Path Integral Representations on the Complex Sphere

TL;DR

This paper explores path integral representations for coordinate systems on the complex sphere S3C, providing explicit solutions, wave-function expansions, and closed-form kernels for multiple orthogonal coordinate systems.

Contribution

It enumerates 21 orthogonal coordinate systems on the complex sphere and derives explicit path integral representations for most, including wave-function expansions and closed-form Green functions.

Findings

Explicit path integral representations for 21 coordinate systems.

Closed-form expressions for kernels and Green functions.

Wave-function expansions for solutions.

Abstract

In this paper we discuss the path integral representations for the coordinate systems on the complex sphere S3C. The Schroedinger equation, respectively the path integral, separates in exactly 21 orthogonal coordinate systems. We enumerate these coordinate systems and we are able to present the path integral representations explicitly in the majority of the cases. In each solution the expansion into the wave-functions is stated. Also, the kernel and the corresponding Green function can be stated in closed form in terms of the invariant distance on the sphere, respectively on the hyperboloid.