An accurate power series expansion for path integrals on curved manifolds

Abstract

A general and simple framework for treating path integrals on curved manifolds is presented. The crucial point is expanding the exponent of the propagator of general diffusion processes in a power series in time. The expansion coefficients are determined by recursive relations and can be analytically evaluated to any desired level of accuracy int. The treatment is both theoretically and numerically advantageous with respect to the other path integral methods known in the literature. Its power is illustrated on two exactly solvable models. The propagator obtained is shown to be much more accurate over a broad range oft than the standard short time approximation. In view of its numerical application this means significant reducing the number of time steps that are required to evaluate a path integral.