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arXiv:2105.13996

On the Unconditional Validity of J. von Neumann's Proof of the Impossibility of Hidden Variables in Quantum Mechanics

C. S. Unnikrishnan

Published May 27, 2021• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2105.13996

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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.
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.
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.
I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. I also identify similar oversight in the GHZ variant of Bell's theorem, invalidating its claim of having found an inconsistency in the premisses of the argument by EPR for completing quantum mechanics. Conceptually, the oversight in both Bell's theorem and its GHZ variant traces back to the oversight in von~Neumann's theorem against hidden variable theories identified by Grete Hermann in the 1930s.
I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. I also identify similar oversight in the GHZ variant of Bell's theorem, invalidating its claim of having found an inconsistency in the premisses of the argument by EPR for completing quantum mechanics. Conceptually, the oversight in both Bell's theorem and its GHZ variant traces back to the oversight in von~Neumann's theorem against hidden variable theories identified by Grete Hermann in the 1930s.
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\mathbb{R}R implies the value f(r) for the quantity f(R)f(\mathbb{R})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.
I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. I also identify similar oversight in the GHZ variant of Bell's theorem, invalidating its claim of having found an inconsistency in the premisses of the argument by EPR for completing quantum mechanics. Conceptually, the oversight in both Bell's theorem and its GHZ variant traces back to the oversight in von~Neumann's theorem against hidden variable theories identified by Grete Hermann in the 1930s.