Von Neumann's 'No Hidden Variables' Proof: A Re-Appraisal

TL;DR

This paper re-examines von Neumann's proof against hidden variables, clarifying that it rules out deterministic dispersion-free models but does not address nonlocal hidden variable theories like Bohm's.

Contribution

It clarifies von Neumann's original argument, showing it excludes only certain types of hidden variables, contrary to common interpretations influenced by Bell.

Findings

Von Neumann's proof excludes dispersion-free hidden variable models.

Bell's interpretation misrepresents von Neumann's original argument.

The proof does not rule out nonlocal hidden variable theories like Bohm's.

Abstract

Since the analysis by John Bell in 1965, the consensus in the literature is that von Neumann's 'no hidden variables' proof fails to exclude any significant class of hidden variables. Bell raised the question whether it could be shown that any hidden variable theory would have to be nonlocal, and in this sense 'like Bohm's theory.' His seminal result provides a positive answer to the question. I argue that Bell's analysis misconstrues von Neumann's argument. What von Neumann proved was the impossibility of recovering the quantum probabilities from a hidden variable theory of dispersion free (deterministic) states in which the quantum observables are represented as the 'beables' of the theory, to use Bell's term. That is, the quantum probabilities could not reflect the distribution of pre-measurement values of beables, but would have to be derived in some other way, e.g., as in Bohm's theory, where the probabilities are an artefact of a dynamical process that is not in fact a measurement of any beable of the system.