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American Journal of Physics

Von Neumann's Hidden-Parameter Proof

James S. Albertson

Published Aug 1, 1961• 16 citations
Publisher

Abstract

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.

DOI
10.1119/1.1937816

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European Journal of Physics•Nov 13, 2025•1 citation
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.
Abstract Von Neumann’s book, The Mathematical Foundations of Quantum Mechanics, is reviewed in its entirety. The absence of the Dirac delta function in von Neumann’s treatment of quantum mechanics is understood from the historical perspective. Basically, the delta function was not on a firm mathematical foundation at the time. Von Neumann introduced the concept of the density matrix and we discuss its application to his proof of “no hidden parameters (variables)” in quantum mechanics. His proof is often maligned as being incorrect. Von Neumann showed that the existence of hidden parameters would require a reworking of the entire structure of quantum mechanics.
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.
Introduction In his Mathematical Foundations of Quantum Mechanics , John von Neumann presented a proof that the quantum statistics cannot be recovered from probability distributions over “hidden” deterministic states that assign definite premeasurement values to all physical quantities. Von Neumann's proof has been dismissed as “silly” by John Bell and by David Mermin, who writes: Many generations of graduate students who might have been tempted to try to construct hidden-variables theories were beaten into submission by the claim that von Neumann, 1932, had proved that it could not be done. A few years later (see Jammer, 1974, p. 273) Grete Hermann, 1935, pointed out a glaring deficiency in the argument, but she seems to have been entirely ignored. Everybody continued to cite the von Neumann proof. A third of a century passed before John Bell, 1966, rediscovered the fact that von Neumann's nohidden-variables proof was based on an assumption that can only be described as silly—so silly, in fact, that one is led to wonder whether the proof was ever studied by either the students or those who appealed to it to rescue them from speculative adventures.
In this paper I provide a detailed history of von Neumann’s “No Hidden Variables” theorem, and I argue it is a demonstration that his axiomatization mathematically captures a salient feature of the statistical interpretation (namely, that hidden variables are incompatible). I show that this reading of von Neumann’s theorem is obvious once one recalls several contextual factors of his work. First, his axiomatization was what I call a Hilbert-style axiomatic completion; indeed, it developed from work initiated by Hilbert (and Nordheim). Second, it was responsive to specific mathematical and theoretical problems faced by Dirac and Jordan’s statistical transformation theory (then called ‘quantum mechanics’). Third, the axiomatization was essentially completed already in his 1927 papers, at least concerning the status of hidden variables, and this would have been obvious to the audience for those papers. Thus, the theorem’s statement and proof were only necessary when the material was presented for a general mathematical audience, i.e., in his 1932 Mathematical Foundations of Quantum Mechanics . With this reading in mind, his claim that quantum mechanics was in “compelling logical contradiction with causality” appears as a straightforward consequence of his theorem. I conclude by reassessing the theorem’s broader historical and scientific significance.
Abstract Hidden variables are hypothetical parameters that, if known, would deterministically specify quantum measurement outcomes, aiming to restore determinism and causality typical of classical mechanics. This work provides a modern and pedagogical exposition of John von Neumann’s 1932 proof on the incompatibility of hidden variables with quantum mechanics. We detail his two main conclusions: (1) dispersion-free ensembles (as defined within the framework of hidden variable theories) are incompatible with quantum mechanics; and (2) the quantum state provides a complete description of physical reality. We critically analyze the key assumptions in von Neumann’s proof, highlighting later challenges—particularly John Bell’s critique and the development of contextual hidden-variable theories by Kochen and Specker. This discussion clarifies von Neumann’s historical role as a pioneer in this debate, while showing how modern hidden-variable theories, such as Bohmian mechanics, remain compatible with quantum predictions. Conversion tables are included between this article’s notation and von Neumann’s original book.