WIGNER FUNCTION IN LIOUVILLE SPACE - A CANONICAL FORMALISM

The Wigner-Weyl (WW) phase-space formulation of quantum mechanics is discussed within the Liouville-space formalism, where quantum operators A are viewed as vectors, represented by L kets /A >>, on which act "superoperators"; the scalar product is << A/B >> = TrA+B. With every operator A, we associate commutation and anticommutation superoperators A- and A+, defined by their actions on any operator B as A-B = h-1 [A,B], A+B = 1/2 (AB + BA). The WW representation corresponds to the choice of a special basis in Liouville space, namely, the eigenbasis of the position and momentum anticommutation superoperators q+ and p+ (where [q,p] = ih). These, together with the commutation superoperators q- and p-, form a canonical set of superoperators, [q+,p-] = [q-,p+] = i (the other commutators vanishing), as functions of which all other superoperators can be expressed. Weyl ordering is expressed as f (q,p) Weyl ordering = f (q+,p+) 1. A generalization of Ehrenfest's theorem is obtained.