Moyal’s Characteristic Function, the Density Matrix and von Neumann’s Idempotent

In the Wigner-Moyal approach to quantum mechanics, we show that Moyal’s starting point, the characteristic function M(τ, θ) = ∫ ψ∗(x)ei(τp̂+θx̂)ψ(x)dx, is essentially the primitive idempotent used by von Neumann in his classic paper “Die Eindeutigkeit der Schrödingerschen 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. This means it exactly reproduces the results of the quantum formalism. The “distribution function” F (X,P, t) is not a probability distribution. It is a 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.