Quantum systems with external electromagnetic fields: The large mass asymptotics

Abstract

The large mass asymptotics of the quantum evolution problem for a system of charged particles that mutually interact through scalar fields and couple to an arbitrary time-varying external electromagnetic field is rigorously described. If K(x,t; y,s;m) denotes the coordinate space propagator (time evolution kernel) of this system, the singular perturbation behavior of K as mass m→∞ is expressed in terms of a gauge invariant asymptotic expansion. In terms of the external fields and interparticle interactions, this expansion provides a nonperturbative approximation for the propagator K that is valid for all particle coordinates x, y and for finite time displacements t−s. For the class of analytic scalar and vector fields that are defined as Fourier transforms of time-dependent measures, the existence of this asymptotic series for K in powers of (m)−1 is established for both real and complex masses. Explicit bounds for the error term are obtained and a manifestly gauge invariant transport recurrence relation is derived that uniquely determines all the coefficient functions of the asymptotic series. The small time asymptotic expansion of K is shown to be embedded within the large mass expansion.