MOYAL QUANTUM-MECHANICS - THE SEMICLASSICAL HEISENBERG DYNAMICS

The Moyal description of quantum mechanics, based on the Wigner-Weyl i somorphism between operators and symbols, provides a comprehensive pha se space representation of dynamics. The Weyl symbol image of the Heis enberg picture evolution operator is regular in (h) over bar and so pr esents a preferred foundation for semiclassical analysis. Its semiclas sical expansion ''coefficients,'' acting on symbols that represent obs ervables, are simple, globally defined (phase space) differential oper ators constructed in terms of the classical now. The first of two pres ented methods introduces a cluster-graph expansion for the symbol of a n exponentiated operator, which extends Groenewold's formula for the W eyl product of two symbols and has (h) over bar as its natural small p arameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of ''quantum trajecto ries.'' Their Green function solutions construct the regular (h) over bar down arrow 0 asymptotic series for the Heisenberg-Weyl evolution m ap. The second method directly substitutes such a series into the Moya l equation of motion and determines the (h) over bar coefficients recu rsively. In contrast to the WKB approximation for propagators, the Hei senberg-Weyl description of evolution involves no essential singularit y in (h) over bar, no Hamilton-Jacobi equation to solve for the action , and no multiple trajectories, caustics, or Maslov indices. (C) 1995 Academic Press, Inc.