The Quantum Mechanics of Minds and Worlds JEFFREY ALAN BARRETT OXFORD UNIVERSITY PRESS www.com THE QUANTUM MECHANICS OF MINDS AND WORLDS www.com TItis book has been printed digitally and produced in a standard specification in order to ensure its continuing availability OXFORD UNIVERSITY PRESS Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Jeffrey A. Barrett 1999 The moral rights of the author have been asserted Database right Oxford University Press (maker) Reprinted 2003 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization.
Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer ISBN 0-19-924743-9 www.com For Martha, Thomas, and Jacob www.com Alexander wept when he heard from Anaxarchus that there was an infinite number of worlds; and his friends asking him if any accident had befallen him, he returns this answer: 'Do you not think it a matter of lamentation that when there is such a vast multitude of them, we have not yet conquered one?' (Plutarch, On the Tranquillity ofMind) www.com PREFACE T HIS book is about the quantum measurement problem, Hugh Everett Ill's proposed resolution, and some of the attempts to understand how it was supposed to work. While there is a brief review of the standard for- mulation of quantum mechanics (complete with a description of a two-slit experiment!) and a short appendix describing the Hilbert-space formal- ism, it is assumed that the reader already knows something about how quantum mechanics works and is comfortable with at least some of the mathematics. There is, in my opinion, no better introduction to the ways of quantum mechanics than David Albert's book Quantum Mechanics and Experience. One might also want to work through a careful presentation of the theory that includes a more detailed description of the mathemat- ical formalism.
Dirac's Principles of Quantum Mechanics is the classic introductory text (and I use Dirac's notation throughout this book). My favourite advanced introduction is Gordon Baym's Lectures on Quantum Mechanics. John Wheeler and W. Zurek's Quantum Theory and Measurement is the standard anthology on the measurement problem.
I have tried to refer to page numbers in this anthology whenever possible. Many conversations with friends and colleagues contributed to this book; in particular, I should like to thank Wayne Aitken, Frank Arntzenius, Guido Bacciagallupi, Jeffrey Bub, Rob Clifton, Michael Dickson, Richard Healey, Meir Hemmo, Peter Lewis, Barry Loewer, Pen Maddy, Brad Monton, Laura Reutsche, Simon Saunders, and Brian Skyrms. I am espe- cially indebted to David Albert for many enlightening discussions over the past several years-anyone familiar with Albert's work will immediately recognize his influence on the way that I think about quantum mechanics. I should also like to thank the anonymous referees who read this book in manuscript form-their comments were invaluable in putting together the final version.
Finally, I should like to thank Ryan Barrett, whose excellent work produced the final figures. It was a pleasure working with the Oxford University Press editors-they were careful, smart, and patient. This book was supported by a University of California President's Research Fellowship, and most of it was written while I was a Visit- ing Fellow at the University of Pittsburgh Center for the History and Phil- osophy of Science in 1996-7. I should like to thank both universities for their kind support.com CONTENTS List offigures xiv 1 A brief introduction 1 1.1 A textbook example: The two-slit interference experiment 2 1.2 Another textbook example: Spin properties of spin-! systems 8 1.3 A more exotic example: The curious behaviour of neutral K mesons 11 1.4 The measurement problem 14 2 The standard formulation of quantum mechanics 18 2.1 The foundations of a new theory 18 2.2 The collapse of the wave function 22 2.3 Von Neumann's formulation of quantum mechanics 30 2.1 The standard theory 31 2.2 A summary of the theory 37 2.4 How the theory works 38 2.6 Von Neumann's psychophysical parallelism and Wigner's friend 47 2.7 The measurement problem (again) 54 3 The theory of the universal wave function 56 3.1 What's wrong with von Neumann's theory? 56 3.2 Other formulations ofquantum mechanics and their problems 59 3.4 The fundamental relativity of states 66 3.5 The appearance of phenomena 70 3.6 The deduction of subjective appearances 75 3.7 The mechanics of macroscopic systems 83 3.8 What are branches? 86 3.9 Interpreting Everett 90 4 The bare theory and determinate experience 92 4.1 The bare theory 94 4.2 The suggestive properties 95 4.4 The account of experience 110 www.com Xll Contents 4.5 Problems with the bare theory 114 4.3 No account of statistical results 117 4.4 No general account of determinate results 119 5 Selecting a branch 121 5.1 Bohm's theory without the trajectories 122 5.3 Surreal trajectories and the persistance of memory 132 5.4 The failure of covariance 140 5.5 Position as the preferred physical property 144 5.6 The limiting properties in the context of Bohm's theory 146 6 Many worlds 149 6.1 The splitting-world interpretation 149 6.2 Traditional and real problems 154 6.1 Too many worlds? 154 6.2 The feeling of splitting 158 6.3 Compatibility with other physical theories 159 6.4 What it takes to be a world 160 6.7 The preferred-basis problem 173 6.3 Many worlds without splitting 179 7 Many minds 185 7.1 How many minds? 185 7.1 The single-mind theory 186 7.2 The many-minds theory 192 7.2 The auxillary dynamics 197 7.1 The transcendental approach 198 7.2 The Bohm-Bell-Vink dynamics 202 7.5 Correlations without correlata 217 8 Many histories 221 8.1 Interference effects and the environment 222 8.2 The sense in which it is difficult to distinguish pure states from mixtures 224 www.com Contents xiii 8.3 Decoherence and determinate perceptions 227 8.4 Gell-Mann and Hartle's many-histories approach 232 8.6 Does the environment select the right determinate quantity? 242 9 The determinate-experience problem 245 Appendices 249 A The Hilbert-space formalism 249 B A concrete example of an EPR experiment in the 252 context of the Bare theory References 255 Index 263 www.com LIST OF FIGURES 1.1 Two-slit setup.2 A open and B open.3 What should happen and what does happen.
(a) The A- or B- distribution. (b) The interference distribution.4 Two-path setup.5 Probability of observing iO at t if the particle is iO at to.1 Einstein's Solvay experiment.2 The x-spin setup.4 Wigner's-friend setup.1 A relatively tame Eiffel-Tower trajectory on Bell's Everett (?) theory.2 How the probability current moves a single particle.3(a) A simple recording in Bohm's theory. M's wave packet moves to down if and only if P travels the -} x -path.3(b) The two-particle configuration in a two-particle wave packet.4 Crossing-paths setup.5 Crossing paths with position record.6 The same in configuration space. (a) First half of the experiment.
(b) Second half of the experiment.7 We try to record P's position in M's x-spin.8 The same in configuration space.9 An EPR experiment in Bohm's theory.10 Two EPR experiments in configuration space. (a) A meas- ures first.11 A difference between Bohm's theory and the bare theory.1 Global states and local states.2 How a connection rule might thread possible local states into worlds.3(a) A repeated x-spin measurement in a splitting-worlds theory.com List offigures xv 6.3(b) A repeated x-spin measurement in a many-threads theory.1 A randomly jumping mind.2 Talking to mindless hulks.3 A mental dynamics without mindless hulks.1 How environmental correlations destroy simple interfer- ence effects. (a) The interference distribution. (b) With a wire loop at A.com 1 A BRIEF INTRODUCTION THE standard theory of quantum mechanics, as formulated by P.
Dirac and John von Neumann, is in one sense the most successful physical theory ever-no other theory has ever made such accurate empirical pre- dictions. It is all the more impressive because what it successfully predicts, the behaviour of the basic constituents of all physical things (electrons, protons, neutrons, photons, etc.), is often wildly counter-intuitive. There is, however, a problem. If one tries to understand the standard formulation of quantum mechanics as providing a complete and accurate framework for the description of all physical interactions, then it soon becomes evi- dent that the theory is at least ambiguous, and, on a less charitable reading, one might even conclude that it is logically inconsistent.
This is known as the quantum measurement problem. Hugh Everett Ill's formulation of quantum mechanics and the various reconstructions of his theory that have appeared since are all attempts to solve the measurement problem. But before considering possible solutions to the measurement problem, it is important to be clear about exactly what the problem is. The basic constituents of matter, when left to themselves, behave in a way that is apparently nothing like the behaviour of the middle-sized objects (chairs, coins, cars, cats, etc.) that form the bulk of our experience and the basis for our physical intuitions.
Because their behaviour is so counter-intuitive, any empirically adequate theory, any theory that makes the right empirical predictions for the experiments that we have performed so far, is bound to be itself counter-intuitive. The standard theory of quan- tum mechanics is certainly counter-intuitive. But that is not the problem. Rather, the problem is that the standard theory cannot be taken to provide a complete and accurate physical description of the odd behaviour that it is supposed to describe.
Our most careful observations suggest that the basic constituents of matter behave in a fundamentally random way. They also suggest that the basic constituents of matter sometimes behave like particles and sometimes like waves. The particle-like behaviour of fundamental par- ticles is seen in such phenomena as cloud-chamber tracks and marks www.com 2 A briefintroduction on photographic film. This particle-like behaviour agrees well with the physical intuitions we have developed from our experience with middle- sized physical systems.
The wave-like behaviour offundamental particles (and of other simple, well-isolated physical systems) is seen in interfer- ence phenomena. This wave-like behaviour of matter is very different from what one would expect from middle-sized physical systems-one might, for example, expect a particle (or any other physical object) to have a determinate position and to follow a determinate trajectory, not to spread out like a wave on a pond. While one might lament the loss of classical determinism, it is the dual behaviour of matter that is really puzzling. Particles (and other simple, well-isolated systems) seem to behave one way when no one is look- ing (the odd quantum wave-like way) and another way when someone is.
This dual behaviour is represented in the standard formulation of quantum mechanics by two dynamical laws: one law describes the evolution of a physical system when no one is looking and the other describes the evolu- tion of the system when someone does. These two dynamical laws and the criterion for when each obtains is the ultimate source of the measurement problem in the standard theory. There are two stock examples of quantum interference effects that we will return to in various forms throughout the book. One ofthese is the two- slit experiment and the other is Wigner's Stem-Gerlach experiment.
Both are discussed below, followed by a somewhat more exotic example.