On the geometrical representation of the path integral reduction Jacobian: The case of dependent coordinates in the description of the reduced motion

TL;DR

This paper derives a geometric expression for the path integral reduction Jacobian in scalar particle quantization on Riemannian manifolds with group actions, focusing on dependent coordinates in the reduced motion.

Contribution

It provides a new geometric formulation of the reduction Jacobian for dependent coordinates in path integral quantization on manifolds with symmetry.

Findings

Derived the reduction Jacobian using scalar curvature of the original manifold.

Extended the geometric representation to dependent coordinate cases.

Connected the Jacobian to the scalar curvature of the principal fiber bundle.

Abstract

The geometrical representation of the path integral reduction Jacobian obtained in the problem of the path integral quantization of a scalar particle motion on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group has been found for the case when the local reduced motion is described by means of dependent coordinates. The result is based on the scalar curvature formula for the original manifold which is viewed as a total space of the principal fibre bundle.