THE UNIV£RSITY OF WESTERN ONTARIO SERIES IN PHILOSOPHY OF SCIENCE A SERIES OF BOOKS ON PHILOSOPHY OF SCIENCE, METHODOLOGY, AND EPISTEMOLOGY PUBLISHED IN CONNECTION WITH THE UNIVERSITY OF WESTERN ONT ARlO PHILOSOPHY OF SCIENCE PROGRAMME Managing Editor J. LEACH Editorial Board J. PEARCE VOLUME 3 www.com JEFFREY BUB University of Western Ontario, Ontario, Canada, and Institute for the History and Philosophy of Science, Tel Aviv University, Israel THE INTERPRETATION OF QUANTUM MECHANIC '. REIDEL PUBLISHING COMPANY DORDRECHT- HOLLAND / BOSTON- U.com Library of Congress Catalog Card Number 74-76479 Cloth edition: ISBN 90 277 0465 1 Paperback edition: ISBN 90 277 0466 X Published by D.
Reidel Publishing Company, P. Box 17, Dordrecht, Holland Sold and distributed in the U., Canada, and Mexico by D. Reidel Publishing Company, Inc. 306 Dartmouth Street, Boston, Mass.
All Rights Reserved Copyright © 1974 by D. Reidel Publishing Company, Dordrecht, Holland No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher Printed in The Netherlands by D. Reidel, Dordrecht www.com TABLE OF CONTENTS PREFACE VII I. The Statistical Algorithm of Quantum Mechanics 1 I.
Early Formulations 3 III. Hilbert Space 8 IV. The Statistical Algorithm 15 V. Generalization of the Statistical Algorithm 24 VI.
The Problem of Completeness 32 I. The Classical Theory of Probability and Quantum Mechanics 32 II. Uncertainty and Complementarity 36 III. Hidden Variables 46 III.
Von Neumann's Completeness Proof 49 IV. Lattice Theory: The Jauch and Piron Proof 55 V. The Imbedding Theorem of Kochen and Specker 65 VI. The Bell-Wigner Locality Argument 72 VII.
Resolution of the Completeness Problem 84 VIII. The Logic of Events 92 I. Classical Logic 93 III. Imbeddability and Validity 108 X.
The Statistics of Non-Boolean Event Structures 119 XI. The Measurement Problem 128 XII. The Interpretation of Quantum Mechanics 142 BIBLIOGRAPHY 151 INDEX OF SUBJECTS 153 www.com PREFACE This book is a contribution to a problem in foundational studies, the problem of the interpretation of quantum mechanics, in the sense of the theoretical significance of the transition from classical to quantum mechanics. The obvious difference between classical and quantum mechanics is that quantum mechanics is statistical and classical mechanics isn't.
Moreover, the statistical character of the quantum theory appears to be irreducible: unlike classical statistical mechanics, the probabilities are not generated by measures on a probability space, i. by distributions over atomic events or classical states. But how can a theory of mechanics be statistical and complete? Answers to this question which originate with the Copenhagen inter- pretation of Bohr and Heisenberg appeal to the limited possibilities of measurement at the microlevel. To put it crudely: Those little electrons, protons, mesons, etc., are so tiny, and our fingers so clumsy, that when- ever we poke an elementary particle to see which way it will jump, we disturb the system radically - so radically, in fact, that a considerable amount of information derived from previous measurements is no longer applicable to the system.
We might replace our fingers by finer probes, but the finest possible probes are the elementary particles them- selves, and it is argued that the difficulty really arises for these. Heisen- berg's y-ray microscope, a thought experiment for measuring the posi- tion and momentum of an electron by a scattered photon, is designed to show a reciprocal relationship between information inferrable from the experiment concerning the position of the electron and information concerning the momentum of the electron. Because of this necessary information loss on measurement, it is suggested that we need a new kind of mechanics for the microlevel, a mechanics dealing with the disposi- tions for microsystems to be disturbed in certain ways in situations defined by macroscopic measuring instruments. A God's-eye view is rejected as an operationally meaningless abstraction.com VIII THE INTERPRETATION OF QUANTUM MECHANICS Now, it is not at all clear that the statistical relations of quantum mechanics characterize a theory of this sort.
After all, the genesis of quantum mechanics had nothing whatsoever to do with a measurelnent problem at the microlevel, but rather with purely theoretical problems concerning the inadequacy of classical mechanics for the account of radiation phenomena. Bohm and others have proposed that the quantum theory is incomplete, in the sense that the statistical states of the theory represent probability distributions over 'hidden' variables. Historically, then, the controversy concerning the completeness of quantum mechan- ics has taken this form: A majority view for completeness, understood in the sense of the disturbance theory of measurement, and a minority view for incompleteness. An interpretation of quantum mechanics should show in what funda- mental respects the theory is related to preceding theories.
I propose that quantum mechanics is to be understood as a 'principle' theory, in Einstein's sense of the term. The distinction here is between principle theories, which introduce abstract structural constraints that events are held to satisfy (e. classical thermodynamics), and constructive theories, which aim to reduce a wide class of diverse systems to component systenls of a particular kind (e. the molecular hypothesis of the kinetic theory of gases).
For Einstein, the special and general theories of relativity are principle theories of space-time structure. I see quantum mechanics as a principle theory of logical structure: the type of structural constraint introduced concerns the way in which the properties of a mechanical system can hang together. The propositional structure of a system is represented by the algebra of idempotent magni- tudes - characteristic functions on the phase space of the system in the case of classical mechanics, projection operators on the Hilbert space of the system in the case of quantum ·mechanics. Thus, the propositional structure of a classical mechanical system is isomorphic to the Boolean algebra of subsets of the phase space of the system, while the logical structure of a quantum mechanical system is represented by the partial Boolean algebra of subspaces of a Hilbert space.
In general, this is a non- Boolean algebra that is not imbeddable in a Boolean algebra. As principle theories, classical mechanics and quantum mechanics specify different kinds of constraints on the possible events open to a physical system, i. they define different possibility structures of events.com PREFACE IX This view arises naturally from the Kochen and Specker theory of partial Boolean algebras, which resolves the completeness problem by properly characterizing the category of algebraic structures underlying the statistical relations of the theory. Kochen and Specker show that it is not in general possible to represent the statistical states of a quantum mechanical system as measures on a classical probability space, in such a way that the algebraic structure of the magnitudes of the system is pre- served.
Of course, the statistical states of a quantum mechanical system can be represented by measures on a classical probability space if the algebraic structure of the magnitudes is not preserved. But such a re- presentation has no theoretical interest in itself in this context. The variety of hidden variable theories which have been proposed all involve some such representation, and are interesting only insofar as they intro- duce new ideas relevant to current theoretical problems. Invariably, the reasons proposed for considering a new algebraic structure of a specific kind are plausibility arguments derived from some metaphysical view of the universe, or arguments which confuse the construction of a hidden variable theory of this sort with a solution to the completeness problem.
I reject the Copenhagen disturbance theory of measurement and the hidden variable approach, because they misconstrue the foundational problem of interpretation by introducing extraneous considerations which are completely unmotivated theoretically, and because they stem from an inadequate theory of logical structure. With the solution of the completeness problem, all problems in the way of a realist interpretation of quantum mechanics disappear, and the measurement problem is exposed as a pseudo-problem. The short bibliography lists only works directly cited, and since the sources of the ideas discussed will be obvious throughout, I have not thought it necessary to introduce explicit references in the text, except in the case of quotations.com CHAPTER I THE STATISTICAL ALGORITHM OF QUANTUM MECHANICS I. REMARKS Classical mechanics - Newton's general theory of motion developed and articulated by Euler, Lagrange, and Hamilton - describes the temporal evolution of a mechanical system in terms of the change in certain ap- propriate physical magnitudes (e.
energy, angular momentum, etc.), which are represented as real-valued functions on a 'phase space' X, a linear (vector) space parametrized by generalized position and mo- mentum coordinates, the phase variables. For a free particle, the phase space is 6-dimensional, with position coordinates ql, q 2' q3, representing the location of the particle in space, and corresponding momentum coordinates Pl,P2,P3. The classical mechanical equations of motion, Hamilton's equations: dqi oH dpi oH -=- -=-- (i = 1, 2, 3) dt 0Pi dt Oqi determine a trajectory in phase space, given the initial values of the variables ql, Q2' Q3; Pl,P2,P3· The quantity H, the Hamiltonian, is a function of the phase variables and characterizes the particular system involved. Since the physical magnitudes are functions of the phase variables, the values of these quantities are defined for every point on the phase trajectory of the system, and are determined for all time via Hamilton's equations by any point on the trajectory, i.
by an assign- ment of values to the 'canonically conjugate' sets of phase variables Ql, Q.2, Q3 andpl,P2,P3· Such an assignment of values - the specification of a point X=(Ql,Q2,Q3;Pl,P2,P3) in phase space - is a classical mechanical state. Electromagnetic phenomena were incorporated into this scheme by the Faraday-Maxwell theory of fields, a field being something like a mechan- ical system with a continuous infinity of phase variables. This extension of classical mechanics began to collapse towards the end of the 19th www.com 2 THE INTERPRETATION OF QUANTUM MECHANICS century. The quantum theory was conceived in 1900 with Max Planck's solution to the 'ultra-violet catastrophe': Planck proposed that electro- magnetic radiation is emitted and absorbed in discrete 'quanta', each energy quantum being proportional to the frequency of the radiation.
The birth of quantum mechanics followed after a gestation period of 25 years, in the dual form of Schrodinger's wave mechanics and the Heisenberg-Born-Jordan matrix mechanics. Schrodinger demonstrated the equivalence of the two theories, and a unified 'transformation theory' was developed by Dirac and Jordan on the basis of Born's probabilistic interpretation of the wave function. In Section II, I sketch the basic ideas behind matrix mechanics and wave mechanics, presenting these theories as different algorithms for generating the set of possible energy values of a system. I discuss von Neumann's critique of the Dirac-Jordan transformation theory, and show that matrix and wave mechanics are equivalent in the sense that they represent formulations of a mechanical theory in terms of different realizations of Hilbert space.
The exposition in this section follows von Neumann. My purpose is to show the origin of the Hilbert space for- mulation of quantum mechanics. I develop the geometry of Hilbert space in Section III. I introduce the notion of a Hilbert space as a vector space over the field of complex numbers, with a scalar product which defines the metric in the space.
The core of this chapter is Section IV. Quantum mechanics incorporates an algorithm for assigning probabilities to ranges of values of the physical magnitudes. I introduce this algorithm in the elementary form applicable to the finite-dimensional case. Essentially, probabilities are generated by statistical states according to a certain rule.
The 'pure' statistical states are represented by the unit vectors in Hilbert space; the physical magni- tudes are represented by operators· associated with orthogonal sets of unit vectors, corresponding to the possible 'quantized' values of the magnitudes. These orthogonal sets of unit vectors function like Cartesian coordinate systems in a Euclidean space.