View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by D-Scholarship@Pitt TIME, SYMMETRY AND STRUCTURE: A STUDY IN THE FOUNDATIONS OF QUANTUM THEORY by Bryan W. Roberts Submitted to the Graduate Faculty of the Department of History & Philosophy of Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2012 UNIVERSITY OF PITTSBURGH HISTORY & PHILOSOPHY OF SCIENCE This dissertation was presented by Bryan W. Roberts It was defended on May 30th, 2012 and approved by John Earman, PhD, History & Philosophy of Science John D. Norton, PhD, History & Philosophy of Science Robert Batterman, PhD, Philosophy Laura Ruetsche, PhD, Philosophy (University of Michigan) Giovanni Valente, PhD, Philosophy Dissertation Advisors: John Earman, PhD, History & Philosophy of Science, John D. Norton, PhD, History & Philosophy of Science ii Copyright c by Bryan W. Roberts 2012 iii TIME, SYMMETRY AND STRUCTURE: A STUDY IN THE FOUNDATIONS OF QUANTUM THEORY Bryan W. Roberts, PhD University of Pittsburgh, 2012 This dissertation is about the sense in which the laws of quantum theory distinguish between the past and the future. I begin with an account of what it means for quantum theory to make such a distinction, by providing a novel derivation of the meaning of \time reversal." I then show that if Galilei invariant quantum theory does distinguish a preferred direction in time, then this has consequences for the ontology of the theory. In particular, it requires matter to admit \internal" degrees of freedom, in that the position observable generates a maximal abelian algebra. I proceed to show that this is not a purely quantum phenomenon, but can be expressed in classical mechanics as well. I then illustrate three routes for generating quantum systems that distinguish a preferred temporal direction in this way. Keywords: philosophy of physics, quantum theory, time, time reversal, T violation. iv TABLE OF CONTENTS PREFACE ......................................... viii 1.0 INTRODUCTION .................................1 1.1 The meaning of time reversal in quantum mechanics.............2 1.2 When quantum theory is time reversal invariant...............4 1.3 When classical theory is time reversal invariant...............5 1.4 When quantum theory is not time reversal invariant.............6 1.5 The mathematical structure of quantum theory...............7 2.0 THREE MYTHS ABOUT TIME REVERSAL ............... 11 2.1 Introduction................................... 11 2.2 First Stage: Why T is Unitary or Antiunitary................ 16 2.3 Second Stage: Why T is Antiunitary..................... 19 2.4 Third Stage: Position and Momentum.................... 21 2.5 Conclusion.................................... 28 3.0 DOES QUANTUM TIME HAVE A PREFERRED DIRECTION? .. 30 3.1 Introduction................................... 30 3.2 Basic Structures................................. 31 3.3 Galilei Invariance................................ 36 3.4 A minimalist ontology............................. 37 3.5 Time reversal invariance............................ 39 3.6 Discussion of the T -Theorem.......................... 44 3.7 Conclusion.................................... 47 v 4.0 WHEN WE DO (AND DO NOT) HAVE A CLASSICAL ARROW OF TIME ......................................... 49 4.1 Introduction................................... 49 4.2 What does not underpin classical TRI.................... 55 4.3 What does underpin classical TRI....................... 60 4.4 Conclusion.................................... 67 5.0 THREE ROUTES TO T-VIOLATION .................... 68 5.1 Introduction................................... 68 5.2 T -violation and Curie's Principle....................... 69 5.3 T -violation and Kramers degeneracy..................... 76 5.4 A template for T -violation........................... 80 5.5 Generalization to continuous observables................... 84 5.6 Conclusion.................................... 88 6.0 CONCLUDING REMARKS .......................... 90 6.1 Summary.................................... 90 6.2 Open Questions................................. 91 APPENDIX A. JAUCH'S LEMMA AND A CLASSICAL ANALOGUE . 94 A.1 Jauch's lemma................................. 94 A.2 A classical analogue of Jauch's lemma.................... 98 APPENDIX B. UNIQUENESS OF THE TIME REVERSAL OPERATOR FOR AN ELEMENTARY PARTICLE WITH SPIN-1=2......... 103 APPENDIX C. BIBLIOGRAPHY ......................... 106 vi LIST OF FIGURES 1.1 Galilei boost covariance says that if the same experiment is performed at dif- ferent constant velocities, it will produce the same results............5 1.2 A particle detection experiment can assign a true or false value to the propo- sition, \There is a particle in the region ∆.” It is true if the particle is in ∆ (left), and false otherwise (right)..........................8 2.1 If two pure states are orthogonal, and hence mutually exclusive, then we assume this fact is independent of the direction of time.................. 18 2.2 An involution is a transformation that is the identity operator (up to an arbi- trary phase) when applied twice, T 2 = c .................... 22 2.3 Time reversal transforms a system in the spatial region ∆ in the same way that it does a system in the translated region ∆ − a............... 25 3.1 The projection operator E∆ has eigenvalue 1 when an experimental detection occurs in the spatial region ∆ (left), and eigenvalue 0 when it does not (right). 33 3.2 The operators Ut represent time translation by a duration t, and are isomorphic to the group (R; +)................................. 33 3.3 Time reverse of a ball rolling down an inclined plane............... 39 4.1 (a) Force field and (b) phase diagram for a harmonic oscillator......... 50 4.2 Time reversal of the harmonic oscillator, in two steps. The first (p 7! −p) is a mirror flip about the q axis; the second (t 7! −t) reverses the order of events. 52 4.3 A particle experiencing the force F = x × x_ with positive initial position and velocity........................................ 58 vii PREFACE What follows are four papers on the nature of time reversal invariance in quantum theory. Each of them (Chapters2,3,4 and5) can be read independently. Chapter3 involved formulating a precise expression of a theorem due to Josef Jauch, and Chapter4 required working out a classical analogue of the theorem. These results are given in AppendixA, and also stand alone. I owe a great debt of gratitude in this work to the exceptional supervision of John Earman and John D. Norton. Their feedback was wonderfully detailed, and their encouragement invaluable. I had the particular pleasure of spending the 2011-2012 academic year working in the Center for Philosophy of Science with John Norton. Those who have visited the Center during his tenure as director will know the pleasure of John wandering into the room grinning, working excitedly through a problem full of colorful diagrams, and then bouncing out with some offhand wise-crack. I wouldn't trade those experiences for the world, and I count myself a better philosopher because of them. I owe more thanks to many other philosophers. Harvey Brown, Craig Callender, and David Malament have my special thanks for very kindly hosting me during my visits, and for many lovely and helpful discussions about this work. For their helpful comments at various stages I also thank my committee members Robert Batterman, Laura Ruetsche and Giovanni Valente, as well as Peter Distelzweig, Bal´azsGyenis, James Ladyman, Christoph Lehner, Wayne Myrvold, Michael Tamir and David Wallace. Thanks also to my parents, for their wonderful support. Above all, I would like to thank my wife, Alma, for her unending support and encouragement during this journey. This would not have been possible without her. I received generous financial aid during the writing of this dissertation from the University viii of Pittsburgh, the Andrew Mellon Foundation, and the Center for Philosophy of Science. The dissertation was greatly improved thanks to a Doctoral Dissertation Research Improvement Grant (#1058902) from the National Science Foundation, which allowed me to undertake a research stay at the University of California, Irvine and the University of California, San Diego during the Winter of 2011. This work also benefited from the generosity of the Wesley Salmon Foundation, which allowed me to present many of these results at the University of Oxford and the University of Bristol in the Spring of 2011. ix Alice sighed wearily. `I think you might do something better with the time,' she said, `than waste it in asking riddles that have no answers.' `If you knew Time as well as I do,' said the Hatter, `you wouldn't talk about wasting it. It's him.' `I don't know what you mean,' said Alice. `Of course you don't!' the Hatter said, tossing his head con- temptuously. `I dare say you never even spoke to Time!' Lewis Carroll, Alice's Adventures in Wonderland x 1.0 INTRODUCTION Philosophy and physics are not, historically speaking, two separated islands of thought. New- ton's Principles of Natural Philosophy was a groundbreaking work of both physics and philos- ophy. Einstein's philosophical ideas were inextricably entwined with his physics. Today, the disciplines of physics and philosophy are often separated in distinct academic departments. However, there remain problems of common interest for which the border between the two is blurred. These problems are often deep, calling for methodologies