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

TL;DR

This paper explores path integration in spaces of constant curvature across various dimensions and geometries, emphasizing the invariance of propagators and Green functions and their interrelations through coordinate transformations.

Contribution

It provides a unified formulation of path integrals and propagators in different curved spaces, highlighting invariant quantities and identities between coordinate systems.

Findings

Explicit spectral solutions are formal in most coordinate systems.

Propagators and Green functions depend on invariant distances.

Identities connect path integral representations across coordinate systems.

Abstract

In this paper path integration in two- and three-dimensional spaces of constant curvature is discussed: i.e.\ the flat spaces $\bbbr^2$ and $\bbbr^3$, the two- and three-dimensional sphere and the two- and three dimensional pseudosphere. The Laplace operator in these spaces admits separation of variables in various coordinate systems. In all these coordinate systems 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.