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 \ensuremath{\Vert}A^>>, on which act ``superoperators''; the scalar product is A^\ensuremath{\Vert}B^>>=TrA${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$B^. With every operator A^, we associate commutation and anticommutation superoperators A${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$ and A${\mathrm{^}}^{+}$, defined by their actions on any operator B^ as A${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$B^=${\mathrm{\ensuremath{\Elzxh}}}^{\mathrm{\ensuremath{-}}1}$[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^]=i\ensuremath{\Elzxh}). These, together with the commutation superoperators q${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$ and p${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$, form a canonical set of superoperators, [q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$]=[q${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$,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.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 \ensuremath{\Vert}A^>>, on which act ``superoperators''; the scalar product is A^\ensuremath{\Vert}B^>>=TrA${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$B^. With every operator A^, we associate commutation and anticommutation superoperators A${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$ and A${\mathrm{^}}^{+}$, defined by their actions on any operator B^ as A${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$B^=${\mathrm{\ensuremath{\Elzxh}}}^{\mathrm{\ensuremath{-}}1}$[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^]=i\ensuremath{\Elzxh}). These, together with the commutation superoperators q${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$ and p${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$, form a canonical set of superoperators, [q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$]=[q${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$,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.

Physical Review AJan 1, 199187 citations

Quantum phase superoperator

Masashi Ban

A unitary superoperator that represents a quantum phase is introduced by means of a density-matrix formalism of quantum mechanics. In this formalism, a quantum phase can be treated without using any of the additional techniques that are indispensable for defining a unitary phase operator when conventional phase operator methods are used. Furthermore, a phase-space distribution function for number and phase variables is introduced by means of the eigenoperator of the phase superoperator, and the basic properties of the distribution function are investigated.A unitary superoperator that represents a quantum phase is introduced by means of a density-matrix formalism of quantum mechanics. In this formalism, a quantum phase can be treated without using any of the additional techniques that are indispensable for defining a unitary phase operator when conventional phase operator methods are used. Furthermore, a phase-space distribution function for number and phase variables is introduced by means of the eigenoperator of the phase superoperator, and the basic properties of the distribution function are investigated.

Physical Review AMar 1, 199511 citations

Potential Operators

WORLD SCIENTIFIC eBooksJan 1, 20220 citations

{)}_{\mathrm{Weyl}}

ordering=f(q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{+}$)1^. A generalization of Ehrenfest's theorem is obtained.