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Soviet Physics Uspekhi

WHY IT IS IMPOSSIBLE TO INTRODUCE HIDDEN PARAMETERS INTO QUANTUM MECHANICS

Alexander I. Akhiezer, R.V. Polovin

Published Apr 30, 1973• 10 citations
Publisher

Abstract

CONTENTS 1. The Statistical Character of the Behavior of Microobjects 500 2. The Mathematical Foundations of Quantum Mechanics 501 3. The Measurement Process 502 4. The Hypothesis of Parameters 503 5. Interference of Probabilities and Hidden Parameters 505 6. Von Neumann's Proof of von Neumann's Theorem 507 7. A Proof of von Neumann's Theorem which does not use the Postulate of Additivity of Incompatible Observables 508 8. Classical and Quantum Logic 509 References 511

DOI
10.1070/pu1973v015n04abeh004999

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CONTENTS 1. The Statistical Character of the Behavior of Microobjects 500 2. The Mathematical Foundations of Quantum Mechanics 501 3. The Measurement Process 502 4. The Hypothesis of Parameters 503 5. Interference of Probabilities and Hidden Parameters 505 6. Von Neumann's Proof of von Neumann's Theorem 507 7. A Proof of von Neumann's Theorem which does not use the Postulate of Additivity of Incompatible Observables 508 8. Classical and Quantum Logic 509 References 511
Introduction To know the quantum mechanical state of a system implies, in general, only statistical restrictions on the results of measurements. It seems interesting to ask if this statistical element be thought of as arising, as in classical statistical mechanics, because the states in question are averages over better defined states for which individually the results would be quite determined. These hypothetical ‘dispersion free’ states would be specified not only by the quantum mechanical state vector but also by additional ‘hidden variables’ – ‘hidden’ because if states with prescribed values of these variables could actually be prepared, quantum mechanics would be observably inadequate. Whether this question is indeed interesting has been the subject of debate. The present paper does not contribute to that debate. It is addressed to those who do find the question interesting, and more particularly to those among them who believe that ‘the question concerning the existence of such hidden variables received an early and rather decisive answer in the form of von Neumann's proof on the mathematical impossibility of such variables in quantum theory.’ An attempt will be made to clarify what von Neumann and his successors actually demonstrated. This will cover, as well as von Neumann's treatment, the recent version of the argument by Jauch and Piron, and the stronger result consequent on the work of Gleason. It will be urged that these analyses leave the real question untouched.
Because the mathematical form of von Neumann's hidden-parameter proof is not well known, its scope has sometimes been unjustifiably narrowed by commentators. The proof is here reproduced in the more familiar Dirac calculus and its precise import made clearer. In particular, it is emphasized that the proof proceeds independently of both the basic statistical assumption of quantum mechanics: 〈R〉av = 〈φ|R|φ〉, and the extension of this assumption to mixtures: 〈R〉av = Tr(UR), where U = Σn|φn〉wn〈φn|. Mention is also made of some recent criticisms of the von Neumann proof.
This article reviews the history of J. von Neumann’s analysis of hidden variables in quantum mechanics and the subsequent analysis by others. In his book The Mathematical Foundations of Quantum Mechanics, published in 1932, von Neumann performed an analysis of the consequences of introducing hidden parameters (hidden variables) into quantum mechanics. He arrived at two principal conclusions: first, hidden variables cannot be incorporated into the existing theory of quantum mechanics without major modifications, and second, if they did exist, the theory would have already failed in situations where it has been successfully applied. This analysis has been taken as an “incorrect proof” against the existence of hidden variables, possibly due to a mistranslation of the German word prufen. von Neumann’s so-called proof isn’t even wrong as such a proof does not exist, but it is an examination of the limitations imposed by internal consistency of the Hilbert space formulation of the theory. One of the earliest attempts to eliminate uncertainty, by D. Bohm, requires a major modification of quantum mechanics (observables are not represented by Hermitian operators), which supports von Neumann’s first principal conclusion. However, testing the Bohm theory requires constructing a physically impossible initial state. As such, the theory has no experimental consequences, so W. Pauli referred to it as an “uncashable check”. As there are no observable consequences, the Bohm theory is possibly a counterexample to von Neumann’s second conclusion that hidden variables in particular would have already led to a failure of the theory.
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.