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 $\mathbf{A}$ has an operator $\hat{A}$ then $f(\mathbf{A})$ has the operator $f(\hat{A})$". 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.
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.
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.
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.
Previously suggested hidden time interpretation of quantum mechanics allows to reproduce the same predictions as standard quantum mechanics provides, since it is based on Feynman many - paths formulation of QM. While new experimental consequences of this interpretation are under investigation, some advantages can be enumerated. (1) The interpretation is much field theoretic - like in classical sense, so it is local in mathematical sense, though quantum (physical) non-locality is preserved. (2) The interpretation is based on one type of mathematical objects, rather than two different (Hilbert space vectors and operators). (3) The interpretation, as it was argued, overcomes the problem of hidden variables in a radically new way, with no conflict to Bell's theorem. Recently an important argument against hidden variables - like formulations of quantum theory was risen - "no protocol" argument. It is argued in the paper, that hidden time interpretation successfully overcomes this argument.
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 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.
Hidden variable theories, ‘the hidden parameter’ as von Neumann called them, entered the quantum scene soon after the formalization of quantum mechanics as we know it today. That formalization was led by front-line physicists like Born, Dirac, Heisenberg, Jordan, and Schrödinger. Attention to hidden variable theories garnered strength following the criticisms towards quantum mechanics formulated by Einstein and colleagues in 1935. That criticism, stating that quantum mechanics was ‘not complete,’ as a theory, led Bohm, in the 1950s, to formulate hidden variable theories. In turn, Bell in 1964 introduced his famed Bell theorem that provided a test for the existence of hidden variable theories. This chapter examines the history of the futile search, for experimental proof, for the existence of these theories.