On the Relationship between the Moyal Algebra and the Quantum Operator Algebra of von Neumann

TL;DR

This paper explores the deep connection between Moyal's phase space formulation of quantum mechanics and von Neumann's non-commutative operator algebra, revealing that the Moyal distribution function corresponds to the quantum density matrix.

Contribution

It establishes a formal relationship showing that Moyal's distribution function is equivalent to the quantum density matrix, linking phase space and operator formalisms of quantum mechanics.

Findings

The distribution function F(P,X,t) is the quantum density matrix.

Coordinates X and P represent mean coordinates of phase space cells.

Non-commutative structure influences the symplectic geometry of quantum phase space.

Abstract

The primary motivation for Moyal's approach to quantum mechanics was to develop a phase space formalism for quantum phenomena by generalising the techniques of classical probability theory. To this end, Moyal introduced a quantum version of the characteristic function which immediately provides a probability distribution. The approach is sometimes perceived negatively merely as an attempt to return to classical notions, but the mathematics Moyal develops is simply a re-expression of what is at the heart of quantum mechanics, namely the non-commutative algebraic structure first introduced by von Neumann in 1931. In this paper we will establish this relation and show that the "distribution function", F(P,X,t) is simply the quantum mechanical density matrix for a single particle. The coordinates, X and P, are not the coordinates of the particle but the mean co-ordinates of a cell structure (a `blob') in phase space, giving an intrinsically non-local description of each individual particle, which becomes a point in the limit to order $\hbar^2$. We discuss the significance of this non-commutative structure on the symplectic geometry of the phase space for quantum processes.