Path Integrals on Riemannian Manifolds with Symmetry and Stratified Gauge Structure

TL;DR

This paper develops a framework for path integrals on Riemannian manifolds with symmetry, including singular quotient spaces, by decomposing the integral into components related to group representations and stratification geometry.

Contribution

It introduces a method to analyze path integrals on manifolds with symmetry and singularities using stratification geometry and representation theory.

Findings

Path integrals decompose into rotational, vibrational, and holonomy factors.

The approach handles orbifold singularities without assuming free group actions.

Reduced path integrals are classified by irreducible unitary representations of the symmetry group.

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 a 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.