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${\mathrm{^}}^{\mathrm{°}}$B^. With every operator A^, we associate commutation and anticommutation superoperators A${\mathrm{^}}^{\mathrm{-}}$ and A${\mathrm{^}}^{+}$, defined by their actions on any operator B^ as A${\mathrm{^}}^{\mathrm{-}}$B^=${\mathrm{ħ}}^{\mathrm{-}1}$[A^,B^], A${\mathrm{^}}^{+}$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${\mathrm{^}}^{+}$ and p${\mathrm{^}}^{+}$ (where [q^,p^]=iħ). These, together with the commutation superoperators q${\mathrm{^}}^{\mathrm{-}}$ and p${\mathrm{^}}^{\mathrm{-}}$, form a canonical set of superoperators, [q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{\mathrm{-}}$]=[q${\mathrm{^}}^{\mathrm{-}}$,p${\mathrm{^}}^{+}$]=i (the other commutators vanishing), as functions of which all other super- operators can be expressed. Weyl ordering is expressed as f(q^,p^${)}_{\mathrm{Weyl}}$ ordering=f(q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{+}$)1^. A generalization of Ehrenfest’s theorem is obtained.