Von Neumann's No Hidden Variable Theorem

TL;DR

This paper examines von Neumann's no hidden variable theorem, highlighting how the phase space formulation of quantum mechanics allows hidden variables by modifying certain assumptions, especially regarding operator ordering.

Contribution

It identifies the specific assumption violation in phase space quantum mechanics that permits hidden variable theories, challenging von Neumann's original no hidden variable conclusion.

Findings

Phase space formulation allows dispersion free ensembles.

The violation of assumption I enables hidden variables.

Operator ordering tracking is key to the violation.

Abstract

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 $\mathbb{R}$ implies the value f(r) for the quantity $f(\mathbb{R})$. 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.