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- Publisher
- American Physical Society (APS)
- Copyright
- Copyright © 1991 The American Physical Society
- ISSN
- 1094-1622
- DOI
- 10.1103/PhysRevA.43.44
- Publisher site
- See Article on Publisher Site
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 ^>>=Tr A ^ ° 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 ^= ħ - 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 ħ). 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.
Journal
Physical Review A – American Physical Society (APS)
Published: Jan 1, 1991