## Proceedings of the International Conference on Geometry, Integrability and Quantization

### Path Integrals on Riemannian Manifolds with Symmetry and Stratified Gauge Structure

Shogo Tanimura

#### 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.

#### Article information

Dates
First available in Project Euclid: 12 June 2015

https://projecteuclid.org/ euclid.pgiq/1434145487

Digital Object Identifier
doi:10.7546/giq-3-2002-431-441

Mathematical Reviews number (MathSciNet)
MR1884865

Zentralblatt MATH identifier
1090.58502

#### Citation

Tanimura, Shogo. Path Integrals on Riemannian Manifolds with Symmetry and Stratified Gauge Structure. Proceedings of the Third International Conference on Geometry, Integrability and Quantization, 431--441, Coral Press Scientific Publishing, Sofia, Bulgaria, 2002. doi:10.7546/giq-3-2002-431-441. https://projecteuclid.org/euclid.pgiq/1434145487