On the geometric representation of the path integral reduction Jacobian in the path integral for interacting systems: The case of dependent coordinates in the description of reduced motion on the orbit space

TL;DR

This paper derives a geometric representation of the path integral reduction Jacobian for a mechanical system with dependent coordinates, using scalar curvature and Christoffel symbols in a nonholonomic basis, relevant for quantizing interacting particles.

Contribution

It provides a new geometric formulation of the path integral reduction Jacobian for systems with dependent coordinates, extending previous results to a broader class of models.

Findings

Derived a geometric representation of the reduction Jacobian.

Connected the Jacobian to scalar curvature of the original manifold.

Utilized a nonholonomic basis for calculations.

Abstract

A geometric representation is found for the previously obtained path integral reduction Jacobian in Wiener-type path integral when quantizing a model mechanical system, which is used to describe the motion of two interacting scalar particles on a product manifold (a smooth compact finite-dimensional Riemannian manifold and vector space) with a given free isometric action of a compact semisimple Lie group. The reduction Jacobian we are dealing with was obtained for the case when, as in gauge theories, dependent coordinates are used to locally describe the reduced motion. As in our similar works, the result is based on the scalar curvature formula for the original manifold which is viewed as a total space of the principal fiber bundle. The calculation of the Christoffel symbols and scalar curvature was performed in a special nonholonomic basis, also known as the horizontal lift basis.