Path Integrals on Riemannian Manifolds with Symmetry and Stratified Gauge Structure

Abstract

We study a quantum system in a Riemannian manifold $M$ on which a Lie group $G$ acts isometrically. The path integral on $M$ is decomposed into a family of path integrals on quotient space $Q = M/G$ and the reduced path integrals are completely classified by irreducible unitary representations of $G$. It is not necessary to assume that the action of $G$ on $M$ is either free or transitive. Hence the quotient space $M/G$ may have orbifold singularities. Stratification geometry, which is a generalization of the concept of principal fiber bundle, is necessarily introduced to describe the path integral on $M/G$. Using it we show that the reduced path integral is expressed as a product of three factors; the rotational energy amplitude, the vibrational energy amplitude, and the holonomy factor.