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^=^-1[A^,B^], A^⁺B^=1/2(A^B^+B^A^). 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^]=i{}). 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 super- operators 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.