Moyal bracket with manifest (affine) symplectic covariance

Moyal’s equation for a Wigner function of a quantum system with (Wigner-transformed) Hamiltonian is where the Moyal bracket is a binary operator on the space of functions over phase space. Unfortunately, it is often written down mysteriously as

(1)  

where the arrows over partial derivatives tell you which way they act, i.e., . This only becomes slightly less weird when you use the equivalent formula , where “” is the Moyal star product given by

(2)  

The star product has the crucial feature that , where we use a hat to denote the Weyl transform (i.e., the inverse of the Wigner transform taking density matrices to Wigner functions), which takes a scalar function over phase-space to an operator over our Hilbert space. The star product also has some nice integral representations, which can be found in books like Curtright, Fairlie, & ZachosThe complete 88-page PDF is here.^a, but none of them help me understand the Moyal equation.

A key problem is that both of these expressions are neglecting the (affine) symplectic symmetry of phase space and the dynamical equations. Although I wouldn’t call it beautiful, we can re-write the star product as

(3)  

where is a symplectic index using the Einstein summation convention, and where symplectic indices are raised and lowered using the symplectic form just as for Weyl spinors: and , where is the antisymmetric symplectic form with , and where upper (lower) indices denote symplectic vectors (co-vectors).

With this, we can expand the Moyal equation as

where we can see in hideous explicitness that it’s a series in the even powers of and the odd derivates of the Hamiltonian and the Wigner function . Furthermore, as we see it quickly reduces to the Poisson bracket .