Phase Space Formulation of Quantum Mechanics as an Hidden Variables Theory

TL;DR

This paper shows that the phase space formulation of quantum mechanics can be viewed as a hidden variables theory with position and momentum as hidden variables, offering new insights into quantum non-commutativity and dispersion.

Contribution

It demonstrates that the phase space formulation is a hidden variables theory and identifies the assumption that led von Neumann to exclude such theories.

Findings

Phase space formulation allows dispersion free ensembles.

The assumption linking functions of operators to operators does not hold in phase space.

Provides new insights into the relation between dispersion and non-commutativity.

Abstract

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.