A Survey of Hidden-Variables Theories

372 Pages·1973·35.994 MB·English

by  F. J. Belinfante and D. Ter Haar (Auth.)

A SURVEY OF
HIDDEN-VARIABLES
THEORIES
BY
F. J. BELINFANTE
Department of Physics,
Purdue University, Lafayette,
Indiana, U.S.A.
PERGAMON PRESS
OXFORD . NEW YORK . TORONTO
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First edition 1973
Library of Congress Cataloging in Publication Data
Belinfante, Frederik Jozef·
A survey of hidden-variable theories·
(International series of monographs on natural
philosophy, v. 55)
Includes bibliographical references.
1. Mathematical physics. 2. Variables
(Mathematics) 3. Quantum theory. I. Title.
QC20.B42 1973 530.1'5 72-13519
ISBN 0-08-017032-3
Printed in Hungary
PREFACE
THIS work was started in 1970 with the idea of writing three papers for publication in
ajournai of the type of American Journal of Physics, explaining the ideas, merits, and failures
of hidden-variables theories in a way understandable to those who have had in wave me­
chanics an introductory course only and who are not particularly mathematically minded.
As the work outgrew the size of journal articles, it is here presented as a book in three
parts.
The author's interest was triggered when in the Fall of 1969 Clauser, Home, Shimony,
and Holt published their plans of performing an experiment of which they claimed that it
would decide unambiguously whether "local" hidden-variables theory or whether quantum
theory is wrong, depending upon whether the measured value of a certain quantity Δ would
be found to be positive or negative. The author then found that the proof of Δ ^ 0 for this
kind of hidden-variables theories could be much simplified. (See Section 4.7 of Part III of
this book.)
While lecturing on this, he was made aware of the existence of a different kind of hidden-
variables theories (called in this book "theories of the first kind") which cannot be disproved
by the experiments proposed by Clauser et al. without simultaneously disproving quantum
theory, as these theories and quantum theory would make identical predictions. On the
other hand, the literature was full of claims that hidden-variables theories would not be
possible at all. The kind of hidden-variables theories that were shown to be impossible was,
of course, somewhat different from the "first" kind mentioned above, as well as from the
"second" ("local") kind that Clauser et al. planned to prove or disprove experimentally.
The "impossible" kind of hidden-variables theories is called here the "zeroth" kind.
While the impossibility of theories of the zeroth kind can be shown without performing
new experiments, the question whether theories of the first or of the second kind agree or
disagree with the behavior of nature is a physical question, and can be anwered only by
experiments hunting for the effects in disagreement with quantum theory which these theories
predict. That also theories of the first kind may predict such effects was first considered
seriously by Papaliolios (1967).
While most of this book is a review (partially a summary, partially an elaboration) of
the work of others, it contains some novel points. Some of the so-called proofs of impossi­
bility of hidden-variables theories are based on results of Kochen and Specker and of
Gleason, which here, by sacrificing some mathematical rigor, are presented in Chapter 3
of Part I in a more understandable form, with proofs even simpler than those proposed
a few years ago by Bell.
XV
XVI PREFACE
In Chapter 2 of Part II, Bohm's 1952 hidden-variables theory (in which the particles'
positions are the hidden variables) is applied to a minimum-uncertainty wave packet for
clarifying the difference in interpretation of free-particle motion in this theory and in ordi­
nary wave mechanics. The application of this theory of Böhm to photons starts from a
reformulation of conventional quantum electrodynamics in which one avoids the use of
creation and annihilation operators, and one describes the photon field by applying the
principles of familiar wave mechanics to certain harmonic oscillators describing the field.
This formulation of pure quantum-theoretical electrodynamics has the advantage of giving
a clearer description of finite-size photon wave packets in x, j,z-space than the more usual
treatment of quantum electrodynamics gives. An account of this treatment is given in
Appendix H of Part II, and in Section 2.4-2.6 of Part III its connection is shown with
Akhiezer-Berestetskii's treatment of photon wavefunctions.
For those acquainted with Dirac's relativistic wave mechanics, it is shown in Section
2.15 of Part II why it is not well possible to generalize Bohm's 1952 theory to a relativistic
theory.
In Chapters 3 and 4 of Part II the intimate relation between the theories of the first kind
constructed by Wiener and Siegel on the one hand, and by Böhm and Bub on the other
hand, is shown. Therefore both theories predict the same deviations from quantum theory
that could in principle be observed experimentally if these theories would be correct.
Not only are the experiments of Papaliolios discussed, which failed to find evidence for the
existence of these effects, but in Chapter 5 of Part II some additional experiments of a similar
nature are proposed, and it is predicted what outcome such experiments should have if these
theories would be any good. This outcome would depend here upon the type of sto­
chastic chaos that would exist for the polarizations of light emitted by the light source.
The work of Gleason and of Kochen and Specker discussed in Part I is usually regarded
as proving that ambiguity enters into the application of the theories of Wiener-Siegel and
Bohm-Bub upon measurements of observables with degenerate eigenvalues. In Section 3.7
and Appendix S of Part II it is shown how a suggestion made in 1969 by Tutsch (see Sec­
tion 3.6 of Part II) can remove these ambiguities and then lead to a formula suggested
already in 1953 by Wiener and Siegel. However, in Appendix K of Part II it is shown how
application of this rule of Tutsch may sometimes lead to paradoxical results.
In Sections 4.3-4.4 of Part III the experiments performed by Freedman and Clauser and
by Holt are discussed. Their analysis in Sections 2.7-2.9 of Part III using photon wave-
functions is kept so simple that it will be understandable also to those for whom the fuller
discussion of the meaning of these photon wavefunctions in Sections 2.4-2.6 of Part III is
too mathematical. The latter discussion was inserted primarily for justifying the use of two-
photon waves with quantum number 7=1 for Holt's experiments by qualifying the claim
found in the literature that two-photon wavefunctions with 7=1 would not exist. This
claim is found not to be valid for the experiments of Holt, although it is valid for experiments
using photons from annihilation of electron pairs, as in the experiments of Bleuler and
Bradt, Wu and Shaknov, Langhoff, and of Kasday, Ullman, and Wu, discussed in Chapter 5
of Part III.
In Chapter 3 of Part III explicit examples of theories of the second kind are given, which
may serve as jnpclels for which tjie validity may be verified o fthe Cteyser-Horjie-Shimony
PREFACE Xvil
inequality Δ <Ξ 0. The derivation of — 1 <Ξ Δ ^ 0 for these theories given in Section 4.7
of Part III is more direct and therefore more easily understandable than the derivation given
by Clauser et al. themselves, who started from an inequality proved by Bell (see Section
4.9 of Part III).
In conclusion, Chapter 6 of Part III ends with an expression of opinion about the present
status of hidden-variables theory.
It is hoped that this book may acquaint many, in the first place, with the ingenious
attempts made by many trying to construct a (crypto)deterministic theory "explaining"
quantum theory or "correcting its shortcomings"; in the second place, with the possibility
of experimental verification or falsification of such theories ; and, finally, with the refusal
of nature, in experiments thus far performed, to give any indication of submitting itself to
the regulations which such theories would impose upon nature's otherwise in the small
unpredictable behavior. (See, however, footnote 25b on page 290.)
HTV z
ACKNOWLEDGEMENTS
I GRATEFULLY acknowledge conversations and correspondences with many workers in
the hidden-variables field which have helped me to clarify my understanding and often to
remove misunderstandings on various aspects of the theory. Among those who have been
especially helpful are: J. Bub, J. F. Clauser, R. A. Holt, M. A. Home, J. M. Jauch, L. R.
Kasday, C. Papaliolios, P. Pearle, A. Shimony, A. Siegel, J. H. Tutsch, and E. P. Wigner.
I also acknowledge financial support of this work by the National Science Foundation by
its grants GP-9381 and GP-29786.
2* xix
FOREWORD TO PART I
THE primary aim of this book in three parts is an attempt to make the literature on hidden
variables understandable to those who are confused by the original papers with their contro­
versies and often excessive use of mathematical methods unknown to the average reader
of physics papers. It is aimed at those who have an open mind for trying out novel ideas
but are also willing to discard any results that do not stand up against criticism based upon
physical rather than philosophical argument. Physicists call a theory satisfactory if (1) it
agrees with the experimental facts, (2) it is logically consistent, and (3) it is simple as com­
pared to other explanations. Our stress on these physical considerations about the theory
means that (in Parts II and III) we will pay attention to the question how these theories
may be tested experimentally. In fact, the author's interest in hidden-variables theories was
kindled only when recently he became aware of the possibility of such experimental tests.
On the other hand, we do not want to ignore the metaphysical implications of the theory.
In this Part I we review the motives which have led different people to developing different
types of hidden-variables theories. The quest for determinism led to theories of the first
kind; the quest for theories that look like causal theories when applied to spatially separated
systems that interacted in the past led to theories of the second kind. The latter contradict
quantum theory, and experiments are underway for verifying who is right: the quantum
theorist, or the hidden-variables theorist of the second kind.
Theories of the first kind are not that easily distinguishable experimentally from quantum
theory. The hidden variables, according to theories of the first kind, have a distribution
of values which tends irreversibly and rapidly to an equilibrium distribution, much like
a velocity distribution of molecules in a gas tends to a Maxwell distribution. Quantum theory
then follows when the equilibrium distribution of hidden variables is reached. It is, however,
possible to willfully and predictably perturb such an equilibrium distribution. When this is
done, deviations from quantum theory should occur according to theories of the first
kind. Preliminary experiments trying to detect such deviations have had negative results.
One might say that this seems to contradict the validity of hidden-variables theory of the
first kind. However, another possible explanation is that the perturbed hidden-variable
distribution relaxed to its equilibrium distribution before the experiment was able to meas­
ure the predicted deviations from quantum theory.
Various authors have tried to prove that hidden-variables theories would be impossible.
These attempts seem strange in view of the fact that several hidden-variables theories do
3
4 FOREWORD TO PART 1
exist. We investigate in this Part I the question what causes the apparent discrepancy.
It turns out that each of these authors defined hidden-variables theory by some set of postu­
lates which then later they proved to be self-contradictory. Such "theories" defined by a self-
contradictory set of postulates, we call theories of the zeroth kind. Their impossibility is
obvious.
It is then interesting to study which properties postulated by these authors made these
theories impossible. We then should be careful, in developing more realistic hidden-variables
theories (of the nonzeroth kind), to avoid the postulates that caused the troubles. In partic­
ular, we shall discuss in Part II how hidden-variables theories of the first kind circumvent
these postulates. Some methods for achieving this are indicated already in this Part I.
One of the major pitfalls to be avoided in constructing a hidden-variables theory lies in
what the hidden variables can or cannot predict about future measurements. The work of
Gleason shows that in general it is impossible to predict uniquely on the basis of any given
hidden variables, whether or not the measurement of a quantity A will lead to the result
A = A„ where A is an eigenvalue of A, or to predict in case of a reproducible measurement
9 n
whether or not the wavefunction ψ by the measurement will be reduced to the eigenfunction
φ of A . What hidden-variables theories do predict is, from a given complete orthonormal
η op
set of eigenfunctions {<£,} describing the possible outcomes of a reproducible measurement,
to which eigenfunction φ will ψ be reduced when that measurement is made. The difference
η
seems subtle, but lies in the fact that the choice of φ from the set {φ,} may be changed into
η
the choice of a différent final state of the measurements when the orthonormal set {0J in
Hilbert space is rotated around φ . Such a rotation is understood to correspond to a change
η
of the experimental setup of the measurement, but this change may be unperceivably small
when the rotation in Hilbert space is in first approximation merely a change of a basic set
of eigenfunctions for some degenerate eigenvalue of the observable measured. Moreover,
this degenerate eigenvalue may be different from the eigenvalue A which the measurement
n
would have found before the change was made.
As Gleason's proof of the result mentioned above is rather abstract, we derive his result
from the more understandable work of Kochen and Specker, which discusses a special case
of Gleason's ideas and which is very revealing. We give a much simplified proof of Kochen
and Specker's result, simpler yet than a proof of Gleason's result given by Bell. Also at
various other places in our survey, we clarify points by novel approaches that simplify the
reasoning. Our main aim here is making the conclusions more easily understandable or
easier to conceive visually. We do not aim at utmost mathematical rigor, and would prefer
a plausibility argument to a lengthy rigorous proof.
FOREWORD TO PART II
IN PART I we defined hidden-variables theories of the first kind as hidden-variables theories
constructed with the following two primary purposes in mind: (1) the theory shall not be
self-contradictory, and (2) the theory shall yield all results of pure quantum theory for all
ensembles of physical systems in which the distribution of the hidden-variable values is a
certain equilibrium distribution. By (1), these theories differ from theories of the zeroth
kind, which we discussed in Part I ; by (2), they differ from theories of the second kind, which
we will discuss in Part HI of this book.
Part II starts with a discussion of Bohm's theory of 1951, which is largely a reinterpretation
of ordinary, nonrelativistic wave mechanics as a hidden-variables theory. A typical achieve­
ment of Bohm's theory is that it allows us in a two-slit interference experiment to tell
through which slit a particular particle reached the screen, without destroying the inter­
ference pattern.
The main trouble with this theory of Böhm is that its relativistic generalization meets
difficulties.
Next we will discuss the theory of Wiener and Siegel. From this theory we obtain the
theory of Böhm and Bub by first omitting from the Wiener-Siegel theory all parts that are
irrelevant for its application and by then adding Bohm-Bub's ideas about additional terms
in the Schrödinger equation effective only during measurements. We will, in this connection,
also consider some of Tutsch's more general, ideas related to the latter point.
Finally, we shall discuss possible experiments aimed at finding deviations from quantum
theory which theories of the first kind predict for measurements made in rapid succession.
We shall calculate the effects predicted by the theories of Wiener and Siegel and of Böhm
and Bub, and by a variation thereof. Some of these experiments have already been performed
in 1967 by Papaliolios, and gave negative results, which impose an upper limit upon the
relaxation time in which in these experiments a purposely perturbed hidden-variables
distribution returned to an equilibrium distribution.
We discuss a method of removing the ambiguity of the set {φ^ of possible measurement
results in the case of degenerate observables, but we find that even thus there are cases
where application of the theory of Wiener-Siegel or Bohm-Bub leads to paradoxes.
81
FOREWORD TO PART III
WHEN the wavefunction of a composite system is not factorizable, the behaviors of its
parts are correlated. In quantum theory this leads to results that seem paradoxical when
considered from a point of view that assumes local causal behavior of elementary particles
and quanta, if among the quantities correlated there are some that are not commutative.
The resulting "nonlocality paradox" (which is a side aspect of the Einstein-Podolsky-Rosen
paradox) could be solved by assuming the existence of hidden variables "of the second kind"
only if one is willing to accept deviations from quantum theory already at the quasistatic
level at which theories "of the first kind" would agree with quantum theory.
Most applications of theories of the second kind considered in Part III deal with a pair of
photons emitted from a single source. In the quantum-theoretical treatment of such prob­
lems, considerations of angular momentum and of photon spin are useful. For photons
passing more than one polarization filter, quantum theory and reasonable hidden-variables
theories should predict Malus's cos2 law. For a pair of photons analyzed by separate filters,
both types of theory lead to polarization correlations, but these correlations are for hidden-
variables theories of the second kind different from what they are for quantum theory. This
is demonstrated in Chapter 3 by some simple model theories, and is generally proved in
Chapter 4, in which we also discuss the crucial experiments of Clauser and of Holt which
are to decide which type of theory is the correct one.
In Chapter 5 we consider experiments in which the photon pair arises from positon-
negaton annihilation. For photons that hard, no polarization filters are known. Therefore
the polarization dependence of Compton scattering is used for an indirect statistical observ­
ation of polarization. The experimental results suffice for disproving an idea of Furry that
nonfactorizable wavefunctions of composite systems by an alteration of quantum theory
would go over into mixtures of factorizable wavefunctions when the parts of the system
become macroscopically separated from each other. Also, the experiments disprove a certain
type of hidden-variables theories of the second kind, but they cannot disprove arbitrary
hidden-variables theories of the second kind.
In Chapter 6 we review briefly some results of all three parts of this survey and conclude
that hidden-variables theory at this stage appears to be unacceptable. A number of attitudes
one can take in view of this fact are enumerated.
243