Generalized Moyal structures in phase-space, master equations and their classical limit: I. General formalism

Generalized Wigner and Weyl transformations of quantum operators are defined and their properties, as well as those of the algebraic structure induced on phase-space, are reviewed. Using such transformations, quantum linear evolution equations are given a phase-space representation; particularly, the general master equation of the Lindblad type generating quantum dynamical semigroups. The resulting expressions are better suited for deriving quantum corrections, taking the classical limit and for a general comparison of classical and quantum systems. We show that under quite general conditions, the classical limit of this master equation exists, is independent of the particular generalized Wigner transformation used and is an equation of the Fokker–Planck type (i.e. with nonnegative-definite leading coefficient) generating a classical Markov semigroup.