Gauge invariant asymptotic expansion of schrödinger propagators on manifolds

Abstract

A semiclassical technique is developed for approximating the time evolution of quantum systems on semi-Riemannian manifolds in the presence of external electromagnetic fields. In particular, a geometrically covariant and gauge covariant formal asymptotic expansion of the Schrödinger propagator is derived. The theory is applicable not only to spinless nonrelativistic quantum systems with constraints, but also to general-relativistic scalar field theory through the Schwinger-DeWitt equation. The basic classical objects in this approach are the geodesics of the manifold. The expansion—in the derivatives of the gravitational and electromagnetic fields—is valid when both propagator arguments lie in a geodesically convex region. The zeroth order approximation consists of the wkb propagator for free quantum evolution on the manifold, modified by a gauge-integral phase factor which carries all the gauge structure of the exact propagator. Higher order expansion coefficients are gauge invariant solutions of a one-dimensional integral recurrence relation derived from geodesic transport. By using a Jacobi field and a Green function associated with the geodesic deviation equation, the expansion coefficients can be put into an explicit form. The first one is computed in detail and manifestly displays the effects of gravity and electromagnetism, both separately and in combination. The expansion derived here generalizes the well-known short time expansion in several ways.