Semiclassical properties of Wigner functions

The following conjecture is shown to be plausible: Consider a sequence of quantum theories with different values of ℎ. In each one, let Z be a non-degenerate self-adjoint operator, having a 'classical limit' as ℎ→0. Let z' be an eigenvalue of Z in the vicinity of a fixed real number z (independent of ℎ) and let W(p,q;z') be the corresponding Wigner function. Then, if we subject W(p,q;z') to a coarse graining over a scale >> ℎ, the result, in the limit ℎ→0, is proportional to the classical distribution δ[Z(p,q)-z]. (orig.)