Abstract
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^>>=TrAB^. With every operator A^, we associate commutation and anticommutation superoperators A and A, defined by their actions on any operator B^ as AB^=[A^,B^], AB^=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^]=iħ). 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^ ordering=f(q,p)1^. A generalization of Ehrenfest’s theorem is obtained.
- Received 18 June 1990
DOI:https://doi.org/10.1103/PhysRevA.43.44
©1991 American Physical Society