preferred 167

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temporal direction that can be used to define temporal evolution.15 Third, in general, we cannot assume any hidden observer, external to the system (i.e., to the space-time (region) under consideration), that could tune the parameters entering such formulation, thus “driving” the system toward a phase transition; this is even more evident if we think of a geometrogenesis phase transition to be associated with the very early universe. On this basis, we need to conclude that geometrogenesis cannot be seen as an example of diachronic emergence, but as another kind of synchronic one. Better still, since time simply disappears from the fundamental quantum gravity description of the world, geometrogenesis is another kind of achronic emergence as all the other kinds of emergence of time in quantum gravity that we discussed above. This is the conclusion that seems to be forced upon us at the fundamental level. But can geometrogenesis be given a temporal interpretation, at least in an approximate, partial sense? We have discussed how this possibility can be realized in classical general relativity and then in a quantum version of general relativity. We have also seen how the same could be possible, even if quantum gravity is based on more radically non-spatiotemporal entities, and gravitational physics in terms of continuum fields and space-time is akin to its hydrodynamic description. Indeed, a viable strategy to give an approximate temporal description of geometrogenesis could be to extend to this context the same relational strategy for the definition of time that we have seen at play in other contexts. First, we can certainly try to reformulate the effective continuum dynamics that is emergent from quantum gravity in a relational language in terms of some internal “clock” variable (itself only an emergent quantity from some underlying subset of degrees of freedom of the theory). Using this, we can follow the evolution of the system toward the early universe. If at some point in this “backward evolution” we encounter either the phenomenological effects or the structural features that we can associate with a geometrogenesis phase transition, then we would be allowed to state that such phase transition has indeed taken place in the early universe, at the beginning of our cosmic history. This cosmological scenario is being investigated in detail, for example, in the context of group field theory condensate cosmology [76, 77, 97].  This absence also makes it challenging to define the notion of “equilibrium” states in quantum gravity, lacking the usual definition and construction. However, this challenge can be met. 15

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But can we then follow the same evolution across the transition? The answer depends on whether the relational reformulation of quantum gravity dynamics remains valid across the transition. This consideration, and the following, also applies if the relational strategy is applied in the very initial formulation of the theory, recasting it in a “deparametrized” form before quantization. Such a strategy, analogous to the one applied extensively in quantum general relativity, has also been investigated in the (tensorial) group field theory approach. For example, one could think of relying on this deparametrized form for setting up a non-equilibrium description of the system, in terms of tackling the issue of quantum gravity phase transitions as dynamical processes. If the relational reformulation does remain valid across the transition, we have a geometrogenesis phase transition that, in a precise sense, can be given a temporal characterization, since it somehow spares time itself from disappearance.16 In general, we should not expect this to be the case. Rather, we should expect that the very conditions that allow for a relational rewriting of the dynamics of the theory fail to be valid in such extreme conditions. The universe, in its spatiotemporal description and time itself, would dissolve at geometrogenesis.17 We would be left with the fundamental description of the same universe provided by the underlying non-spatiotemporal quantum gravity theory. As we wrote above for the hydrodynamic-like transition in quantum gravity, and also in this case, the temporal description of the process leading to the disappearance of time, that is, ceases to be valid exactly when such disappearance takes place. There was no diachronic emergence across the two descriptions. There can be, at best, a diachronic (i.e., in terms of temporal change) approach toward the disappearance of time, and a  The situation would then be very close to the emergent universe scenario in string gas cosmology. 17  Moreover, in the context of this discussion of quantum gravity phase transitions and geometrogenesis, we should recall and stress the remark we have made about the two possible perspectives on the new non-spatiotemporal entities of quantum gravity. If one deems them as mathematical tools only and instead maintains an ontology of continuum fields, the non-geometric and non-spatiotemporal phases would have no reason to be considered physical or philosophically interesting. The technical challenges for the quantum gravity approach (e.g., the need to study the continuum phase diagram of the theory, to identify the geometric phases, understand their physics, and to characterize when one has a phase transition) would remain the same. But the conceptual ones would not. And the questions about whether geometrogenesis is a physical, if not temporal, process would be meaningless, since the non-geometric phase would not be physical or “real” in any way. 16

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diachronic account from the emergence of time onward. Once more, this is where the analogy with ordinary atoms and fluids, which live in spacetime, fails and we are left in search of new intuitions.

6  Concluding Remarks The search for new intuitions about a timeless universe, one in which time is an emergent notion and its emergence has the complex, multifaceted nature that we have discussed, is an important part of the search for a new understanding of time in contemporary physics. It should be clear from our discussion that this search can only be successful if it is a joint effort of mathematicians, theoretical and experimental physicists, and philosophers because the issues to be solved to achieve such understanding are conceptual as much as they are physical and mathematical. Any more solid grasp on the hidden richness of time that we could only glimpse at this stage will be an exciting reward for this collective effort.

References [1] D. Oriti, Disappearance and emergence of space and time in quantum gravity, Stud. Hist. Phil. Sci. B 46 (2014) 186. [2] D. Oriti, The Bronstein hypercube of quantum gravity, in Beyond Spacetime, N. Huggett, K. Matsubara and C. Wüthrich, eds., pp. 25–52, Cambridge University Press, Cambridge (2020). [3] D. Oriti, Levels of spacetime emergence in quantum gravity, in Philosophy beyond Spacetime: Implications from Quantum Gravity, N. Huggett, B. LeBihan and C. Wüthrich, eds., Oxford University Press, Oxford (2018). [4] D. Oriti, The universe as a quantum gravity condensate, C.R. Phys. 18 (2017) 235. [5] C. Rovelli, The layers that build up the notion of time, this volume, 2105.00540. [6] R. Frigg and S. Hartmann, Models in science, in The Stanford Encyclopedia of Philosophy, E. N. Zalta, ed., Metaphysics Research Lab, Stanford University (2020). [7] J. Ladyman and D. Ross, Every Thing Must Go: Metaphysics Naturalized, Oxford University Press, Oxford (2007). [8] M. Morganti and T. E. Tahko, Moderately naturalistic metaphysics, Synthese 194 (2017) 2557.

170

T h e C o m p l e x Ti m e l e s s E m e r g e n c e o f Ti m e i n Q u a n t u m G r a v i t y

  [9] J. Butterfield and C. Isham, On the emergence of time in quantum gravity, in The Arguments of Time, J. Butterfield, ed., pp. 111–168, Published for the British Academy by Oxford University Press (1999). [10] J. Butterfield, Less is different: Emergence and reduction reconciled, Found. Phys. 41 (2011) 1065. [11] J. Butterfield, Emergence, reduction and supervenience: A varied landscape, Found. Phys. 41 (2011) 920. [12] M. A. Bedau and P. Humphreys, Emergence: Contemporary Readings in Philosophy and Science, The MIT Press (03, 2008), 10.7551/mitpress/ 9780262026215.001.0001. [13] J. Butterfield and N. Bouatta, Emergence and reduction combined in phase transitions, URL http://philsci-archive.pitt.edu/8554/. Contribution to the Frontiers of Fundamental Physics (FFP 11) Conference Proceedings, February 2011. [14] R. W. Batterman, Emergence, singularities, and symmetry breaking, Found. Phys. 41 (2011) 1031. [15] R. Batterman, Critical phenomena and breaking drops: Infinite idealizations in physics, Stud. Hist. Philos. Sci. Part B: Stud. Hist. Philos. Mod. Phys. 36 (2004) 225. [16] E. Castellani, Reductionism, emergence, and effective field theories, Stud. Hist. Philos. Sci. Part B: Stud. Hist. Philos. Mod. Phys. 33 (2000) 251. [17] D. J. Chalmers, Strong and weak emergence, in The Re-Emergence of Emergence: The Emergentist Hypothesis from Science to Religion, P. Davies and P. Clayton, eds., Oxford University Press (2006). [18] A. Rueger, Physical emergence, diachronic and synchronic, Synthese 124 (2000) 297. [19] D. Yates, Emergence, in Encyclopaedia of the Mind, H. Pashler, ed., pp. 283– 287, SAGE Reference (2013). [20] O. Sartenaer, Flat emergence, Pac. Philos. Q. 99 (2018) 225. [21] R. C. Bishop and G. F. R. Ellis, Contextual emergence of physical properties, Found. Phys. 50 (2020) 481. [22] N. Huggett and C. Wuthrich, Introduction: The emergence of spacetime, in Out of Nowhere: The emergence of Spacetime in Quantum Theory of Gravity (2021). [23] K. Crowther, As below, so before: “synchronic”and “diachronic”conceptions of spacetime emergence, Synthese 198 (2020) 7279. [24] S. Baron, The curious case of spacetime emergence, Philos. Stud. 177 (2019) 2207. [25] B. L. Bihan and N. S. Linnemann, Have we lost spacetime on the way? narrowing the gap between general relativity and quantum gravity, Stud. Hist. Philos. Sci. Part B: Stud. Hist. Philos. Mod. Phys. 65 (2019) 112.

171

Ti m e a n d S c i e n c e

[26] N. Huggett and C. Wüthrich, The (a)temporal emergence of spacetime, Philos. Sci. 85 (2018) 1190. [27] B. L. Bihan, Space emergence in contemporary physics: Why we do not need fundamentality, layers of reality and emergence, Disputatio 10 (2018) 71. [28] K. Crowther, Appearing Out of Nowhere: The Emergence of Spacetime in Quantum Gravity, Ph.D. thesis, University of Sydney, 2014. [29] R. W. Batterman, Hydrodynamics versus molecular dynamics: Intertheory relations in condensed matter physics, Philos. Sci. 73 (2006) 888. [30] R. W. Batterman, The tyranny of scales, in The Oxford Handbook of Philosophy of Physics, R. W. Batterman, ed., pp. 255–286, Oxford University Press (2013). [31] O. Pooley, Substantivalist and relationalist approaches to spacetime, in The Oxford Handbook of Philosophy of Physics, R. Batterman, ed., Oxford University Press (2013). [32] C. Hoefer, Absolute versus relational spacetime: For better or worse, the debate goes on, Br. J. Philos. Sci. 49 (1998) 451. [33] E. Curiel, On the existence of spacetime structure, Br. J. Philos. Sci. (2014) axw014. [34] D. Giulini, Some remarks on the notions of general covariance and background independence, Lect. Notes Phys. 721 (2007) 105. [35] M. Gaul and C. Rovelli, Loop quantum gravity and the meaning of diffeomorphism invariance, Lect. Notes Phys. 541 (2000) 277. [36] C. Rovelli, ““Forget time”, Found. Phys. 41 (2011) 1475. [37] C. Rovelli, What is observable in classical and quantum gravity?, Class. Quant. Grav. 8 (1991) 297. [38] D. Marolf, Almost ideal clocks in quantum cosmology: A Brief derivation of time, Class. Quant. Grav. 12 (1995) 2469. [39] C. Rovelli, Partial observables, Phys. Rev. D 65 (2002) 124013. [40] B. Dittrich, Partial and complete observables for canonical general relativity, Class. Quant. Grav. 23 (2006) 6155. [41] V. Husain and T. Pawlowski, Time and a physical Hamiltonian for quantum gravity, Phys. Rev. Lett. 108 (2012) 141301. [42] T. Thiemann, Reduced phase space quantization and Dirac observables, Class. Quant. Grav. 23 (2006) 1163. [43] K. Giesel and T. Thiemann, Scalar material reference systems and loop quantum gravity, Class. Quant. Grav. 32 (2015) 135015. [44] K. Giesel and A. Herzog, Gauge invariant canonical cosmological perturbation theory with geometrical clocks in extended phase-space: A review and applications, Int. J. Mod. Phys. D 27 (2018) 1830005. [45] C. Kiefer, Conceptual problems in quantum gravity and quantum cosmology, ISRN Math. Phys. 2013 (2013) 509316.

172

T h e C o m p l e x Ti m e l e s s E m e r g e n c e o f Ti m e i n Q u a n t u m G r a v i t y

[46] C. Kiefer, Quantum gravity: Whence, whither?, in Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unified Framework, pp. 1–13, 2012. [47] T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2007), 10.1017/CBO9780511755682. [48] P. A. Hoehn, A. R. H. Smith and M. P. E. Lock, Trinity of relational quantum dynamics, Phys. Rev. D 104 (2021) 066001. [49] E. Castro-Ruiz, F. Giacomini, A. Belenchia and C. Brukner, Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems, Nat. Commun. 11 (2020) 2672. [50] F. Giacomini, Spacetime Quantum Reference Frames and superpositions of proper times, Quantum 5 (2021) 508. [51] C. Rovelli and L. Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys. B 442 (1995) 593. [52] C. J. Isham, Canonical quantum gravity and the problem of time, NATO Sci. Ser. C 409 (1993) 157. [53] M. Bojowald, P. A. Hohn and A. Tsobanjan, Effective approach to the problem of time: general features and examples, Phys. Rev. D 83 (2011) 125023. [54] D. Marolf, Solving the problem of time in mini-superspace: Measurement of Dirac observables, Phys. Rev. D 79 (2009) 084016. [55] E. Alesci and F. Cianfrani, Quantum reduced loop gravity: Semiclassical limit, Phys. Rev. D 90 (2014) 024006. [56] J. W. Barrett, R. J. Dowdall, W. J. Fairbairn, H. Gomes, F. Hellmann and R. Pereira, Asymptotics of 4d spin foam models, Gen. Rel. Grav. 43 (2011) 2421. [57] S. K. Asante, B. Dittrich and J. Padua-Arguelles, Effective spin foam models for Lorentzian quantum gravity, Class. Quant. Grav. 38 (2021) 195002. [58] I. Agullo and P. Singh, Loop quantum cosmology, in Loop Quantum Gravity: The First 30 Years, A. Ashtekar and J. Pullin, eds., pp. 183–240, WSP (2017). [59] H. Sahlmann, Loop quantum gravity: A short review, in Foundations of Space and Time: Reflections on Quantum Gravity, pp. 185–210, 1 (2010). [60] A. Perez, The spin foam approach to quantum gravity, Living Rev. Rel. 16 (2013) 3. [61] D. Oriti, Group Field Theory and Loop Quantum Gravity, 8 (2014). [62] R. Loll, Quantum gravity from causal dynamical triangulations: A review, Class. Quant. Grav. 37 (2020) 013002. [63] R. Gurau and J. P. Ryan, Colored tensor models: A review, SIGMA 8 (2012) 020. [64] D. Oriti, The microscopic dynamics of quantum space as a group field theory, in Foundations of Space and Time: Reflections on Quantum Gravity, pp. 257–320, 10 (2011).

173

Ti m e a n d S c i e n c e

[65] T. Krajewski, Group field theories, PoS QGQGS2011 (2011) 005. [66] S. Surya, The causal set approach to quantum gravity, Living Rev. Rel. 22 (2019) 5. [67] M. Blau and S. Theisen, String theory as a theory of quantum gravity: A status report, Gen. Rel. Grav. 41 (2009) 743. [68] H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, Space-time structures from IIB matrix model, Prog. Theor. Phys. 99 (1998) 713. [69] B. Bahr and S. Steinhaus, Hypercuboidal renormalization in spin foam quantum gravity, Phys. Rev. D95 (2017) 126006. [70] B. Dittrich, The continuum limit of loop quantum gravity — a framework for solving the theory, in Loop Quantum Gravity: The First 30 Years, A. Ashtekar and J. Pullin, eds., pp. 153–179 (2017). [71] A. Eichhorn, T. Koslowski and A. D. Pereira, Status of backgroundindependent coarse-graining in tensor models for quantum gravity, Universe 5 (2019) 53. [72] S. Carrozza, Flowing in group field theory space: A review, SIGMA 12 (2016) 070. [73] M. Finocchiaro and D. Oriti, Renormalization of group field theories for quantum gravity: new scaling results and some suggestions, Front. Phys. 8 (2021) 649. [74] A. Pithis and M. Sakellariadou, Group field theory condensate cosmology: An appetizer, Universe 5 (2019) 147. [75] S. Gielen and L. Sindoni, Quantum cosmology from group field theory condensates: A review, SIGMA 12 (2016) 082. [76] D. Oriti, L. Sindoni and E. Wilson-Ewing, Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates, Class. Quant. Grav. 33 (2016) 224001. [77] L. Marchetti and D. Oriti, Effective relational cosmological dynamics from quantum gravity, JHEP 05 (2021) 025. [78] D. Oriti and X. Pang, Phantom-like dark energy from quantum gravity, 2105.03751. [79] E. Wilson-Ewing, A relational Hamiltonian for group field theory, Phys. Rev. D 99 (2019) 086017. [80] S. Gielen and A. Polaczek, Generalised effective cosmology from group field theory, Class. Quant. Grav. 37 (2020) 165004. [81] S. Gielen, Frozen formalism and canonical quantization in (group) field theory, 2105.01100. [82] C. Delcamp and B. Dittrich, Towards a phase diagram for spin foams, Class. Quant. Grav. 34 (2017) 225006. [83] A. Kegeles, D. Oriti and C. Tomlin, Inequivalent coherent state representations in group field theory, Class. Quant. Grav. 35 (2018) 125011. [84] B. Bahr, B. Dittrich and M. Geiller, A new realization of quantum geometry, Class. Quant. Grav. 38 (2021) 145021. 174

T h e C o m p l e x Ti m e l e s s E m e r g e n c e o f Ti m e i n Q u a n t u m G r a v i t y

[85] T. Koslowski and H. Sahlmann, Loop quantum gravity vacuum with nondegenerate geometry, SIGMA 8 (2012) 026. [86] S. Surya, Evidence for a phase transition in 2D causal set quantum gravity, Class. Quant. Grav. 29 (2012) 132001. [87] A. G. A. Pithis and J. Thürigen, Phase transitions in TGFT: Functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models, JHEP 12 (2020) 159. [88] J. Ambjørn, J. Gizbert-Studnicki, A. Görlich, J. Jurkiewicz, N. Klitgaard and R. Loll, Characteristics of the new phase in CDT, Eur. Phys. J. C 77 (2017) 152. [89] T. Konopka, F. Markopoulou and L. Smolin, Quantum graphity, hep-th/0611197. [90] D. Oriti, Group field theory as the microscopic description of the quantum spacetime fluid: A New perspective on the continuum in quantum gravity, PoS QG-PH (2007) 030. [91] M. L. Mandrysz and J. Mielczarek, Ultralocal nature of geometrogenesis, Class. Quant. Grav. 36 (2019) 015004. [92] L. Smolin, The self-organization of space and time, Philos. Trans. R. Soc. Lond. A 361 (2003) 1081. [93] R. Percacci, The Higgs phenomenon in quantum gravity, Nucl. Phys. B 353 (1991) 271. [94] R. H. Brandenberger, String gas cosmology after Planck, Class. Quant. Grav. 32 (2015) 234002. [95] J. Magueijo, L. Smolin and C. R. Contaldi, Holography and the scale-invariance of density fluctuations, Class. Quant. Grav. 24 (2007) 3691. [96] J. Mielczarek, Big Bang as a critical point, Adv. High Energy Phys. 2017 (2017) 4015145. [97] L. Marchetti and D. Oriti, Quantum fluctuations in the effective relational GFT cosmology, Front. Astron. Space Sci. 8 (2021) 683649.

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Chapter 7

T IME A ND D U R ATI ONS I N RE L AT I VIS T I C PHY SI CS Marc Lachièze-Rey Université Paris VII, Paris, France [email protected]

What time is it? How much time does this trip take? It is clear that the word time takes different meanings in these two questions: datation and duration. The confusion remains without consequence in Newtonian physics (NP) because of the narrow link between the two notions, a link that permits the construction of “time” as their synthesis. The situation is very different in Einsteinian physics (EP), where duration (renamed “proper duration”) is perfectly well defined but datation is not.

1  Datations and Durations: Time and Proper Times Datations and durations are different entities: datations concern events and durations concern histories, making careful distinction necessary.1  See Voyager dans le temps: la physique moderne et la temporalité, Marc Lachièze-Rey, Seuil Sciences Ouvertes, 2013. 1

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1.1  The basis: events, space-time, histories The notion of event is intuitive and corresponds to its usual view: something which occurs instantaneously, with no duration. In Newtonian language, this means at some position in space (location) and at a position in time (date). In Einsteinian language, this becomes “a position in spacetime.”2 Events occur everywhere in space-time: emission, reception, or collision of particles at the microscopic scale; stellar explosions, or collisions between celestial bodies at the macroscopic one. Space-time appears as the set of events, or rather of virtual events (i.e., locations where something may happen): [virtual] events are identified with points of space-time. The notion of history also corresponds to our intuition: a continuous sequence of events, with one initial and one final. We can also call it a process. Geometrically, it appears as a continuous sequence of points in space-time, thus a line segment, with initial and final points-events. Any observer, any object, passes through successive events; in geometrical terms, he/it occupies successive points in space-time, which describe a line called his worldline. My life is a history called my worldline. My progression along my worldline is what I usually call the “flow of time.” From the point of view of relativity, a physical object identifies with its worldline. 1.2  Measuring time? Both in NP or in EP, any history (thus a worldline segment) has a duration, better called its proper duration in EP. An observer, a clock, can always measure (in principle) the duration of his own history and it is a fundamental fact that, both in NP and in EP, any “time measurement” reduces to a duration measurement. A clock, a watch, any kind of instrument “measuring time,” is in fact a chronometer that registers the duration of its own history between two clicks,3 an initial and a final event. For a  By an abuse of language, one may call “Newtonian space-time” the reunion of space and time. But it differs fundamentally from Einsteinian space-time since it is not a metric manifold (see below): it admits two distinct and unrelated metrics, one for time and one for space. 3  The clock generally involves a periodic phenomenon (mechanical, electronic, atomic ...) characterized by its period, defined as the duration between two oscillations. Measuring the duration is counting these periods. A physiological rhythm (respiration, heartbeats, digestion, etc.) is in fact a typical duration. Similarly, our “mental” or “spiritual” time is also 2

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usual watch, the first click-event is the introduction of an initial date and hour (mise à l’heure), given from outside; the second click-event is the “reading of time.” An actualization mechanism automatically adds the measured value to the formerly introduced date (or hour). Thus, the second is a unit of duration4 rather than a “unit of time,” just like a meter is a unit of length and not a unit of space. In fact, duration happens to be the only time-related measurable quantity. And what an observer calls time is nothing but his experience of the durations in his history, which he calls “proper time” (see below). That a clock measures the proper duration of its own history (rather than “time”) is sometimes referred to as the clock hypothesis, and is at the basis of Einsteinian theories. But in the years following the publication of special relativity, it was rejected by some philosophers and physicists (in particular by Henri Bergson) who intended to reintroduce the possibility of the existence of time. In NP, all time-related notions, including time itself, are derived from durations (see below). In EP, proper duration is the only valid timerelated notion. 1.3  Proper times Despite the appellation, the “proper times” involved in EP do not have the properties of time. There are as many unrelated proper times as there are observers, or clocks. Each one is valid for one particular observer only: a convenient way to rename his proper durations along his particular worldline, after the choice of an arbitrary origin on it. My proper time, for instance, runs along my own worldline exclusively, and nowhere else in space-time. In the language of EP, this is not a “time function” (see below). It qualifies my own events only, those occurring to me. But I cannot use it to date, or to classify, events occurring elsewhere in space-time. Of course, I can observe these events but my observation of an event (e.g., the reception of a luminous signal from it) is a different event, occurring on my own worldline. This is at the origin of relativistic shifts (see below). constructed from fundamental durations: reading a book, performing a calculation, thinking, and so on. 4  According to La Treizième Conférence 2énérale des poids et mesures (1967), the legal definition is 9,192,631,770 times the period of a specific radiation of the cesium atom. A period is the proper duration between two successive maxima of this radiation.

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My proper time, your proper time, and the infinite collection of the proper times of other observers (or objects) are different quantities, defined on different supports (the corresponding worldlines). There is no way to compare them, except in very specific situations. NP is based on the assumption (the Newtonian ansatz5) that the proper times of all observers can be synchronized, and this is at the basis of the construction of Newtonian time. However, this synchronization appears to be impossible in the real world, and this is what forbids the introduction of time.

2  Time From Chroincidence A number is associated with each history: its duration (proper duration in EP). NP assumes a fundamental property, for which I introduce the neologism “chroincidence,” According to chroincidence, two histories that share the same initial event and the same final event have the same duration. Far from being a logical necessity, this “Newtonian ansatz” corresponds to intuition and appears reasonable. It is a basis for the construction of Newtonian time. Assuming chroincidence, an arbitrary choice of origin allows us to define a unique datation from the durations; it assigns a date (a number) to each event, in such a way that the duration of a history equals the difference between the dates of its final and its initial events. This defines it without ambiguity, and the ordering of the dates fixes chronology as a total order relation between events, from which follow simultaneity, anteriority, etc. “Time” appears as the composite notion resulting from the harmonious synthesis of datation, duration, chronology, all resulting from the primary notion of duration.

3  No Chroincidence in the Real World Chroincidence, however, does not hold in the real world. This is now a perfectly well-established experimental fact,6 as will be shown below. It is perfectly taken into account by EP, and its direct manifestations are  See Voyager dans le temps, op. cit.  Of course, these results were not available to Einstein, for whom the only indication was the constancy of the velocity of light. 5 6

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usually analyzed in terms of “relativistic effects,” in particular “temporal delays” and “temporal shifts” (delays and shifts, for short), which I present in the next sections. Although the previous construction of time does not apply, a given space-time admits an infinity of time functions in general,7 at least locally. Every observer can make his personal choice from among them, and use it to classify events. He will generally require, for convenience, that his choice roughly coincide with his proper time along his own worldline, so that it will reproduce such and such aspect of time for him.8 This may be compared to a store whose objects for sale have no price indicated: every client is free to fix the prices at his convenience, thereby creating a “price function” in the same way that an observer creates his time function in space-time (that he calls “time”). Different clients have different lists of prices (time functions); different observers have different (and contradictory) chronological classifications of the events. As a typical example, our universal time is chosen to coincide as much as possible with the proper times of the terrestrial observers, along their own worldlines. Strictly speaking, this cannot hold for more than one unique observer, but this approximation is largely sufficient for social life. Another example is provided by cosmic time, whose construction is allowed by the high degree of regularity of our big bang cosmological models.9 It provides convenient labeling and classification of the cosmic events although it is not a measurable quantity.10 In any case, it is a mathematical fact that no space-time at all (even the flat Minkowski space-time of special relativity) admits a time function whose lapse coincides with durations. This means that there is no time function that clocks can measure.11 This contrasts strongly with our  A time function is a space-time function that mimics a datation in the sense that it assigns a number to each event (with the constraint that its gradient remains timelike everywhere). A space-time either admits no (global) time function or it admits an infinity of them, whose indications are mutually incompatible. 8  However, treating a time function like time usually leads to logical contradictions, like assigning different dates to the same event. 9  This applies to the wider class of Friedman–Lemaître models, which obey the cosmological principle; see L’âge de l’univers, Marc Lachièze-Rey, Humensis 2021. 10  Its evaluation requires the knowledge of the complete geometry of space-time. No clock rigorously indicates cosmic time. No measured duration coincides with the lapse of cosmic time (see L’âge de l’univers, op. cit.). 11  Except for a very limited class of specific clocks. 7

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intuition and is probably the main difficulty for tackling the relativistic world and EP, whose main merit is to provide a description of the world without time.

4  Relativistic Delays A relativistic delay is a direct consequence of the violation of chroincidence: the durations of two stories sharing the same initial and the same final events (thus between two fixed events) have different measured values. Such delays, mainly observed in clock desynchronization experiments, are now measured with a very high precision (see below). These experiments are modern realizations of a famous thought experiment called the “Langevin twins paradox.” Although it is attributed to Paul Langevin, a colleague and a friend of Einstein, the story is slightly more subtle (During, 2014), but it provides a good popular illustration. Although contrary to our intuition, it presents absolutely no paradoxical character: only attempts to analyze it using the notion of time generate logical contradictions. Delays may be of kinematical or gravitational origin.

5  A Twin Story Let me first present a simple version of the “paradox.” It involves two twins, R (rest) and T (travel). They share their common life up to their 20th anniversary, which is the common initial event of their two subsequent different stories: R remains sedentary on Earth, while T starts to travel in space at high velocity. At some moment, T turns back and lands back on Earth, where he meets R again. The meeting is the common final event of their two different stories. The delay is manifest as a difference in their ages: T is 30 years old and R is 40 years old: a delay of 10 years! To answer former objections, each of the twins has experienced an entirely normal “flow of time,” and entirely normal durations. The flow of his proper time corresponds to his psychological time, to his physiological rhythms (respiration, heart rate, etc.), and also to his clock measurements as usual.

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6  Delay as a Geometrical Effect This delay finds a simple interpretation in the geometrical context of space-time. In EP (both in special and general relativity), and in contradiction with NP, space-time admits a unique metric. Very generally in geometry, the role of a metric is to assign a metric interval to any line segment. In our usual geometry (and in NP), space admits the Euclidean metric and the corresponding metric interval is what we call length. Also in NP, time admits a different metric whose metric interval is duration. The situation is entirely different in EP, where space-time admits a unique metric whose metric interval is of mixed nature (neither entirely spatial nor entirely temporal), and it is the (proper) duration for a history.12 Here lies probably the most fundamental difference between NP and EP: two separate and disconnected metrics in NP, for space and for time; a unique metric g in Einsteinian space-time. This is the origin of relativistic delays. Different curves (histories) in space-time, joining the same two fixed points (an initial and a final event), have different metric intervals (durations): this is the failing of the chroincidence property. It is the fourdimensional analogue of the familiar situation in three-dimensional space: different curves (paths) joining the same two fixed points have different metric intervals (lengths). In space, among the lines joining two points, the straight one minimizes the length. In space-time, among the histories joining two events, the straight one maximizes13 the duration. The principle of inertia states that the worldline of an inertial observer is a straight line. A non-inertial worldline (like that of the twin T) presents accelerations; that is, velocity changes, which are represented as angles in the spatiotemporal description. This explains the asymmetry between the two twins: R remains at rest on Earth and never accelerates, he is inertial and his worldline is a straight line, whereas T accelerates, at least for taking off, for turning back, and for landing.14  The same metric interval also defines lengths, but that does not matter here.  “Maximizes” rather than “minimizes.” The difference is due to the Lorentzian character of the space-time metric (see, e.g., Marc Lachièze-Rey, Cosmology: A First Course, Cambridge University Press, 1995). 14  If space-time is curved, “straight line” should be replaced by “geodesic,” its generalization. The principle of inertia states that the worldline of an inertial observer is a geodesic. 12 13

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Because it is caused by velocity differences, the delay is called “kinematical time delay.” Both special and general relativity give a perfect account of it. But general relativity, in addition, predicts “gravitational time delays.” They appear when the two histories explore regions of space-time with different curvatures, which are different values of the gravitational potential. Both effects have been well observed.

7  A Crucial Experiment The first direct observation of the relativistic delay occurred in 1971. The physicist Joseph C. Hafele and the astronomer Richard E. Keating placed cesium atomic clocks (rather than twins) on board different airplanes. One plane circled the Earth, from west to east. Its velocity (around 830 km/h) almost compensated for Earth’s rotation, so that it remained almost at rest with respect to the fixed stars. A second plane flew in the reverse direction, and a third one remained on ground, at the Naval US Observatory. This makes three different stories sharing the same initial event (the common taking off) and the same final event (the common landing). The measured delays (around minus 50 ns for east; around 270 ns for west) were in accordance with the general relativity predictions, at 10% precision of the measurements, confirming that chroincidence does not hold in the real world. They combined the kinematical effect (velocities and accelerations) and the gravitational one (differences in gravitational potential with altitude). Similar experiments followed. One focused on the gravitational effect, minimizing the kinematical contribution thanks to a reduction in the planes’ velocity to around 500 km/h. The observed gravitational delay of 50 ns was in accordance with the general relativity predictions, at 1.6% precision. Moreover, the spectral shifts (see below) were recorded during one of the Hafele–Keating flights by sending periodic laser impulsions to the plane, which were reflected and then recorded on ground. This confirmed their relationship with the delays (see below): red and blue shifts do not compensate during the whole flight, as they would according to NP.

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8  The Shapiro Delay The Shapiro delay (Shapiro, 1964) represents another manifestation of the same effect. A light ray travels from a source (another planet) to an observer on Earth. A massive object (the Sun) between source and observer induces a space-time curvature, which modifies the path of the light ray (as during the 1919 eclipse). As a result, the light signal reaches the observer later than it would in the absence of a massive object, resulting in a kind of gravitational lensing effect. An undeformed light ray and a deformed one, starting simultaneously, would not reach the observer at the same moment in his history, but with a Shapiro delay. For a light ray tangent to the Sun, the expected delay is about 200 microseconds. The effect was first confirmed (with about 20% precision) using a radar signal sent from Earth and reflected by a planet beyond the Sun: Mars, Venus, or Mercury. The delay occurred when the Sun passed between the planet and Earth, and then disappeared. Better precision was obtained later (in 1976) thanks to the reflection by the Viking landing probes on Mars. The Shapiro delay can also be observed outside the Solar System. This occurs in a binary system if the trajectory of the signal emitted by a body (in particular, the periodic one from a neutron star) approaches the massive companion before reaching the observer. This was observed in systems of two neutron stars, PSR B1913+16 in 1984 and PSR B1534+12 later, and also in the mixed system PSR J1614-2230, where the signal from the neutron star is deflected by the mass of the white dwarf companion.

9  Blue- and Redshifts Relativistic shifts (“temporal shifts”) are a second kind of relativistic effects, which apply also to durations. A shift does not express a dilatation, or a contraction, of a duration, but the difference between the durations of two different stories (which in this case do not share the same initial and final events). Most often, this is a spectral shift: a particular case

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where the durations under consideration are the typical periods of some electromagnetic radiation.15 The shift is defined as the relative difference between the proper duration of a process (a history) and the proper duration of the observation of the process by an observer, which is another history. Typically, an observer observes the history H of a source that emits radiation, from its initial event I to its final event F; for instance, a process occurring on a distant planet, in which case both I and F occur on that planet. The proper duration d of H can be measured by any clock on the planet, but not by the observer. The latter, located on Earth (for instance), receives electromagnetic signals, and his observation is another history, H’, concerning him (the observer) and belonging to his own worldline. It is made of the successive receptions of the signals sent from the events of H. It begins at event I’, which is the reception of the signal emitted by event I; it ends at F’, the reception of the signal emitted by F. Using his own clock, the observer measures the duration d’ of the story H’. The shift is defined as the relative difference z = (d’ – d) / d. Very often, the shift is a spectral shift. This means that the duration d identifies with the period Temitted of the emitted (usually electromagnetic) radiation. The spectral shift is z = (Treceived – Temitted) / Temitted, where Treceived is the received period. This appears as a generalization of the usual Doppler effect. One distinguishes kinematical and gravitational shifts in the same way as delays are. Both occur in many astronomical observations, usually mixing the two types. This is, for instance, the case for the well-known cosmological spectral redshift, since the emission by a source (a remote galaxy) occurred in our causal past, when the gravitational potential was different because of cosmic expansion. It plays a fundamental role in cosmology.16

10  The Kinematical Shift The familiar Doppler–Fizeau effect is a typical example of spectral shift, although usually not analyzed as a temporal shift. It occurs with sonic  The period of a radiation is the duration between two successive maxima of the periodic phenomenon corresponding to that radiation. 16  Note that the observed spectral shift of a galaxy may include an additional Doppler component due to its proper velocity, in addition to the expansion component (see, e.g., Cosmology: A First Course, op. cit.). 15

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waves and is quite easy to understand: when the source of a periodic signal (the sound, with period Temitted) is approaching me (the observer), the emitted sound requires less and less time to reach me. When analyzed in the NP context, I receive sonic pulses with intervals shorter than if the source were at rest; this means that the shorter period Treceived is equivalent to higher frequency. The spectral shift is defined as z = (Treceived – Temitted) / Temitted. This lies entirely outside the context of relativistic physics, and the non-relativistic shift z is easily calculated like (the radial component of) the source velocity, in units of the sound velocity. For a receding source, the effect is reversed. The same applies to light, and more generally to electromagnetic radiation, although it is not apparent in life. Its correct description requires EP, which brings two crucial differences. Firstly, both in special and general relativity, this kinematical effect is not proportional to the radial component of the relative velocity: its linear approximation (only) equals the radial velocity of the source (in units of c), but higher-order contributions are present. Secondly, there is an additional contribution (also kinematic) that depends on the transverse component.

11  The Einstein Effect In addition to kinematical shifts, there is also a gravitational one: the Einstein effect, described by general relativity only. Like a kinematical shift, it expresses the difference between the proper durations of an observed history, and that of the observation of this story. But it depends on the different space-time curvatures (or gravitational potentials) that the two histories explore. In familiar language, a clock monitored by a remote observer seems to click more slowly when it (the observed clock) lies near a massive object. The effect increases with the gravitational potential field, and it becomes extreme in the vicinity of a black hole. Like the kinematical shift, it is usually observed as a spectral shift. 11.1  The black hole case A black hole is an extreme space-time configuration where such effects reach maximum intensity, making it impossible to define time functions. A black hole is surrounded by a horizon. Nothing can escape from inside the horizon. In its vicinity, the space-time curvature is strong and an external source is observed with a strong gravitational redshift. The latter 187

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increases when the source is closer to the horizon (where the gravitational potential reaches the maximal possible value), and tends to infinity when it reaches it (although the horizon is at finite proper distance, a very counterintuitive fact). This means that radiation emitted on the horizon will never be detected. We may, for instance, imagine a cosmonaut in free fall towards the black hole horizon, sending signals every hour (of his clock). We receive the signals more and more slowly, and we will never receive the infinitely redshifted signal emitted when he (the cosmonaut) reaches the horizon. Spectral shifts have been recently measured in the proximity of the supermassive invisible black hole (4 million solar masses) in the center of our galaxy. The 2020 Nobel Prize in Physics was given to the two astronomers Reinhard Genzel (Max Planck Institute, near Munich) and Andrea Ghez (University of California, Los Angeles), who observed, during the last 20 years, stars orbiting the black hole with strong velocities, of thousands (up to 7000) of km/s. In particular, the Gravity instrument of the ESO VLT telescope provided the excellent resolution17 required to examine the star called S2, at the perihelion (the closest to the black hole) of its 16-year elliptic orbit. This allowed the nonlinear character of the kinematical redshift as predicted by EP, as well as the gravitational redshift, to be demonstrated for the first time.

12  Spectral Redshift and Energy A spectral shift is a case of temporal shift, but it can also be analyzed in terms of energy. The famous relation E = ћ/T associates the period T of an electromagnetic radiation with the energy of the photons carrying it: a spectral shift means a difference between the energy of the received photon and that of the same photon at emission. A redshift appears as an energy loss by the photon: it must struggle against a recession velocity (kinematical), or climb a potential well (Einstein effect). This means non-conservation of energy in space-time. Its conservation in NP results from the time translational invariance (time is a conjugate quantity of energy). The absence of this fundamental symmetry in relativistic space-time18 implies no energy conservation.  Thanks to interferences between the signals of the four telescopes.  Except in the presence of a timelike killing vector field: it expresses translational invariance with respect to a time function, which thus plays, in some respects, a role similar to time. 17 18

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13  Observations of Einstein effect 13.1  Einstein effect on Earth The Einstein effect was first tested on Earth by exploiting the differences in potential due to altitude. General relativity predicts a shift z = 1.1 × 10–16 per meter: a clock at high altitude seems to be quicker than at lower altitude (this is of course an illusion since a clock really always keeps exactly the same rhythm). In the 1960s, the American physicists Robert Pound and Glen Rebka used the height of the Harvard University tower (22.6 m). They applied the recently discovered Mössbauer effect to nuclear spectroscopy of the gamma rays emitted by the Fe57 isotope of iron. General relativity was confirmed. The precision increased in further experiments and reached 1% in a 1964 experiment (Pound and Snider, 1964). A larger altitude difference between emitter and receptor increases the shift. In 1976, a Scout D rocket was launched by Vessot and Levine (the Gravity Probe A experiment), with an H maser atomic clock on board. Its frequency was compared with that of a clock on ground, with a 0.007% precision. Today, the 10−18 precision of optical clocks permits a 10–5 precision for Einstein effect measurements, even for altitude differences as small as 1 cm. Recently (in 2020), researchers from the Tokyo University used the 450 m of the Tokyo tower to measure the weak shift (z = 5 × 10–14) with high precision permitted by strontium 87 optical clocks. Beyond additional confirmation of general relativity, the combination of high precision and robustness offers practical opportunities to detect modifications in the terrestrial gravitational potential (i.e., space-time curvature) due to internal inhomogeneities or motions. This opens the field to new applications in geodesy, geophysics, hydrology, oceanography, and seismology, including the possibility of previsions in these fields. 13.2  Redshift from the Sun In 1908, Einstein predicted that the difference in gravitational potential between Sun and Earth surfaces should generate a shift of around 2 × 10–6 (z c = 630 m/s19). This was roughly confirmed in 1962, but the  Since at lowest order, a velocity v generates a shift z = v/c, it is often convenient to express a shift in velocity units, i.e., by the expression zc. 19

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observations were blurred by the strong Doppler shifts generated by the surface motions due to solar granulation. Quite surprisingly, the most precise results to date come from spectral analysis of the light of the solar disk as reflected by the Moon (González Hernández et al., 2020). They confirm general relativity with a precision of around 1%. 13.3  From Sirius A strong Einstein effect is expected on condensed objects like white dwarfs. The star Sirius (in fact, Sirius A) is a main sequence star, like the Sun, at 8.6 light years from us. It belongs to a binary system, and its invisible companion Sirius B was recognized as a white dwarf in 1915. The gravitational redshift measurement of the spectral lines of Sirius B was one of the three classical tests of general relativity. It was first accomplished in 1925 by Walter Adams at Mt. Wilson observatory, following recommendations by Eddington. They obtained z = 6.3 × 10–5 with the hydrogen Balmer line, but with a low precision because of light contamination by Sirius A. Most recent observations have led to a better estimation, around 3 × 10-4 (z c around 80 km/s), which allowed mass estimations for the white dwarf (Greenstein et al., 1985). More compact objects, like neutron stars or black holes, are expected to give rise to stronger gravitational shifts. This has been tentatively detected recently in the binary system 4U 1916-053, which comprises an ordinary star and a neutron star in orbital motion, with a period of 50 min. The spatial X-ray observatory Chandra made spectroscopic observations of the X-ray emission by this system. Redshifts as strong as 10–3 (zc around 300 km/s) have been detected in the absorption lines by iron and silicon elements. Much stronger than usual Doppler shifts, they were interpreted as gravitational redshifts from gaseous matter of the accretion disk around the neutron star, in the vicinity of its surface, thus in its strong gravitational potential (Trueba et al., 2020). 13.4 Clocks In 2010, an experiment (Chou et al., 2010) compared two optical atomic clocks based on aluminum ions and the authors observed “time dilation” (their own expression), due to kinematic as well as gravitational effects. Their measurements can be interpreted as both delays and shifts (see below). 190

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The two clocks are physically distant but connected by an optical fiber. A kinematical effect results from their comparison when one is moved with a velocity of less than 10 m/s. The observed desynchronization of the clock signals can be interpreted as a kinematical shift. But the moving clock is in fact accomplishing round trips, whose durations differ from those of the “trips” of the clock at rest: the moving clock’s history is analogous to that of the traveling twin, so that the difference in durations can be interpreted as a delay. The gravitational effect is also measured for an altitude displacement (of one clock) of 33 cm only, as a fractional frequency change of 4.1 × 10−17. According to the authors, improvement in and extensions of this technique open the field to new applications as detailed above, as well as to space-based tests of fundamental physics. 13.5  The GPS system The current use of the GPS system can also be considered as a permanent confirmation that time cannot exist. Each satellite among the 24 of the NAVSTAR constellation carries a cesium atomic clock and emits radio signals with information concerning its position. Their important velocities (3.87 km/s) generate relativistic kinematical shifts, and in particular a transverse Doppler shift of 8.3 × 10−11. It directly affects the signal, cumulating to 7 microseconds/day. The stronger gravitational shift, 5.3 × 10−10, generates a delay of 45 microseconds/day. The total amounts to 38 microseconds/day; in case of a non-relativistic treatment, this would generate an error in position of about 15 km/day (a 1 microsecond shift corresponds to a 300 m error in positioning).

14  Linking Delays and Shifts As mentioned above, some experimental results can be interpreted simultaneously as relativistic delays or shifts (as, for instance, during one of the Hafele–Keating flights reported above). In fact, both are manifestations of the same effect. Coming back to the twins, one may imagine that T, the traveling one, is sending periodic signal to his brother at rest, with a fixed period Pemitted. During the first part of his trip, T is receding so that R receives the signals at period Preceived, with a positive redshift z  = Preceived/Pemitted – 1 > 0. During the back part, T is 191

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approaching and R receives a blueshifted signal with z = Preceived / Pemitted – 1 < 0. NP would predict that the red- and blueshifts are reduced to their linear parts, so that they would compensate during the whole journey: the sum of the emitted periods would coincide with the sum of the received ones, ensuring chroincidence, with a zero delay. In EP, however, the Doppler shift formula differs; the shifts do not compensate (only their linear parts do). This explains why kinematic delays are quadratic in velocity. In this sense, the delay appears as the residual effect of the shifts.

15  What Remains? The ultimate notion of temporal flavor that remains in EP is proper duration, “proper time” being simply another name for it. This is what people experience (psychologically, physiologically, etc.) and call their flow of time; this is what clocks measure. Although the world of NP is structured by time, the space-time in EP is structured by its causal structure. This is the collection of the causal links between events, expressing the physical possibility that one event may or not influence another. It codes the constancy of the velocity of light, and the impossibility of surpassing it.20 This causal structure plays a fundamental role in EP, which in some sense takes the place of that played by time in NP. But it is defined using the spatiotemporal metric,21 not from a temporal one. Thus, it has no direct links with time functions, proper times, or durations. The space-time of general relativity may be considered a causal structure, to which is superposed an additional “scaling function,” which allows for defining proper durations (the only notion with a temporal flavor). One may speculate (as in “causal theories” or “conformal theories”) that the latter is superfluous, and only appears as an emergent property of the causal structure.

References Chou, C. W., Hume, D. B., Rosenband, T. and Wineland, D. J. (2010). Optical Clocks and relativity, Science 329, 1630–1633.  It is a partial order relation in EP (not a total order as in NP); causal past and causal future are well defined in EP, but the present is not. 21  It corresponds to its conformal part. 20

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During, E. (2014). Langevin ou le paradoxe introuvable, Revue de métaphysique et de morale 2014/4 (N° 84), pp. 513–527 (https://www.cairn.info/revue-demetaphysique-et-de-morale-2014-4-page-513.htm). González Hernández, J. I., Rebolo, R., Pasquini, L., et al. (2020). The solar gravitational redshift from HARPS-LFC Moon spectra. A test of the General Theory of Relativity, A&A (arXiv_2009.10558). Greenstein, J. L., Oke, J. B. and Shipman, H. (1985). On the redshift of Sirius B, Quart. J. Roy. Astron. Soc. 26, 279–288. Pound, R. V. and Rebka, Jr. G. A. (1959). Gravitational red-shift in nuclear resonance, Phys. Rev. Lett. 3(9), 439–441; Pound, R. V. and Rebka, G. A. (1960). Apparent weight of photons, Phys. Rev. Lett. 4, 337–341. Pound, R. V. and Snider, J. L. (1964). Effect of gravity on nuclear resonance, Phys. Rev. Lett. 13(18), 539–540. Shapiro, I. I. (1964). Fourth test of general relativity, Phys. Rev. Lett. 13, 789–791 (https://doi.org/10.1103/PhysRevLett.13.789). Trueba, N., Miller, J. M., Fabian, A. C., Kaastra, J., Kallman, T., Lohfink, A., Proga, D., Raymond, J., Reynolds, C., Reynolds, M. and Zoghbi, A. (2020). A redshifted inner disk atmosphere and transient absorbers in the ultracompact neutron star X-ray binary 4U 1916-053, Astrophys. J. Lett. (arXiv: 2008.01083).

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Chapter 8

P ROB L E M O F T IM E: LI E THEO RY S UF F IC E S TO RESOLVE I T Edward Anderson Institute for the Theory of STEM and Foundational Questions Institute (fqXi), USA [email protected]

The Problem of Time is a tangle of fundamental questions held to be foundational for much of Quantum Gravity. This is due to conceptual gaps between, on the one hand, General Relativity, and on the other hand, other observationally confirmed paradigms of Physics: Newtonian Physics, Special Relativity, Quantum Mechanics, and Quantum Field Theory. Multiple facets of the Problem of Time were first pointed out by Wheeler, DeWitt, and Dirac over 50 years ago. These were subsequently gathered and classified by Kuchař and Isham. They argued that the lion’s share of the problem consists of facet interferences. They also posed the question of in which order to approach the facets. We reinterpret these facets as Background Independence aspects. At the local classical level, we show moreover that these form two copies of Lie Theory with a Wheelerian 2-way route between them. This resolves facet ordering. It also accounts for facet interferences. Closure turns out to be central. Closure is modeled by Lie brackets, Lie algebras, and Lie algebroids. It works via the Generalized Lie Algorithm (broadening Dirac’s Algorithm for constrained systems). Elsewhere in our resolution, the following mathematics occurs. Lie derivatives model Relational aspects. Lie brackets algebra commutants cover Observable aspects. Lie brackets algebra Deformation leading to 195

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Lie Rigidity underlies aspects of Constructability, such as of spacetime from space. Categorically analogous structures then phrase quantum counterparts of all aspects. In this way, a fundamental tangle of physically and philosophically interesting questions has attained a concrete mathematical understanding.

1 Introduction Each observationally established paradigm of Physics entails its own conceptualization of time and space. The ensuing conceptual gap is chiefly concentrated between the following. General relativity (GR; Sections 2.4 and 8.2.6) lies on one side. On the other side unfold other observationally confirmed paradigms of Physics: Newtonian Physics (Section 2.1, with relational critique in Section 2.2), Special Relativity (SR; Section 2.3), Quantum Mechanics (QM), and Quantum Field Theory (QFT). This gap leads to a series of conceptual problems, often referred to as the Problem of Time [18, 22, 23, 44, 45, 65, 68, 76]. This is meant in a multifaceted sense, since the above gap entails multiple differences in temporal notions. Setting out to explain the Problem of Time requires maintaining a clear distinction between the spacetime (S) perspective on the one hand and “space-time split, dynamics and canonical” (C) perspectives on the other. Due to this, we need to outline (Sections 2.1–2.6) how these paradigms differ among themselves regarding notions of all three of time, space, and spacetime. Presenting these physical differences requires mathematical exposition at the level of diffeomorphisms and Killing vectors (Sections 2.4–2.6). Sections 2.1–2.3 build up simpler temporo-spatial notions in preparation for this. Most facets were first envisaged over 50 years ago by Wheeler [22], DeWitt [23], and Dirac [7, 8, 12, 18]. It took 25 years for the full conceptual content of the Problem of Time1 to be assembled into Kuchař and Isham’s [44, 45] classification of facets. Attempting to extend the resolution of a first Problem of Time facet to include a second facet was found to have a strong tendency to interfere with the first facet’s resolution. Due to this, Isham and Kuchař argued that  As well as subject areas, let us capitalize “Problem of Time,” “Background Independence,” and their respective facets, aspects, and strategies. This is, firstly, to render them easier to spot and parse in the text and, secondly, to disambiguate from colloquial use of any of their constituent words. 1

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the lion’s share of the Problem of Time consists of facet interferences. They also posed a longstanding problem: in which order to approach the facets [44, 45, 47]? The Problem of Time constitutes a conceptual subset [22, 76] of the difficulties by which Quantizing Gravity is hard [18, 23, 37, 40, 50, 52, 57, 63]. Resolving Kuchař and Isham’s Problem of Time facets can be reconceptualized as [68, 76] implementing Background Independence2 aspects. Verily, the above gap more generally shows up between those paradigms of Physics that are Background-Dependent and those that are BackgroundIndependent. At the local classical level, this model of Background Independence can furthermore be recognized [86, 88] as Lie Theory [4, 19, 20, 31, 64, 69, 75]. Thus, Lie Theory provides a Local Theory of Background Independence [86, 88] that constitutes a Local Resolution of the Problem of Time [76, 79, 80, 81, 83, 84, 85, 87, 88] at the classical level. A tangle of fundamental problems thus reduces to a new application of a basic and well-known branch of Mathematics. This development is both interesting3 and renders this subject much more widely accessible. Indeed, this is a clarifying enough development that presenting it ab initio is highly preferable to stating up front what Kuchařs and Isham’s facets are. Providing the first such account is the current article’s purpose. More precisely, the facets are reduced to, firstly, two copies of Lie Theory corresponding to S and C, and secondly, a Wheelerian two-way route [22, 28] between these. That Lie Theory’s parts exhibit nonlinearly ordered interplay for well-understood reasons explains some facet interferences and facet ordering. Other facet interferences are explained by, firstly, the C and S versions being entitled to appear at different points in the study (the two copies have, on occasion, previously been confused or conflated) and, secondly, by the fact that previous literature did not entertain multiple facets within  In the current article, Background Independence is considered at the Differential Geometry level of mathematical structure. See Section 8 and, e.g., [16, 60, 76] for more about Background Independence. 3  These fundamental problems have been of longstanding interest not only to Physicists but also to Philosophers of Physics. The main point for these to take home is that there has been a substantial leap in the mathematical identification of these problems. From this, why many of these fundamental problems appear together and interact with each other has become technically clear. For the separate matter of experimentalists benefiting from making careful distinction between notions of time, see rather, e.g., [59, 61] for the Newtonian, SR and quantum part and [76] for extension to, and combination with, GR. 2

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a single overarching mathematical formulation (which Lie Theory and mild recategorization [76] combine to provide). We set the scene for our Lie-Theoretic resolution using Lie derivatives [9, 42, 69] to model Relationalism [34, 48, 66, 76, 79, 80, 82, 88] in Section 3. The main part of our program turns out to be Closure [18, 43, 81, 83] (Section 4). In fact, a key point in resolving the Problem of Time is as follows: Relationalism’s philosophically rooted considerations ([2, 3, 49] outlined in Section 2.2) rapidly lose centrality to Algebra from the advent of Closure onward. Modeling Closure requires, in more detail, Lie brackets [20, 64, 75], Lie algebras [15, 64], and Lie algebroids [56]. It also requires the Generalized Lie Algorithm [86, 88] (of which Dirac’s Algorithm for constrained systems [8, 12, 18, 43] is a well-known subcase). This is followed by two separate developments. The first involves a slight extension of Lie’s Integral approach to Geometrical Invariants [4, 71] so as to incorporate observables [7, 47, 72, 76, 84, 88] (Section 5). The second is the Lie algebra cohomology [19, 73] approach to Constructability [22, 54, 70, 76, 85, 88] (Section 6). Section 7 follows this up with the “Wheelerian 2-way route” [22] between the C and S copies. This consists of a further Constructability: of spacetime from space. The reverse route pins a Lie algebroid interpretation [67, 76] on GR’s Refoliation Invariance [29, 45]. In Section 8, we conclude by pointing to our new local classical understanding’s global and quantum consequences. We also keep track via Fig. 8.16 of the conceptions and names of facets and aspects from prior to our Lie-Theoretic reconception of the subject. This is both for readers familiar with previous literature and for identifying these structures as subcases that drop out of the current article’s analysis.

2  Time and Background in Each Paradigm of Physics 2.1  Space and time in Newtonian Physics Newton [1] conceived of an absolute time and an absolute space. Both of these are infinite, continuous, and imperceptible. Both act and yet cannot be acted upon. The conventional mathematical models for these are as follows. For absolute time, the real line with 1-d Euclidean distance is a “time 198

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Figure 8.1   Space and time in Newtonian Physics.

metric” (Fig. 8.1.a). For absolute time, one has the real line 3 with 1-d Euclidean distance, while absolute space is modelled by Euclidean space 3 with a 3-d Euclidean metric, as interpreted with a fixed origin O and fixed axes A (Fig. 8.1.b). Past, present, and future take intuitive forms in Newtonian Physics (Fig. 8.1.c). Each present instant, which also embodies the Newtonian Paradigm’s notion of simultaneity [59], is labeled by a value of Newtonian absolute time. The Galilean transformations (Fig. 8.1.d) underlie Galilean Relativity. This interrelates the inertial frames of Newtonian Mechanics — that is, the frames privileged by Newton’s First Law applying therein. From Newton’s point of view, these frames are either at rest in absolute space or moving uniformly through it along a straight line. The freedom of Fig. 8.1e is also available in practice. Already in Newtonian Physics, time and energy are conjugate quantities. This leaves Hamiltonians carrying connotations of time, a feature we subsequently make use of. 2.2  Relational critique The immovable external character of the absolute space and time assumed in the Newtonian Paradigm has been challenged by relationalists, such as Leibniz [2] and Mach [3]. In fact, Newton [1] has already provided relative notions of space and time. While he saw these to be practical, he also regarded them as secondary. 199

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An alternative “Relational Paradigm” regarding these as primary could start as follows. Physics is solely concerned with relations between entities (matter and force mediators) that act testably and are actable upon [21]. Anything that does not comply with these criteria is held to be a physical non-entity (such as absolute space and absolute time). We consider the following Relational Principles. Leibniz’s Space Principle: Space is the order of coexisting things [2]. Mach’s Space Principle [3]4: “No one is competent to predicate things about absolute space and absolute motion. These are pure things of thought, pure mental constructs that cannot be produced in experience. All our principles of mechanics are, as we have shown in detail, experimental knowledge concerning the relative positions of motions and bodies.” Leibnizian Time(lessness) Principle: There is no time at the primary level for the Universe as a whole [2, 48, 76]. Mach’s Time Principle [3]: “It is utterly beyond our power to measure the changes of things by time. Quite the contrary, time is an abstraction at which we arrive through the changes of things.” See, e.g., [41, 49] for further “absolute versus relational motion debate” considerations. The current article does not dwell on the current subsection’s generalities, moving instead to a sharp mathematical implementation of these principles in Section 3. 2.3  Special Relativity (SR) Electromagnetism exhibits not Galilean but Lorentzian Relativity. This now includes boost transformations for passing from a rest frame to one moving (without loss of generality: Fig. 8.2a) in the x-direction with constant velocity v. The time coordinate itself transforms nontrivially here (unlike for Galilean Relativity: Fig. 8.1d). Subsequently demanding that Mechanics and Electromagnetism share the same Relativity led to the next historical discovery: Einstein’s SR. (Neither Leibniz nor Mach had any actual theories implementing their principles.) Thus, Einstein now faced the dilemma of whether the shared Relativity should be Galilean or Lorentzian, i.e., involving infinite  This is not to be confused with Mach’s Principle for the Origin of Inertia ([49], Section 3 of [76]), or with anything else called Mach’s Principle in other works (of which, e.g., [49, 57] have an extensive selection). 4

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Figure 8.2   Space, time, and spacetime in SR.

or finite propagation speed: the speed of light c. Figure 8.2e enlarges this to a more logically complete trilemma. Consistency with Nature turned out to require the finite case, entailing correcting Newtonian Mechanics so as to exhibit Lorentzian Relativity as well. Including space- and time-translations, the Lorentz group of Lorentz transformations (boosts and spatial rotations) extends to form the Poincaré group Poin(4) of Fig. 8.2b. In SR, space and time are co-geometrized as flat Minkowski spacetime 4. This carries the Minkowski metric η , of signature (–+++) and whose line element can always be put into the standard form of Fig. 8.2c. Postulating a spacetime metric builds in observers’ ability to measure lengths and times. Let us contrast this spacetime perspective of SR with the separate geometrizations of space and time of Newtonian Physics. One of SR’s notions of time takes the form of 1 of the 4 coordinates on 4. Nonzero vectors in the 3 model of space always have positive norms. In contrast, there are three types of nonzero vector X in 4 as per  Fig. 8.2d: timelike, null, and spacelike. The existence of these three types of spacetime vector is central to the physical interpretation of SR (Fig. 8.2e). In particular, massive and massless particles follow timelike and null worldlines, while no physical form of particles follows spacelike 201

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worldlines. Finally, in Relativity it is the further distinct notion of proper time that models time as measured by each observer’s clock, integrating along their past worldline. Passage from Newtonian Physics to SR, however, has just traded one set of absolute structures — absolute space and absolute time — for another: absolute Minkowski spacetime 4. 4 still acts while not being actable upon, while SR’s new class of global inertial frames is not explained by SR itself [55]. Further differences include that Newtonian absolute simultaneity has been replaced by SR’s relative simultaneity [59]. In the process, the kinds of surfaces that carry significance have changed qualitative type. On the one hand, they are spatial planes of absolute simultaneity in Newtonian Physics. On the other hand are the null cones in SR along which the free motion of light occurs. Shared Relativity moreover implies that all other massless particles’ free motion occurs on the very same null cones, which are thus universal. Furthermore, massive particles are only permitted to travel from a spacetime point — an event — into the interior of the future null cone of that event. In this way, Relativity possesses a nontrivial conceptualization of causality [36]. For instance, in Fig. 8.2e–0) events on and within the future null cone of event p in 4 are the only ones that can receive signals from p and thus be influenced by event p. Finally, passing to SR gives a fundamental standing to c, which is subsequently indispensable in formulating GR. Ordinary QM largely takes after Newtonian Physics as regards its notions of time and space. However, QM’s evolution and collapse postulates attribute further roles to time. QFT’s notions of time largely take after SR’s. We return to a few temporal issues specific to QM and QFT in Section 2.5. 2.4  GR from the spacetime (S) perspective SR fails to include Gravitation. This can be remedied by passing to curved spacetime modeling [36, 55], in which only a local “freely falling” notion of inertial frames is supported. In this GR setting, space and time remain cogeometrized. Our model for this is now a differentiable manifold M; this still carries a (–+++) signature metric, denoted here by γ [30, 35, 36, 55].5 By this everywhere locally reducing to SR’s η (and minimal coupling), the other laws of Physics locally take their SR form.  This has determinant γ and Ricci scalar curvature R[γγ].

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Figure 8.3   Spacetime (S) perspective on General Relativity.

GR retains a notion of coordinate time (though this is now a more general coordinate), and of proper time as well. GR also retains notions of timelike, spacelike, and null. 4’s timelike and null straight lines, as followed by massive and massless particles, repectively, are however bent by the gravitational field in GR. The curves that are followed by relatively-accelerated free-falling particles ensue. These curves are, in the absence of other forces, geodesics. Causality continues to play a major role [36] in the spacetime formulation of GR. One difference is that now matter and gravity affect the larger-scale causal properties. The null cone structure is thereby dynamical (Fig. 8.3b). Simultaneity in GR closely parallels that of SR [59]. A suitable action principle underlying the spacetime formulation of GR6 is Einstein’s and Hilbert’s (Fig. 8.3c). This variationally implies Einstein’s field equations of GR. GR is not just a replacement of 4’s flat geometry with a generally curved notion of geometry that encodes a Relativistic Theory of Gravitation. For it also represents a freeing of Physics from absolute or Background-Dependent structures. This is a transition between the following two kinds of situations. The first involves “actors performing on a stage,” of which Newtonian Physics and SR are examples. The second concerns “material blobs deforming ambient spacetime somewhat like a rubber sheet” by energy–momentum sourcing spacetime curvature in accord with Einstein’s field equations of GR.  We give this for GR in vacuo for simplicity, with the now-customary cosmological constant term Λ included. Furthermore, our presentation can be extended to include a sufficient set of matter fields [28, 76] to describe Nature. 6

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This is a large conceptual change compared to Section 2.3’s moderate gains from trading Newtonian absolute structures for SR ones. Some further context for this freeing is that, while GR was not originally set up to directly implement our above selection of Mach principles (or others), GR can nonetheless be recast to manifest ours, as outlined in Section 3. Metrics being symmetric, γ has a priori 4 × 5/2 = 10 degrees of freedom per spacetime point. Furthermore, we consider the space of spacetimes [40] on a given M, which we denote as in Fig. 8.3d), and spacetime diffeomorphisms as per Fig. 8.3e). Because of the spacetime diffeomorphisms, 4 of γ  γ ’s 10 degrees of freedom are unphysical, leaving us with 6. The quotient of Fig. 8.3f is a resulting arena of physical significance: the space of spacetime geometries: equivalence classes of spacetimes on M modulo the corresponding diffeomorphisms [26, 40, 76].7 2.5  Killing vectors Those diffeomorphisms that are also isometries are picked out by Killing vector solutions ξ of Killing’s equation [5, 9, 25, 36] (Fig. 8.3g). This equation is phrased in terms of the Lie derivative: a simple and well-known purely Differential-Geometric notion of derivative [9, 25]. For example, we can rederive Eucl(3) and Poin(4) by the d and 4 cases of solving Killing’s equation. Killing vectors furthermore underlie many temporal and other background structures in Newtonian Physics and SR. GR is addtionally ultimately about generic solutions [30], which possess no Killing vectors. In passing to GR, the Euclidean group of Mechanics and Ordinary QM and the Poincaré group of SR and QFT are superseded by the following. In some situations, the trivial isometry group of a generic spacetime steps up. In other situations, the much less understood group of spacetime diffeomorphisms Diff (M) enters. Some time-related applications are affected by GR spacetimes not in general having privileged frames [36]. Others are more concretely affected by there being no timelike Killing vector (timelike as judged by the metric). That M4 has a privileged class of times is underlain by this. In this way, much of the structure that many SR and QFT calculations are based upon is lost. This affects, for instance, vacuum uniqueness,  We use bold mathfrak leading letters for spaces of entities so as to immediately avoid confusion between these and the entities themselves. 7

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positive-negative mode splits, particle species labeling by Poin(4)representations... 2.6  Dynamical and Canonical formulation of GR (C perspective) GR is additionally not only a theory of 4-d spacetime, but also a dynamics [13, 28, 74, 76] of 3-d space. Dynamics entails heterogeneous treatment of space and time, involving, in particular, evolution in time. SR’s spacetime 4 itself neither evolves in time nor plays the timeless role of configuration. In contrast, from a Relativistic point of view, spacetime contains both notions of spatial configuration and of time (Fig. 8.4a). 3-d space is modeled here by a fixed topological manifold ∑ carrying an evolving positive-definite 3-metric h.8 This has a priori 3 × 4/2 = 6 degrees of freedom per space point. Furthermore, the spacetime metric is split in the manner of ADM [13] (Fig. 8.4b). Given a fixed ∑, the possible h thereon forms the configuration space [23, 26] of Fig. 8.4c). A spatial slice within spacetime is standardly characterized not only by its induced metric h but also by its extrinsic curvature K. The Einstein–Hilbert action (Fig. 8.3a) gives, under the split in Fig. 8.4b, the well-known ADM action (Fig. 8.4d). The ensuing C perspective [13, 22, 23, 28, 76] emphasizes constraints, evolution, and canonical machinery. Examples of the last of these include Poisson brackets and Hamiltonians, which are both useful toward Quantum Theory. Contrast this with the S perspective’s emphases. GR’s momenta p turn out [13] to be closely related to K, as per Fig. 8.4e: a tensor-density version with a trace term K = tr(K) subtracted off. The ADM action yields two constraint equations. First, the momentum constraint9 M (Fig. 8.4f) arises from variation with respect to the shift β (Fig. 8.4b). Second, the Hamiltonian constraint H (Fig. 8.4g) arises here from variation with respect to the lapse α (Fig. 8.4b). Interpretation of these constraints is of great significance in both conceiving of, and (at least locally) resolving, the Problem of Time. That of M is straightforward: it corresponds to diffeomorphism invariance of the spatial geometry on ∑. That is, to Diff (∑ ∑) playing the role of a physically redundant group of transformations (compare gauge  h has components hij, determinant h, covariant derivative D, and Ricci scalar curvature R = R[h]. 9  We jointly highlight all constraints by using upper-case calligraphic font leading letters. 8

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Figure 8.4   Dynamical and Canonical (C) perspective on General Relativity.

groups’ role in Gauge Theory). By this, Diff (∑ ∑)-invariant information in spatial 3-metrics — spatial 3-geometries — constitutes less redundant configurations for GR (Fig. 8.4h). This accounts for Wheeler coining the more specific term Geometrodynamics [17, 22, 28] for GR’s dynamical formulation. Spatial 3-geometries have 6 – 3 = 3 degrees of freedom per space point. The corresponding configuration space is Wheeler’s [17, 22] superspace (Fig. 8.4h), as further studied, e.g., in [26, 51, 62, 74]. Interpretation of H is tougher (see Sections 3.2 and 4, and [88]). Quantizing GR’s constraints gives something like Fig. 8.4i–j). The quantum momentum constraint, as depicted, has its momenta ordered to the right [27]. In the quantum Hamiltonian constraint, ‘ ’ refers to caveats 206

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with choice of kinematical quantization [38], need for regularization, and operator-ordering ambiguities. The operator ordering displayed is the Laplacian one [11].

3  Relationalism as Implemented by Lie Derivatives 3.1  State Space Relationalism Let us consider both S and C perspectives for a given system, G. We refer to each’s incipient notion of spacetime or space as its carrier, Carrier. The corresponding incipient states and state spaces [6, 22, 33, 39, 40] are denoted as in Fig. 8.5a,b). State Space Relationalism consists of the following postulates: SSR-1) Background Independence precludes a background carrier metric and any background internal structures pertaining to fields. SSR-2) Allow for a symmetry group10 of physically redundant transformations G to act on Carrier and fields thereon. We concentrate on the case of continuous transformation groups G acting upon the state space state. These take the form of a connected Lie group G. Involving G in this way is a further example of Physics’ practical need for mathematically-convenient descriptions that make use of  physically-redundant states. In each case, a redundant state space state

Figure 8.5   State Space Relationalism. 10

 Also known as automorphism group among mathematicians.

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ensues (Fig. 8.5g). Configurational Relationalism [34, 66] and Spacetime Relationalism [76] are subcases of State Space Relationalism. In some models, it is possible to work directly on reduced state spaces  (i.e., with G quotiented out). In other models, it is not, due to state  state ’s topology or geometry being complicated or unknown. The following indirect implementation remains universally available, however (at least formally). Let us first generalize from states to arbitrary objects O (from column 1 to column 2 in Fig. 8.5). This is so as to include notions of distance, information, correlation, quantum operators, etc. If some O is not G-invariant, it can be subjected to a group action (Fig. 8.5d). This sends the space of objects Obj to the bundle P(O Obj, G). This looks to be a step in the wrong direction as regards freedom from G. This is, however, doubly compensated for by following up with a move that uses all of G, which we denote by Sg∈G. With this, we descend to the quotient space in Fig. 8.5g. We thus have overall the G-Act, G-All procedure (Fig. 8.5f). This formally succeeds in producing G-invariant objects from our initial objects. A useful archetype of G-All operation is the widely encountered group averaging. The current article’s main applications, however, extremize over G instead, which we denote by Eg∈G. In the continuous Lie group case, the G-Act move we consider is an infinitesimal Lie derivative correction (Fig. 8.5e). For M further equipped with some level of structure σ , solving the corresponding generalized Killing equation [9, 25] of Fig. 8.5g) determines for (M, σ ) the corresponding generalized Killing vectors g. Aside from the metric structure case of the Killing equation itself, other s covered include similarity, conformal, affine, or projective structure. In each case, the solutions form a Lie algebra aut(Carrier, σ ) of the corresponding geometrical automorphisms. This is the Lie algebra g infinitesimally corresponding to our G. Velocities entering our actions are now to receive G-corrections [42]. Solving the carrier space’s generalized Killing equation gives moreover a universal prescription [78] for the form that these G-corrections are to take. Next, extremize the resulting action (Fig. 8.5d) over the group’s auxiliaries g to complete the G-Act G-All implementation. G-generators G ensue. Configurational Relationalism thus acts as a Constraint Provider: an underlying principle that produces constraints [22, 48]. This paragraph’s procedure is well-named Best Matching [34, 48, 66, 76, 80, 82, 88]; its output are shuffling constraints Shuffle. 208

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3.2  Temporal Relationalism Adopting a split space-time approach requires considering not only the above Configurational Relationalism but also the following Temporal Relationalism that rests upon Leibnizian timelessness. TR-1) Include no extraneous times or extraneous timelike variables [66]. TR-2) Include no label times λ either [66].11 Let us implement our postulates at the level of the Principles of Dynamics actions. This requires not the usual difference combination of kinetic term T and potential term V (Fig. 8.6b) but a product combination (Fig. 8.6c). We furthermore use configuration–change variables (Q, dQ)  or (Q, Q′), where dot rather than configuration–velocity variables (Q, Q) and dash are differentiation with respect to t and λ , respectively. This is so as to avoid velocities’ temporal connotations. We additionally require our action to be homogeneous-linear in the change variables. Our firstprinciple formulation is presented in Fig. 8.6c.3). Furthermore, we can model our changes as Lie derivatives £dQ (in close parallel to [42]). This aligns with how we model the other two types of Relationalism. Various increasingly familiar (albeit decreasingly Temporally Relational) intermediaries feature in Figs. 8.6c.1-2. These attest to the physical equivalence between the standard formulation of Physics and our Temporally Relational formulation. Figure 8.6c.1 involves not an external time but a label time λ ; velocities are here with respect to label time: Q′. Furthermore, this label time is meaningless by the interchange displayed in Fig. 8.6c.1). This does not comply with TR-2), but Fig. 8.6c.2 deals with this. Meaningless label time λ does not feature at all in this action. Our final action’s homogeneous linearity in Lie changes implies — by updating [66, 88] Dirac’s argument [18] concerning our first action’s Manifest Reparametrization Invariance — that at least one primary constraint must ensue. Temporal Relationalism thus joins Configurational Relationalism in being a Constraint Provider. Our homogeneousquadratic kinetic terms yield quadratic constraints. Sections 3.4 and 3.5 show that this recovers familiar-looking equations, which are now, however, placed on Temporally Relational foundations.  Labels are alias parameters. That some notions of time are reparametrizable (Fig. 8.6.a) is with reference to this. 11

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210 Figure 8.6   Temporal Relationalism. W: = E − V is the potential factor, where E is the total energy. M is the kinetic metric.

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3.3  Combining Temporal and Configurational Relationalism Temporal Relationalism implementation (TRi) can moreover be applied at all levels of the Principles of Dynamics (and various sequels: foliations, canonical quantization, path integrals [76], etc.). Using TRi versions avoids [76] “facet interferences” as one moves away from formulations in terms of actions to any formulation. For instance, TRi reformulations include new auxiliary variables (Fig. 8.7) and differential almostHamiltonians (Fig. 8.10). Using TRi versions does not, however, alter the Lie content of the formulation. This very largely saves us from having to present TRi versions here (see [76, 82, 83, 84, 85] for details). Example 1) Against a historical backdrop lacking in viable relational alternatives to Newton’s absolute Mechanics, we outline a Relational Mechanics [34, 66] in column 1 in Fig. 8.7. CR-1) fails here, since the usual absolutely interpreted flat Euclidean metric is clearly visible in the construction of the kinetic term. Example 2) SR correspondingly fails STR-1) by having a Minkowski spacetime background. STR-2) holds, however, at the level of SR possessing Poincaré invariance. Example 3) These shortcomings are resolved by superseding both of these models by a GR model. See Fig. 8.3a-f) and the last column of Fig. 8.7 for S and C formulations of this, respectively. While the Einstein–Hilbert action needs no correction terms, other parts of the S formulation are nontrivial in this regard. For instance, its perturbation theory [58] involves Lie derivative corrections so as to implement a point identification map. Also, a Measure Problem [45, 76] occurs at the Quantum level. The incipient configurations, configuration space, redundant group, and quotient space are as per Fig. 8.7.2) (a reissue of part of Fig. 8.4). Corrections are now to be not in terms of ADM’s shift vector b but rather of a TRi frame vector F (such that b = F′). These are further packaged into the Lie-Relational action for GR (Fig. 8.2.h). This shows that GR can be rearranged to directly manifest Leibniz and Mach’s time and space principles. [Again, the usual difference-type action — now ADM’s Fig. 8.4c) — will not do. This is now due to the presence of the ADM lapse α : an extraneous timelike variable. See also Fig. 8.7.2g) for a string of more widely known intermediary actions.] 211

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212 Figure 8.7   Relational Mechanics and Relational form of dynamical GR. m is the mass-weighted Euclidean metric with inverse n.

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Figure 8.8   Emergent Machian time.

Using the Lie-Relational action for GR, there is now no lapse α . Yet the usual Hamiltonian constraint H (Fig. 8.4.g) remains encoded, now instead as a primary constraint. As ever, the momentum constraint M is the secondary constraint corresponding to spatial diffeomorphism freedom. The configuration–change variables repackaging of this gives the TRi Thin Sandwich equation. This furthermore coincides as a PDE with the usual Thin Sandwich equation (posed to solve for b to its TRi counterpart being for F˙ ). This coincidence means that older PDE-level work on the GR Thin Sandwich [24, 46] carries over. The (TRi) Thin Sandwich does not end here as it proceeds to construct an adjacent future coating of spacetime. A priori timelessness can furthermore be emergently resolved by Mach’s “time is to be abstracted from change.” This occurs via viewing (ε or) H in this context not as an “energy equation” but conjugately as an equation of time [10, 48, 76]. We thus collectively denote H and ε by Chronos. This indeed rearranges to give an emergent Machian time tem, for instance, as in Fig. 8.8a). This resolution additionally remains expressible in terms of Lie derivatives. That is, “time is to be abstracted from Lie-derivative-change of configuration”. The constant tem(0) included encodes the freedom of choice of “calendar year zero.” Figure 8.8b gives the interplay between emergent time and Configurational Relationalism. tem also implements the further principle [28] of “choose time so that motion is simplest.” This is realized at the level of simplifying the momenta and the field equations.

4  Closure, as Implemented by Lie Brackets 4.1  Lie (and Poisson) algebraic structures Given a source of generators G, we also need to know the relations between them. In our present context, Lie brackets (defined in Fig. 8.9.1-3) 213

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Figure 8.9   Lie and Poisson brackets and algebraic structures.

provide our relations as per Fig. 8.9b. The G here are generally termed structure functions. These are all constant — structure constants — in the case of a Lie algebra, but are at least in part functions in the more general case of a Lie algebroid [56]. We jointly refer to these two cases as Lie algebraic structures. If the derivation property of Fig. 8.9.4 holds as well, such are a fortiori Poisson. 4.2  Generalized Lie Algorithm (GLA) In considering Closure, our starting position is that Relationalism has provided us with some generators G. It turns out, however, that we cannot assume these to be complete or even consistent. Instead, we make Lie brackets out of them to ask whether Closure is possible, proceeding as follows. We preliminarily introduce a notion of weak equality ≈, i.e., equality up to a linear function(al) of the generators. This is in contrast to equality in the usual sense, which we denote as usual by = but refer to as strong equality. Then, given a candidate set of generators, the Generalized Lie Algorithm (GLA) permits the following 6 outcomes [18, 43, 86] among the brackets between them. GLA-1) New generators G ′ arising as integrabilities of already known generators G. GLA-2) Identities: equations reducing to 0 = 0. GLA-3) Inconsistencies: equations reducing to 0 = 1. GLA-4) Rebracketing: passing to Dirac–Lie brackets that absorb any secondclass generators encountered. They are defined to the exclusion of first-class generators: those that brackets-close [18, 86]. GLA-5) “Specifier equations” when in the presence of an appending process. GLA-6) Topological obstructions [50, 37, 52]. 214

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Some readers will have noticed that the GLA described above both parallels and generalizes the more widely known Dirac Algorithm [18] for constrained Hamiltonian systems. The GLA furthermore covers a TRi Dirac Algorithm [66, 70, 76] as well as how to handle Lie generators from the S perspective [88]. Allowing for GLA-3) is a substantial insight of Dirac’s. This gives one’s algorithm selection principle properties, with the capacity to reject candidate sets of generators. The GLA thereby has considerably more power than previous [4] Lie algorithms. Dirac–Lie brackets [88] generalize the more familiar Dirac brackets [18]. The restriction-based version of the proof that Dirac brackets are still geometrically Poisson brackets carries over to Dirac–Lie brackets still being geometrically Lie brackets. The Dirac Algorithm provides a familiar example to clarify what GLA-5) means by an appending process. Here, constraints are appended to the Hamiltonian H via Lagrange multipliers so as to form Fig. 8.10a)’s total Hamiltonian. The TRi version requires appending instead to the differential Hamiltonian dH via cyclic changes so as to form the extended differential almost-Hamiltonian in Fig. 8.10.b. (This is Hamiltonian in the physical variables and yet involves auxiliary variables’ changes instead of their momenta). In each case, GLA-5) indeed consists of equations that specify what forms a priori free appending variables can take. The GLA terminates when one of the following applies [86]. A) It hits an inconsistency. B) It cascades to triviality (no degrees of freedom left). C) It arrives at an iteration that produces no new entities — generator or specifier equations — while retaining some degrees of freedom. The last of these is the termination condition, which renders a candidate theory successful at this stage: consistent and nontrivial. In this case, the final output is a Lie algebraic structure of first-class entities. On some occasions, some of GLA-1-6) and ensuing inconsistencies can be avoided by setting a priori free constants to take fixed values. In this way, a right-hand-side term drops out by picking up a zero factor. This is termed strong vanishing; it plays a crucial role in Sections 6 and 7.1. One can also consider starting with a candidate set of generators, and then Encoding them via Relationalism. Suppose that the GLA gives that a model’s generators arising from Relationalism require further generators so as to close. Then, it may be possible to revisit Relationalism so as to further Encode these integrability constraints. More generally, Provide and Encode moves can be performed recursively. This reflects that Relationalism and Closure fuse. For, algebras in 215

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216 Figure 8.10   Extended Hamiltonians and their TRi counterparts.

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general have not only generators but also relations. In Fig. 8.16f, this fusion is depicted by the two-way arrow or 2-cycle. We are to cycle around here until one of the following occurs: Either Relationalism and Closure are jointly satisfied or the candidate theory is found to be trivial or inconsistent and is thus discarded. The end product of a successful candidate theory’s passage through the GLA is a generators algebraic structure. This consists solely of Lie-firstclass generators closing under Lie (or in a sense more generally Lie–Dirac) brackets. [It could, however, be more generally a generators-and-specifiers algebraic structure.] Upon suitably passing a Dirac-type Algorithm’s test, Best Matching’s Shuffle constraints can now be renamed F lin: first-class linear constraints. Relational Mechanics” generators close as a Lie algebra, SR’s as the Poincaré Lie algebra, GR’s spacetime’s as the spacetime diffeomorphism Lie algebra, and GR’s canonical constraints close as the Dirac algebroid [18, 29, 32] (a Lie algebroid). These can be found in a) to d) of Fig. 8.11 respectively; only nontrivial commutation relations are exhibited. a) form the Euclidean algebra, ε then commuting with everything. b) form the Poincaré algebra. c) form the diff (M) algebra, with involvement of the Lie derivative with respect to an arbitrary spacetime vector; this is an infinitely generated Lie algebra. d.1) similarly forms a diff (∑) subalgebra. d.2) signifies that [32] H is a diff (∑) scalar density. Interpretation of d.3) is far more involved. For instance, d.3) reads that [27] M is an integrability of H. So suppose Diff (∑)-Relationalism were not initially entertained. Then, the (Tri) Dirac Algorithm would enforce it anyway, via this integrability and then an example of the above notion of Encoding. This means that in unreduced GR, one cannot have Temporal Relationalism without the support of Configurational Relationalism. d.3)’s right- hand-side, moreover, contains structure functions h−1(h(x)). So this is the bracket by which GR’s Constraint Closure is indeed in the form of an algebroid [67]. See [29, 32, 88] for further details about interpreting d.3). The above fusion of Relationalism’s incipient outcome and Closure into the generators and relators characterization of Lie (or Poisson) algebraic structures has two separate sequels, as per Sections 5 and 6. This, moreover, leaves Closure being central among Background Independence aspects.

217

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218 Figure 8.11   Examples of Generator Closure (including Constraint Closure).

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5  Assignment of Observables For a given system S’s state space state(S), our first sequel consists of Finding a Function Algebra Thereover [84]. The constituent functions are to be suitably smooth as regards being able to subsequently perform all physically necessary calculations. In the presence of generators, these functions need to zero brackets-commute with these generators (Fig. 8.12a). This could be strongly or weakly zero. The name given to this endeavor in Physics is Assignment of Observables to our system [7, 47, 81]. The current article picks out observables in this article by use of the sans serif font. We also denote the function algebra these form by Obs(state(S)). Moreover, the Jacobi identity of Fig. 8.12b) signifies that finding Liebrackets-commuting notions of observables O is only consistent if the g in question are already known to close. By this, the Problem of Time decouples into a Relationalism-and-Closure subproblem, which has to be solved prior to the Problem of Observables. This is the meaning of Fig. 8.16d,e)’s lower 1-way arrow. While this was already known to Dirac [7], its interpretation as a Problem of Time facet ordering was not remarked upon until [72]. It is often useful to recast each notion of observables defining zero commutation relations as [84] an explicit first-order linear PDE system (Fig. 8.12.d). This is a type of system that Lie [4] considered to be the core of the mathematics that he was developing.12 Furthermore, the wellknown Flow Method [4, 69, 71, 84] is suitable [84] for such systems. More specifically, the Problem of Observables’ PDE system is a slight extension [84] of Lie’s Integral Approach to Geometrical Invariants [4, 71], by which it is clearly also Lie-Theoretic. This extension couples in a trivial ODE in the strong case and yet a nontrivial one in the weak case. Fig. 8.12.c’s Jacobi identity means that one cannot simply seek individual observables that solve the defining brackets. We thus need the entire solution space Obs(state(S)) of the PDE system as equipped with the corresponding brackets [7, 72]. Finally, Assignment of Observables is to be followed up by Expression in terms of Observables of all the directly meaningful quantities in one’s theory. This requires further elimination-type calculations.  Note in particular that Relationalism’s Generalized Killing equation is also a system of this core type. Also, Closure and the next section’s Constructability involve such a system, albeit viewed in reverse. That is, not integrating but differentiating, which is systematic. 12

219

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220 Figure 8.12   Assignment of Observables.

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6  Constructability, as Implemented by Deformations if Rigidity Holds Until the 1960s, local Lie Theory consisted of the parts covered in Sections 3–5.13 A separate useful sequel to Lie brackets Closure has since appeared, as indicated by Fig. 8.16.d-e)’s upper one-way arrow. That is, to examine the extent to which this Closure is maintained under deformations [19] of the generators (Fig. 8.13.a). One possible outcome is that only our original generators maintain consistency. This phenomenon is called Lie Rigidity. This is underlain by Lie algebra cohomology, as per Fig. 8.13.b’s condition. The Poisson version of all this may also apply. So far, examples that have been investigated quite often recover one’s incipient generators as one of a few discrete options from a product of factors, any one of which can vanish strongly. This is a “branching” phenomenon ([43] already observed that the Dirac Algorithm exhibits such a feature, though not yet in the context of Constructability). Rigidity gives a way in which less structure assumed — the deformed generators — can recover specific forms of generators that close. These are isolated points of consistency in an otherwise inconsistent parameter space of deformations. This is what Constructability means. Whenever this happens, new foundations are discovered, whether for one’s undeformed theory by itself, or for multiple theories now jointly founded as a set of branches. Let us first illustrate this with an example of a deformed quadratic generator in Flat Geometry (Fig. 8.13.c,d). Two strongly vanishing roots emerge, which can be identified with Projective Geometry and Conformal Geometry. That is, we have hit upon new joint foundations [77] for Flat Geometry’s dichotomy of “top geometries” supported.14 The GLA moreover finds these by mere consistency of an algorithmic turn-the-handle process: no knowledge of, or “intuition for” Flat Geometry is required. The above example of Constructability is more specifically of space from less structure of space assumed. Additionally, its spacetime from less structure of spacetime assumed parallel holds as well, the underlying  Excepting the key Dirac consistency part of the GLA, which remained confined to the Dirac Algorithm for Constraint Closure. 14  “Top” is meant here in the sense of having a maximally extended set of generators. These two top geometries follow from adding different mutually inconsistent sets of generators to the similarity group’s. Further details of our argument require dimension ≥ 3 [77]. 13

221

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222 Figure 8.13   Constructability of Space from Less Structure of Space Assumed.

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working being unaffected by signature [88]. Both of these are selfConstructabilities. Since spacetime and space each are structures to which primality is often ascribed, both of the above are furthermore primality-maintaining Constructabilities.

7  Wheelerian 2-way Route Between Dynamical and Spacetime Lie Claws 7.1  Spacetime Constructability from Space A further kind of Constructability is moreover primality traversing: that of spacetime from space. Deformations and Rigidity continue to apply here, and “branching” ensues as well. In particular, GR and its spacetime emerge from sparser spatial foundations as one of very few consistent alternatives [54, 70, 76, 85]. In vacuo, Dirac-type Algorithms derive the explicit value of the DeWitt coefficient [23] x = 1 in GR-as-Geometrodynamics’ kinetic term as one of very few roots of an algebraic equation, as per Fig. 8.14a,b). Upon introducing minimally coupled matter, the infinite, finite, or zero shared propagation speed trilemma in Fig. 8.2e emerges from the choices of roots in the additive term in Fig. 8.14c). This includes recovering Einstein’s dilemma of Galilean versus Lorentzian local Relativity as a (Dirac-type subcase of) GLA branching. This furthermore constitutes an example of inclusion of Lie algebras’ contracted limits [31]. This follows from the Dirac-type Algorithm enforcing that each minimally coupled matter field shares its propagation speed with Gravitation, By this, these fields must share null cones with each other. GR can thereby be rederived from Relationalism’s implementation of Leibniz and Mach’s space and time principles. Closure, Deformation, and Rigidity do the work here. (Various mathematical simplicities that limit the deformation of H in Fig. 8.13a also apply.) 7.2  Refoliation Invariance The mathematical form taken by the Poisson bracket of two H’s (Fig. 8.11d.3) is well known to provide a Refoliation Invariance implementation of Foliation Independence: a Background Independence aspect desirable [44] in GR-like theories. This addresses whether, in evolving from an initial spatial slice to a final one, it makes a difference to go via Fig. 8.15a)’s red slice or its purple slice. This corresponds to whether two 223

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224 Figure 8.14   Constructability of Spacetime from Space.

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225 Figure 8.15   Refoliation Invariance of GR (and its categorical generalization posed).

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fleets of observers moving in two different ways between initial and final slices make the same observations upon attaining their common final state. The resolution [29] — of which Fig. 8.15a is the TRi version [76] — is that these must coincide up to a mere diffeomorphism of the final spatial slice. For GR, this follows from Fig. 8.11d.3 itself. To finish establishing a reduction to Lie Theory at the classical level, we need to mention that, firstly, foliations admit a Lie-Theoretic reformulation [56]. Secondly, that Refoliation Invariance is itself an aspect interaction: a type of Closure that is relationally restricted (by closing up to an automorphism of the final object). So the Wheelerian 2-way route between Lie claws is itself Lie-Theoretic. Thus, the whole of Classical Local Theory of Background Independence is. Finally, Fig. 8.15.c depicts a more general version of this aspect — RIO (Reallocation of Intermediary Object) Invariance — as required for our concluding discussions.

8 Conclusion We evoked 4 parts of Lie Theory. (0) Lie derivatives to implement Relationalism. (1) Lie brackets Closure as tested by the Generalized Lie Algorithm (GLA), with Lie algebraic structures as output. (2) Placing Function Algebras over state spaces (which acquires Lie-Theoretic content upon coupling with Closure). (3) Lie-Deforming one’s Lie algebraic structure’s generators, giving Constructability by onset of Lie Rigidity. These form a 4-part Background Independence model, fitting together as in Fig. 8.16c), which reflects the underlying Lie claw arrowgraph of Fig. 8.16.d). 8.1  Recovery of previous literature’s individual Problem of Time facets We now justify the above “Lie-Theoretic implementation of Background Independence” choice of mathematical structures by how previous literature’s Problem of Time facets [44, 45] (Fig. 8.16.a) drop out as subcases (Fig. 8.16.b). Many of the Problem of Time facets were originally envisaged at the quantum level. However, a Local Problem of Time’s facets — i.e., the consistent portion of facets that are neither global nor involve non-uniqueness — all have classical precursors [68, 76]. A) Frozen Formalism Problem Implementing Leibnizian timelessness for the Universe as a whole in the form of Temporal Relationalism for second-order Physics produces a quadratic constraint Chronos (such as H 226

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Figure 8.16   Evolution of nomenclature for Problem of Time facets through to Background Independence aspects, strategic implementations, and Lie-Theoretic interpretation, with their consequent ordering.

or ε). As this choice of name suggests, this is to be interpreted as an equation of time. The quantization of this (e.g., Fig. 8.4g) is then also stationary alias frozen. The Frozen Formalism Problem facet consisted of being surprised why the left-hand side of this is zero where Quantum Theory would lead one to expect to find some time-derivative term like those of Fig. 8.17a.1) 227

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for some notion of time. This is accompanied by the further issue of which time would feature in this role. As some contexts ordinary QM and QFT’s notions of time in this role are Background-Dependent: Newtonian absolute time and underpinned by 4’s timelike Killing vector, respectively. By now, however, this surprise has been repackaged as but a ready consequence of basing Physics on Leibnizian timelessness. Additionally, a means of reconciling this frozenness with the apparent local experience of time has been provided by Mach’s Time Principle that time is to be abstracted from change. This returns an emergent recovery of Newtonian time for Mechanics, and of a proper-time type construct for GR. Classical emergent Machian time does not moreover carry over to the quantum level. Quantum change must be given an opportunity to contribute to the emergent time here. So, for instance, semiclassical [23, 44, 45] emergent Machian time [76] coincides with classical Machian time to zeroth order (heavy-slow degree of freedom h terms). However, fast–light degree of freedom l corrections differ between the classical and semiclassical cases, in accord with Fig. 8.17.a.2’s distinction in functional dependencies. B) Thin Sandwich Problem Our G-Act G-All Method for indirectly implementing Configurational Relationalism’s Best Matching subcase’s GR-as-Geometrodynamics subcase recovers the Thin Sandwich Problem facet [14, 17]. In double contrast to A), this was already traditionally classically posed and most directly associated with the GR momentum constraint M. See Fig. 8.17b.1) for the meaning of Thin Sandwich, in contradistinction to the earlier but ill-posed Thick Sandwich in Fig. 8.17b.2. Its relation to time [44, 45] is described, for instance, in [14]’s title; its end product is furthermore a local slab of GR spacetime immediately adjacent to ∑. C) Functional Evolution Problem Given that both A) and B) are Constraint Providers, we need to check whether the totality of constraints provided close. If Closure is specialized in all of the canonical, Quantum, and Field-Theoretic ways, then it returns the Functional Evolution Problem facet [44, 45], whose conceptual content is in Fig. 8.17c). D)  The (Canonical) Problem of Observables Given a state space, there are physical reasons to also need to take a Function Algebra Thereover 228

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and then to express all physical quantities in terms of elements of this. This is what we identify as the uncoupled aspect, Dirac’s observation [7], that our functions need to commute with the theory’s first-class constraints C now being viewed instead as an aspect interaction (giving a facet interference). Aside from this uncoupling, this facet has largely not needed to evolve in conceptualization or, consequently, in name (column D of Fig. 8.16.a,b). “Problem of Observables” refers to how, for Gravitational Theory in general, observables are rather hard to find. Emphasizing the PDE reformulation of the brackets’ relation between observables and constraints adds to [44, 45, 47]’s exposition. Recognizing this as a slight extension of Lie’s Integral approach to Geometrical Invariants [4, 71] is a more recent [84] development. Fig. 8.17d) provides a quantum counterpart; at the quantum level, observables pick up further subtleties [53, 76]. A) to D) above reflect [44, 45]’s canonical bias in posing facets. Spacetime counterparts of B) to D) make up for this [68, 76], as per the sixth to eighth columns of Fig. 8.16a,b). E)  Spacetime Construction Problem The reverse passage from space to spacetime traditionally received quantum motivation. Dynamical entities undergoing fluctuations is inevitable at the quantum level by the Generalized Uncertainty Principle (Fig. 8.17f). For GR, this amounts to fluctuations of the 3-metric and extrinsic curvature. As Wheeler pointed out [22], these are far too numerous to be embedded within a single spacetime. Therefore, GR-as-Geometrodynamics [22] (or similar canonical formulations) would be expected to take over from spacetime formulations at the quantum level. At the primary level, it is not then clear what happens to notions associated with classical GR spacetime, such as (micro)causality [45], dimensionality, or a continuum. Familiar such notions would arise instead as emergent properties in semiclassical or lower-energy regimes; calculations of this remain hard to complete. It may be somewhat surprising that Classical GR already admits Construction of Spacetime from Space. This is underpinned by cohomologically-codified Rigidity arising from Deformation of the generators. Earlier versions viewed Constructability as “putting families of constraints” through the Dirac Algorithm [32, 54, 70, 76]. This, however, misses out on Constructability’s observed range of successes having an underlying topological explanation. 229

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F)  The Foliation Dependence Problem This has largely not needed to evolve, being both classically posed and a solved problem for GR since 1973 [29]. Refoliation Invariance gives primality-traversing passage from spacetime to space. Here, spacetime is not only sliced up, but physical equivalence between different slicings is also established. As a consequence of paragraph 1 of E), this beautiful geometrical way in which classical GR manages to attain Refoliation Invariance is however not necessarily expected to carry over to the quantum level. 8.2  Facet interferences explained That facet ordering forms a arrowgraph rather than the prior common supposition of some unknown linear ordering accounts for some previous impasses. Adjacent facets in this Lie claw arrowgraph moreover account for a large proportion of facet interferences. The corresponding aspect inter-relations often turn out to be well understood in Lie Theory. Closure is also rather heavily overrepresented among these inter-relations due to its central location. Many other aspect inter-relations involve adapting other aspects to the relational framework by forming TRi and G-Act G-All versions. These are mild recategorizations that do not eject us from Lie Theory. One interrelation of note to which the previous three sentences apply is replacing the Dirac Algorithm by the GLA [88]. The more general TRi [76, 88, 83, 84, 85] also frees us from restrictions peculiar to Relational action principles. Schemes prior to introducing TRi and G-Act G-All — in general enough terms to form a common consistent mathematical formulation for treating all facets — amounted to “forgetting about” these features while considering further facets, enabling Relational facets to creep back in. That Relationalism and Closure constitute a coupled problem is a very basic observation from the point of view of Lie algebraic structures, and yet is absent from much previous Problem of Time literature. Our remark about the pure observables aspect rendering Dirac’s commutant criterion for observables itself to be an inter-relation is relevant again here. So is our viewing Fig. 8.12.b to indicate that Assigning Observables is to postcede Closure. That Constructability is a separate sequel to Closure follows from finding commutants with given generators having little to do with Deformation of the generators. (Though if “branching” occurs, each branch needs its own observables to be computed out.) Finally, recall the argument in Section 2 for Foliation Independence itself being an 230

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inter-relation of Relationalism and Closure, at least in GR-like cases for which it is resolved by Refoliation Invariance. Some other facet interferences were induced by not making or denoting distinctions between S and C copies. This applies in particular to canonical versus spacetime observables. Some cases of this are underlain by not specifying whether the diffeomorphisms under discussion are of spacetime or of space. 8.3  End summary Lie Theory is very familiar [4, 69]. With hindsight, it is, moreover, the expected natural language for formulating the Differential-Geometric level of mathematical structure at which the laws of local Classical Physics are usually modeled. However, since those reviewing this area from 1967 [22, 23] via [44, 45, 65, 68] to 2017 [76] were neither fully aware of nor stated the Lietheoretic coherent whole of Fig. 8.16f), this is certainly both a new and significant mathematical observation to make. That the amount of Lie Theory needed to model the Problem of Time somewhat exceeds its habitually taught subset may account for this observation not having been made before. 8.4  Further outline of global and quantum versions The current article’s upgrade in understanding the local situation substantially advances how to phrase the Global Problem of Time [45, 76, 78, 89] and its underlying Background Independence. Additionally, many features of the Lie claw transcend category. Thus, having realized the Lie claw’s involvement at the classical level strongly suggests how to order facets at the quantum level as well. Here, Lie Mathematics needs to be supplanted by a suitable Operator Algebra [53, 73]. So Lie claw arrowgraph, Lie generator, GLA, Lie algebraic structure Deformation, Rigidity... become Operator Algebra equivalents. These are still interrelated in claw arrowgraph form, and all of our aspects of Background Independence remain poseable in this new arena. With Lie Theory recentering the Problem of Time around (algebraic) Closure, the Problem of Time line of thought in fact just places us on the usual boat as regards what obstructs attaining Quantum Gravity. That is, topological obstruction problems that largely revolve around quantum Closure, which are called anomalies [50, 37, 52]. 231

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At the quantum level, we also need a preliminary Assignment of Observables. This is so as to have a kinematical quantization within which to promote first-class constraints to quantum operators in the first  do not, however, in place. These preliminary unreduced observables U  general commute with Chronos . Nor do they commute with the quantum  lin lin that are also present in a Dirac as first-class linear constraints F  are to be supplanted by dynamiopposed to reduced quantization. The U  cal Dirac observables D at the usual facet-ordering point that the Problem of Observables Starting afresh by posing the quantum Problem of Observables rather than trying to promote classical observables to quantum operators is also recommended. For in general, the quantum constraints depend on operator-ordering, and the quantum brackets algebraic structure is not necessarily isomorphic to the classical one (e.g., by Fig. 8.17c). So, quantities commuting with these are not particularly likely to uplift from the classical counterpart, which knows nothing about the quantum level’s operator ordering or algebraic structure. Also, the flow PDEs for observables become a considerably more involved mathematical problem at the level of quantum operators. Significant “Problem of Time” programs beyond this point shall largely consist of Global Analysis, Topology, and Operator Algebras, rather than

Figure 8.17   Problem of Time facets.

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heuristic Physics or Philosophy. One idea is to concentrate on what can differ between categories: the Comparative Theory of Background Independence. Constructabilities and RIO (the categorical generalization of Refoliation Invariance in Fig. 8.15c) can, on the one hand, be posed in all categories. On the other hand, their affirmation depends in particular on the category in question’s cohomology groups. A Comparative Theory of Bacground Independence thus means to concentrate on what can differ and use this as a selection principle for which levels of mathematical structure permit Background Independent Physics.

Acknowledgments I thank Chris Isham, Przemyslaw Malkiewicz, Don Page, and participants at the Applied Topology Discussion Group, 2020 Problem of Time Summer School and 2021 Global and Combinatorial Methods in Fundamental Physics Summer School for discussions; Jeremy Butterfield, Malcolm MacCallum, Reza Tavakol and Enrique Alvarez for support with my career. Part of this work was done at DAMTP Cambridge, APC Université Paris VII, IFT Universidad Autonoma de Madrid, and Peterhouse Cambridge.

References [1] Newton, I. (1686). Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) (1686). Translated from the Latin by, e.g., I. B. Cohen and A. Whitman (1999). Berkeley, CA: University of California Press. In particular, see therein the Scholium on Time, Place, Space and Motion.   [2] Leibniz, G. W. (1715). The Metaphysical Foundations of Mathematics (1956). Chicago: University of Chicago Press; See also (1715–16). The Leibnitz–Clark Correspondence (1956) ed. H. G. Alexander. Manchester: Manchester University Press.   [3] Mach, E. (1883). Die Mechanik in ihrer Entwickelung, Historisch-kritisch dargestellt Leipzig: J. A. Barth. Translated from the German as (1960). The Science of Mechanics: A Critical and Historical Account of its Development La Salle, Ill: Open Court.   [4] Lie, S., Engel, F. (1888–1893). Theory of Transformation Groups Vols. I to III Leipzig: Teubner; For an English translation with modern commentary of Volume I, see J. Merker (2015). Berlin: Springer, arXiv:1003.3202. 233

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  [5] Killing, W. (1892). Concerning the Foundations of Geometry. J. Reine Angew Math. (Crelle) 109, 121.   [6] Lanczos, C. (1949). The Variational Principles of Mechanics. Toronto: University of Toronto Press.   [7] Dirac, P. A. M. (1949). Forms of Relativistic Dynamics. Rev. Mod. Phys. 21, 392.   [8] Dirac, P. A. M. (1951). The Hamiltonian Form of Field Dynamics. Canad. J. Math. 3 1.   [9] Yano, K. (1955). Theory of Lie Derivatives and its Applications. Amsterdam: North-Holland. [10] Clemence, G. M. (1957). Astronomical Time, Rev. Mod. Phys. 29, 2. [11] DeWitt, B. S. (1957). Dynamical Theory in Curved Spaces. [A Review of the Classical and Quantum Action Principles.]. Rev. Mod. Phys. 29, 377. [12] Dirac, P. A. M. (1958). The Theory of Gravitation in Hamiltonian Form. Proceedings of the Royal Society of London A 246, 333. [13] Arnowitt, R., Deser, S., Misner, C. W. (1962). The Dynamics of General Relativity. In Gravitation: An Introduction to Current Research ed. L. Witten New York: Wiley. arXiv:gr-qc/0405109. [14] Baierlein, R. F., Sharp, D. H., Wheeler, J. A. (1962). Three-Dimensional Geometry as Carrier of Information about Time. Phys. Rev. 126, 1864. [15] Jacobson, N. (1962). Lie Algebras. Chichester: Wiley. Reprinted (1979). New York: Dover. [16] Anderson, J. L. (1964). Relativity Principles and the Role of Coordinates in Physics. In Gravitation and Relativity ed. H-Y. Chiu, W.F. Hoffmann p. 175. New York: Benjamin. [17] Wheeler, J. A. (1964). Geometrodynamics and the Issue of the Final State. In Groups, Relativity and Topology ed. B. S. DeWitt, C. M. DeWitt New York: Gordon and Breach. [18] Dirac, P. A. M. (1964). Lectures on Quantum Mechanics. New York: Yeshiva University. [19] Gerstenhaber, M. (1964). On the Deformation of Rings and Algebras. Ann. Math. 79, 59; Nijenhuis, A., Richardson, R. (1966). Cohomology and Deformations in Graded Lie Algebras. Bull. Amer. Math. 72, 406. [20] Serre, J.-P. (1965). Lie Algebras and Lie Groups. New York: Benjamin. [21] Anderson, J. L. (1967). Principles of Relativity Physics. New York: Academic Press. [22] Wheeler, J. A. (1968). Superspace and the Nature of Quantum Geometrodynamics. In Battelle Rencontres: 1967 Lectures in Mathematics and Physics ed. C. DeWitt, J. A. Wheeler. New York: Benjamin. [23] DeWitt, B. S. (1967). Quantum Theory of Gravity. I. The Canonical Theory. Phys. Rev. 160, 1113. [24] Belasco, E. P., Ohanian, H. C. (1969). Initial Conditions in General Relativity: Lapse and Shift Formulation. J. Math. Phys. 10, 1503. [25] Yano, K. (1970). Integral Formulas in Riemannian Geometry. New York: Dekker. 234

P r o b l e m o f Ti m e : L i e T h e o r y S u f f i c e s t o R e s o l v e I t

[26] Fischer, A. E. (1970). The Theory of Superspace. In Relativity: Proceedings of the Relativity Conference in the Midwest, held at Cincinnati, Ohio June 2–6, 1969 ed. M. Carmeli, S. I. Fickler, L. Witten. New York: Plenum. [27] Moncrief, V., Teitelboim, C. (1972). Momentum Constraints as Integrability Conditions for the Hamiltonian Constraint in General Relativity. Phys. Rev. D 6, 966. [28] Misner, C. W., Thorne, K., Wheeler, J. A. (1973). Gravitation. San Francisco: Freedman. [29] Teitelboim, C. (1973). How Commutators of Constraints Reflect Spacetime Structure. Ann. Phys. N.Y. 79, 542. [30] Hawking, S. W., Ellis, G. F. R. (1973). The Large-Scale Structure of Space-time. Cambridge: Cambridge University Press. [31] Gilmore, R. (1974). Lie Groups, Lie Algebras, and Some of Their Applications. New York: Wiley. Reprinted (2002). New York: Dover. [32] Hojman, S. A., Kuchař, K. V., Teitelboim, C. (1976). Geometrodynamics Regained. Ann. Phys. N.Y. 96, 88. [33] Arnol’d, V. I. (1978). Mathematical Methods of Classical Mechanics. New York: Springer. [34] Barbour, J. B., Bertotti, B. (1982). Mach’s Principle and the Structure of Dynamical Theories. Proc. Roy. Soc. Lond. A 382, 295. [35] O”Neill, B. (1983). Semi-Riemannian Geometry with Applications to Relativity. San Diego: Academic Press. [36] Wald, R. M. (1984). General Relativity. Chicago: University of Chicago Press. [37] Alvarez-Gaumé, L., Witten, E. (1984). Gravitational Anomalies. Nucl. Phys. B 234, 269. [38] Isham, C. J. (1984). Topological and Global Aspects of Quantum Theory. In Relativity, Groups and Topology II ed. B. S. DeWitt, R. Stora. Amsterdam: North-Holland. [39] DeWitt, B. S. (1984). Spacetime Approach to Quantum Field Theory, ibid. [40] Isham, C. J. (1985). Aspects Of Quantum Gravity. Lectures given at Conference: C85-07-28.1 (Scottish Summer School 1985:0001), available on KEK archive. [41] Barbour, J. B. (1989). Absolute or Relative Motion? Vol. 1: The Discovery of Dynamics. Cambridge: Cambridge University Press. [42] Stewart, J. M. (1991). Advanced General Relativity. Cambridge: Cambridge University Press. [43] Henneaux, M., Teitelboim, C. (1992). Quantization of Gauge Systems Princeton: Princeton University Press. [44] Kuchař, K. V. (1992). Time and Interpretations of Quantum Gravity. In Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics ed. G. Kunstatter, D. Vincent, J. Williams. Singapore: World Scientific. Reprinted in (2011). Int. J. Mod. Phys. D 20, 3. 235

Ti m e a n d S c i e n c e

[45] Isham, C. J. (1993). Canonical Quantum Gravity and the Problem of Time. In Integrable Systems, Quantum Groups and Quantum Field Theories ed. L. A. Ibort and M. A. Rodríguez (Kluwer, Dordrecht 1993), gr-qc/9210011. [46] Bartnik, R., Fodor, G. (1993). On the Restricted Validity of the Thin-Sandwich Conjecture. Phys. Rev. D 48, 3596. [47] Kuchař, K. V. (1993). Canonical Quantum Gravity. In General Relativity and Gravitation 1992 ed. R. J. Gleiser, C. N. Kozamah, O. M. Moreschi M. Bristol: Institute of Physics Publishing, gr-qc/9304012. [48] Barbour, J. B. (1994). The Timelessness of Quantum Gravity. I. The Evidence from the Classical Theory. Class. Quant. Grav. 11, 2853. [49] (1995). Mach’s Principle: From Newton’s Bucket to Quantum Gravity ed. J. B. Barbour, H. Pfister. Boston: Birkhäuser. [50] Weinberg, S. (1995). The Quantum Theory of Fields. Vol. II. Foundations. Cambridge: Cambridge University Press. [51] Fischer, A. E., Moncrief, V. (1996). A Method of Reduction of Einstein’s Equations of Evolution and a Natural Symplectic Structure on the Space of Gravitational Degrees of Freedom, Gen. Rel. Grav. 28, 207. [52] Bertlmann, R. A. (1996). Anomalies in Quantum Field Theory. Oxford: Clarendon. [53] Landsman, N. P. (1998). Mathematical Topics between Classical and Quantum Mechanics. New York: Springer–Verlag. [54] Barbour, J. B., Foster, B. Z., ó Murchadha, N. (2002). Relativity Without Relativity, Class. Quant. Grav. 19, 3217, gr-qc/0012089. [55] Rindler, W. (2001). Relativity. Special, General and Cosmological. Oxford: Oxford University Press. [56] Moerdijk, I., Mrčun, J. (2003) Introduction to Foliations and Lie Groupoids. Cambridge: Cambridge University Press. [57] Rovelli, C. (2004). Quantum Gravity. Cambridge: Cambridge University Press. [58] Brandenberger, R. (2004). Lectures on the Theory of Cosmological Perturbations. In Lect. Notes Phys. 646, 127, hep-th/0306071. [59] Jammer, M. (2006). Concepts of Simultaneity. From Antiquity to Einstein and Beyond. Baltimore: Johns Hopkins University Press. [60] Giulini, D. (2007). Some Remarks on the Notions of General Covariance and Background Independence. In An Assessment of Current Paradigms in the Physics of Fundamental Interactions ed. I. O. Stamatescu, Lect. Notes Phys. 721, 105, arXiv:gr-qc/0603087. [61] (2008; 2010). Time in Quantum Mechanics Vols. 1 and 2 ed. G. Muga, R. Sala Mayato, I. Egusquiza Berlin: Springer. [62] Giulini, D. (2009). The Superspace of Geometrodynamics. Gen. Rel. Grav. 41, 785, arXiv:0902.3923. [63] Wald, R. M. (2009). The Formulation of Quantum Field Theory in Curved Spacetime. In Proceedings of the “Beyond Einstein Conference” ed. D. Rowe. Boston: Birkhäuser, arXiv:0907.0416. 236

P r o b l e m o f Ti m e : L i e T h e o r y S u f f i c e s t o R e s o l v e I t

[64] Stillwell, J. (2010). Naive Lie Theory. New York: Springer. [65] Anderson, E. (2011). The Problem of Time. In Classical and Quantum Gravity: Theory, Analysis and Applications ed. V. R. Frignanni. New York: Nova, arXiv:1009.2157. [66] Anderson, E. (2011). The Problem of Time and Quantum Cosmology in the Relational Particle Mechanics Arena, arXiv:1111.1472. [67] Bojowald, M. (2011). Canonical Gravity and Applications: Cosmology, Black Holes, and Quantum Gravity. Cambridge: Cambridge University Press. [68] Anderson, E. (2012). Problem of Time, Annalen der Physik, 524, 757, arXiv:1206.2403. [69] Lee, J. M. (2013). Introduction to Smooth Manifolds 2nd Ed. New York: Springer. [70] Anderson, E., Mercati, F. (2013). Classical Machian Resolution of the Spacetime Construction Problem, arXiv:1311.6541. [71] Olver, P. (1999). Classical Invariant Theory. Cambridge: Cambridge University Press 1999; (2013). Applications of Lie Groups to Differential Equations 2nd Ed. New York: Springer. [72] Anderson, E. (2014). Beables/Observables in Classical and Quantum Gravity. SIGMA 10, 092, arXiv:1312.6073. [73] Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P. (2013). Poisson Structures. Berlin: Springer-Verlag. [74] Giulini, D. (2014). Dynamical and Hamiltonian formulation of General Relativity, Chapter 17 of Springer Handbook of Spacetime ed. A. Ashtekar, V. Petkov. Dordrecht: Springer-Verlag, arXiv:1505.01403. [75] Hall, B. C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics 222, New York: Springer. [76] Anderson, E. (2017). The Problem of Time. Quantum Mechanics versus General Relativity. Springer International. Fundam. Theor. Phys. 190, (2019) 1-920 DOI: 10.1007/978-3-319-58848-3. See also https://conceptsofshape.space/ problem-of-time-book-and-main-articles-typos-and-corrections-page/ for this book, [87], and the current article. [77] Anderson, E. (2018). Geometry from Brackets Consistency, arXiv:1811.00564. [78] Anderson, E. (2018). Shape Theories. I. Their Diversity is Killing-Based and thus Nongeneric, arXiv:1811.06516. [79] Anderson, E. (2019). A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism, arXiv 1905.06200. [80] (2019). II. Configurational Relationalism, arXiv 1905.06206. [81] (2019). III. The Other Classical Facets Piecemeal, arXiv 1905.06212. [82] (2019). V. Combining Temporal and Configurational Relationalism for Finite Theories, arXiv:1906.03630; (2019). VI. Combining Temporal and Configurational Relationalism for Field Theories and GR, arXiv:1906.03635. [83] (2019). VII. Constraint Closure, arXiv:1906.03641. [84] (2020). VIII. Assignment of Observables, arXiv:2001.04423 237

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[85] (2019). IX. Spacetime Reconstruction, arXiv:1906.03642. [86] (2019). XIV. Grounding on Lie’s Mathematics, arXiv:1907.13595. [87] Anderson, E. (2019). Lie Theory suffices to understand, and Locally Resolve, the Problem of Time, arXiv:1911.01307. Published in a longer version as: (2021). Lie Theory suffices to Resolve the Local Classical Problem of Time. Geom. Integ. Quant. XXII, ed. I. M. Mladenov, V. Pulov, A. Yoshioka. Sofia: Bulgarian Academy of Sciences; [88] Anderson, E. (2020). Lie Theory Suffices for Local Classical Resolution of Problem of Time: 0. Preliminary Relationalism as implemented by Lie Derivatives, https://conceptsofshape.files.wordpress.com/2020/10/liepot-0-v2-15-10-2020.pdf; 1. Closure, as implemented by Lie brackets and Lie’s Algorithm, is Central., https://conceptsofshape.files.wordpress.com/2020/10/lie-pot1-v2-15-10-2020.pdf; 2.  Observables, as implemented by Function Spaces of Lie Bracket Commutants, https://conceptsofshape.files.wordpress.com/2020/10/ lie-pot-2-v1-15-10-2020.pdf; (2021). 3. Constructability, as implemented by Deformations in the presence of Rigidity, https://conceptsofshape. space/lie-pot-3/ [89] Anderson, E. (2020). Global Problem of Time Sextet. -I. Introduction and Notions of Globality, https://conceptsofshape.files.wordpress.com /2020/12/global-pot-minus-1-v3-04-01-2021.pdf; O. Relational Preliminaries: Generator Provision and Stratification, https://conceptsofshape.files. wordpress.com/2020/12/global- pot-0-v1-21-12-2020.pdf; 1. The central Closure’s Symplectic, Poisson and Foliation Mathematics, (2021). https://conceptsofshape.space/global-pot-1/.

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Chapter 9

VI EW S , VA RI ET Y, A ND C EL ES T I AL S PHER ES Lee Smolin Perimeter Institute for Theoretical Physics, Canada and University of Waterloo, Canada [email protected]

This chapter describes a continuation of the program of causal views, in which the world consists of nothing but a vast number of partial views of its past. Each view is associated with an event and is a representation of the immediate causal past of that event. These consist mainly of processes that transfer energy, momentum, and other charges to it from its past events. There is fundamentally no space or space-time, just a large number of events, which are the causes of events to come. This is a development of energetic causal set theories developed with Marina Cortês. Momentum and energy are fundamental, and are conserved under their transformation from present events to future events. As a result, Minkowski space-time emerges in a way that preserves causal relations. The locality of events as constructed in the emergent space-time is a consequence of the conservation of energy-momentum funda­mentally.

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In this chapter I propose that the views of events can be represented in terms of degrees of freedom on punctured two surfaces — each puncture corresponding to an im­mediate past event. This makes possible versions of the theory that are relativistically invariant.

1 Introduction The first question that physics asks is what the world is ultimately made of. A measure of the failure of positivism is that, after more than a century of dominance, what we really want to know is still the ontology of the world. Newton gave us an ontology of point particles moving in an absolute space, subject to mutual forces at a distance. While Newton knew better, the sheer usefulness of Newtonian dynamics made it hard to disagree; only a few, such as Descartes and Leibniz, succeeded in formulating a viable alternative conceptually — based on the ideas of relationism. But even armed with a more consistent philosophy, none was able to compete with Newton as a paradigm for exact description and prediction. But beginning with Faraday and Maxwell, a new ontology did arise that was able to address the questions raised by action at a distance: the ontology of fields. And since the turn of the 20th century fields — both quantum and classical — have dominated our ontological thinking as well as our practices as physicists. The picture of a universe of fields has inspired many discoveries, but its claims to give a fundamental ontology with a consistent foundation remains aspirational. A few quantum field theories are known to exist rigorously — and most of these live in 1 + 1 dimensions. The few exceptions are highly constrained by supersymmetry. None of the quantum field theories that make up the standard model are rigorous; in fact, our understanding of the physics of quantum electromagnetic fields and nuclei is based on expansions with zero radius of curvature. Lattice gauge theory appears to be in a more hopeful state until we remember that fermions double, making it impossible to describe those field theories that describe nature. There is, then, a clear need to begin to explore the possibility of a new ontology for physics. This paper reports on a modest attempt to provide an alternative ontology that may be sufficiently different, and sufficiently rich 240

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conceptually, that it may have a chance to underlie and complement quantum mechanics, quantum field theory, and general relativity. Called the causal theory of views, this begins somewhat like the causal set theory in that we declare that what is real includes events and causal processes by which they contribute to the catalyzation of new events.1 But our events are both sparser and more ephemeral. They do not exist in any familiar sense (nothing does, if this new proposal is correct); rather, we perceive our events to be moments of transformation. What gets continually processed and transformed is momentum and energy. So, these have no stable existence — apart from the very important fact that they are conserved. An event, while it lasts, is shaped by those impulses from its causal past — and this gives it a view of its near causal past. Each view is brief and incomplete. But the sum of them is the universe. This is an ontology in which the universe continually recreates itself — as all it is the collection of partial views of the causal pasts of the current events. This chapter reports progress in the construction of a theory that is meant to be an ultraviolet completion of general relativity and quantum theory, called the causal theory of view [1,2]. Previous papers have shown how relativistic point particle dynamics (and also, of course, space) and non-relativistic quantum mechanics emerge, in different limits [1, 2]. I present a relativistic theory of causal views. By this, I mean one whose action and equations of motion are invariant under Lorentz transformations on momentum spaces.2 The key to this is a relativistically invariant expression for the dynamics, based on an invariant measure of the difference between the views of two events (Eqs. 49, 17, and 42, respectively, below). The main idea is that we live in a quantum universe made up of nothing but a vast number of partial views of its past. Each view is associated with an event, and is a representation of the immediate causal past of that event. In this theory, space is not fundamental, nor is space-time, although there is a constructive, Bergsonian notion of time, related to the idea of becoming, or the now, as a consequence of the continual  Related models were proposed in [48,47].  Of course, because there is no space-time. Individual solutions including those that could be candidates to describe our own universe, break those and indeed all symmetries. 1 2

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creation of events. Energy, momentum, and other conserved charges are, however, present in the fundamental picture, and each event has so much of each. These are transferred to that event’s antecedents in a way that is consistent with their conservation. These transfers are primarily what an event “can know” about its causal past, and are coded in the views. The collection of such views is the be-able of the theory. There is a phase of the theory in which non-relativistic quantum theory is derived. This is the basis for the claim that these theories are non-local hidden3 variables theories [1, 2, 6, 7]. If space is not present initially, exactly how does it emerge? We make use of a mechanism for the emergence of space first discovered (as far as I know) in the relative locality and DSR papers [49, 50, 6, 7], which are also theories formulated in momentum space. The idea was then developed in the energetic causal set work. The key point is simply that conservation of energy-momentum implies locality in a space-time of the same dimension. We can draw a diagram of every interaction, which shows that processes are embedded in a little shard of space-time. The question is how to get the little shards to line up consistently to define an emergent space-time. Once put that way, we can find the necessary conditions, and construct classes of theories that satisfy them. At the fundamental level, in the absence of space, how does nature decide which interactions are stronger and which are weaker? In [1, 2, 6, 7], a simple answer was proposed: measures of similarity and differences of views provide a suitable replacement for distant or nearby in space. So, if the most important structure in a field theory is the derivative, in our theory the most basic operation is the comparison of views. Roughly speaking, the more similar two views are, the more likely they are to interact. And the more similar they become, the more repulsive the force [6, 7, 49, 50]. To summarize, we propose that each event in the history of the universe is nothing but the information available there about its near-term causal past. The more these views differ, the more diverse the world is.

 Despite various rumors to the contrary, there are not a few non-local hidden variables theories that reproduce predictions of quantum mechanics [8–17]. 3

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The universe, then, is governed by a law that aims to maximize the diversity of its views [2, 29]. This picture is also suggested by the basic twistor duality [20], according to which an event in space-time is dual to the set of light rays that converge there.4 We also may wonder if there is a relationship between these twistor inspired dualities and the celestial sphere [25], recently discussed as a possible structure relevant for quantum grav­ity. We show that the new theory proposed here is a kind of inversion of the structure in asymptotically flat space-times, in which we abolish the external boundary, along with its one fictional global outside observer, and rediscover the inversion of these structures as the microstructure of each event in a quantum universe. The one-line summary of the message of this paper is that the views of celestial spheres, turned inside out by a large conformal transformation,5 from an event, are the right degrees of freedom to describe closed quantum universes. The right dynamics (i.e., the one that recovers non-relativistic quantum mechanics in an appropriate limit) is to extremize the variety or diversity of the views [29]. This is a very old idea; it can be found, for example, in Leibniz’s revolutionary “The Monadology” [19]. Newton, who was Leibniz’s contemporary, had what would be regarded today as a more conventional view: he imagined that the universe is made of mass points flying around at the command of local equations of motion. There is one Observer who lives outside the universe and observes it without being observed. That archaic picture of a universe that can be seen as a whole by an all-powerful asymptotic observer is clearly holding back progress. The universe is not something that exists, that we monitor. To be an observer is, I propose, the same thing as being part of the universe. To observe the universe, you must be a participant in it. By a view, I mean information about the causal past of an event, which is coded in degrees of freedom at the event. But if an event has the puny structure of a “structureless point,” this is no more than the values of a few fields, evaluated at that point. There is not much one can do  However, to incorporate a causal structure we have to break some of the symmetries of twistor space. We hope to return to this in a future publication. 5  [24] 4

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with that. I find that Leibniz’s idea starts to get interesting when we blow each event up to a two-sphere, so that it has directional information built into it. When I use the word “information” here, I do not mean Shannon [51] or von Neumann [52] information. I mean Gregory Batesman information [53], defined as “the difference that makes a difference.”6 Here is the basic idea of Bateson: we consider making a minimal and local change in the value of some physical observable, x at a time, t0 from x1 to x2. A long time, T, later, we compare the two evolutions, the one starting from x2 to that from x1. In many cases, the late-time states are minimally different. It may be impossible to tell the two apart. This is typical for systems in thermal equilibrium. In this case, no information has been conveyed. But there are other cases in which the two states are macroscopically distinct. Think of a light switch. We say that the initial value of x carried information. The change from x1 to x2 was a difference that made a difference.7 So, what I mean by a quantum universe is a continually becoming causally related sequence of views of the causal pasts of events, where the view of event, which is equivalent to the name of the event, is a snapshot of the information arriving from the causal past to the two spheres of directions. Furthermore, the information coded on the S2 about its causal past could be “classical,” such as the direction a certain pulse of energymomentum is arriving from the past, or a more “quantum” description, such as a Bell state expressing entanglement of different regions of the past. Interpretational issues, concerning the nature of time, the structure of a theory of a closed system, and quantum foundations, are discussed in [3–5]. As in other cases, the theory does not dictate its interpretation. In many instances, views on foundational issues are not strongly constrained by formalism, so you can describe these models using the concepts of whichever interpretation you prefer. We proceed to review the main idea of the CTV (Causal Theory of Views).

 Gregory Bateson was an anthropologist and psychiatrist who also invented the concept of the double bind. 7  In statistical mechanics, we speak of damage being done. 6

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2  Summary of the Causal Theory of Views [1, 2] It is important to emphasize that the Causal Theory of Views is first of all a new proposal for the ontology of the physical world. • The universe is a process of continual becomings and transformations, made up of events and causal processes. • There is fundamentally no space, no space-time. • Causal processes carry impulses of energy, momentum, and other charges. This requires taking energy and momentum as primitive, operational concepts that do not need space to define them. Indeed, they have perfectly good operational definitions in terms of calorimeters, photomultiplier tubes, etc. We conjecture an inverse theorem that constructs an emergent dimensionality of space for each conserved quantity. • In all processes involving transfer of energy-momentum, it is conserved. Thus,



p +a J = = p a− J =

∑

p +MaJ

∑

p −MaJ

in M∈Past (J )

out N ∈Fut (J )

(1)

• Events do not exist; they happen. They are initiated by a combination of two or more causal processes. Then, they do one thing, which is to reshuffle the quanta of energy-momentum they receive and send them forth in new causal processes that will initiate a future event. • These transformations and processes conserve energy, momentum and other charges. • There is just one universe, it happens just once. 2.1  A view may be expressed as a quantum state in a celestial sphere We define a four-dimensional relativistic energy-momentum space, M

pa = (pi , p0 = ε ) ∈ M,

i =1, 2, 3 (2)

M is endowed with a Lorentzian metric and connection, which may or may not be related. 245

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The simplest characterization of a view is merely a set of labeled points on a unit punctured S2. This can be coded into an unordered list of punctures, each of which is labeled by the angles from which it approached, (w, w) together with the energy ε. Each view is, in this version,

ν I = {(w , w , ∈)αI }. (3)



If qa is a null one-form representing a photon’s state, it corresponds to a point on the celestial sphere qa = (1 + ww , w + w , −l(w − w), 1 − ww) (4)



Under a Lorentz transformation, qa transforms as a vector density

qa =

1 Λ ba qb (5) |cw + d|2

w and w transform as

w→

aw + b , cw + d

w→

aw + b (6) cw + d

with

ad – bc = 1

(7)

incoming ε > 0 and outgoing ε < 0. We posit that a variable number of impulses pαE interact in the formation of an event, E; this requires that the energy-momenta carried by these impulses that form the event (which are labeled pEa) combine to the single injection of energy and momentum, which endows the event. Thus, the view of an event E is taken from its causal past set, which must include at least the elementary processes, or impulsives, whose combination initiated the event.

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ν E = {pa ∈ past(E)}

(8)

This is governed by a map:

M × M… M → M : φ 1 ⊗ φ 2 … → µ (9)

One can also imagine an event making a probe of the future by sending out a number of photons, so that the total energy is preserved. Thus, the whole action of an event can be understood as a reorganization of the energy-momenta incoming from the past to a new way of dividing the total incoming energy.

M ⊗ M… M → M → M × M : φ 1 ⊗ φ 2 … → µ (10)

3  Relativistic Measures of Differences of Views There are different ways we can express the view of an event; each leads to a different definition of variety, and hence potential energy. They share a common strategy, which is to use the technique of best matching [61, 62] to define the differences between a pair of views. So next, we have a review. 3.1  Best matching and local gauge invariance Our relational philosophy tells us that observables that depend on an arbitrary choice of coordinates cannot be physical. One way to accomplish this is through the method of best matching, developed by Barbour and Bertotti [61, 62]. We want to compare the past causal views of a pair of events, which we will now call E and F, without reference to any frame external to or common to them. We want to treat as identical, views that differ from each other by elements of a symmetry group H. That is, if σ ∈ H does exist such that there are two views E and F such that F = σ ○ E

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we consider the two views identical, as there is nothing in their common internal relations that will distinguish them. We start with a measure of difference, d (E, F), between two views. Then we consider how this measure of difference is affected by the relative orientations, positions, and motions of the two. Thus, what we really want is to vary σ to find the group action on E that minimizes its distinctiveness from F. We call this a best-matching configuration.

Drel(E, F) = D(E, σ * ○ F)dmin(12)

where σ represents an action of group H on the space of possible views. and σ * is the element of H that minimizes the difference between the two views. There are a number of different strategies for doing this comparison. Each gives us a definition of difference on the set of punctures on the S2, which is to say, in the space of views. 3.2  The view as an unordered set of incoming energy-momentum The simplest view of an event E is an unordered set of nE incoming energy-momentum vectors,

vE = {qaEα …} (13)

Different events receive different numbers of pulses, nE. We next find a measure of the difference among pairs of these that is invariant under Lorentz transformations and permutations. Let’s start by comparing two light-like energy-momentum vectors. We use the Minkowski metric on momentum space

1 D[qa , pb ] = − (qa − pa )2 = qa pb g ab (14) 2

This is Lorentz invariant, non-negative when they are both null incoming or both null outgoing. It vanishes when pa = qa, so it gives meaning to how different the two null vectors are. However, this comparison makes use of a common reference frame. That is, when we use the form (14), we are implicitly thinking of 248

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comparing two views, made by a single observer. This is why we apply the same Poincaré transformation to both of them. Suppose we are in touch with a large number of observers scattered in space-time. From time to time, they each take a snapshot of their past view and send them to us. But an evil child has gotten into the collection and stripped them all of their labels, so what is left is an unlabeled and randomly ordered pile of snapshots. Our job is to reconstruct the partial order of the views in order to have the most likely history of our universe, which is an unordered collection of snapshots. Best matching is a technique for doing just this. It gives a Poincaré invariant way to give the snapshots a partial order in terms of the differences and similarities of these views. To do this, we need the freedom to Poincaré transform either one, and take the minimum of the resulting differences

D[qa pb ]b.m. = ((σ ⋅ qa )pb g ab|minimum value over group action σ (15)

For example. we can use σ to bring the two momenta onto alignment, so the best matched value is q0p0 if they are massive, 0 if they are massless. Now let us have n incoming energy-momenta in each view. We arbitrarily label the energy-momentum vectors belonging to the same view by indices α and β . The best-matched value of the distinction is

D {qαa } {pbβ }|

b . m.

=

∑

π ( α , β… )

(σ ⋅ qσa ⋅α ), pbβ  g ab …

best match

(16)

We evaluate the difference measure over all possible permutations of one set, and for each permutation, we scan the group on that side, looking for a set of pairings and orientations that lead to the greatest matching. We then define the value of the best matched [61, 62] to be the minimal difference between the two views, compared over all Poincaré transformations and all permutations. In the case that the number of snapshots or incoming impulses in each view is not equal, we consider all ways to distribute the smaller set among the larger.

D(E, F ) = D {qαa } {pbβ }

b . m.

=

∑

π ( α , β… )

249

(σ ⋅ qσa ⋅α ), pbβ  g ab…

bm m2

(17)

Ti m e a n d S c i e n c e

3.3 The view as a state in a boundary Chern–Simons theory When we go from classical theory to quantum theory, the values of the degrees of freedom are promoted to functionals of possible values. So, in the “classical” theory we have just described, the incoming causal processes arrive from the past at an event, E, from a direction specified by a point (w, w) on the S2. In the previous picture, this corresponds to a null vector in Minkowski space, and the full view from E is the unordered collection of these null vectors. The best-matching classes of the collections of null vectors give, as we just showed, a language for views suitable for constructing relativistic equivalence classes of views. If we take the non-relativistic limit, that set of vectors reduces to a set of timelike vectors, which were the basis for demonstrating the emergence of non-relativistic quantum mechanics in [1, 2]. Next, we construct a way to recognize the views that allows us to keep more topological information. The basic idea is that the set of null directions (w, w, ε)I now become punctures, which we then use to parameterize a set of Hilbert spaces on the punctured two spheres. Note that I will call the first correspondence “classical” and the second “quantum,” but they are both elements of the construction of a classical theory. The second class of states assigns to each state a wavefunction ψ(z) on the punctured S2. From a non-relativistic perspective, we could expand these functions in terms of representations of SO(3). This would seem to bring us back to the SU(2) spin networks of LQG [38].

(w , w ) on the S2 → ( j , m) (18)

But we can do more. With the understanding that the conformal two spheres carry a representation of the 3 + 1 Lorentz group, whose action is isormophic to the 2 dimensional conformal group, we may expand the wavefunctions in terms of S L(2, C) representation labeled spin networks. There are both compact and noncompact representations. But we shouldn’t stop here. To encode the magnitude of the energymomentum impulses, we need to represent the energy ε in terms of a further extension to the Poincaré group representation theory.

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We want to add the translations so the ε’s are now indicating how the states are elements of a representation of translations. So we have gone from

(w , w , ε ) on the S2 → ( j , m, ε ) (19)

which are representation labels for the 3 + 1 dimensional Poincaré group. Therefore, we may have a spin-foam model constructed from the representation theory of Poincaré (1, 3), as described in [63]. The events are now intertwiners of Poincaré (1, 3), with a specified number of incoming and outgoing zero-mass states. We each have a view, or views, one for each moment. This is precisely information about our recent causal past, projected onto a sphere in momentum space. Our view is in fact a two-spheres on which is projected photons and other quanta coming from our causal past. So, it is very natural to formulate a theory of views, as we have sketched it here, as a theory in which each event is blown up to a sphere. 3.4  The ladder of dimensions Indeed, this idea sits right at the center of some of the most important ideas about quantum space-time geometry to have been studied these last few decades. I am referring to that very influential concept in quantum gravity and combinatorial topology, which is called the ladder of dimensions. The basic idea, enunciated by Crane [31], is as follows. The top dimension — in our case 4 — is a topological quantum field theory on a four-manifold M. This will be a BF theory for some Lie group (or quantum group), G. We are interested in the case where M = Σ x R. There are no non-dynamical fields on M-because the partition function of BF theory can be shown to be independent of the choice of triangulation used to regulate it. So, there are very few bulk physical observables. Down one dimension, however, there are induced physical degrees of freedom living on the boundary,

∂(∑ × R) = (∂ ∑ ) × R + ∑ + − ∑ − (20) 251

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These are described by a G Chern–Simons theory on the threedimensional boundaries [34]. The two manifold Σ may also have boundaries. Or we may introduce two-dimensional boundaries by choosing a surface B that splits Σ into two. The axioms of Atiyah and Segal [72] for topological quantum field theory posit that for each embedding of a two-dimensional surface B into a compact three manifold, Σ, that splits that three manifold into two halves, Σ+ and Σ– there is a Hilbert space, HB. For every topological three manifold Γ+ that has B as its boundary, i.e., B = ∂Σ+, there is a state ψ ∈ HB. Let us assume that B is oriented, so it has two sides, B+ and B–. There is a conjugate map, † † : B+ → B–(21) We can use † to construct an inner product, given a state |ψ > ∈ HB+, and a dual state, < Φ|> ∈ HB– , the inner product is naturally defined, < ψ| ϕ > ∈ C

(22)

This is then an invariant of the topology of Σ because it is invariant under the choice of the splitting surface, B. Let us call this structure the ladder theory. The ladder theory is not quantum gravity — nor is it a model of quantum gravity in four spacetime dimensions. All of its degrees of freedom and physical observables live on the three-dimensional boundary. There is what has been called boundary observable algebra [34]. Remarkably, quantum gravity in three dimensions is a topological quantum field theory (TQFT), as was first discovered by Ponzano and Regge [74]. Even more remarkably, there are several ways that the ladder theory may be disordered or constrained to yield a quantization of general relativity in four space-time dimensions with three-dimensional boundaries. 3.5  Quantum gravity and spin networks One way to do this is to disorder the partition function by embedding spin networks for G into the three-dimensional bulk [32, 33], Σ.8 These end  A G spin network is an abstract graph, whose edges are labeled by irreducible representations of G and whose vertices are labeled by intertwiners of the incident representations. When embedded in a manifold as just described, it is also called a spin network. 8

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with punctures on the 2 dimensional boundary. These inherit the labels on the spin network graphs, which are G representations. That is to say, we extend the previous correspondence. Now we assign a Hilbert space, HBpa,ja , to every punctured two surface, Bpa,ja. A state in that Hilbert space is assigned to every embedding of a spin network in Σ+ that ends on the punctures, matching irreducible representations, j-up to topological deformations of the embeddings that leave the punctures fixed. Given the correspondence arising from QG between representations and intertwiners of SU(2) and area and volume, these states have natural interpretations as triangulated three geometries. When G is chosen as a local symmetry or gauge group for general relativity, the resulting partition function can be shown to be a quantization of general relativity [26, 34, 35]. These are called spin-foam models. One of the things that is wonderful about them is that one can continue to use boundary observables. In some cases, they inherit new meanings from the map between the TQFT and gravity. For example, the nonAbelian electric fluxes now carry geometrical observables such as areas and volumes. In some cases, the partition functions have ambiguities; this can happen because there are physical observables that depend on chirality, which is not represented by the one-dimensional structures of the spin networks. These require modification of the spin networks. The edges of the spin networks may be blown up into tubes. There are new degrees of freedom, which are winding numbers of the tubes. The end points of those lines, on the spatial boundaries, which were points, are now blown up into circles. To represent these, we must extend G to give it structure that can transform non-trivially under diffeomorphisms of these circles as well as under parity transformations. When G is compact, we can extend it to a quantum group at the root of unity. Or we extend the group to a loop algebra (or Kac–Moody algebra), the group of mappings of a circle into a Lie group. The Virasoro group, which is the group of mappings of circles onto circles, centrally extended, acts at those circles because it is a subgroup of the extension of G, giving us new physical degrees of freedom, which can be represented by a CFT, such as the WZW theory. This structure fits nicely into the triangulations of four-dimensional manifolds that we use in quantum gravity. The dual of the triangulation is a graph in a three-dimensional bulk. The tetrahedra form the bulk and dual to each tetrahedron is a three-dimensional spin network. Each 253

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tetrahedron is dual to an intertwiner. Its four triangles are each dual to a shared boundary of two tetrahehedra, which in turn is dual to an edge connecting the node in each tetrahedra. Every four-simplex is bounded by 5 tetrahedra. That four-simplex then represents an event in which n tetrahedra are replaced by the 5 – n tetrahedra. In this way, we get a quantum theory of gravity based on a Lorentz group gauge symmetry, G = SL(2, C) or SO(3,1). This is close, but it is not exactly what we want. The models based on dynamical causal structure need to label the punctures by relativistic four momenta, paI. So do the celestial spheres. There are at least two ways to do this: • Extend the gauge group to the Poincaré group.   This requires a good dose of the infinite dimensional unitary representations of the Lorentz and Poincaré groups, so we postpone further discussion of it to a later publi­cation. • Extend the group to the de Sitter or Anti de Sitter group.

G = SL(2, C) → SU(2, 2)

(23)

Construct the theory for the de Sitter group. Then, take Λ to zero, if needed. 3.6  The Chern–Simons boundary action The second road is technically simpler, so we start with that: we work in a spinorial version of the MacDowell–Mansouri formulation of general relativity [26]. Everything we do, we work with Lorentz space-time. This is a small modification of a B ˄ F theory [27] for the double cover of the Desitter (or Ads) group: SU (2, 2). The novelty of this formalism is that we add a twist to the idea that general relativity is a constrained topological field theory by making that theory a gauge theory of SU(2, 2), where the constraints that introduce local degrees of freedom break the gauge group down to SL(2,C). As a result, the frame field one-form eAA’ is expressed as components SU ( 2 , 2 ) of the SU(2, 2) connection, in SL( 2 ,C ) that exists because SU(2, 2) is broken spontaneously into SL(2, C); i.e., the metric geometry is a Higgs field — an order parameter that marks the spontaneous breaking of SU(2, 2) to SL(2, C). 254

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The implications of this extension can be worked out in any firstorder version or extension of general relativity. For the convenience of the reader who may not be familiar with the exotic Plebanski formalism [35], and is more likely to have met the Palatini variables [36], we work here with the latter variables. The theory is based on an SU(2, 2) connection, AIJ, which decomposes as AAB = A(AB) = AAB     chiral SU(2)L connection             AA′B′ = A(A′B′) = AA′B′   chiral SU(2)R connection             AAA′ = Λ eAA′     the frame field one-forms

(24) (25)

We see that this approach requires a non-vanishing cosmological constant, Λ9. The corresponding components of the curvatures are F AB = dA + A 2 + Λe AA′ ∧ eAB′ (26)

  F A′B′ = dA A′B′ + A A′B′ ∧ AΛe AA′ ∧ eAB′ (27)

F A′A′ = Λ (de AA′ + ABA ∧ e BA′ + ABA′′ ∧ e AB′ ) = D ∧ e A′A′ = T

A′A′

(28)

where TAA’ is the torsion tensor. The action of SU(2, 2) BF topological theory is   1 IJ 1 k B ∧ FIJ − 2 F IJ ∧ F IJ  + ∫ YCS (SU (2 , 2)) 2  Σ ×R g e  ∂Σ ×R 4π 

S = −l ∫ = −l ∫

Σ ×R

−

1 BAB ∧ FAB − BA′B′ ∧ FA′B′ − BAB′ ∧ FAB′ − 1n2 + BAA′ ∧ DeA′A′  g2 

 4 4 1 AB ( F ∧ F AB + F A′B′ ∧ FA′B′ + De A′A′ ∧ DeAA′ ) + Λe 4  2 + 2  2 e e  g

+∫

Σ× R

k k k AA′ (YCS (SU (2)L ) − (YCS (SU (2)R ) + e ∧ DeAA′ ) 4π 4π 4π

(29)

The boundary term is

 The I, J... = (AA’) = (010’1’) are four component Dirac spinor indices, and we are taking advantage of the two to one map between SO(5) and SU(2, 2). 9

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SBBCS =

k 4π

∫

S2 × R

(YCS (SU )(2)L ) − YCS (SU (2)R ) + e AA′ ∧ DeAA′ ) (30)

One breaks the symmetry by adding Lagrange multipliers, which enforce BAB = e AA′ ∧ eAB′ (31)



After we break the symmetry by implementing these constraints, the action is then the Palatini action plus topological and boundary terms. SPalatini = −l∫

∂Σ × R

+

 1 4  AA′  2 4 B A′A B′ 4  g 2 + e 2  {e ∧ e A′ ∧ FAB − e ∧ eA ∧ FA′B′ } − ∧ e  g 2 + e 2 

1 AA′ e2 B ∧ De AA′ + ( F AB ∧ FAB + F A′B′ ∧ FA′B′ + (De AA′ ∧ DeAA′ )) 2 4 g

+∫

∂Σ × R

k k AA′  k  YCS (SU (2)L − YCS (SU (2)R + e ∧ DeAA′    4π 4π 4π

     (32) Let us look at the first variation and make sure the action is functionally differentiable.

δS=

∫ [(EOM)

AB

]δ A AB + (EOM )A ′B ′ δ A A ′B ′ + (EOM )AA ′ δ e AA ′

M

 1    1  k   1 AB  2  Σ AB −  2 −  FAB  ∧ δ A +  2 Σ A ′B ′    g 2 e π g       ∂M   k  1   1  k − 2  δ AAA ′ ∧ D ∧ eAA ′ −  2 −  FA ′B ′  ∧ δ A A ′B ′ +  e 2π  2 π e   

+

  

∫

(33)

For the action to be functionally differentiable, the boundary term of the variation must vanish. As was first shown in [34, 37, 39], there are a number of ways to accomplish this. There are subtleties in each signature. Here, we discuss only the basics of the Lorentzian signature [37].

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There is a set of boundary conditions that leave the edge modes for ε , aAB and aA’B’, (the pull back of the forms into the timelike or null boundaries) all free, AA’

1 k  1 Σ AB =  2 −  FAB (34) e 2π  g2



1 k  1 Σ A′B′ =  2 −  FA′B′ 2 e 2π  g



k  1 0 =  2 −  D ∧ eAA′ e 2π 

(35)

This affirms that we must have a boundary term that includes the energy and momentum, i.e., one based on the Poincaré or de Sitter groups. The translations are generated by energy and momentum; thus, we can label the punctures by energy and momentum, and complete the picture of a causal theory of views based on celestial spheres. We then have on each event’s S2 a set of n = nin + nout punctures, of which nin are incoming and nout outgoing. The quantum field on each S2pa,ja‘s is based on a Hilbert space Hnin , nout (σσ )I …, where σbar means σ with a bar on it, as you did. We work with a left-right symmetric spin network basis [37], where the edges are labeled by representations of SUL(2), SUR(2). The irreducible reps (?) are labeled by (jL, jR, IL, IR). The reality conditions impose that the physical representations are balanced jL = jR. 3.7  Some remarks The next step is to define a difference operator Dˆ (E , F ) on

Dˆ : HE ⊗ HF → R+ . (36)

We can use the inner product on Chern–Simons theory to measure similarity. A state of G Chern–Simons theory on the nA punctured S2, notated,

|A, n A ,( y , yi , ri ) > (37) lives in a Hilbert space HnA ( y , yi , ri ). 257

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This unphysical Hilbert space is dependent on the positions on the S ’s of the nA punctures. We note that the punctures play a role analogous to the n-particle trajectories in the particle construction. As in that case, we ask for the best matching to construct the inner product. The Hilbert spaces decompose according to the positive number of punctures nA 2



HSpα , jα =

∫

Sp2α , jα

(dy , dy )HS−punctures (dy , dy ) (38)

We then define the inner product by

< (B, nB rj |A, nA , ri ) >=< B|A > δ n A n B δ r j r k (39)

the positions of the punctures do not matter up to diffeomorphisms of S2- punctures. This is reflected in the braid group symmetry B∇(nA ) acting on the Hilbert space HnA. If group G were compact, these Hilbert spaces would be of finite dimension. As it is for the Lorentzian theory, we probably be constrained to work within the countably infinite unitary representations [73]. The Hilbert spaces decompose according to the positive number of punctures nA

. < A, nA ,( yi , ri )|φ o φ o|B, nB ,( yi , ri ) >=< B|A > δ n A n B δ rbm (40) j rk



< B, nB rj |A, nA , ri >=< B|A > δ nA nB δ rj rk (41)

We can then define the difference in two views: view A on nA punctures, which are in the past of E, and view B on nB punctures. For a quantum system, it is natural to take the inverse of the inner product to be the difference. As in the classical description, we first compare the punctures and then the fields. 2



1 D[E , F] = (42) < ψ Eψ F >

4  The Dynamics If the context is novel, the dynamics will be as conventional as possible. We begin by defining potential energy, then a Hamiltonian. 258

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4.1  The variety as potential energy To make the potential energy, we sum the views — difference D[E, F] over all pairs of causally unrelated events, < >. Because it is based on a best matching procedure, which takes the best value over all pairs in the orbit of the Lorentz group action D[E, F], the result is Lorentz invariant. That a pair is not causally related is also Lorentz invariant. Thus, up to divergent terms, and effects of non-invariant cut-offs, the potential energy is Lorentz invariant.

U=

∑ D( I , J )

bestmatched

I< >J

(43)

The last step is to restrict to events I and J in the past of a third “observer event,” K. D (I,J; K)bestmatched(44)



We then sum over all triples,

ν = ∑

∑

K I J|inPast ( K )

D( I , J ; K )bestmatched (45)

We propose to call this the acausal variety. We propose it as a measure of potential energy.

gU = ν >< = ∑

∑

K I >< J|∈Past ( K )

D( I , J ; K )bestmatched (46)

g is a coupling constant. We notice that the best matched value of a comparison is Lorentz invariant, as it considers the differences scanning over the whole group. Additionally, the criteria we use to pick out which pairs of events to be compared are not altered under the operations defined in. One might want to call potential energy non-local. I prefer to call it a-local because it makes no reference to space. The theory does have an emergent spatial geometry, and relative to that, the potential that derives from the variety remains non-local, as is discussed in [1, 2]. The Hamiltonian is then

H0 = gU (47)

In the non-relativistic limit, defined as kinetic energy dominated by some mass, m, this exactly reproduces Bohm’s potential [9]. 259

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Note that in the non-relativistic limit a kinetic energy term will appear proportional to [1, 2]. In the full relativistic theory the kinetic energy terms are hidden in the constraints. K .E. =



(pi )(p j ) g ij

(48)

M

4.2  The dynamics of difference: the half-integral We are ready now to define the dynamics of the theory. We propose that the dynamics be defined by the following half-integral. The degrees of freedom that we integrate over are the causal structure, given by the partial order structure on the S2 ‘s and the energymomentum transferred in each causal relation. Imposing the constraints as δ functionals in the measure of the half-integral.

Z(J) =



∑ ∏ dp

processes

 I|> J

J aI

 δ (CJI )δ (Q IJ ) ∏ δ (PaI )e iH0 ( p ) (49)  I

The constraints are imposed as constraints in the measure of the half-integral. This is the complete definition of the theory. There is no reference to space or space-time at the level of the fundamental definition of the theory. As a consequence, there is no ћ. There are no non-trivial commutation relations and no uncertainty principle. 4.3  Semiclassical limit: the emergence of Minkowski space-time Now, we introduce Lagrange multipliers to exponentiate each of the constraints. There is a conservation law PaI for each event, so we write

δ (PaI ) = ∫ dzIa e IZI Pα (50) a

I

The next step is finding the equations of motion. We will see that the semiclassical theory is sufficient for understanding how a classical spacetime emerges from this theory,10 10

 We assume the simplest case where the momentum is conserved linearly.

260

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JI

K Take the first variation by paJ of the action and set it to zero.



0=

δ Seff δν = ( z Ja − zKa ) + NpbI g ab + g K (52) K δ paJ δ paJ

where gab is the metric of momentum space. At first, ignore the potential term

g

δν (53) δ pa J K

It then follows that in the limit g → 0,

( z Ja − zKa )( z Jb − zKb ) gab = 0 (54)

We see that the Lagrange multipliers za have become coordinates in an emergent Minkowski space-time. We also see that gab is a conformal metric on space-time and the intervals zaJ - zaK are null. Thus, we see that a conformal metric emerges on space-time, which is just the inverse of the metric on momentum space.

5 Conclusions Any proposal for a new physical ontology faces huge challenges, even greater than those that confront attempts to understand quantum gravity or foundations within a more standard ontology. The causal theory of views was proposed in [1, 2], using elements from previous models of space-time: causal set theory, energetic causal sets, and is influenced by relative locality [49, 50], twistor theory [32], the study of amplitudes, and loop quantum gravity [30]. How is it doing?   1. The framework of the ontology is clear, but there are several subtle issues that are still being studied [3, 4].   2. The be-ables are views of an event, which is what can influence an event from that event’s causal past. We have several versions of this; these models differ by the math­ematical framework used.  3. Fundamentally, there is no space or space-time. Energy and momentum are fundamen­ tal; relativistic space-time emerges as a 261

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consequence of the conservation of energy and momentum. We have seen how this happens in several contexts.   4. The dynamics is based on comparisons of how diverse the views of pairs of events are. The potential energy is related to the variety of the universe, which is the sum of all pairs of their differences.   5. We understand how to derive non-relativistic quantum many-body theory, due to the similarity of the variety and the Bohmian quantum potential.  6. We start by formulating relativistic theories of views. We see how special rel­ativity can emerge readily, making use of the connection between energy-momentum conservation and locality in space-time. But how will general relativity appear? One answer is to introduce parallel transporters into the causal processes, so the energy mo­mentum that arrives need not be that which was sent.   7. The statistical physics of the theory described here is going to be very interesting. The statistics of the possible and actual states or transitions are very different in our context. Preliminary numerical studies by M. Cortês of a simple model show new phases and new phase transitions [41–44]. The equilibrium ensemble is not reached during the lifetime of the universe. As a result, our universe is, we conjecture, very far from ergodic.   8. There is an expectation that underlies or motivates the study of asymptotic structures. Space-time may be really weird when we probe it at very short distances, but if we could travel in an opposite direction, further and further away from us, at longer and longer wavelengths, it is ultimately more of the same, only more so. Our view is quite the opposite; far, far away from us is going to be more and more quantum and, indeed, beyond quantum. There is no reason to expect a classical asymptotia when IR/UV symmetry governs, going to the very very large gets to the same “place” as going to the very very small. Which is to say that if we haven’t yet seen indications of IR/UV symmetry we have not yet begun to study real cosmology.  9. The Causal Theory of Views is also part of another research program, which is based on the idea that completions of backgrounddependent theories eliminate their background dependence by a process that replaces dualities with trialities. This is based on the observation that in many cases of dualities in physics, the duality transformations are based on a fulcrum of background structure. 262

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For example, UV/IR dualities or in general weak/strong dualities leave fixed a scale that defines which is which. Or, the Born dualities are based on a fixed definition of time. In these cases, a deeper theory was found by eliminating the dual pairs’ dependence on non-dynamical background elements by elevating the duality to a triality in which each of the three elements is defined by the interaction of the other two [71]. This motivated my studies of cubic matrix models, which furnish many examples of such duality to triality moves. Indeed, string theory furnishes some beautiful examples of the passage from a background dependent to a background independent formulation, which is cubic. (Think of the construction of actions for strings of the form of S = Tr Φ3). There are also many examples in the mathematical general relativity literature of the special role played by cubic actions, such as the Plebanski action. These cubic formulations are by far the simplest — since the actions are cubic, the equations of motion are all quadratic equations. Many unexpected new results were made possible by adopting one of these cubic formulations. 10. This proposal raises a number of fundamental issues concerning quantum foundations and the nature of time, which are discussed elsewhere [3–5]. Much remains to be done.

Acknowledgments I would like to thank Luca Ciambelli, Anna Knorr, and Sabrina Pasterski for comments on early drafts of this chapter; also Stephon Alexander, Giovanni Amelino-Camelia, Julian Bar­bour, Marina Cortês, Louis Crane, Laurent Freidel, Jaron Lanier, Joao Magueijo, Roberto Mangabeira Unger, Fotini Markopoulou, Carlo Rovelli, and Clelia Verde for collaborations on related projects. I am grateful to David Finkelstein, Lucien Hardy, Roger Penrose, Robert Spekkens, and Antony Valentin for many critical conversations that helped shape this work. This research was supported in part by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province 263

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of Ontario through the Ministry of Research and Innovation. This research was also partly supported by grants from NSERC, FQXi, and the John Tem­pleton Foundation.

References   [1] L. Smolin (2018). The dynamics of difference, arXiv:1712.04799, Foundations of Physics, 48, 121–134.   [2] L. Smolin (2021). Views, variety and quantum mechanics, arXiv:2105.03539.  [3] L. Smolin and C. Verde (2021). The quantum mechanics of the present, arXiv:2104.09945.   [4] L. Smolin, C. Verde and M. Cortês (2021). Physics, time and qualia, Journal of Consciousness Studies, 28(9–10), 36–51.  [5] C. Verde and L. Smolin. There is no measurement problem at present, in preparation.  [6] L. Smolin (2012). A real ensemble interpretation of quantum mechanics, Foundations of Physics, 42, 1239–1261, doi: 10.1007/s10701-012-9666-4 arXiv:1104.2822.   [7] L. Smolin (2016). Quantum mechanics and the principle of maximal variety, arXiv:1506.02938, Foundations of Physics, 46(6), 736–758.   [8] L. de Broglie (1927). La mécanique ondulatoire et la structure atomique de la matière et du rayonnement, Journal de Physique et le Radium, 8(5), 225–241, Bib-code: 1927JPhRa. . . 8. .225D, doi: 10.1051/jphysrad:0192700805022500.   [9] D. Bohm (1952). A suggested interpretation of the quantum theory in terms of hidden variables, I. Physical Review, 85(2), 166–179, Bibcode:1952PhRv...85..166B, doi: 10.1103/PhysRev.85.166. A suggested interpretation of the quantum the­ory in terms of hidden variables, II. Physical Review, 85(2), 180–193, Bibcode: 1952PhRv. . .85..180B, doi: 10.1103/Phys. Rev.85.180. [10] E. Nelson (1966). Derivation of the Schrödinger equation from Newtonian mechanics, Physical Review, 150, 1079; E. Nelson (1985). Quantum Fluctuations, Princeton University Press. [11] S. Alder (2004). Quantum Theory as an Emergent Phenomenon, Cambridge Uni­ versity Press, New York; S. Alder (2002). Statistical dynamics of global unitary invariant matrix models as pre-quantum mechanics, arXiv:hep-th/0206120. [12] A. Starodubtsev (2003). A note on quantization of matrix models, arXiv:hepth/0206097, Nuclear Physics, 674, 533–552. [13] L. Smolin (1983). Derivation of quantum mechanics from a deterministic non-local hidden variable theory, I. The two dimensional theory, IAS preprint; L. Smolin (1986). Stochastic mechanics, hidden variables and gravity, in 264

Vi e w s , Va r i e t y, a n d C e l e s t i a l S p h e r e s

Quantum Concepts in Space and Time, ed. C. J. Isham and R. Penrose, Oxford University Press. [14] L. Smolin (2006). Could quantum mechanics be an approximation to another theory?, arXiv: quant-ph/0609109. [15] L. Smolin (2002). Matrix models as non-local hidden variables theories, arXiv:hep-th/0201031. [16] F. Markopoulou and L. Smolin (2004). Quantum theory from quantum gravity, arXiv:gr­gc/0311059, Physical Review D, 70, 124029. [17] M. J. W. Hall, D.-A. Deckert, and H. M. Wiseman (2014). Quantum phenomena modelled by interactions between many classical worlds, Physical Review X, 4, 041013. [18] F. Markopoulou and L. Smolin (2007). Disordered locality in loop quantum gravity states, arXiv:gr-gc/0702044, Classical and Quantum Gravity, 24, 3813–3824. [19] G. Leibnitz, The Monadology, 1714. [20] R. Penrose and M. A. H. MacCallum (1973). Twistor theory: an approach to the quantisation of fields and space-time, Physics Reports, 6(4), 241–315, Bib­ code:1973PhR…..6..241P, doi: 10.1016/0370-1573(73)90008-2. [21] J. A. Wheeler (1973). From relativity to mutability, in The Physicist’s Conception of Nature, ed. J. Mehra, Dordrecht: Reidel, pp. 202–247; J. A. Wheeler (1983). On recognising law without law, American Journal of Physics, 51, 394–404; J. A. Wheeler (1985). Bohr’s “Phenomenon” and “Law Without Law”, in Chaotic Behaviour in Quantum Systems, ed. G. Casati, Theory and Applications. Proceedings NATO Achievements, pp. 363–378; J. A. Wheeler (1987). How come the quantum?, in New Techniques and Ideas in Quantum Measurement Theory, ed. D. M. Greenberger, New York: New York Academy of Sciences, pp. 304–316. [22] Z. Perjzes (1980). An introduction to twistor particle theory, Hungarian Academy of Sciences. [23] F. Markopoulou (2009). Space does not exist, so time can, arXiv:0909.1861. [24] F. Markopoulou (2000). The internal description of a causal set: what the universe looks like from the inside, Communications in Mathematical Physics, 211, 559–583, doi: 10.1007/s002200050826, arXiv:gr-gc/9811053. [25] L. Donnay, S. Pasterski and A. Puhm (2020). Asymptotic symmetries and celestial CFT, arXiv:2005.08990, doi: 10.1007/JHEP09(2020)176; S. Pasterski, S.-H. Shao and A. Strominger (2016). Flat space amplitudes and conformal symmetry of the ce­ lestial sphere, arXiv:1701.00049, doi: 10.1103/ PhysRevD.96.065026. [26] S. W. MacDowell and F. Mansouri (1977). Unified geometric theory of gravity and supergravity, Physical Review Letters, 38(14), 739–742, Bibcode:1977PhRvL..38..739M, doi: 10.1103/PhysRevLett.38.739. [27] L. Smolin and A. Starodubtsev (2003). General relativity with a topological phase: an action principle, arXiv:hep-th/0311163. 265

Ti m e a n d S c i e n c e

[28] Y. Ling and L. Smolin (2001). Holographic formulation of quantum supergravity, Physical Review D, 63, 064010. [29] J. Barbour and L. Smolin (1992). Extremal variety as the foundation of a cosmological quantum theory, arXiv:hep-th/9203041. [30] C. Rovelli and L. Smolin (1995). Discreteness of area and volume in quantum gravity, Nuclear Physics B, 442, 593, gr-qc/9411005 (erratum Nuclear Physics B, 456, 753–754, 1995); C. Rovelli and L. Smolin (1995). Spin networks and quantum gravity, Physical Review D, 52, 5743–5759, gr-gc/9505006. [31] L. Crane (1995). Clock and category: is quantum gravity algebraic, arXiv:gr-gc/9504038, Journal of Mathematical Physics, 36, 6180–6193, doi: 10.1063/1.531240; L. Crane (1994). Topological field key to quantum gravity, in Knots and Quantum Gravity, ed. J. C. Baez, Clarendon, Oxford, J. W. Barrett (1995). Quantum gravity as topological quantum field theory, arXiv:gr-qc/

9506070, Journal of Mathematical Physics, 36, 6161–6179. [32] R. Penrose and M. A. H. MacCallum (1973). Twistor theory: an approach to the quantisation of fields and space-time, Physics Reports, 6(4), 241–315, Bib­ code:1973PhR…..6…241P, doi: 10.1016/0370-1573(73)90008-2. [33] C. Rovelli and L. Smolin (1995). Spin networks and quantum gravity, Physical Review D, 52, 5743–5759. [34] L. Smolin (1995). Linking topological quantum field theory and nonperturbative quantum gravity, Journal of Mathematical Physics, 36, 6417. [35] J. F. Plebaski (1977). On the separation of Einsteinian substructures, Journal of Mathematical Physics, 18, 2511–2520. [36] A. Palatini (1919). Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rendiconti del Circolo Matematico di Palermo (1884– 1940), 43, 203–212 [English translation by R. Hojman and C. Mukku in P. G. Bergmann and V. de Sabbata (eds.) Cosmology and Gravitation, Plenum Press, New York, 1980]. [37] L. Smolin (2000). A holographic formulation of quantum relativity, Physical Review D, 61, 084007, arXiv:hep-th/980191, DW6333. [38] C. Rovelli and L. Smolin (1995). Discreteness of area and volume in quantum gravity, Nuclear Physics B, 442, 593, gr-qc/9411005 (erratum Nuclear Physics B, 456, 753–754, 1995). [39] Y. Ling and L. Smolin (2001). Holographic formulation of quantum supergravity, Physical Review D, 63, 064010, arXiv:hep­th/0009018. [40] L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin (1987). Spacetime as a causal set, Physical Review Letters, 59, 521–524. [41] M. Cortês and L. Smolin (2014). The universe as a process of unique events, Physical Review D, 90, 084007, arXiv:1307.6167 [gr-qc]. [42] M. Cortês and L. Smolin (2014). Quantum energetic causal sets, Physical Review D, 90, 044035, arXiv:1308.2206 [gr-qc].

266

Vi e w s , Va r i e t y, a n d C e l e s t i a l S p h e r e s

[43] M. Cortês and L. Smolin (2016). Spin foam models as energetic causal sets, Physical Review D, 93, 084039, arXiv:1407.0032, doi: 10.1103/ PhysRevD.93.084039. [44] M. Cortês and L. Smolin (2018). Reversing the Irreversible: from limit cycles to emergent time symmetry, Physical Review D, 97, 026004, arXiv:1703.09696. [45] L. Smolin and C. Verde (2021). The quantum mechanics of the present, arXiv:2104.09945. [46] L. Smolin (2020). The place of qualia in a relational universe, Philarchiv, https://philarchive.org/rec/SMOTPO-3. [47] F. Markopoulou (2000). Quantum causal histories, Classical and Quantum Gravity, 17, 2059–2072, doi: 10.1088/0264-9381/17/10/302, arXiv:hep-th/9904009. [48] C. Furey. Notes on algebraic causal sets, unpublished notes (2011); Cambridge Part III research essay (2006). [49] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, and L. Smolin (2011). The principle of relative locality, Physical Review D, 84, 084010, arXiv:hepth 1101.0931. [50] L. Freidel and L. Smolin (2016). Gamma ray burst delay times probe the geometry of momen­ tum space, Foundations of Physics, 46(6), 736–758, arXiv:hep-th/arXiv:1103.5626, doi: 10.1007/s 10701-016-9994-x. [51] C. Shannon (1948). A mathematical theory of communication, The Bell System Technical Journal, 27, 379–423. [52] J, Von Neumann (1932). Mathematische Grundlagen der Quantenmechanik, Berlin: Springer, ISBN 3-540-59207-5; J. Von Neumann (1955). Mathematical Foundations of Quantum Mechanics, Princeton University Press, ISBN 978-0-691-02893-4. [53] G. Bateson (1970). Form, substance and difference, The Nineteenth Annual Korzybski Memorial Lecture, delivered January 9, 1970, under the auspices of the Institute of General Semantics. Reprinted from the General Semantics Bulletin, No. 37, 1970. Accessed 23 May 2021. [54] L. Smolin (2015). Nonlocal beables, International Journal of Quantum Foundations, 1, 100–106, arXiv:1507.08576. [55] J. Barbour and B. Bertotti (1982). Mach’s principle and the structure of dynamical theories, Proceedings of the Royal Society (London) A, 382, 295. [56] R. M. Unger and L. Smolin (2015). The Singular Universe and the Reality of Time, Cambridge University Press. [57] L. Smolin (2013). Time Reborn, Houghton Mifflin Harcourt, Penguin and Random House Canada. [58] W. M. Wieland (2015). New action for simplicial gravity in four dimensions, Classical and Quantum Gravity, 32, 015016, arXiv:1407.0025. [59] S. Alexander, W. J. Cunningham, J. Lanier, L. Smolin, S. Stano­jevic, M. W. Toomey, and D. Wecker (2021). The autodidactic universe, arXiv:2104.03902

267

Ti m e a n d S c i e n c e

[60] J. Barbour and B. Bertotti (1982). Mach’s principle and the structure of dynamical theories, Proceedings of the Royal Society A, 382(1783), 295–306; J. Barbour (1974). Relative-distance Machian theories, Nature, 249, 328–329; J. Barbour and B. Bertotti (1977). Gravity and inertia in a Machian framework, Il Nuovo Cimento B, 38, 1–27. [61] For a good introduction, see F. Mercati (2018). Shape Dynamics: Relativity and Relation­ alism, Oxford University Press, doi: 10.1093/oso/ 9780198789475.001.0001. [62] J. Barbour and B. Bertotti (1982). Relational concepts of space and time, British Journal for the Philosophy of Science, 33, 251; Proceedings of the Royal Society (London) A, 382, 295; J. B. Barbour (1986). Leibnizian time, Machian dynamics, and quantum gravity, in Quantum Concepts in Space and Time, eds. R. Penrose and C. J. Isham, Oxford University Press, Oxford. [63] S. Banerjee (2019). Null infinity and unitary representation of the Poincare Group, Journal of High Energy Physics, 01, 205, arXiv:1801.10171 [hep-th]. [64] L. Smolin (2000). M theory as a matrix extension of Chern-Simons theory, Nuclear Physics B, 591, 227–242, arXiv:hep-th/0002009. [65] L. Smolin (2001). The exceptional Jordan algebra and the matrix string, arXiv:hep-th/0104050. [66] L. Smolin (2000). The cubic matrix model and the duality between strings and loops, arXiv:hep­th/0006137. [67] Y. Ling and L. Smolin (2000). Eleven dimensional supergravity as a constrained topological field theory, arXiv:hep-th/0003285. [68] E. Livine and L. Smolin (2002). BRST quantization of matrix Chern-Simons theory, arXiv:hep­th/0212043. [69] Y. Yargic, J. Lanier, L. Smolin, and D. Wecker (2022). A cubic matrix action for the standard model and beyond, arXiv:2201.04183. [70] S. Alexander, W. J. Cunningham, J. Lanier, L. Smolin, S. Stanojevic, M. W. Toomey, and D. Wecker (2021). The autodidactic universe, arXiv:2104.03902. [71] L. Smolin (2017). Extending dualities to trialities deepens the foundations of dynamics, International Journal of Theoretical Physics, 56, 221–231, (special issue in memory of David Finkelstein), arXiv:1503.01424. [72] G. Segal (2001). Topological structures in string theory. Philosophical Transactions of the Royal Society A, 359(1784), Bibcode:2001RSPTA.359.1389S, doi: 10.1098/rsta.2001.0841; A. Pressley and G. Segal (2003). Loop Groups (Oxford Mathematical Monographs), New ed. Clarendon Press, Oxford. [73] J. Moussouris (1983). Quantum models of space-time based on recoupling theory, D. Phil thesis, St. Cross College, Oxford. [74] G. Ponzano and T. Regge (1968). Semiclassical limit of Racah coefficients, in Spectroscopic and Group Theoretical Methods in Physics, ed. F. Bloch, pp. 1–58. North-Holland.

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Chapter 10

S C I E N T IF I C C OS M O G O NY, THE T IM E IN Q UA NT U M R ELATI VI STI C PHY S I C S Gilles Cohen-Tannoudji* and Jean-Pierre Gazeau** *Université Paris-Saclay, France [email protected] **Université Paris Cité, France [email protected]

1 Introduction It is widely agreed upon that, at least in the physical sciences, the problem of time in science mainly lies in the issues of quantum gravity, namely the ones of the relation between quantum physics and general relativity, and this is the point of view that will be followed in this chapter of the present book. Historically, the main difficulty of the reconciliation of these two theories was that, at their onset, they looked drastically incompatible; whereas general relativity was remarkably well founded conceptually, but had only a few very specific observational implications, quantum physics, on the other hand, had more and more phenomenological implications at all scales of matter sciences, but continuously suffered from interpretive conceptual difficulties, especially about the problem … of time. Remember that time in quantum mechanics is the parameter associated with the unitary evolution operator having the Hamiltonian operator 269

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of the system as generator. This time is a concept built from the ratio action/energy. The Hamiltonian is self-adjoint with a spectrum bounded below (no infinitely negative energy!). Hence, the time parameter cannot be raised to the status of a quantum observable, an issue already pointed out by Pauli [1] in 1926. Time remains in the hands of the observer. Moreover, the unitarity of the evolution operator breaks at each (quantum) measurement. As noticed by Heisenberg [2], “the discontinuous change in the quantum state takes place ... because it is the discontinuous change in our knowledge ... that has its image in the discontinuous change of the state.” Whereas quantum measurement formalism describes the measurement interaction between the “pointer” and the system one intends to measure, nothing describes or explains the event called reduction or collapse of wave function or quantum state. Actually, the latter is similar to the outcome of a game, with urn, with dice, or whatever: God does play dice with the universe! There is nothing more to understand beyond the sidération (the fact of being staggered) provoked by the event that happens. The non-kinematical time is made of these sidérations, and this time has an arrow and is always observerdependent. Time arrow captures the uniqueness and distinctive character of sequential events whereas time’s cycle provide these events with another kind of meaning by evoking lawfulness and predictability. [3]

A contrario, the kinematical time is the inverse of a frequency. It is a concept built from the ratio distance/speed. It is viewed as a geometrical coordinate in any 4-dimensional space-time. The fact that its canonically conjugate variable is a (Fourier) frequency allows for building its quantum counterpart as a self-adjoint operator, with a spectrum the real line. This kinematical time operator has informational content within the framework of signal analysis. It is canonically conjugate to the frequency operator, and no Planck constant h is involved here (see [4] and [5] for details). Now, it turns out that, rather recently, important encouraging events occurred in the domain of a possible reconciliation of quantum physics and general relativity. These events occurred in both the experimental and the conceptual contexts. In the experimental or observational area, we note the discovery in 2012 of the Brout–Englert–Higgs boson, the

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cornerstone of the standard model of elementary particles and fundamental interactions, and the spectacular progresses of observational astrophysics represented by the establishment of a robust cosmological standard model, known as the concordance cosmology or ΛCDM (for Λ, the cosmological constant, Cold Dark Matter), or the first observations of gravitational waves. Also, in a more theoretical context, we note what has been called the reflexive, or quantum-informational turn of the interpretation of quantum theory and of general relativity. What we intend to show, in the present chapter, is to try and take advantage of these advances and to show that if LCDM and the standard model of elementary particles and fundamental interactions can be considered compatible without needing the introduction of any ad hoc field or particle, or hidden variables, then quantum and relativistic physics can be, at least phenomenologically, reconciled to underly what we call scientific cosmogony, the embedding framework in which the problem of time in science can be put in a way that is more tractable than ever. It is in a book intended for a wide audience, published in 1946, that Georges Lemaître, considered as one of the founders of modern cosmology, presented his cosmological model that he names the hypothesis of the primordial atom and qualifies as an essay of cosmogony [6]. As a Catholic priest, he was very eager to draw a clear separation line between what he names cosmogony and considers as a theory of the universe, or as the natural beginning of the universe, and the domain of theology. In the fourth chapter of his book, titled “Cosmogonic Hypotheses”, he presents a brief survey of cosmogonic assumptions proposed by famous thinkers before the scientific revolution that occurred at the beginning of the twentieth century, such as Buffon, Laplace, and mainly Kant, whose contribution is analyzed in most detail: “The theory of Kant is not just a theory of the formation of the solar system; it is a cosmogonic theory with the full meaning of the word, a theory of the universe” [6, 129]. Moreover, after having mentioned the conception that Kant had about matter, he adds that Kant “dares to say: — give me matter, and I would do a world out of it —, what is to say, I would show you how a world can come out of it.” [6, 121] At the end of this chapter, in a transition to the last chapter about the primordial atom, he concludes: “I could not do better than by repeating, while transposing it, the word of Kant: — give me an atom, and I would do from it the universe.” [6, 146].

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In the last chapter, the conclusion of his book, he defines [6, 174] “the object of a cosmogonic theory as the search of ideally simple initial conditions from which could have resulted, through the interplay of known forces, the current world in its full complexity.” [6, 176] This is the definition that we shall adopt in the present chapter for scientific cosmogony, as an embedding framework to understand another aspect of time in quantum physics,1 besides its role as an evolution parameter mentioned above. As a support of this hypothesis, we shall firstly, in the second section, review the present assets of the cosmological standard model and summarize the main outcomes of the work that we have recently published [7] in order to reconcile this new cosmological standard model with the standard model of particle physics and fundamental interactions. Then, in the third section, we shall discuss the light these outcomes could shed on the problem of time in cosmology.

2  The New Standard Model of Cosmology 2.1  A short historical survey In about one century, cosmology, the science of the universe as a whole, has known essentially three stages:   (i) a first one, just after the publication by Einstein of the general theory of relativity, and of his first cosmological model, involved important debates between Einstein and de Sitter about the principle of the relativity of inertia and the role of the cosmological term or cosmological constant;  (ii) it is only after the observational confirmation of the expansion of the universe theoretically predicted by Lemaître, that cosmology became a scientific discipline based on a genuine standard model, the so-called HBBM (for Hot Big Bang Model), assuming the vanishing

 The hypothesis of the primordial atom is the title of the last text by Georges Lemaître in the collection of five texts that constitutes the book; this text and the one untitled the expansion of space are the two chosen by Jean-Pierre Luminet in the book he published, together with a text of Alexandre Friedman, under the title Essays of cosmology (and not cosmogony): A. Friedman and G. Lemaître Essais de cosmologie preceded by L’invention du big bang by J.-P. Luminet, Paris: Le Seuil, Sources du savoir, 1997. 1

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of the cosmological constant, able to account for most of observations at extra-galactic and cosmic scales, but facing severe shortcomings due to the existence of the singularity implied by the big bang assumption; (iii) finally, at the turn of the twenty-first century, thanks to more and more accurate measurements of the cosmological microwave background (CMB), cosmology, equipped with the new cosmological standard model named ΛCDM, entered its third stage, which marks a spectacular comeback of the cosmological constant and, as we think to have shown in [7], raises some hopes to solve the shortcomings of the HBBM.

ΛCDM is not the first cosmological model involving a three-stage cosmology; actually, Georges Lemaître’s cosmological model, based on the hypothesis of the primordial atom, already included three stages2. The first stage called by Georges Lemaître the “rapid expansion phase” corresponding to the explosion of the primordial atom that he also called “a single quantum,” to stress that he assumed that the first stage of his cosmogony relied on quantum physics. The second stage corresponded essentially to Einstein’s cosmological model involving the “cosmic repulsion” induced by the cosmological constant, temporarily equilibrating the effects of gravitation, an effect that eventually leads to the third stage of expansion, the one that is now observed. 2.2  The assets of ΛCDM The new cosmological standard model involves assumptions translating the cosmological principle (isotropy and homogeneity of matter-energy distribution at cosmic scales). Its assets are fourfold:   (i) The rediscovery of the cosmological constant about which de Sitter has stressed that it is unavoidable if one wants not to abandon the foundational principle of the relativity of inertia;  (ii) The validation of the scenario of primordial inflation, which somehow replaces the singularity implied by the big bang assumption, and was hitherto lacking credibility;

 In his analysis of the book of Lemaître, Jean-Pierre Luminet calls this three-stage cosmology, the “hesitant universe.” 2

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(iii) The concordance of all the methods used nowadays to determine distances, namely, the CMB, the supernovae, and the “baryonic acoustic oscillations” (BAO); (iv) The discovery of two dark components, i.e., with no interactions except for gravitation, of the distribution of matter-energy at cosmic scales, namely the dark energy and the dark matter, which together, today amount to about 95% of the material content of the universe. 2.3  Summary of the outcome of ref. [7] The three stages of ΛCDM can be seen in Fig. 10.1 taken from [7], in which the Hubble radius is plotted versus the scale parameter.3 This mode of representation of the cosmological evolution, showing two time-dependent observables, one versus the other, has the merit of making the time implicit. If one is interested in the time issue, it is a matter of interpretation to answer the question: what is the time that is implicit in this figure? That is the question that we shall answer in Section 3. For the moment, let us briefly review the results obtained in [7], in which the interested reader will find detailed explanations and relevant references. The so-called primordial inflation stage is one of the major assets of ΛCDM thanks to the high accuracy measurement of the CMB: the fluctuations of order 10–5 in the temperature and polarization distributions cannot be explained by any standard model contribution, but only through some beyond the standard model (BSM) contribution, likely occurring at an energy of order 1015–16 GeV, which was called the inflation scenario, and now known as the bouncing scenario [8]. A great merit of LCDM is to have strongly validated this scenario, which was hitherto lacking any credibility. It is in the expansion stage (from point β to point ψ in Fig. 10.1) that the issue of the compatibility of the particle physics (SM) and of cosmology standard (CSM) models is addressed in modern cosmology. In fact, it is this issue that distinguishes ΛCDM from the Lemaître primordial atom model: thanks to what we know now from the standard model of particle physics, and has been clearly confirmed by the discovery in 2012 of the BEH (Higgs) boson, before the electroweak symmetry breaking that occurred at point γ , all the particles of the SM are assumed to be massless  The scale parameter is a dimensionless quantity equal to the time-dependent radius of the universe divided by its present day value. 3

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Figure 10.1  Hubble radius L(a) = H–1(a)(c = 1) is plotted versus the scale factor a(t) in logarithmic scale.

and to acquire their actual mass thanks to the Higgs mechanism, which rules out the primordial atom of Lemaître who imagined it as a superheavy nucleus (a superheavy “isotope of the neutron,” as he said). The outcome of [7], where the colorless confinement or hadronization occurring at point δ plays a central role, is briefly summarized in the following: “The reconciliation of the two standard models does not need recourse to any ad hoc fields, particles, or hidden variables” and “Dark matter is interpreted as a gluonic Bose-Einstein condensate.”

3  The Problem of Time in Cosmology 3.1  The “extra-mundane time” of de Sitter If one wants to reconcile the standard models of particle physics and cosmology, one is faced with the problem of time in cosmology, which is well summarized in this quotation of de Sitter that clearly implies the existence of two time dimensions: The three-dimensional world must, in order to be able to perform ‘motions’, i.e., in order that its position can be a variable function of the 275

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time, be thought movable in an ‘absolute’ space of three or more dimensions (not the time-space x, y, z, ct). The four-dimensional world requires for its ‘motion’ a four- (or more-) dimensional absolute space, and moreover an extra-mundane ‘time’ which serves as independent variable for this motion. [9]

In fact, if one wants to think the time evolution of the universe as a whole, one has to imagine oneself as being an observer whose proper time would be the extra-mundane time suggested by de Sitter, and who would observe the universe from the outside, what we called in [7] (Section 4.2) [10], [11] “a local tangent observer.” Now it has been shown by Connes and Rovelli [12] that such a global time noted, as τ in [7], called thermal time, as opposed to the local time, noted as t in [7], can be defined in a way that does not contradict the principles of general relativity (although it is induced by the state of the universe as a whole, it is not an “absolute time”). Now it turns out that the gedanken experiment that consists in imagining that one can observe the universe from the outside seems to be precisely what has been performed with the measurement of the CMB: removing everything that can interfere or blur the radiation in the foreground is nothing but observing from the outside the universe after it emitted the radiation, some four hundred thousand years after the “big bang.” It is the performance of this gedanken experiment that led to the discovery of the dark components of the full matter-energy content of the present day universe, the sum of the dark energy related to the cosmological constant Λ (when it is put in right hand side of Einstein’s equation), and the dark matter also related to Λ (but when it is put back in the left hand side): In the measurement of the CMB radiation, it is crucial to get rid of the light that is emitted in the foreground in order to obtain the original map of the CMB radiation. This can be done using known technics, but once this is done, one is faced with the problem of the gravitational lensing possibly distorting the path of light between its emission and its arrival at the detector. To solve this problem, it has been possible to use a technique that has been already used to get information about the dark matter present in some very heavy super clusters of galaxies: such clusters may induce a gravitational lensing potential distorting (and possibly multiplying) the image of a galaxy situated far behind the cluster; correlating the distorted observations one has been able to produce a map

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of the dark matter present in the cluster or in its halo that induces the lensing. The success of this technique has been considered as a proof of the presence of dark matter at extra galactic scales. Using this technique for the full sky distribution of the CMB with both the measurements of the temperature and of the polarization of the radiation, the Planck experiment has been able to yield two outcomes essential for the establishment of the cosmological standard model: on the one hand, the original map of the CMB, not distorted by the lensing, that can be used as the input data in simulations, and, on the other, a full sky map of the gravitational lensing potential, which we interpret in as the (anti-de Sitter) world matter identified with dark matter. [7]

Actually, this interpretation had been already proposed by in a previous work [13], but it was lacking the firm justification that came from another previous work [14], according to which the four-dimensional anti-de Sitter space-time viewed as a manifold embedded in fivedimensional ambient flat space with metrics (+ + – – –), provides quantum cosmogony with a 5D pseudo-Euclidean kinematics [15] in which the 5th dimension is temporal, the extra-mundane time. Such a kinematics, suited for a Newtonian phenomenology, considers the five universal constants, namely Planck’s constant h, the vacuum velocity of light c, Newton’s constant GN, Boltzmann’s constant kB, and the cosmological constant Λ, the landscape of which is schematized in Fig. 10.2. It is in the phenomenological stage of matching the standard models that the problem of time is addressed in [7]: When applied in the full expansion phase, the cancellation of the cosmological constant contribution by this anti-de Sitter world matter [leading to an effective dark universe with a vanishing total active mass] amounts to replace in our quantum cosmology, the local time t by the global time τ. One could say that considering these two times amounts to a complexification of the time, and that t and τ are complex conjugate variables: if the densities in our quantum cosmology are analytic functions depending on the global time τ they do not depend on its complex conjugate, namely the local time t.

3.2  A beginning of the world before the beginning of space and time In a short article [16] published in Nature, as a comment after a statement by Sir Arthur Eddington saying that “philosophically, the notion of a 277

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Figure 10.2   The landscape of quantum cosmogony involving the three theoretical frameworks that are kinematics, phenomenology, and dynamics. The intersections of these three theoretical frameworks determine seven sectors: a kinematical one; three phenomenological ones, in which is proposed the matching, free of any ad hoc adjunction, of the standard models of particle physics and of cosmology; and three purely theoretical ones, of which two are well established (quantum field theory and general relativity), and a third one (covariant quantum field theory) that is currently in a program stage.

beginning of the present order of Nature is repugnant to him,” Georges Lemaître, on the one hand, makes precise what he means with the primordial atom, “a unique atom, the atomic weight of which is the total mass of the universe” and, on the other hand, formulates what appears to be the problem of time in cosmology: If the world has begun with a single quantum, the notions of space and time would altogether fail to have any meaning at the beginning; they would only begin to have a sensible meaning when the original quantum had been divided into a sufficient number of quanta. If this suggestion is correct, the beginning of the world happened a little before the beginning of space and time. [emphasis ours] 278

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As we said above, the problem with Georges Lemaître’s cosmology is that his hypothesis about the primordial atom contradicts what we know now about the emergence of mass in the early universe: the initial state of the universe is more likely the ground state of the universe rather than a superheavy nucleus. Such an initial state can be considered in the scientific cosmogony sketched above. Its five universal constants consist, together with the velocity of light c, in four elementary quanta: i) Two quanta relative to matter, namely the quantum of action equal to Planck’s constant h, the quantum of entropy (or of information) equal to Boltzmann’s constant kB; ii) Two quanta relative to space-time, namely a quantum of space area G Ap = c3N , and a quantum of curvature equal to Λ, the cosmological constant. On the one hand, the two pairs of elementary quanta, relative both to matter and to space-time, allow articulating in quantum cosmogony the two times which may have different origins, namely the local dynamical time with the global thermal time, and, on the other hand, they allow defining the nothingness: there is nothing in the whole universe with action smaller than h, there is nothing with an entropy smaller than kB, nothing with a surface area smaller than AP and nothing with a curvature smaller than Λ. With such a definition of the nothingness, the ideally simple initial state of the scientific cosmogony can be identified with the quantum vacuum, which is a well-defined particular state of the Fock’s space, the Hilbert space of covariant quantum field theory [17], in which the occupation numbers of all the involved quantum fields are equal to zero, but which is not the nothingness, but, as Pascal told, lies halfway between matter and nothingness.4 Such was the idea that guided us in [7] towards our interpretation of dark matter in the standard cosmological model: Actually, the “effective dark universe” which, as said above, is supposed to provide the cosmological standard model with a quantum vacuum or a  “D’où l’on peut voir qu’il y a autant de différence entre le néant et l’espace vide, que de l’espace vide au corps matériel ; et qu’ainsi l’espace vide tient le milieu entre le matière et le néant.” Translation: “From which it can be seen that there is the same difference between nothingness and empty space as between empty space and matter; and thus, that empty space lies halfway between matter and nothingness.” Pascal, Œuvres complètes, La Pléiade, ed. 1998, 384. 4

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ground state, cannot be thought of as an “empty spacetime” in which some objects would move, but rather as a medium in which occur vacuum polarization events, what Gürsey calls scintillation events,5 namely events each consisting in the virtual creation of a particle-antiparticle pair, followed, a short time later, by its annihilation. If the particles of the pair are fermions, such event would contribute to a negative curvature (normal de Sitter) world matter, a sort of a baryonic Fermi-Dirac Sea, at the exterior of the Hubble horizon, and if they are bosons, the event would contribute to a positive curvature (anti-de Sitter) world matter, a gluonic Bose-Einstein condensate, at the interior of the Hubble horizon.

4  As a Conclusion As a tribute to Steven Weinberg (1933–2021), we want to quote his conception of quantum field theory that can lead to a possible reformulation of the Lemaître hypothesis of the primordial atom: In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields. In a relativistic theory the wave function is a functional of these fields, not a function of particle coordinates. Quantum field theory hence led to a more unified view of nature than the old dualistic interpretation in terms of both fields and particles. [19]

Put together, Fig. 10.3, which shows the role of cornerstone that the Higgs boson plays in the standard model of particle physis and fundamental interactions, and Figure 10.4, which schematizes the matching of the two standard models, suggest that one could say that going from the Lemaître model to the current CSM just needs to replace the primordial atom by … the Higgs boson!

 The mass scintillation model imagined by Gürsey, is comparable with the steady state cosmology of Bondi [The Steady-State Theory of the Expanding Universe, Monthly Notices of the Royal Astronomical Society (vol. 108, no 3, pp. 252–270)] in which the creation, at constant density of matter-energy, induces the expansion of the universe. 5

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Figure 10.3   The BEH (Higgs) boson as the key stone of the SM. The blue lines represent the couplings of the various fields, those coupled to the BEH, including the BEH itself acquiring mass through the BEH mechanism; those not coupled to it either remain massless or keep their mass if any. (Figure inspired from [18]).

Figure 10.4  Schematized illustration of the matching of the standard model of particle physics (SM) and the cosmological standard model (ΛCDM), thanks to a minimal “beyond the standard models” (BSM) assumption, according to [5].

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References   [1] Pauli, W. E. The general principles of wave mechanics, in Encylopedia of Physics, Vol. V, edited by S. Flugge, Berlin: Springer, 1958, p. 60.   [2] Mermin, N. D. quoting Heisenberg, What’s bad about this habit, Phys. Today, 2009, 62, 8–9.   [3] Gould, S. J. Time’s Arrow, Time’s Cycle: Myth and Metaphor in the Discovery of Geological Time, 1st ed., Cambridge: Harvard University Press, 1988.   [4] Gazeau, J-P., and Habonimana, C. Signal analysis and quantum formalism: quantizations with no Planck constant, in Landscapes of Time-Frequency Analysis, Vol. 2, Applied in Numerical and Harmonic Analysis series, New York: Springer International Publishing, 2020. arXiv:2001.04916 [quant-ph]   [5] Cohen-Tannoudji, G., Gazeau, J-P., Habonimana, C., and Shabani, J. Quantum models à la Gabor for space-time metric, 2022, 24(6), 835.   [6] Lemaître, G. L’hypothèse de l’atome primitif — Essai de cosmogonie, Préface de Ferdinand Gonseth, Neuchâtel: Editions du Griffon, 1946.   [7] Cohen-Tannoudji, G., and Gazeau, J-P. Cold dark matter: a gluonic Bose Einstein condensate in anti-de Sitter space time, hal-03419703, 2021.   [8] Bergeron, H., Czuchry, E., Gazeau, J.-P., and Małkiewicz, P. Quantum Mixmaster as a model of the Primordial Universe, Universe, 2020, 6, 7.   [9] de Sitter, W. On the relativity of inertia. Remarks concerning Einstein’s latest hypothesis, in KNAW, Proceedings, 19 II, 1917, Amsterdam, pp. 1217–1225. [10] Mickelsson, J., and Niederle, J. Contractions of representations of de Sitter groups, Commun. Math. Phys., 1972, 27, 167–180. [11] Garidi, T., Huguet E., and Renaud, J. De Sitter waves and the zero curvature limit, Phys. Rev. D, 2003, 67, 124028. [12] Connes, A., and Rovelli, C. Von Neumann algebra automorphisms and timethermodynamics relation in general covariant quantum theories, Class. Quant. Grav., 1994, 11, 2899–2918. [13] Cohen-Tannoudji, G. The de Broglie universal substratum, the Lochak monopoles and the dark universe, Ann. Fond. Louis Broglie, 2019, 44, 187–209. [14] Gazeau, J-P. Mass in de Sitter and anti-de Sitter universes with regard to dark matter, Universe, 2020, 6, 66. https://www.mdpi.com/2218-1997/6/5/66 [15] Bacry, H., and Lévy-Leblond, J.-M. Possible kinematics, J. Math. Phys., 1968, 9, 1605. [16] Lemaître, G. The beginning of the world from the point of view of quantum theory, Nature, 1931, 3210, 127. [17] Rovelli, C. Quantum Gravity (Cambridge Monographs on Mathematical Physics), Cambridge: Cambridge University Press, 2015. [18] Gürsey, F. Reformulation of general relativity in accordance with Mach’s principle, Ann. Phys., 1963, 21, 211–242. 282

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[19] Weinberg, S. What is Quantum Field Theory, and What Did We Think It Is? Historical and Philosophical Reflections on the Foundations of Quantum Field Theory, at Boston University, March 1996. https://arxiv.org/abs/ hep-th/9702027v1 [20] Cohen-Tannoudji, G., and M. Spiro, M. Le boson et le chapeau mexicain, Paris: Gallimard (Folio Essais), 2013.

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Chapter 11

P T S Y MM ETRY Carl M. Bender Washington University in St. Louis, USA [email protected]

Energy is perhaps the most fundamental physical quantity. When an experimental physical measurement is performed, such as the position or velocity of a particle, what is actually measured is energy. To observe the particle, one must detect the photons (i.e., the light) emitted by the particle, and these photons deposit energy in the recording device. Although the number system used to describe theories of modern physics includes both real and complex numbers, it is accepted that the only possible outcome of any energy measurement is a real number. Indeed, the dials and meters on laboratory devices are labeled with real numbers only. In modern physics, the behavior of a physical system is described in terms of a fundamental quantity called the Hamiltonian. The Hamiltonian in quantum mechanics is referred to as the energy operator because any energy measurement of the system will return one of the eigenvalues of the Hamiltonian. When we say that a system is quantized, we mean that the energy of the system cannot be arbitrary, as in classical physics, but rather is restricted to be one of these eigenvalues, and these eigenvalues must be real. Like a plane-geometry course based on Euclid’s axioms, an introductory quantum mechanics course begins with a list of widely accepted

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postulates. Almost all of these postulates are physical in character, and they read as if they were formulated by physicists. For example, they include such physically intuitive requirements as causality, stability, and conservation of probability. However, one postulate stands out; it is different from all of the others because it is highly mathematical in character. This is the requirement that the Hamiltonian describing a physical system be Hermitian. (This means that if the Hamiltonian is represented as a matrix, the matrix remains invariant under combined complex conjugation and matrix transposition.) This fancy and powerful mathematical requirement guarantees that any measurement of the energy of a physical system will return a real number. Perhaps even more interesting is that Hermiticity also guarantees that physical probability is conserved in time. This means that the probability of finding a particle, such as an electron, somewhere in the universe does not change in time; an electron cannot magically appear or disappear. However, one might wonder whether it is possible to replace the abstract mathematical condition of Hermiticity with a simpler and more intuitive physical condition that would also guarantee the reality of an energy measurement and the conservation of probability. The purpose of this article is to demonstrate that there is indeed such a condition. This simple physical condition is called PT symmetry. Here, T represents the operation of time reversal and P represents the operation of space reflection.

1  The Condition of PT Symmetry Let us begin by explaining the scientific basis of PT symmetry. In the early days of modern physics at the end of the 19th century and the beginning of the 20th century, it became clear that Newton’s theory of classical mechanics was inadequate. Newton’s theory had to be replaced by a new theory of mechanics that was consistent with Einstein’s theory of special relativity. The theory of special relativity recognizes that because the speed of light is measured to be the same in all inertial frames of reference, all physical theories must be invariant under Lorentz transformations. A Lorentz transformation is defined as a mapping of one space-time point (x, y, z, t) into another space-time point (x’, y’, z’, t’) that preserves the quantity x2 + y2 + z2 – t2. (This quantity defines what is known as the 286

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light-cone, and must remain constant and unchanged because the speed of light is the same in all inertial frames of reference.) If we think of the space-time point (x, y, z, t) as a four-component vector, then mapping this vector into the vector (x’, y’, z’, t’) is done by matrix multiplication, and the set of all such real 4 × 4 matrices is called the Lorentz group. Einstein’s theory of special relativity implies that the Lorentz group is the fundamental symmetry group of the natural world. The parity operator P is an element of the Lorentz group because it reflects (changes the sign of) the spatial coordinates x, y, and z, P: (x, y, z, t) → (–x, –y, –z, t), and thus leaves x2 + y2 + z2 – t2 invariant. Also, the time-reversal operator T is an element of the Lorentz group because it reflects (changes the sign of) the time coordinate t, T: (x, y, z, t) → (x, y, z, –t), and again leaves x2 + y2 + z2 – t2 invariant. The PT operator changes the sign of all four components (that is, the space and time coordinates) of the space-time vector: PT: (x, y, z, t) → (–x, –y, –z, –t), and thus the PT operator is also an element of the Lorentz group. The Lorentz group has a remarkable mathematical structure. It consists of four distinct parts, as indicated in Fig. 11.1. The first part, as shown in the upper-left part of Fig. 11.1, is called the proper orthochronous Lorentz group (POLG). The POLG contains all the elements of the Lorentz group that are continuously connected to the identity transformation, which is the transformation that leaves the vector (x, y, z, t) unchanged. The second part of the Lorentz group, as shown in the upper-right portion of Fig. 11.1, consists of all the Lorentz transformations in the POLG combined with (multiplied by) space reflection P. The third part of the Lorentz group (lower-left portion of Fig. 11.1) consists of all the Lorentz transformations in the POLG combined with time reversal T. The fourth part of the Lorentz group consists of all the Lorentz transformations in the POLG combined with PT. 287

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Figure 11.1  Schematic picture of the Lorentz group. As shown, the set of all possible real Lorentz transformations consists of four disconnected parts.

It was originally believed that the physical world was invariant under all of the Lorentz transformations, schematically shown in Fig. 11.1. However, in 1957, the Nobel Prize was awarded to T. D. Lee and C. N. Yang for their theoretical prediction that parity symmetry is not a symmetry of the universe [1]. This means that an experiment can be designed that can distinguish between a left-handed laboratory and a right-handed laboratory (that is, a laboratory reflected in space)! Indeed, such an experiment was actually performed by C. S. Wu at Columbia University [2], and she was awarded the very first Wolf Prize in 1978 for this work. Subsequently, J. W. Cronin and V. L. Fitch were awarded the 1980 Nobel Prize for their demonstration that time reversal T is also not a symmetry of nature [3]. Their experiment demonstrated that a laboratory traveling forward in time is experimentally distinguishable from an identical laboratory that is traveling backward in time! We conclude from this work that the Lorentz transformations in the upper-right and lower-left portions of the Lorentz group illustrated in Fig. 11.1 are not fundamental symmetries of the universe. These discoveries raise an extraordinarily interesting question: What about the lower-right side of Fig. 11.1? That is, if we have two identical laboratories, except that one laboratory is left-handed and traveling backward in time while the other is right-handed and traveling forward in time, are they physically distinguishable? That is, is the combined P and T reflection (space-time reflection) a fundamental symmetry of nature? 288

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2  Complex Number System To address this question, we must introduce a complex number system. Mathematicians have known for centuries that the system of real numbers is incomplete and that it has a natural and inescapable completion that includes complex numbers. Let us begin with the simplest number system, namely, the positive integers 1, 2, 3, … It is evident that this number system is incomplete because if we attempt to perform basic arithmetic operations with these numbers, we are immediately forced to enlarge the number system to include new numbers. For example, if we ask, “what is the result of subtracting 5 from 5?”, “we are forced to include the number 0 in the number system, and if we ask, what is the result of subtracting 5 from 2?”, we are forced to append the negative integers to the number system. If we then ask, “what is the result of dividing 3 by 5?”, we are forced to include the rational numbers. Next, if we ask, “what is the square root of 5?”, we are forced to include real irrational algebraic numbers (real numbers that solve algebraic equations, in this case the solution to the equation x2 = 5). Finally, if we ask, “what is the ratio of the circumference of a circle to the diameter of a circle?”, we are forced to extend the number system to include transcendental numbers, such as p. (Transcendental numbers are new irrational numbers that are not solutions to algebraic polynomial equations such as x2 = 5.) We have now filled out and completed the set of all real numbers. But we are still not done. We can now ask, “what is the square root of a negative number?” For example, what is the solution to the equation x2 = –5? The result cannot be a real number because the square of a real number cannot be negative. Thus, we are forced to introduce yet another new number i into our number system, whose defining property is i2 = –1. The number i is called an imaginary number, as opposed to a real number, because its square is negative. Finally, we define the set of all complex numbers as a linear combination of any two real numbers a and b, where b multiplies the imaginary number i. Every complex number z can be written in the simple form z = a + ib. 289

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We have now finished constructing the complex number system: It is a profound mathematical theorem of complex analysis that the system of complex numbers, as we have constructed it, is complete; the solution of any algebraic equation can be written in this form a + ib, and no additional numbers need to be included in this number system.

3  A Fundamental Symmetry of Complex Numbers Now that the complex numbers have been defined, we can immediately see that the complex numbers exhibit a simple symmetry. The imaginary number i is defined above by the property that its square is –1. However, the number –i also has the property that its square is –1. Because of this, we cannot distinguish whether the complex number system is built from the fundamental imaginary number i or from –i. Indeed, all mathematical theorems that are based on the complex number system remain valid, independent of, and insensitive to the choice of the sign of i. This mathematical operation i → –i has a surprising physical interpretation in quantum mechanics. E. Wigner showed that in quantum mechanics the time-reversal operator T changes the sign of the imaginary number i. There are two ways to see why this is so. First, we observe that if we change the sign of i in the Schrödinger equation i y’(t) = H y(t), we must compensate for this sign change by changing the sign of the time variable t. Second, we recall that the position operator x and the momentum operator p obey the Heisenberg operator algebra xp – px = i. This equation is fundamental to the theory of quantum mechanics and is the basis of the Heisenberg uncertainty principle. Since the momentum p changes sign under time reversal but the position x does not, it follows that i must change sign to preserve the Heisenberg algebra. An immediate consequence of this is that the combination ix does not change sign under PT reflection. Thus, we see that the combination ix is PT invariant, even though it is not Hermitian. 290

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4  Complex Numbers and PT Symmetry Let us now return to the structure of the Lorentz group illustrated in Fig. 11.1. Remember that the Lorentz group, as pictured in Fig. 11.1, consists of all real 4 × 4 matrices (that is, 4 × 4 matrices whose 16 elements are all real numbers) that leave the quantity x2 + y2 + z2 – t2 unchanged. What happens if we relax the restriction that the matrix elements must be real and allow them to be complex while keeping the restriction that the quantity x2 + y2 + z2 – t2 must remain unchanged? The answer is that the resulting complex Lorentz group consists of two and not four disconnected parts. There is a continuous path through the complex Lorentz group from the upper-left to the lower-right portions of the real Lorentz group shown in Fig. 11.1, and there is also a continuous path from the upper-right to the lower-left parts of the real Lorentz group. Thus, the extension of the real number system to the complex number system forces us to include PT reflection as one of the fundamental symmetries of the universe. Of course, complex numbers are used ubiquitously in the formulation of the theories of modern physics. However, we now see that a profound consequence of extending the real numbers to the complex numbers leads immediately to the conclusion that PT symmetry is a fundamental natural symmetry.

5  Extension of PT Symmetry to CPT Symmetry There is a further profound consequence of extending the real Lorentz group to the complex Lorentz group, namely, the proof of the famous CPT theorem of particle physics. The properties and interactions of elementary particles are derived from quantum field theory, which is a branch of physics that emerges from combining the assumptions of special relativity and quantum mechanics. If one makes several crucial assumptions about the nature of particle interactions, such as locality, positivity of the particle spectrum, and Hermiticity of the Hamiltonian, then one can prove the CPT theorem. The proof of this theorem is long and difficult and fills an entire book [4]. Roughly speaking, this theorem states that by extending the real Lorentz group to the complex Lorentz group, and following a continuous path from the upper-left to the lower-right part of Fig. 11.1, an elementary particle undergoes a remarkable change from the original particle to its corresponding anti-particle. 291

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The conclusion is that there is a remarkable fundamental discrete symmetry in nature: The physical laws that one observes in a laboratory made from ordinary matter are the same as the physical laws that one would observe if this laboratory were reflected in space, were traveling backward in time, and were built out of antimatter. It is important to distinguish this CPT theorem from the assumption of PT symmetry. In this chapter we are proposing to avoid making the powerful and highly mathematical axiom that the Hamiltonian must be Hermitian and to replace this axiom with the much weaker and more physical requirement that the Hamiltonian be PT symmetric. Our assumption that the Hamiltonian is PT symmetric is based only on the geometrical properties of the Lorentz group and the elementary mathematical properties of complex numbers. Because the assumption that the Hamiltonian is PT symmetric is much weaker than the assumption that the Hamiltonian is Hermitian, we will see that there are many new kinds of Hamiltonian systems that we can consider and analyze that were never considered before. The systems described by these new Hamiltonians have surprising and unexpected physical properties, and beginning in 2008, many of these properties were observed and verified in hundreds of beautiful experiments performed in laboratories around the world.

6  What is Symmetry and Why is it Useful? We have used the term symmetry repeatedly, but we have not yet given a precise explanation of this often misunderstood concept. It is important to clarify what is meant when we say that a physical system possesses symmetry. A physical system is described by an equation or a set of equations. We say that such a system possesses symmetry if the equations remain invariant (unchanged) when the symmetry transformation is performed. Let us consider a simple example. The differential equation y’’(t) = y(t), where t is a time variable, has two symmetries. First, this equation remains invariant under the time-reversal operation t → –t, which is a discrete symmetry like PT symmetry. Second, it remains invariant under time translation t → t + a, where a is an arbitrary constant; this is a continuous symmetry because a is arbitrary. 292

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While this equation is symmetric under each of these transformations, we emphasize that the solutions to this equation need not be. To illustrate, observe that one solution to this equation is y(t) = et. Under time reversal, this solution becomes e–t, and under time translation this solution becomes et+a, which can be rewritten as Aet, where A = ea is a constant. Thus, it is clear that the solution et is not invariant under either of these symmetries. While this equation is time-reversal symmetric, there is another solution, sinh(t), that is actually antisymmetric under time reversal. Furthermore, there are no solutions that are invariant under time translation, except for the trivial solution y(t) = 0. Thus, at first it may appear that knowing that an equation has symmetry is not useful. However, this is not the case. Symmetry provides us with a powerful tool: When an equation has symmetry, we can use the symmetry operation to generate new solutions from old solutions. For example, if we begin with just the one solution et and perform time reversal, we get the new solution e–t, and if we then make use of time translation, we get the most general solution to the original differential equation: y(t) = Aet + Be–t, where A and B are arbitrary constants, and we have used the linearity of the equation to combine all of the solutions into a single expression. Thus, from just one solution, we have used symmetry to generate all possible solutions to the original equation. Furthermore, symmetry can be used as a powerful tool to simplify the form of a difficult equation, and such simplification may provide a route to finding solutions to the equation. We do not elaborate on such procedures here except to stress that when an equation possesses symmetry, this symmetry can often be used in two possible ways, not only to construct new solutions from already known solutions, but also to construct transformations that enable one to discover previously unknown solutions to the equation.

7  An Elementary PT-symmetric Classical Physical System It is quite easy to construct physical systems that are PT symmetric. For example, let us imagine a box and that something, such as water or 293

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Figure 11.2   A box with gain.

Figure 11.3   A pair of boxes with balanced gain and loss. Such a system is PT symmetric.

perhaps electrons or some form of energy, is flowing into the box (see Fig. 11.2). Because something is flowing into the box, we say this box has gain, and we remark that because something is flowing into the box, whatever is inside the box is increasing in amount, and thus the system is not in equilibrium. Next, imagine that the system consists of two boxes, one box with gain and another box with a balanced amount of loss (Fig. 11.3). This system is also not in equilibrium, but now it is PT symmetric. This is because under the operation of space reflection P we reflect the boxes about a point in space midway between the boxes; under time reversal T, whatever is flowing in is now flowing out and whatever is flowing out is now flowing in. 294

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Figure 11.4   The two components of the PT-symmetric system in Fig. 11.3 coupled by a pipe. If the pipe has a large enough diameter, then the stuff flowing into the box with gain and the stuff flowing out of the box with loss can be balanced, and the entire system can be in dynamic equilibrium. The sudden onset of dynamic equilibrium occurs at a critical value of the coupling strength g.

Now let us connect the boxes with a pipe that allows the stuff in the box with gain to flow into the box with loss (see Fig. 11.4). We now say that the boxes are coupled, and we use a parameter g, called a coupling strength, to represent the ease with which the stuff in the gain box can flow into the loss box. If we imagine that the stuff in the boxes is just water, then parameter g is a measure of the diameter of the pipe that couples the boxes. Such a system has two possibilities: If the diameter of the pipe is not large enough to equalize the flow of the water, then the system will not be in equilibrium. However, if we gradually increase the diameter of the pipe, we will eventually reach a critical diameter (that is, a critical value of the coupling strength g) at which the system is in dynamic equilibrium. When the system is in equilibrium, the amount of water in each box remains constant in time, and the water flowing into the gain box is balanced by the water flowing through the pipe and entering the loss box, and in turn the water flowing into the loss box is compensated by the water flowing out of the loss box.

8  An Elementary PT-symmetric Quantum-mechanical Physical System Now, let us see what happens if the box-and-pipe system is quantized. First, consider the system in Fig. 11.2. The Hamiltonian for this extremely simple system is just a 1 × 1 matrix: 295

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H = [a + ib], where a and b are real parameters. This Hamiltonian is not Hermitian because H does not remain invariant when we take the complex conjugate and transpose of H. The Schrödinger equation for this system is just the first-order differential equation i y’(t) = H y(t): i y’(t) = (a + ib) y(t). The wave function y(t) represents the probability amplitude for the stuff in the box and is a measure of how much stuff there is in the box as a function of time. The solution to the Schrödinger equation is y(t) = C e–iat + bt, and we see immediately that if b is negative, the amount of stuff in the box decreases in time (like radioactive decay), but if b is positive, then the amount of stuff in the box increases in time (like a bank account that is accruing interest). The Hamiltonian for the combined two-box system pictured in Fig. 11.3 is 0   a + ib H combined =  a − ib   0 and the Hamiltonian for the coupled two-box system in Fig. 11.4 is g   a + ib . H coupled =  a − ib   g Note that neither of these two-box Hamiltonians is Hermitian. However, it is easy to check that both of these Hamiltonians are PT symmetric. To do so, we verify that both Hamiltonians remain unchanged under the effect of combined time reversal T (where we interchange the sign of i) and space reflection P (where we interchange the two boxes). This interchange is performed by the parity (space reflection) matrix 0 1 P= . 1 0 296

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An interesting problem is to determine the quantum energy levels of the coupled-box system. To do so, we must find the eigenvalues of the Hamiltonian for the coupled system. These eigenvalues are the roots of the equation det(Hcoupled – IE) = E2 – 2aE + a2 + b2 – g2 = 0, which is called the secular equation. This quadratic formula tells us that there are two roots (which means that the system has two energy levels): E = a ± (g2 – b2)1/2. We see immediately that if the coupling strength g is greater than b, then the energies are real, but if g is less than b, then the energies are complex. There is a critical value of g, namely g = b, at which there is a transition from complex energies to real energies. This transition from complex energies to real energies is a feature of virtually all PT-symmetric quantum-mechanical systems. At this transition point, the corresponding classical system goes from a state of dynamic instability to a state of dynamic equilibrium. Conventional Hermitian quantum-mechanical systems never exhibit such a transition because their energies are always real. However, this feature of PT-symmetric systems is almost universal, and it is especially interesting that this critical transition, which can be interpreted as a phase transition (such as the melting of ice or the boiling of water), is easy to observe in the laboratory; it has been seen and recorded in a multitude of experiments.

9  First Experiments Involving PT-symmetric Optical Systems The PT phase transition was first observed at the University of Arkansas and the University of Central Florida in groundbreaking experiments on PT-symmetric optical systems [5, 6]. In these experiments, two wave guides (light channels) were used (see Fig. 11.5). One wave guide had gain and the other had loss. This is an experimental realization of the schematic diagram in Fig. 11.4. (Light can be pumped into the wave guide with gain by shining a laser on it and light can be extracted from the wave guide with loss by coating the surface of the wave guide with a 297

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Wave guide with gain Direction of light propagation Wave guide with loss

Figure 11.5   Realization of the schematic gain-loss illustration in Fig. 11.4 in optics experiments using wave guides [5, 6]. In these experiments, two coupled wave guides, one with loss and the other with gain, were used to create and study a PT-symmetric optical system. In these famous classic experiments and in many subsequent experiments involving other types of optical systems, as well as electronic systems and mechanical systems, the PT phase transition was clearly and definitively observed.

light-absorbing metal such as chromium or simply by relying on natural loss due to material absorption. The coupling of the wave guides is achieved by allowing them to touch.) When I visited the laboratory at the University of Arkansas, where the wave guides were fabricated and the first of these optical experiments was performed, I was allowed to take an informal picture of the experimental apparatus. This picture is reproduced in Fig. 11.6.

10  Constructing Model Non-Hermitian PT-symmetric Hamiltonians Recall that the combination ix is PT invariant because i changes sign under time reversal T and x changes sign under space reflection P. Thus, we can begin with any Hermitian PT-symmetric Hamiltonian and deform this Hamiltonian into the non-Hermitian domain by adding ix or multiplying by ix. Let us begin with a simple example of a quantum harmonic oscillator. The quantum harmonic oscillator is defined by the simple quadratic Hamiltonian H = p2 + x2. This Hamiltonian is both Hermitian and PT symmetric. We can deform this Hermitian Hamiltonian into the non-Hermitian domain by adding the quantity eix, where e is an arbitrary real parameter called a deformation parameter. The new Hamiltonian has the form 298

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Figure 11.6   Experimental apparatus at the University of Arkansas, where the first experiment involving a PT-symmetric optical system was performed. (This picture was taken by the author of this article during a visit to the University of Arkansas.)

H = p2 + x2 +eix. The time-independent Schrödinger equation associated with this Hamiltonian is –y’’(x) + x2 y(x) + eix y(x) = E y(x), and this equation is accompanied by the boundary condition y(±∞) = 0. The solution to this equation gives the following formula for the eigenvalues En: En = 2n + 1 + e2/4 (n = 0, 1, 2, 3, …). It is an amazing and unexpected result that all of the eigenvalues of the Hamiltonian H are real, even though this Hamiltonian is not Hermitian. 299

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Next, we construct an elaborate PT-symmetric model Hamiltonian by using a multiplicative, rather than an additive, deformation of the harmonic-oscillator Hamiltonian: H = p2 + x2(ix)e (e real). Figure 11.7 displays the energy eigenvalues (energy levels) of this Hamiltonian as functions of the deformation parameter e. The plot in Fig. 11.7 was published in [8], and this remarkable plot launched the study of non-Hermitian PT-symmetric Hamiltonians. This figure shows that for positive e the energy levels are all real and discrete. However, for negative e the energy levels coalesce in pairs and seem to disappear. At the points where the energies coalesce, often called exceptional points, the energy levels become complex. The region e < 0 is called

Figure 11.7   Energy levels of the Hamiltonian H = p2 + x2(ix)e plotted as functions of e. These energy eigenvalues are all real for positive e, but they become complex for negative e. Only the real parts of the eigenvalues are shown. There is a PT phase transition at e = 0, and this transition occurs exactly at the point at which the Hamiltonian is conventionally Hermitian.

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the region of broken PT symmetry and the region e > 0 is called the region of unbroken PT symmetry. As we saw in our discussion of coupled boxes with loss and gain, the region of unbroken PT symmetry corresponds physically to a system in physical equilibrium. In this region the phases of physical states of the Hamiltonian oscillate in time, but the norms of the states are time independent. However, in the broken PT-symmetric region, where the energy levels are complex, the system is not in equilibrium; the norms of some quantum states grow in time and the norms of other states decay in time. Such physical behavior is familiar in quantum mechanics; in some quantum-mechanical systems particles undergo radioactive decay and disappear in time, while the number of decay products correspondingly increases in time. The reality of the energy levels in Fig. 11.7 for e > 0 is a nontrivial consequence of PT symmetry and is extremely difficult to prove mathematically. It took several years before a rigorous proof of reality was finally published by P. Dorey, C. Dunning, and R. Tateo [9].

11  Some Other PT-symmetric Hamiltonians We emphasize that at e = 0 there is a transition in Fig. 11.7 between a region of complex energy levels and a region of entirely real energy levels, and it is precisely at the boundary point between these two regions that the Hamiltonian is Hermitian. Evidently, there are infinitely more nonHermitian PT-symmetric Hamiltonians than Hermitian Hamiltonians, and typically the Hermitian Hamiltonians reside at this PT boundary. It is easy to find other examples of PT-symmetric Hamiltonians; the recipe is to construct a PT-symmetric deformation of a Hermitian Hamiltonian. To illustrate, let us begin with the Hermitian Hamiltonian representing the quartic anharmonic oscillator H = p2 + x4. Then, just as we did for the harmonic-oscillator Hamiltonian, we construct a PT-symmetric deformation of this Hamiltonian by multiplying by ix raised to the power e: H = p2 + x4(ix)e (e real).

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Figure 11.8   Energy levels of the PT-symmetric Hamiltonian H = p2 + x4(ix)e plotted as functions of real e. As in Fig. 11.7, the energy levels are all real for positive e and complex for negative e. There is a phase transition at e = 0, and this transition occurs at the point at which the Hamiltonian is Hermitian. This plot has an interesting feature indicated by the dashed line; the higher-energy levels here oscillate gently back and forth about the vertical line situated at e = –1. This oscillation feature is too small to be seen in this graph.

The energy eigenvalues of this deformed Hamiltonian are plotted in Fig. 11.8 and, as before, we see that there are two regions, a region of unbroken PT symmetry when e > 0, where the energies are real, and a region of broken PT symmetry when e < 0, where the energies are complex. Also, at the boundary point e = 0 between these two regions, the Hamiltonian is conventionally Hermitian. As another example, we can begin with the Hermitian Hamiltonian H = p4 + x2, and then construct the PT-deformed Hamiltonian

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Figure 11.9   Energy levels of the PT-symmetric Hamiltonian H = p4 + x2(ix)e plotted as a function of real e. Again, as in Figs. 7 and 8, the energy levels are all real for positive e, complex for negative e, and there is a phase transition at e = 0, at which the Hamiltonian is Hermitian.

H = p4 + x2(ix)e (e real). Once again, we see that the energy levels are real when e > 0, complex when e < 0, and that there is a PT transition at e = 0 (see Fig. 11.9).

12  Conservation of Probability Hermitian Hamiltonians define quantum theories as having real energy levels, and for isolated physical systems in equilibrium, this is an essential property. We have seen that PT-symmetric Hamiltonians in the unbroken PT-symmetric regime also possess this crucial reality property. However, quantum theories defined by Hermitian Hamiltonians possess a second

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important property called unitarity, which is the technical term for the conservation of probability. This means, for example, that a particle, such as an electron or a proton, cannot disappear or come into existence. (Of course, in a scattering experiment, this is precisely what can happen, but such a system is not isolated and is described by a non-Hermitian Hamiltonian.) The crucial question to be answered is whether a quantum system described by a PT-symmetric non-Hermitian Hamiltonian also exhibits the property of unitarity. (If it does not, then unfortunately, while such a Hamiltonian may be mathematically interesting, it must be rejected on physical grounds as a possible or plausible description of physical reality.) The definitive answer to this question is that a quantum system defined by a PT-symmetric Hamiltonian does possess the property of unitarity in the region of unbroken PT symmetry. Establishing this property was not easy; the property of unitarity was established [10] several years after the study of PT-symmetric Hamiltonians began in 1998. A detailed discussion can be found in [11]. Here, we give a brief summary of the argument. The key point is that every PT-symmetric quantum-mechanical Hamiltonian has a new and hidden symmetry if the PT symmetry is unbroken (that is, if all of the energy levels are real). This symmetry is called C symmetry. Mathematically, this symmetry strongly resembles the operation of charge conjugation — that is, changing ordinary matter into antimatter — and this is why we use the symbol C to represent this symmetry. C symmetry is a mirror symmetry in the sense that the operation of C reflection followed by another C reflection has no effect. Analogously, C symmetry resembles charge conjugation in the sense that changing matter into anti-antimatter and then repeating this process has no effect. In mathematical terms, we express the fact that the operations of parity (space reflection) P, time-reversal T, and C reflection are all mirror symmetries by the algebraic equations P2 = 1, T2 = 1, C2 = 1. We also know that the order in which space and time are reversed is irrelevant; this is expressed by the algebraic equation PT = TP. Similarly, the order of PT reflection and C reflection is also irrelevant: CPT = PTC. Finally, the statement that C is a symmetry of the Hamiltonian H means that the algebraic order of operators H and C is irrelevant: HC = CH. To summarize, we found three simultaneous equations satisfied by the new symmetry operator C: C2 = 1, CPT = PTC, HC = CH. 304

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One might think that it would be easy to solve this simple-looking system of three simultaneous equations to determine the exact form of operator C. However, the simple form of these equations is deceptive, and they have only been solved exactly for elementary Hamiltonians. In general, one can only find an approximate solution for the C operator. Nevertheless, these equations are extremely powerful, and they can be used to establish a crucial feature of PT-symmetric theories: In a region of unbroken PT symmetry (in which the energy levels are real), the quantum theory defined by a PT-symmetric Hamiltonian conserves probability (see, for example, [10,11]).

13  Laboratory Studies and Some Practical Applications of PT Symmetry The field of PT-symmetric quantum theory is exciting to researchers because it provides infinitely many new physically as well as mathematically interesting theories to study, and these theories have remarkable and often unexpected properties. In some cases, these new theories may have practical and commercial consequences, and in others, PT symmetry may provide a route to a deeper understanding of the fundamental nature of the universe. From the early studies of PT-symmetric optical systems [6], it was recognized that PT-symmetric optical components provide powerful new tools for controlling the flow of light. Subsequent studies using wave guides and lasers have shown that this is indeed the case. In a number of studies of PT-symmetric lasers, the phenomenon of unidirectional invisibility has been observed [12, 13], and the PT phase transition has been observed in systems of coupled optical microcavities in which a beam of light is trapped in a disk-shaped solid-state material [14]. It may well be possible to have PT-symmetric optical components that use light beams in computers rather than wires to speed up the rate of computation. PT-symmetric electrical systems have also been studied in great detail. The PT phase transition has been observed in superconducting wires [15,16] and in electronic oscillators [17]. Additionally, beautiful work has been done in the Electrical Engineering Department at Stanford University, which shows that PT-symmetric electronic circuits can be used efficiently to charge the batteries in electric cars while they are driving [18]. Additionally, there have been studies of PT-symmetric atomic diffusion 305

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[19], PT-symmetric microwave cavities [20], PT-symmetric mechanical systems [21], PT-symmetric nuclear magnetic resonance [22], and PT-symmetric nonlinear waves [23]. Interestingly, the equations of fluid mechanics are PT symmetric, and fluid instabilities, such as the rapid formation of a thundercloud in the atmosphere, are nothing but a PT phase transition [24]. As we have said repeatedly, one of the noteworthy properties of PT-symmetric quantum-mechanical systems is the PT phase transition, where the energy levels for the system go from being real to being complex, and the system ceases to be in dynamic equilibrium. Conventional Hermitian Hamiltonians do not exhibit such a transition. It has recently been discovered that at this PT transition point, it is possible to make highly accurate laboratory measurements of the physical properties of the system. This remarkable feature of PT-symmetric systems is called enhanced sensing [25, 26].

14  Future Possible Theoretical Applications of PT Symmetry It is not difficult to construct PT-symmetric analogues of conventional physical theories, and such analogues often have remarkable and unexpected properties that may help explain some experimental observations. We list three examples below: (1) Conventional quantum electrodynamics (the quantum theory of electrically charged particles) is regarded as an incomplete theory because it lacks a theoretical property known as asymptotic freedom. In contrast, PT-symmetric quantum electrodynamics is asymptotically free and, therefore, stands alone as a complete quantum-mechanical theory. Moreover, in the PT-symmetric analogue of quantum electrodynamics, we find that like charges attract and unlike charges repel [27]. This is the exact opposite of what happens in conventional electrodynamics! Similarly, in the case of PT-symmetric gravity, we find that two masses repel instead of attracting one another (as in the case of conventional gravity). This may eventually offer a theoretical explanation for why the expansion of the universe is accelerating. (2) Modern theories of particle physics suffer from well-known problems; namely, the appearance of infinities. These infinities can be removed by using a technique called renormalization, but renormalizing often introduces other serious problems, namely, instabilities 306

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and ghost states (states of negative probability). However, we find that these new problems can often be eliminated by using the techniques of PT-symmetric quantum theory. For example, in the currently accepted theory of particle physics, which is known as the grand unified field theory, renormalization causes serious instability and leads to the catastrophic prediction that the ground state of the system is unstable and will eventually decay. This happens because the process of renormalization makes the Hamiltonian nonHermitian, but fortunately, the renormalized Hamiltonian is PT symmetric. As a result, the theory remains stable and the ground state does not decay [28]. (3) Neutrinos are elementary particles that were first discovered in experiments on radioactivity. Neutrinos are often produced in nuclear decay reactions, and neutrinos coming from nuclear reactors and from the sun have been detected. Neutrinos are mathematically similar to electrons in that they both obey the famous Dirac equation, but unlike electrons, neutrinos do not carry electric charge. Moreover, they are much lighter than electrons, and it was originally thought that neutrinos, like photons (particles of light), had no mass at all. More recent experimental studies of neutrinos indicate that there are three different species of neutrinos and that neutrinos can undergo species oscillation; that is, in a beam of neutrinos coming from a nuclear reactor or from the sun, the admixture of neutrino species can change. It is generally believed that the existence of neutrino oscillations implies that neutrinos are not massless, but rather have extremely small but nonzero masses. PT-symmetric quantum theory suggests a simpler and, we believe, more elegant possibility — the PT-symmetric Dirac equation suggests that one may have species oscillations and still have massless neutrinos [29, 30]. Interestingly, recent measurements in Germany have halved the upper bound on the mass of the neutrino associated with the electron [31]. The study of PT-symmetric physical systems is only two decades old, but it has become a highly active area of theoretical and experimental study and investigation. There are currently well over 6000 papers written on this subject, and this number is growing rapidly. There have been more than 100 conferences and workshops entirely devoted to this exciting subject, and scores of student theses have been written. While fundamental and naturally occurring PT-symmetric systems have not yet been 307

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found, this area of research has already provided much mathematical and physical insight into the physical world, and we are optimistic that rapid progress will continue to be made.

Acknowledgments The author thanks the Simons Foundation, the Alexander von Humboldt Foundation, and the U.K. Engineering and Physical Sciences Research Council for partial financial support.

References   [1] Lee, T. D., and Yang, C. N. (1956). Question of parity conservation in weak interactions. Physical Review 104(1), 254–258.   [2] Wu, C. S., Ambler, E., Harward, R. W., Hoppes, D. D., and Hudson, R. P. (1957). Experimental test of parity conservation in beta decay. Physical Review 105, 1413–1415.   [3] Christenson, J. H., Cronin, J. W., Fitch, V. L., and Turlay, R. (1964). Evidence for the 2p decay of the K20 meson. Physical Review Letters 13(4), 138–140.   [4] Streater, R. F., and Wightman, A. S. (1964). PCT, Spin and Statistics, and All That. Princeton University Press.  [5] Guo, A., Salamo, G., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G., and Christodoulides, D. N. (2009). Observation of  PT-symmetry breaking in complex optical potentials. Physical Review Letters 103, 093902.   [6] Ruter, C. E., Makris, K. G., El-Ganainy, R., Christodoulides, D. N., Segev, M., and Kip, D. (2010). Observation of parity-time symmetry in optics. Nature Physics 6, 192–195.   [7] Bender, C. M. (2016). PT symmetry in quantum physics: from a mathematical curiosity to optical experiments. Europhysics News 47/2, 17–20.  [8] Bender, C. M., and Boettcher, S. (1998). Real spectra in non-Hermitian Hamiltonians having PT symmetry. Physical Review Letters 80, 5243–5246.  [9] Dorey, P., Dunning, C., and Tateo, R. (2001). Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics. Journal of Physics A 34(28), 5679–5704. [10] Bender, C. M., Brody, D. C., and Jones, H. F. (2002). Complex extension of quantum mechanics. Physical Review Letters 89, 270401. [11] Bender, C. M. et al. (2018). PT Symmetry: In Quantum and Classical Physics. World Scientific.

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[12] Lin, Z., Ramezani, H., Eichelkraut, T., Kottos, T., Cao, H., and Christodoulides, D. N. (2011). Unidirectional invisibility induced by PT-symmetric periodic structures. Physical Review Letters 106, 213901. [13] Feng, L., Ayache, M., Huang, J., Xu, Y.-L., Lu, M. H., Chen, Y. F., Fainman, Y., and Scherer, A. (2011). Nonreciprocal light propagation in a silicon photonic circuit. Science 333, 729–733. [14] Peng, B., Ozdemir, S. K., Lei, F., Monifi, F., Gianfreda, M., Long, G. L., Fan, S., Nori, F., Bender, C. M., and Yang, L. (2014). Parity-time-symmetric whispering-gallery microcavities. Nature Physics 10(5), 394–398. [15] Rubinstein, J., Sternberg, P., and Ma, Q. (2007). Bifurcation diagram and pattern formation of phase slip centers in superconducting wires driven by electric currents. Physical Review Letters 99, 167003. [16] Chtchelkatchev, N., Golubov, A., Baturina, T., and Vinokur, V, (2012). Stimulation of the fluctuation superconductivity by PT Symmetry. Physical Review Letters 109, 150405. [17] Schindler, J., Li, A., Zheng, M. C., Ellis, F. M., and Kottos, T. (2011). Experimental study of active LRC circuits with PT symmetries. Physical Review A 84, 040101(R). [18] Assawaworrarit, S., Yu, X., and Fan, S. (2017). Robust wireless power transfer using a nonlinear parity-time-symmetric circuit. Nature 546, 387–391. [19] Zhao, K. F., Schaden, M., and Wu, Z. (2010). Enhanced magnetic resonance signal of spin-polarized Rb atoms near surfaces of coated cells. Physical Review A 81, 042903. [20] Bittner, S., Dietz, B., Gunther, U., Harney, H. L., Miski-Oglu, M., Richter, A., and Schafer, F. (2012). PT symmetry and spontaneous symmetry breaking in a microwave billiard. Physical Review Letters 108, 024101. [21] Bender, C. M., Berntson, B., Parker, D., and Samuel, E. (2013). Observation of PT phase transition in a simple mechanical system. American Journal of Physics 81, 173. [22] Zheng, C., Hao, L., and Long, G. L. (2013). Observation of a fast evolution in a parity-time-symmetric system. Philosophical Transactions of the Royal Society A 371, 2012.0053. [23] Yu, F. (2017). Dynamics of nonautonomous discrete rogue wave solutions for an Ablowitz-Musslimani equation with PT-symmetric potential. Chaos 27, 023108. [24] Qin, H., Zhang, R., Glasser, A. S., and Xiao, J. (2019). Kelvin-Helmholtz instability is the result of parity-time symmetry breaking. Physics of Plasmas 26, 032102. [25] Chen, P.-Y., Sakhdari, M., Hajizadegan, M., Cui, Q., Cheng, M.-C., El-Ganainy, R., and Alù, A. (2018). Generalized parity-time symmetry condition for enhanced sensor telemetry. Nature Electronics 1, 297–304.

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[26] Xiao, Z., Li, H., Kottos, T., and Alù, A. (2019). Enhanced sensing and nondegraded thermal noise performance based on PT-symmetric electronic circuits with a sixth-order exceptional point. Physical Review Letters 123, 213901. [27] Bender, C. M., and Milton, K. A. (1999). A nonunitary version of massless quantum electrodynamics possessing a critical point. Journal of Physics A 32, L87. [28] Bender, C. M., Hook, D. W., Mavromatos, N. E., and Sarkar, S. (2016). PT-symmetric interpretation of unstable effective potentials. Journal of Physics A 49, 45LT01. [29] Jones-Smith, K. and Mathur, H. (2010). Non-Hermitian quantum Hamiltonians with PT symmetry. Physical Review A 82, 042101. [30] Ohlsson, T. (2016). Non-Hermitian neutrino oscillations in matter with PT-symmetric Hamiltonians. Europhysics Letters 113, 61001. [31] Aker, M. et al. (2019). Improved upper limit on the neutrino mass from a direct kinematic method by KATRIN (KATRIN Collaboration). Physical Review Letters 123, 221802.

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Chapter 12

EX P E RIM E NTA L EVI DENCE F OR T I M E RE VER S AL VI O LATI ON David G. Hitlin California Institute of Technology Pasadena, USA [email protected]

1 Introduction The many conceptual aspects of the subject of time discussed in this volume have as their foundation our experimental knowledge of the description of the sequence of events in our four-dimensional world, a world with three spatial dimensions, and a fourth, the temporal dimension, marking the sequence of events in the evolution of a physical system. Modern physical theories, based on quantum mechanics and quantum field theory, manifestly treat the time dimension as the parameter governing the dynamic evolution of a physical system. In field theory, the time evolution of a system is described by a Lagrangian (more properly, a Lagrangian density) from which the equations of motion of the system in spacetime are derived. The study of the symmetries of the Lagrangian of a system, related to the conservation or lack thereof of a physical quantity by Noether’s theorem (Noether, 1918), is a principal approach to understanding the organization of a theory. In particular, an understanding of the behavior of the Lagrangian under the discrete symmetry operations of parity P, charge conjugation C, and time reversal T has proven to be crucial to the development of the Standard

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Model of elementary particle physics. It is, however, important to distinguish between spatial transformations such as parity, described by unitary operators, and temporal transformations governing the time evolution of a system, such as T, described by an anti-unitary operator (Wigner, 1932). A further important distinction involves the individual terms in the Standard Model Lagrangian, which represent the different known forces: the nuclear or strong force, described by quantum chromodynamics, with gluons as the force carriers, the electroweak force, a unification of the previously separate electromagnetic force carried by the photon, and the weak force carried by heavy W ± and Z0 bosons. The Standard Model does not incorporate gravity, but the potential for observing effects of violation of Lorentz invariance or of CPT symmetry due to gravity has aroused much interest (Kosteleckȳ, 2004). Incorporating Lorentz and CPT violations into extensions of general relativity or related theories of gravitation can produce effects that can be incorporated into extensions of the Standard Model that may be experimentally accessible. The T operator corresponds to changing the sign of time t to –t in quantities or expressions in dynamical equations involving velocity, momentum, or angular momentum. Elementary particle physics provides us with opportunities to test whether time-reversal invariance, i.e., the T operator, is in fact a symmetry of the Lagrangian that describes the universe of particles we observe: the Standard Model, as well as to search for violations of T symmetry that could provide evidence of physics beyond the Standard Model. A careful definition of measurable effects is central to an understanding of the T operation (Bohm, 1996; Arntzenius, 2009; Roberts, 2017; Barashevsky, 2020). The arrow of time, the increase of entropy in the evolution of macroscopic physical systems, including our universe, due to the second law of thermodynamics, is clearly not symmetric in the interchange of t and –t, but this is not a violation of T symmetry. Decays of unstable particles, a prime object of study in elementary particle physics, are also, in general, time-asymmetric processes due to the irreversibility of initial conditions; particle decay comparisons are nonetheless a primary tool in time reversal studies, although arriving at a universally agreed definition of appropriate conditions has proved controversial. There is, however, a clear experimental test of time reversal invariance in the weak interactions, which will be described in some detail in this article. 312

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There have been many excellent treatments of the theoretical and experimental aspects of the study of time reversal invariance, among them (Sachs, 1987; Streater, 1964; Roberts, 2015; Sozzi, 2020; Schubert, 2015; Bernabéu, 2015); I will not attempt to be as comprehensive as these, but rather will review what has been accomplished experimentally in testing the relevant concepts. There are limits on the possible size of T invariance violation in the strong interactions and clear evidence for the observation of the violation of time reversal symmetry in the electroweak sector, in particular, in the decays of neutral mesons. This article will summarize the current experimental situation of tests of time reversal invariance.

2  Experimental Studies of T Reversal Symmetry 2.1  Detailed balance Both classical mechanics and classical electrodynamics are time-reversal symmetric theories. This implies a reciprocity relation: the probability of an initial state i being transformed into a final state f is the same as the probability of an initial state identical to f, but with momenta and spins reversed, transforming into a state i also having momenta and spins reversed. T invariance is a sufficient, but not a necessary, condition for this to hold; thus, a failure of the reciprocity relation is an unambiguous signal for T violation. Reciprocity implies detailed balance, i.e., a connection between the differential cross sections for reactions a + b → c + d and c + d → a + b, but detailed balance may hold even if there is T violation (Henley, 1957). Detailed balance in the strong interactions has been demonstrated most precisely in the comparison of the differential cross sections for the reactions 27Al + p  24 Mg + α (Blanke, 1983), which are found to agree to within ± 0.5%, consistent with T invariance, and implying an upper limit on a time-reversal-violating amplitude in the reaction of ≥5 × 10–4 at 80% confidence level. This is the strongest limit on the violation of time-reversal symmetry in the strong interactions. 2.2  Neutrino oscillations In principle, neutrino oscillations could be exploited as a test of timereversal invariance, but it has not yet been possible to fulfill the experimental conditions required to perform the test, which would consist of 313

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comparing the transformation of neutrino species a into species b with the reversed process, e.g., by comparing reactions in which νm beams produce outgoing νe, and vice versa. This would require appropriate accelerators to produce intense beams of the complementary species νe at high energies, as well as adequate treatment of the fact that the matter in which the neutrino beams would propagate from production to detection is not CPT symmetric (Henley, 2011). There are two other situations in elementary particle physics in which one can experimentally probe for violation of T invariance: (1) searching for a permanent elementary electric dipole moment of a fermion, and (2) studying decays of neutral mesons in which it is possible to exchange initial and final states. 2.3  Permanent electric dipole moments A permanent electric dipole moment, or EDM, is even under C, but odd under P and T transformations. Thus, a non-zero EDM would be manifest evidence for T (and P) violation. Standard Model weak interactions can produce a very small EDM in loop diagrams via the CabibboKobayashi-Maskawa (CKM) mechanism (Kobayashi, 1973); the Standard Model estimates for various EDMs are shown in Table 12.1. The q term of quantum chromodynamics, due to instanton1 effects (Peccei, 1977), a P-odd, T-odd term in the Lagrangian due to spontaneous symmetrybreaking could, via the strong interactions, produce, for example, a nonzero EDM of the neutron, proton, or atom. The experimental search for an EDM typically employs the simultaneous application of external electric and magnetic fields. The approach varies in experiments with neutrons, charged elementary particles, and atoms or molecules. But the basic idea is to search for an alteration in the Larmor precession frequency of a magnetic dipole in a magnetic field when a parallel electric field is reversed, which would be caused by an EDM. Many new physics models can produce EDMs at tree or one-loop levels that are substantially larger than the Standard Model values. This exciting prospect has spawned many new EDM searches, but to date no

 An instanton is a topological solution of a four-dimensional quantum field theory, appearing in both the electroweak and QCD sectors. Through the axial anomaly, they explain the mass of Goldstone bosons. 1

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Figure 12.1   The effect of parity P (in this figure, reflection about the equator) and timereversal T transformations on an elementary fermion with a permanent magnetic moment m and a permanent electric dipole moment (EDM) d. Since d is P-odd and T-even, while m is P-even and T-odd, the existence of a non-zero EDM would violate P and T.

Table 12.1   Compilation of electric dipole moment (EDM) predictions in the Standard Model compared with current experimental limits EDM

Standard Model (e cm)

Electron

≈10

–44

Reference (Pospelov, 2014)

Limit (e cm)

Reference

1.1 × 10 (90% CL)

(Andreev, 2018) ThO

1.3 × 10–28 (90% CL)

(Cairncross, 2017) HfF+

–29

Muon

≈10–35

Scaled from e EDM (Babu, 2001)

1.8 × 10–19 (95% CL)

(Bennett, 2009)

Neutron

≈(1-6) × 10–32

(Seng, 2015)

1.8 × 10–26 (90% CL)

(Abel, 2020)

≈4 × 10–34

(Chupp, 2019)

7.4 × 10–30 (95% CL)

(Graner, 2017)

1.5 × 10–27 (95% CL)

(Allmendinger, 2019)

199

Hg atom

129

Xe atom

evidence has been found, although limits have become more stringent. The 2020 Particle Data Group compilation (Pich, 2020) of the most stringent EDM limits is shown in Table 1, together with the predictions of the Standard Model. The current limits on EDMs can be interpreted to limit the anomalous q term to a value