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

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

The primary motivation for Moyals 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-, FP,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.

Author: **B. J. Hiley**

Source: https://archive.org/