Integrability, Coordinate Systems, and Separation of Variables in Path Integrals

TL;DR

This paper reviews how separation of variables is applied in path integrals for integrable quantum systems, including free motion and potential forces, and suggests exploring parametric coordinate systems on homogeneous spaces.

Contribution

It provides a summary of separation of variables in path integrals for integrable systems and proposes systematic study of parametric coordinate systems on homogeneous spaces.

Findings

Separation of variables aids in solving path integrals for integrable quantum systems.

Application to free motion and motion with potential forces.

Proposal for systematic investigation of coordinate systems on homogeneous spaces.

Abstract

In this contribution I summarize the achievements of separation of variables in integrable quantum systems from the point of view of path integrals. This includes the free motion on homogeneous spaces, and motion subject to a potential force, and I would like to propose systematic investigations of parametric coordinate systems on homogeneous spaces.