Path integral on a manifold with a non-free group action

TL;DR

This paper extends a path integral measure factorization method to manifolds with nonfree group actions, analyzing quantum motion and measure invariance issues in such geometric settings.

Contribution

It introduces a generalized measure factorization approach for path integrals on manifolds with nonfree group actions, linking solutions on original and reduced manifolds.

Findings

Path integral measure becomes non-invariant under reduction.

Relation between original and reduced path integrals is established.

Application to quantum scalar particle motion on manifolds.

Abstract

The method of the factorization of the path integral measure, based on a nonlinear filtering equation, is extended to the case of a nonfree isometric action of the compact semisimple unimodular Lie group on a smooth compact Riemannian manifold. The method is applied to the path integral which describes the "quantum" motion of the scalar particle on this manifold. The relation between path integral representing the solution of the parabolic equation on initial and reduced manifold is derived. It is shown that reduction reduction leads to the non invariance of the path integral measure.