The Modal Interpretation of Quantum Mechanics

379 Pages·1998·13.808 MB·English

by  Dennis Dieks, Pieter E. Vermaas (auth.), Dennis Dieks, Pieter E. Vermaas (eds.)

THE MODAL INTERPRETATION OF QUANTUM MECHANICS
THE WESTERN ONTARIO SERIES
IN PHILOSOPHY OF SCIENCE
A SERIES OF BOOKS
IN PHILOSOPHY OF SCIENCE, METHODOLOGY, EPISTEMOLOGY,
LOGIC, HISTORY OF SCIENCE, AND RELATED FIELDS
Managing Editor
WILLIAM DEMOPOULOS
Department 0/ Philosophy, University o/Western Ontario, Canada
Managing Editor 1980-1997
ROBERT E. BUTTS
Late, Department 0/ Philosophy, University o/Western Ontario, Canada
Editorial Board
JOHN L. BELL, University o/Western Ontario
JEFFREY BUB, University 0/ Maryland
ROBERT CLIFTON, University 0/ Pittsburgh
ROBERT DiSALLE, University o/Western Ontario
MICHAEL FRIEDMAN, Indiana University
WILLIAM HARPER, University o/Western Ontario
CLIFFORD A. HOOKER, University o/Newcastle
KEITH HUMPHREY, University o/Western Ontario
AUSONIO MARRAS, University o/Western Ontario
JURGEN MITTELSTRASS, Universitiit Konstanz
JOHN M. NICHOLAS, University o/Western Ontario
ITAMAR PITOWSKY, Hebrew University
GRAHAM SOLOMON, Wilfrid Laurier University
VOLUME 60
THEMODAL
INTERPRETATION
OF QUANTUM
MECHANICS
Edited by
DENNIS DIEKS and PIETER E. VERMAAS
Utrecht University, The Netherlands
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6135-3 ISBN 978-94-011-5084-2 (eBook)
DOI 10.1007/978-94-011-5084-2
Printed on acid-free paper
AlI Rights Reserved
© 1998 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1998
Softcover reprint of the hardcover 1s t edition 1998
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permis sion from the copyright owner.
TABLE OF CONTENTS
DENNIS DIEKS AND PIETER E. VERMAAS / Preface vii
DENNIS DIEKS AND PIETER E. VERMAAS / Introduction 1
MICHAEL DICKSON AND ROB CLIFTON / Lorentz-Invariance in
Modal Interpretations 9
DENNIS DIEKS / Locality and Lorentz-Covariance in the Modal
Interpretation of Quantum Mechanics 49
JASON ZIMBA AND ROB CLIFTON / Valuations on Functionally
Closed Sets of Quantum Mechanical Observables and Von
Neumann's 'No-Hidden-Variables' Theorem 69
PIETER E. VERMAAS / The Pros and Cons of the Kochen-Dieks
and the Atomic Modal Interpretation 103
NICK REEDER / Projection Operators, Properties, and Idempotent
Variables in the Modal Interpretations 149
GUIDO BACCIAGALUPPI / Bohm-Bell Dynamics in the Modal
Interpretation 177
MATTHEW J. DONALD / Discontinuity and Continuity of Definite
Properties in the Modal Interpretation 213
LAURA RUETSCHE / How Close is "Close Enough" 223
JEFFREY BUB / Decoherence in Bohmian Modal Interpretations 241
MEIR HEMMO / Quantum Histories in the Modal Interpretation 253
PAUL BUSCH / Remarks on Unsharp Quantum Observables,
Objectification, and Modal Interpretations 279
HARVEY BROWN, MAURICIO SUAREZ AND GUIDO BACCIAGALUPPI /
Are 'Sharp Values' of Observables Always Objective Elements
of Reality? 289
BRADLEY MONTON / Quantum-Mechanical Self-Measurement 307
JEFFREY A. BARRETT / The Bare Theory and How to Fix It 319
FRANK ARNTZENIUS / Curiouser and Curiouser: A Personal
Evaluation of Modal Interpretations 337
PREFACE
More than seventy years after the inception of quantum mechanics people are
still complaining that the subject has not been satisfactorily interpreted. Al
though the complaint is justified, there is no doubt that considerable progress
has been made. In fact, a consensus seems to be developing among researchers
in the field that an adequate interpretation of the quantum mechanical for
malism should and can be given, provided it is stripped of the notorious pro
jection postulate and provided measurements are treated as ordinary physical
interactions, i.e. no special role is assigned to the concept of measurement.
Modal interpretations of quantum mechanics form a fairly broad class of
interpretations of the kind just mentioned. They have been developed during
the nineteen eighties and nineties, and therefore are not yet as well known as
some of their older competitors. But interest is growing, and from 12 to 14
June 1996 the first international conference specifically devoted to the modal
interpretation of quantum mechanics took place at Utrecht University. The
present volume consists of the papers presented at that conference, plus an
introduction and a critical appraisal.
The material collected here explains what gave rise to the modal interpre
tations and what problems they solved (in particular the infamous measure
ment problem). In addition, several papers focus on problems that still have
to be solved. The volume thus not only introduces the modal approach and
provides an overview of established results, but it also takes the reader to the
forefront of current research in the subject. The collection will therefore be
very useful to anyone interested in the interpretation of quantum mechanics.
Students and researchers in the foundations and philosophy of science and in
theoretical physics will benefit particularly from this book in which the con
ceptual and foundational relevance of technical considerations receives full
attention.
We wish to thank Pieter Kok, Hanneke Pasveer and Sheila McNab for
their expert help in preparing the final form of the manuscript. Our thanks
also go to the speakers at the conference, and to Frank Arntzenius, for their
contributions.
Dennis Dieks and Pieter Vermaas, Utrecht University.
vii
DENNIS DIEKS AND PIETER E. VERMAAS
INTRODUCTION
1 MOTIVATION: THE INTERPRETATIONAL PROBLEMS
Since the very first beginnings of quantum mechanics there has been much
argument and debate about the interpretation of the theory. Three main
themes can be discerned in the discussions: the status of indeterminism, the
role of measurements, and locality. The three are not unrelated.
Born was the first to propose that the wavefunction should be used to
calculate probabilities. Initially, Born thought that these probabilities were
completely analogous to those occurring in classical statistical mechanics:
fractions in an ensemble of physical systems, e.g. particles, in which each
individual system is characterized by values of the usual classical magnitudes
(positions, velocities, etc.). The probability 11,b(xWdx, for example, should in
that case be seen as the fraction of particles with positions in a neighborhood
dx around x. Soon, however, a classical interpretation turned out to be very
problematical. Interference phenomena show that the wave aspect of the
theory goes much deeper than Born originally supposed: one cannot simply
assume that 1,b corresponds to an ensemble consisting of tiny corpuscles with
different positions and velocities.
But if the idea that 1,b represents such an ensemble of different systems
is rejected, what is the meaning of the probabilities? If 11,b12 does not refer
to the fraction of systems with properties in a certain range, to what does it
refer? The answer that has become standard is that quantum probabilities
are probabilities of measurement outcomes. According to this standard view
quantum mechanics is not about the properties of physical systems, but is
about the relative frequencies of results in repeated measurements. This
idea was worked out in the Copenhagen interpretation according to which
it is impossible to attribute definite properties to quantum systems outside
a measurement context. Terms like position and momentum can be used to
describe only the whole of the quantum object and the macroscopic measuring
device.
Although it becomes possible in this way to give a clear meaning to the
probabilities in the theory - they are fractions in ensembles of possible mea
surement outcomes - new interpretational problems inevitably arise. Mea
surement has now become a central concept in the interpretation. But what
makes measurements so special that they become the sole basis for the at
tribution of definite properties? Is it not true that a measurement is just
a physical interaction, namely one in which a correlation is established be
tween an object system and a measuring device? The latter intuition cannot
D. Dieks fj P.E. Vermaas (eds.). The Modal Interpretation of Quantum Mechanics, 1·7.
© 1998, Kluwer Academic publishers.
2 DENNIS DIEKS AND PIETER E. VERMA AS
be accommodated in a simple way by the standard view, for it treats in
teractions between quantum systems in a fundamentally different way from
measurements. Von Neumann gave the difference a firm place in the mathe
matical formalism by distinguishing between two kinds of evolution: in mea
surements "collapses of the wavefunction" take place, leading to eigenstates
of the observables that become definite, whereas in ordinary physical inter
actions the total wavefunction of the interacting systems evolves unitarily.
It is sometimes said that the difference reflects the distinction between mi
croscopic interactions on the one hand and interactions with a macroscopic
measuring device on the other. But this only adds to the problems: in phys
ical theory or in our conceptions about the constitution of matter there is
nothing to suggest a clear-cut distinction between microscopic and macro
scopic. Such a well-defined and sharp transition is needed, however, if the
microscopic/macroscopic dichotomy is to give a physical explanation of the
occurrence of collapses.
If the collapse of the wavefunction is taken seriously, one of the conse
quences is that measurements can have non-local effects. In situations of the
Einstein-Podolsky-Rosen-type, in which two quantum systems are described
by an entangled state, a measurement made on one system will steer the
other system into the state that is correlated to the collapsed state of the
first system (the "relative state"). In other words, the collapse is not con
fined to the system on which the measurement is directly performed, but also
takes place in correlated systems, completely independently of the distance
between the systems. According to Schrodinger this kind of non-locality is
the characteristic feature of quantum mechanics.
The aforementioned non-locality in the formalism of quantum mechanics
would disappear if the collapse of the wavefunction (the projection postulate)
could be avoided. But at first sight it seems very difficult to do without the
collapses. The reason is that in a good measurement some result will cer
tainly be found; and after a result has materialized, it seems obvious that its
probability has become 1. The state should therefore become the associated
eigenstate, for this is the only state that makes the found result certain. In
addition, a repetition of the measurement will, in ideal circumstances, repro
duce the same result (it is assumed that no disturbance takes place between
the measurements and that the quantum system survives the first measure
ment unharmed). That is, in an ideal measurement the probability becomes
1 that one specific result will be found again and again in subsequent mea
surements. The probability interpretation of the wavefunction again appears
to force us to assume that the first measurement induces a transition to an
eigenstate of the measured observable, corresponding to the result that was
found.
The latter argument for the occurrence of collapses is unconvincing, how
ever. It is well-known from the literature on the so-called many-worlds inter
pretation of quantum mechanics that the agreement between the results of
INTRODUCTION 3
consecutive measurements can be accounted for without collapses. The crux
is that the uncollapsed total wavefunction of objects and measuring devices
contains information about the correlations between measurement outcomes.
Without collapses quantum mechanics is still able to predict that the con
ditional probability that an ideal measurement will yield a certain result is
1, given that an immediately preceding ideal measurement yielded the same
result.
2 THE MODAL SOLUTION
Considerations of the kind just mentioned prompted Bas van Fraassen !1,2,3}
in the nineteen-seventies to join his colleague at Yale University Henry Marge
nau in the attempt to get rid of non-locality via the rejection of the projection
postulate. Van Fraassen went further than Margenau, however, and proposed
a number of new ideas that cast new light on the role of indeterminism and
that offered prospects of completely obliterating the difference between mea
surements and physical interactions.
A crucial point in van Fraassen's approach is the distinction he made be
tween the quantum state t/J and what he called the value state. The quantum
state is the usual quantum mechanical wavcfunction (vector in Hilbert space),
whose time evolution is governed by thc SchrOdinger equation. But this state
is no longer seen as a mere instrument for making predictions about measure
ment results: it is given an interpretation in terms of "states of affairs" , facts
about the world. The proposed interpretation thus makes a step in the direc
tion of scientific realism, in the sense of "telling a story about how the world
is according to quantum mechanics" (Van Fraassen himself did not endorse
scientific realism but combined the interpretation with his "constructive em
piricism" according to which one can remain agnostic about the truth of the
unobservable part of the account a theory gives of the world - the existence of
such an account is indispensable, however, also for constructive empiricists).
The most important new idea concerns the exact relation between the quan
tum state and what actually is the case. This relation is assumed to contain
an element of modality, in the following sense: the quantum state tells us
what may be the case, i.e. which physical properties systems may possess.
At this point it should be becoming clear why there can be a role for an
additional state, the value state: the quantum state only specifics what pos
sibilities there arc. The value state is introduced to represent what actually
is the case. For example, if a particle has a definite energy, the value state is
the energy eigenstate belonging to the appropriate eigenvalue; the quantum
state will generally not be an energy eigenstate, because it just is the state
that follows, by SchrOdinger evolution, from the earlier state or the particle
and the other systems with which it interacts. The decisive assumption is
that quantum mechanics, as a dynamical theory, applies only to the quan-
4 DENNIS DIEKS AND PIETER E. VERMAAS
tum state. The quantum state satisfies the dynamical evolution equation and
therefore always evolves unitarily, without collapses.
Because the quantum state is not considered to reflect what actually oc
curs, the need for the projection postulate disappears. Measurements can
be treated as ordinary physical interactions, represented in the Hamiltonian
that governs the unitary evolution. To see more clearly how this works, the
reader should reflect on how the quantum state evolves in a measurement
evolution according to the von Neumann measurement scheme. All possible
outcomes of the measurement will be represented in the final quantum state,
by terms in the total superposition. Of course, only one of these outcomes
will in fact be realized. But because it is the task of the value state, and
not of the quantum state, to represent the actual outcome, it cannot be a
ground for complaint against the quantum state that it contains all possibil
ities. The quantum state tells us what could be (have been) the case, also
after the measurement. Obviously, the relation between quantum state and
value state must be probabilistic.
Van Fraassen dubbed his proposal the modal interpretation of quantum
mechanics. The idea behind this term is that the quantum state 'I/J is the
basis for modal statements, i.e. statements about what possibly or necessarily
is the case.
Of course, something more has to be said about exactly which value states
are possible, given a quantum state. Here van Fraassen proposed that all value
states in the range of a system's density operator W (obtained from the total
'I/J by partial tracing) are possible; and he further stipulated that at the end of
measurement interactions, characterized by specific interaction Hamiltonians,
the probabilities of the various possible value states are given by the Born
probability distribution.
Initially, there was little response to van Fraassen's proposals. Neverthe
less, it seems that the time was ripe for ideas of this kind: in the eighties
several authors published ideas that in retrospect can be regarded as elabo
rations of or variations on van Fraassen's modal themes. Kochen, Healey and
Dieks put forward modal interpretations in which the biorthogonal (Schmidt)
decomposition of the quantum state of a system plus its environment fixes
which physical magnitudes are definite-valued. In terms of density operators
W, the idea is that the magnitudes that possess definite values are the ones
represented by observables which have exactly the same projectors in their
spectral decomposition as W itself. Later Bub put forward a rival scheme, in
which one privileged observable is singled out as definite-valued a priori, and
in which the maximal algebra of other observables is determined that, given
the quantum state, can also be definite without the problem of Kochen and
Specker contradictions.
These proposals have given rise to a considerable amount of research.
Vermaas and Dieks extended the biorthogonal decomposition version of the
modal interpretation so that it became able to specify correlations between