Path Integrals on Riemannian Manifolds with Symmetry and Induced Gauge Structure

TL;DR

This paper develops a general framework for path integrals on Riemannian manifolds with symmetry, reducing them to quotient spaces and classifying them via group representations, applicable to complex inhomogeneous spaces.

Contribution

It introduces a comprehensive formulation of path integrals on manifolds with non-free group actions, incorporating stratification geometry and covering all inequivalent quantizations.

Findings

Path integrals reduce to quotient space integrals classified by group representations.

The formulation applies to inhomogeneous spaces with stratification geometry.

Boundary conditions are established at singular points in the quotient space.

Abstract

We formulate path integrals on any Riemannian manifold which admits the action of a compact Lie group by isometric transformations. We consider a path integral on a Riemannian manifold M on which a Lie group G acts isometrically. Then we show that the path integral on M is reduced to a family of path integrals on a quotient space Q=M/G and that 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 our formulation is applicable to a wide class of manifolds, which includes inhomogeneous spaces, and it covers all the inequivalent quantizations. To describe the path integral on inhomogeneous space, stratification geometry, which is a generalization of the concept of principal fiber bundle, is necessarily introduced. Using it we show that the path integral is expressed as a product of three factors; the rotational energy amplitude, the vibrational energy amplitude, and the holonomy factor. When a singular point arises in $ Q $, we determine the boundary condition of the path integral kernel for a path which runs through the singularity.