Path Integration and Separation of Variables in Spaces of Constant Curvature in Two and Three Dimensions

Abstract

In this paper path integration in two- and three-dimensional spaces of constant curvature is discussed: i.e. the flat spaces R2 and R3, the two- and three-dimensional sphere and the two- and three-dimensional pseudosphere. We are going to discuss all coordinates systems where the Laplace operator admits separation of variables. In all of them the path integral formulation will be stated, however in most of them an explicit solution in terms of the spectral expansion can be given only on a formal level. What can be stated in all cases, are the propagator and the corresponding Green function, respectively, depending on the invariant distance which is a coordinate independent quantity. This property gives rise to numerous identities connecting the corresponding path integral representations and propagators in various coordinate systems with each other.