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Annals of the Japan Association for Philosophy of Science

On the Problem of Hidden Variables in Algebraic Quantum Field Theory

Yuichiro Kitajima

Published Jan 1, 2006• 5 citations
Publisher

Abstract

There have been many discussions about whether all observational propositions in nonrelativistic quantum mechanics can be taken as simultaneously definite without contradiction. Von Neumann, Jauch and Piron, and Bell have shown that not all observational propositions in nonrelativistic quantum mechanics can be interpreted in this way. In the present paper, I examine these theorems from an operator algebraic point view in order to apply these theorems to algebraic quantum field theory. Then I point out that not all observational propositions in local algebras can be taken as definite, in addition to nonrelativistic quantum mechanics.

DOI
10.4288/jafpos1956.15.25

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Progress of Theoretical Physics•Sep 1, 1969•2 citations
There have been many discussions about whether all observational propositions in nonrelativistic quantum mechanics can be taken as simultaneously definite without contradiction. Von Neumann, Jauch and Piron, and Bell have shown that not all observational propositions in nonrelativistic quantum mechanics can be interpreted in this way. In the present paper, I examine these theorems from an operator algebraic point view in order to apply these theorems to algebraic quantum field theory. Then I point out that not all observational propositions in local algebras can be taken as definite, in addition to nonrelativistic quantum mechanics.
Based partly on proving that algebraic relativistic quantum field theory (ARQFT) is a stochastic Einstein local (SEL) theory in the sense of SEL which was introduced by Hellman (1982b) and which is adapted in this paper to ARQFT, the recently proved maximal and typical violation of Bell's inequalities in ARQFT (Summers and Werner 1987a–c) is interpreted in this paper as showing that Bell's inequalities are, in a sense, irrelevant for the problem of Einstein local stochastic hidden variables, especially if this problem is raised in connection with ARQFT. This leads to the question of how to formulate the problem of local hidden variables in ARQFT. By giving a precise definition of hidden-variable theory within the operator algebraic framework of quantum mechanics, it will be argued that the aim of hidden-variable investigations is to determine those classes of quantum theories whose elements represent a statistical content that cannot be reduced in a given way. In some particular way to be stated, a proposition will be stated which distinguishes quantum field theories whose statistical content cannot be reduced without violating some relativistic locality principle.
The question of the possible existence of hidden variables is re-examined. It is shown that hidden variables can exist only if every proposition (yes-no experiment) is compatible with every other one. It is further shown that this property is in contradiction with empirical facts. The theorem which leads to this conclusion is a strengthening of the theorem of von Neumann on the same subject. The question is raised whether there exist perhaps quantum mechanical systems which admit approximate dispersion-free states.
In this paper, we examine an argument of Jauch and Piron, which aims to prove the impossibility of hidden variables underlying the quantum theory, on the basis of certain assumptions that are weaker than those used by von Neumann for the same purpose. We show that, while the assumptions of Jauch and Piron are in fact weaker than those of von Neumann, the net result is that they actually prove nothing new at all. The conclusions of Jauch and Piron concerning the nonexistence of hidden variables are indeed seen to follow from a false assumption; i.e., that the impossibility of propositions that describe simultaneously the results of measurements of two noncommuting observables is an "empirical fact." Actually, it is shown that this assumption follows if and only if one first assumes what the authors set out to prove; i.e., that the current linguistic structure of quantum mechanics is the only one that can be used correctly to describe the empirical facts underlying the theory.
The study of hidden-variable interpretations of quantum mechanics was transformed by two papers of Bell. In his 1966 paper in Reviews of Modern Physics [1] he analysed the argument of von Neumann [2] and the more powerful arguments of Jauch and Piron [3] and of Gleason [4], all leading to similar conclusions about the nonassignability of simultaneous values to all the observables of a nontrivial quantum system; and he showed that these arguments do not exclude what may be called “contextualistic” hidden-variable theories, in which the value of an observable O is allowed to depend not only upon the hidden state λ, but also upon the set C of compatible observables measured along with O . In Bell's 1964 paper in Physics [5] (written later than the other, in spite of the earlier publication date) he gave a new kind of argument against a hidden-variable interpretation of quantum mechanics, by showing that no hidden-variables theory which satisfies a certain condition of locality can agree with all the statistical predictions of quantum mechanics. The main purpose of this lecture is to discuss an experiment, designed by Clauser, Horne, Holt and myself [6], which is based on Bell's proof of the incompatibility of quantum mechanics and local hidden-variable theories. Before doing this, however, I wish to make several comments.
Noncontextual hidden variables theories, assigning simultaneous values to all quantum mechanical observables, are inconsistent by theorems of Gleason and others. These theorems do not exclude contextual hidden variables theories, in which a complete state assigns values to physical quantities only relative to contexts. However, any contextual theory obeying a certain factorisability condition implies one of Bell's Inequalities, thereby precluding complete agreement with quantum mechanical predictions. The present paper distinguishes two kinds of contextual theories, ‘algebraic’ and ‘environmental’, and investigates when factorisability is reasonable. Some statements by Fine about the philosophical significance of Bell's Inequalities are then assessed. INTRODUCTION In most of the discussions of Bell's Inequalities little notice is taken of the historical circumstances of their discovery. J. S. Bell was told by J. Jauch of the work of A. Gleason [1957] which showed that simultaneous values cannot be assigned to all observables of a quantum mechanical system in a way that respects their algebraic structure (except in the simple case in which the relevant Hilbert space has dimension less than three). Gleason's theorem thus precludes a type of hidden variables theory that has come to be called ‘noncontextual’. The question of agreement between the statistical predictions of quantum mechanics and those of a noncontextual hidden variables theory does not even arise, since such a theory cannot be consistently formulated.
The problem of hidden variables, purely conceptual at the start, has been early influenced by the practical aims of “causal theories” towards a mathematical scheme broader than the quantum formalism. Thus several authors have examined if a new formalism, built in the same spirit as that of quantum mechanics, however more general, might admit a determination by means of hidden parameters, and concluded negatively. In the present paper the two aspects (conceptual and practical) are considered separately. It first contains a study of the unrestricted proofs (Von Neumann, Jauch and Piron), given for the non-existence of hidden parameters. They are focussed on the dispersion-free states bound to them. Thus a preliminary stage consists in analyzing what properties these states should have. A method is given here for displaying these; they are shown to have an artificial character, which fits neither Von Neumann's proof nor that of Jauch and Piron and other authors (Misra). Thus, these proofs do not establish the inexistence of hidden variables, but only that of dispersion free states built more or less on the pattern of the quantum states. Suggestions are then proposed on the probable features of hidden parameters. They lead us to regard as improbable the hope of building a new deterministic scheme to be substituted for the quantum formalism. Thus, the determinism which can be maintained appears as purely conceptual. The question of a “deeper level” and the relation between determination and pictorial description are discussed briefly.