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

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.Comment: 17 page

On geometrical representation of the Jacobian in a path integral reduction problem

The geometrical representation of the Jacobian in the path integral reduction problem which describes a motion of the scalar particle on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group is obtained. By using the formula for the scalar curvature of the manifold with the Kaluza--Klein metric, we present the Jacobian as difference of the scalar curvature of the total space of the principal fibre bundle and the terms that are the scalar curvature of the orbit space, the scalar curvature of the orbit, the second fundamental form of the orbit and the square of the principle fibre bundle curvature.Comment: 8 page

Reduction of path integrals for interacting systems: The case of using dependent coordinates in the description of reduced motion on the orbit space

We consider a reduction procedure in Wiener-type path integral for a finite-dimensional mechanical system with a symmetry representing the motion of two interacting scalar particles on a manifold that is the product of the total space of the principal bundle and a vector space. By analogy with what is done in gauge theories, the local description of the reduced motion on orbit space is carried out using dependent coordinates. The factorization of the measure in the path integral, which is necessary for the reduction, is based on the application of the stochastic differential equation of the optimal nonlinear filtering from the theory of stochastic processes. The non-invariance of the measure in the path integral under the reduction is shown. The Jacobian of the reduction is generated by the projection of the mean curvature vector field of the orbit onto the submanifold, which is used to determine the adapted coordinates in the principal fiber bundle associated with the problem under study.Comment: 38 pages, the final formula of the reduction Jacobian has been revised, some typos correcte

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

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.Comment: 41 pages, Expressions for some Christoffel symbols have been revised, typos and text have been correcte