  In the Wigner-Moyal approach to quantum mechanics, we show that Moyal's
starting point, the characteristic function $M(\tau,\theta)=\int
\psi^{*}(x)e^{i(\tau {\hat p}+\theta{\hat x})}\psi(x)dx$, is essentially the
primitive idempotent used by von Neumann in his classic paper "Die
Eindeutigkeit der Schr\"odingerschen Operatoren". This paper provides the
original proof of the Stone-von Neumann equation. Thus the mathematical
structure Moyal develops is simply a re-expression of what is at the heart of
quantum mechanics and reproduces exactly the results of the quantum formalism.
The "distribution function" $F(X,P,t)$ is simply the quantum mechanical density
matrix expressed in an $( X,P)$-representation, where $X$ and $P$ are the mean
co-ordinates of a cell structure in phase space. The whole approach therefore
clearly has little to do with classical statistical theories but is a
consequence of a non-commutative nature of the theory.
