von Neumann in 1932 was the first to outline the possible non existence of dispersion free ensembles in quantum mechanics, and he used also such basic evidence to give a preliminary proof on incompatibility between quantum mechanics and (non-contextual) hidden variables theory.In the present paper we give a detailed theoretical elaboration on the manner in which such fundamental subject could be explored at perceptive and cognitive level in humans.We also discuss a general design of the experiment that we have in progress so to give direct indications to other researchers engaged in such field.
von Neumann in 1932 was the first to outline the possible non existence of dispersion free ensembles in quantum mechanics, and he used also such basic evidence to give a preliminary proof on incompatibility between quantum mechanics and (non-contextual) hidden variables theory.In the present paper we give a detailed theoretical elaboration on the manner in which such fundamental subject could be explored at perceptive and cognitive level in humans.We also discuss a general design of the experiment that we have in progress so to give direct indications to other researchers engaged in such field.
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.
The impossibility of making quantum mechanics deterministic by adding ‘hidden variables’ was discussed by J. von Neumann in 1932. His proof of the impossibility of dispersion-free ensembles was criticized as being logically circular, by Grete Hermann soon after, and as fundamentally flawed, by John Bell in 1964. Bell’s severe criticism and an explicit counter-example are generally considered as the definitive demonstration that J. von Neumann’s proof is inadequate. I show, without using the formalism of quantum mechanics, that J. von Neumann’s central assumption of the linear additivity of the expectation values, even for incompatible (noncommuting) observables, is a necessary consequence that follows directly from the fundamental conservation laws of physical quantities, applicable universally. With this result, J. von Neumann’s result is fully resurrected. I identify the physical reason why dispersion-free ensembles cannot exist. After discussing the conflict of Bell’s counter-example with the factual mechanics, I highlight the result that the local hidden variable theories, for which the Bell’s inequalities were derived, are grossly incompatible with the fundamental conservation laws. In a scrutiny of the vital difference between the ‘causal incompleteness’ of quantum mechanics, discussed by J. von Neumann, and the ‘representational incompleteness’ discussed by Einstein, Podolsky, and Rosen, I prove the striking demarcating result that a multi-particle system that is considered causally completed can remain EPR-incomplete. I identify the wave nature of the action as the reason behind the irreducible quantum dispersion and the impossibility of dispersion-free ensembles.
The impossibility of theories with hidden variables as an alternative and replacement for quantum mechanics was discussed by J. von Neumann in 1932. His proof was criticized as being logically circular, by Grete Hermann soon after, and as fundamentally flawed, by John Bell in 1964. Bell's severe criticism of Neumann's proof and the explicit (counter) example of a hidden variable model for the measurement of a quantum spin are considered by most researchers, though not all, as the definitive demonstration that Neumann's proof is inadequate. Despite being an argument of mathematical physics, an ambiguity of decision remains to this day. I show that Neumann's assumption of the linear additivity of the expectation values, even for incompatible (noncommuting) observables, is a necessary constraint related to the nature of observable physical variables and to the conservation laws. Therefore, any theory should necessarily obey it to qualify as a physically valid theory. Then, obviously, the hidden variable theories with dispersion-free ensembles that violate this assumption are ruled out. I show that it is Bell's counter-example that is fundamentally flawed, being inconsistent with the factual mechanics. Further, it is shown that the local hidden variable theories, for which the Bell's inequalities were derived, are grossly incompatible with the fundamental conservation laws. I identify the intrinsic uncertainty in the action as the reason for the irreducible dispersion, which implies that there are no dispersion-free ensembles at any scale of mechanics. With the unconditional validity of its central assumption shown, Neumann's proof is fully resurrected.
Von Neumann use 4 assumptions to derive the Hilbert space (HS) formulation of quantum mechanics (QM). Within this theory dispersion free ensembles do not exist. To accommodate a theory of quantum mechanics that allow dispersion free ensemble some of the assumptions need be modified. An existing formulation of QM, the phase space (PS) formulation allow dispersion free ensembles and thus is qualifies as an hidden variable theory. Within the PS theory we identify the violated assumption (dubbed I in the text) to be the one that requires that the value r for the quantity R implies the value f(r) for the quantity f(R). We note that this violation arise due to tracking within c-number hidden variable theory of the operator ordering involved in HS theory as is required for a 1-1 correspondence between the theories.
An hidden variable (hv) theory is a theory that allows globally dispersion free ensembles. We demonstrate that the Phase Space formulation of Quantum Mechanics (QM) is an hv theory with the position q, and momentum p as the hv. Comparing the Phase space and Hilbert space formulations of QM we identify the assumption that led von Neumann to the Hilbert space formulation of QM which, in turn, precludes global dispersion free ensembles within the theory. The assumption, dubbed I, is: "If a physical quantity A has an operator A^ then f(A) has the operator f(A^)". This assumption does not hold within the Phase Space formulation of QM. The hv interpretation of the Phase space formulation provides novel insight into the interrelation between dispersion and non commutativity of position and momentum (operators) within the Hilbert space formulation of QM and mitigates the criticism against von Neumann's no hidden variable theorem by, virtually, the consensus.
In his book \textit{The Mathematical Foundations of Quantum Mechanics}, published in 1932, J. von Neumann performed an analysis of the consequences of introducing hidden parameters (hidden variables) into quantum mechanics. He showed that hidden variables cannot be incorporated into the existing theory of quantum mechanics without major modifications, and concluded that if they did exist, the theory would have already failed in situations where it has been successfully applied. von Neumann left open the possibility that the theory is not complete, and his analysis for internal consistency is the best that can be done for a self-referenced logical system (Gödel's theorem). This analysis had been taken as an ``incorrect proof" against the existence of hidden variables. von Neumann's so-called proof isn't even wrong as such a proof does not exist. One of the earliest attempts at a hidden variable theory was by D. Bohm, and because there were no experimental consequences, W. Pauli referred to it as an ``uncashable check." To our knowledge, a successful hidden variable extension to quantum mechanics with testable consequences has not yet been produced, suggesting that von Neumann's analysis is worthy of rehabilitation, which we attempt to provide in a straightforward manner.