A PHASE-SPACE TECHNIQUE FOR THE PERTURBATION EXPANSION OF SCHRODINGER PROPAGATORS

A perturbation theory for Schrödinger and heat equations that is based on phase-space variables is developed. The Dyson series representing the evolution kernel is described in terms of two basic classical quantities: the free classical motion along flat space geodesies and the Green function for the Jacobi operator in phase space. Further, for problems with Abelian interactions it is demonstrated that the perturbation theory may be summed to all orders yielding an exponentiated connected graph description for the evolution kernel. Connected graph representations provide an efficient method of constructing various semiclassical approximations wherein expansion coefficients are directly determined by explicit cluster integrals. This type of application is discussed for the case of Schrödinger and heat equations with external electromagnetic fields. Detailed expressions for coefficients are obtained for both the gauge invariant large mass expansion as well as the short time Schwinger-DeWitt expansion. Finally it is shown how to apply this phase-space method so that it incorporates a recently proposed covariant perturbation theory. © 1995 American Institute of Physics.