Path Integrals in Curved Spaces

Abstract

In this paper we present a simplification of the path integral solution of the Schrödinger equation in terms of coordinates which need not be Cartesian. After presenting the existing formula, we discuss the relationship between the distance and time differentials. Making this relationship precise through the technique of stationary phase, we are able to simplify the path integral. The resulting expression can be used to obtain a Hamiltonian path integral. Finally, we comment on a similar phenomenon involving differentials in the Itô integral.