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 ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$, 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.

70 pages, AmSTeX, DESY 93 - 141 (mailer corrupted file, and truncated it)

References in corpus (1)

• Path Integral Solution of a Class of Explicitly Time-Dependent Potentials

Cited by in corpus (6)

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• How the Geometric Calculus Resolves the Ordering Ambiguity of Quantum Theory in Curved Space
• Time-Slicing Path-integral in Curved Space
• Path Integration on Darboux Spaces
• Path Integration on Hermitian Hyperbolic Space