Lý thuyết Lượng tử là Hiện tượng Trồi lên: Cơ học Thống kê Mô hình Ma trận, Tiền thân của Lý thuyết Trường Lượng tử (S.L. Adler)

QUANTUM THEORY AS AN EMERGENT PHENOMENON The Statistical Mechanics of Matrix Models as the Precursor of Quantum Field Theory Quantum mechanics is our most successful physical theory. However, it raises conceptual issues that have perplexed physicists and philosophers of science for decades. This book develops a new approach, based on the proposal that quantum theory is not a complete, final theory, but is in fact an emergent phenomenon aris- ing from a deeper level of dynamics. The dynamics at this deeper level is taken to be an extension of classical dynamics to non-commuting matrix variables, with cyclic permutation inside a trace used as the basic calculational tool.

With plausible assumptions, quantum theory is shown to emerge as the statistical thermodynam- ics of this underlying theory, with the canonical commutation–anticommutation relations derived from a generalized equipartition theorem. Brownian motion cor- rections to this thermodynamics are argued to lead to state vector reduction and to the probabilistic interpretation of quantum theory, making contact with recent phe- nomenological proposals for stochastic modifications to Schrödinger dynamics. A D L E R received his Ph. degree in theoretical physics from Princeton.

He has been a Professor in the School of Natural Sciences at the In- stitute for Advanced Study since 1969, and from 1979 to 2003 held the State of New Jersey Albert Einstein Professorship there. Adler’s research has included seminal papers in current algebras, sum rules, perturbation theory anomalies, and high energy neutrino processes. Adler has also done important work on neutral current phenomenology, strong field elec- tromagnetic processes, acceleration methods for Monte Carlo algorithms, induced gravity, non-Abelian monopoles, and models for quark confinement. For nearly twenty years he has been studying embeddings of standard quantum mechanics in larger mathematical frameworks, with results described in this volume.

Adler is a member of the National Academy of Sciences, and is a Fellow of the American Physical Society, the American Academy of Arts and Sciences, and the American Association for the Advancement of Science. He received the J. Sakurai Prize in particle phenomenology, awarded by the American Physical So- ciety, in 1988, and the Dirac Prize and Medal awarded by the International Center for Theoretical Physics in Trieste, in 1998.com QUANTUM THEORY AS AN EMERGENT PHENOMENON The Statistical Mechanics of Matrix Models as the Precursor of Quantum Field Theory STEPHEN L. ADLER Institute for Advanced Study, Princeton www.com    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , UK Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www.

Adler 2004 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2004 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.com To Sarah Brett-Smith, with love and admiration www.com Contents Acknowledgements page x Introduction and overview 1 1 The quantum measurement problem 2 2 Reinterpretations of quantum mechanical foundations 6 3 Motivations for believing that quantum mechanics is incomplete 9 4 An overview of this book 13 5 Brief historical remarks on trace dynamics 18 1 Trace dynamics: the classical Lagrangian and Hamiltonian dynamics of matrix models 21 1.1 Bosonic and fermionic matrices and the cyclic trace identities 21 1.2 Derivative of a trace with respect to an operator 24 1.3 Lagrangian and Hamiltonian dynamics of matrix models 27 1.4 The generalized Poisson bracket, its properties, and applications 29 1.5 Trace dynamics contrasted with unitary Heisenberg picture dynamics 32 2 Additional generic conserved quantities 39 2.1 The trace “fermion number” N 39 2.2 The conserved operator C̃ 42 2.3 Conserved quantities for continuum spacetime theories 52 2.4 An illustrative example: a Dirac fermion coupled to a scalar Klein–Gordon field 58 2.5 Symmetries of conserved quantities under p F ↔ q F 62 3 Trace dynamics models with global supersymmetry 64 3.1 The Wess–Zumino model 64 3.2 The supersymmetric Yang–Mills model 67 vii www.com viii Contents 3.3 The matrix model for M theory 70 3.4 Superspace considerations and remarks 72 4 Statistical mechanics of matrix models 75 4.1 The Liouville theorem 76 4.2 The canonical ensemble 81 4.3 The microcanonical ensemble 88 4.4 Gauge fixing in the partition function 93 4.5 Reduction of the Hilbert space modulo i eff 100 4.6 Global unitary fixing 106 5 The emergence of quantum field dynamics 117 5.1 The general Ward identity 119 5.2 Variation of the source terms 124 5.3 Approximations/assumptions leading to the emergence of quantum theory 128 5.4 Restrictions on the underlying theory implied by further Ward identities 139 5.5 Derivation of the Schrödinger equation 147 5.6 Evasion of the Kochen–Specker theorem and Bell inequality arguments 151 6 Brownian motion corrections to Schrödinger dynamics and the emergence of the probability interpretation 156 6.1 Scenarios leading to the localization and the energy-driven stochastic Schrödinger equations 157 6.2 Proof of reduction with Born rule probabilities 170 6.3 Phenomenology of stochastic reduction – reduction rate formulas 174 6.4 Phenomenology of energy-driven reduction 175 6.5 Phenomenology of reduction by continuous spontaneous localization 185 7 Discussion and outlook 190 Appendices 193 Appendix A: Modifications in real and quaternionic Hilbert space 194 Appendix B: Algebraic proof of the Jacobi identity for the generalized Poisson bracket 194 Appendix C: Symplectic structures in trace dynamics 198 Appendix D: Gamma matrix identities for supersymmetric trace dynamics models 201 Appendix E: Trace dynamics models with operator gauge invariance 204 www.com Contents ix Appendix F: Properties of Wightman functions needed for reconstruction of local quantum field theory 206 Appendix G: BRST invariance transformation for global unitary fixing 208 References 212 Index 220 www.com Acknowledgements I have many people to thank for their assistance in aspects of this work. The dis- covery by my thesis student Andrew Millard of the conservation of C̃ provided the underpinning for the entire project.

I am greatly indebted to him, and to my other collaborators in the course of parts of this work, Gyan Bhanot, Dorje Brody, Todd Brun, Larry Horwitz, Lane Hughston, Achim Kempf, Indrajit Mitra, John Weckel, and Yong-Shi Wu. I am grateful to Jeeva Anandan, Angelo Bassi, Todd Brun, Lajos Diósi, Larry Horwitz, Lane Hughston, Gerald Goldin, Stanley Liu, Peter Morgan, Philip Pearle, Artem Starodubtsev, and several anonymous publisher’s reviewers, for many insightful comments on the first draft of this book. I am particularly in- debted to Philip Pearle for detailed comments on the Introduction and Chapter 6, to Larry Horwitz for a careful reading of the entire manuscript, and to Todd Brun and Peter Morgan for remarks that led to the present form of Sec. Finally, I wish to thank my wife Sarah for her perceptive support throughout this long project.

I have benefited from conversations and/or email correspondence with a great many others as well; a list (undoubtedly incomplete) includes: Philip Anderson, John Bahcall, Vijay Balasubramanian, Lowell Brown, Jeremy Butterfield, Tian-Yu Cao, Sudip Chakravarty, Freeman Dyson, Sheldon Goldstein, GianCarlo Ghirardi, Siyuan Han, William Happer, James Hartle, Roman Jackiw, Abraham Klein, John Klauder, Pawan Kumar, Joel Lebowitz, Anthony Leggett, James Lukens, G. Mangano, Herbert Neuberger, Ian Percival, Michael Ramalis, Soo- Jong Rey, Lee Smolin, Yuri Suhov, Leo Stodolsky, Terry Tao, Charles Thorn, Sam Treiman, Walter Troost, Steven Weinberg, Frank Wilczek, David Wineland, and Edward Witten. Parts of this book are based on papers that were previously published in Nuclear Physics B (Adler, 1994; Adler and Millard, 1996; Adler, 1997a) and in Physics Letters B (Adler, 1997b; Adler and Horwitz, 2003), and I wish to thank Elsevier Science, Ltd. for permission to use this material.

I similarly wish to thank Institute of Physics Publishing Ltd. for permission to use material originally published in x www.com Acknowledgements xi Journal of Physics A: Math. Finally, I wish to acknowledge the American Physical Society for use of material originally published in papers appearing in the Journal of Mathematical Physics (Adler, Bhanot, and Weckel, 1994; Adler, 1998; Adler and Kempf, 1998; Adler and Horwitz, 1996, 2000) and in Physical Review D (Adler and Wu, 1994; Adler, 2000, 2003a). I also wish to acknowledge the hospitality of the Aspen Center for Physics, and of both the Department of Applied Mathematics and Theoretical Physics and Clare Hall at Cambridge University, as well as my home base at the Institute for Advanced Study in Princeton.

The Albert Einstein Professorship that I held while writing this book was partially funded by the State of New Jersey, and my work is also supported in part by the Department of Energy under Grant No. DE-FG02- 90ER40542.com Introduction and overview Quantum mechanics is our most successful physical theory. It underlies our very detailed understanding of atomic physics, chemistry, and nuclear physics, and the many technologies to which physical systems in these regimes give rise. Addi- tionally, relativistic quantum mechanics is the basis for the standard model of ele- mentary particles, which very successfully gives a partial unification of the forces operating at the atomic, nuclear, and subnuclear levels.

However, from its inception the probabilistic nature of quantum mechanics, and the fact that “quantum measurements” in the orthodox formulation appear to re- quire the intervention of non-quantum mechanical “classical systems,” have led to speculations by many physicists, mathematicians, and philosophers of science that quantum mechanics may be incomplete. Among the Founding Fathers of quantum theory, Einstein and Schrödinger were both of the opinion that quantum mechanics is in some way unsatisfactory, and this view has been amplified in more recent pro- found work of John Bell, among others. In an opposing camp, many others in the physics, mathematics, and philosophy communities have attempted to provide an interpretational foundation in which quantum mechanics remains a complete and self-contained system. Among the Founding Fathers, Bohr, Born, and Heisenberg maintained that quantum mechanics is a complete system, and a number of re- cent proposals have been made to improve upon or to provide alternatives to their “Copenhagen Interpretation.” The debate continues, and has spawned an enormous literature.

While it is beyond the scope of this book to give a detailed review of all the proposals that have been made, to set the stage we give a brief discussion of the measurement problem in Section 1, and we survey some of the current proposals to revise the interpretational foundation of quantum mechanics in Section 2. The rest of this book, however, is based on the premise that quantum mechan- ics is in fact not a complete system, but rather represents a very accurate asymp- totic approximation to a deeper level of dynamics. Motivations for pursuing this track are given in Section 3. The detailed proposal to be developed in this book 1 www.com 2 Introduction and overview is that quantum mechanics is not a complete theory, but rather is an emergent phenomenon arising from the statistical mechanics of matrix models that have a global unitary invariance.

We use “emergent” here in the sense that it is used in condensed matter, molecular dynamics, and complex systems theory, where higher level phenomena (phonons, superconductivity, fluid mechanics, etc.) are seen to arise or “emerge” as the expressions, in appropriate dynamical contexts, of an underlying dynamics that at first glance shows little resemblance to these phenomena. Initial ideas in this direction were developed by the author and col- laborators in a number of papers dealing with the properties of what we termed “generalized quantum dynamics” or, in the terminology that we shall use in this exposition,“trace dynamics.” The purpose of this book is to give a comprehensive review of this earlier work, with a number of significant additions and modifica- tions that bring the project closer to its goal.