arXiv subjects

S. N. Storchak

Publications and source records attributed to S. N. Storchak.

At least 19 recordsLinked to original sources

arXiv2025

An example of path integral reduction for a simple symmetric mechanical system on a product manifold

A path integral reduction procedure in Wiener-type path integrals, based on the approach developed in arXiv:1912.13124, is applied to a simple invariant mechanical system defined on a product manifold with a given free, proper and isometric action of the group SO(2). The Jacobian of the path integral reduction and the integral relation between the path integrals representing the fundamental solutions of the parabolic equations on initial and reduced manifolds are obtained.

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arXiv2024

Notes on path integral reduction in scalar electrodynamics

Based on a method developed earlier for a finite-dimensional mechanical system, the problem of path integral reduction for scalar electrodynamics is considered. Using the Coulomb gauge, the stochastic differential equations for the reduced dynamics on the orbit space are obtained. It is shown that the geometry of the reduced space is completely determined by the behaviour of the material fields. Since the main role in the singular behavior of the reduction Jacobian is played by the mean curvature of the orbit, the final solution of the path inyegral reduction problem in the field system under consideration is possible only by carrying out an adequate regularization of the term that determines the volume of the orbit, and on which the additional correction to the interaction potential completely depends.

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arXiv2023

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.

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arXiv2023

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.

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arXiv2020

Non-zero momentum level reduction in path integrals for dynamical systems with symmetry given on a product manifold consisting of the total space of the principal fiber bundle and a vector space

The case of non-zero momentum level reduction in Wiener path integrals for a mechanical system with symmetry describing the motion of two scalar particles with interaction on a Riemannian product manifold with the given action a compact semisimple Lie group is considered. The original product manifold consists of the vector space and a smooth compact finite-dimensional Riemannian manifold, which, due to the action of the group, can be regarded as the total space of the principal fiber bundle. The integral relation between the path integrals representing the fundamental solutions of the backward Kolmogorov equation defined on the total space of the principal fiber bundle (the original Riemannian product manifold) and the corresponding backward Kolmogorov equation gion the space of the sections of the associated covector bundle is obtained.

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arXiv2020

On the geometric representation of the path integral reduction Jacobian for a mechanical system with symmetry given on a manifold that is a product of the total space of the principal fiber bundle and the vector space

For the Jacobian resulting from the previously considered problem of the path integral reduction in Wiener path integrals for a mechanical system with symmetry describing the motion of two interacting scalar particles on a manifold that is the product of a smooth compact finite-dimensional Riemannian manifold and a finite-dimensional vector space, a geometric representation is obtained. This representation follows from the formula for the scalar curvature of the original manifold endowed by definition with a free isometric smooth action of a compact semisimple Lie group. The derivation of this formula is performed using adapted coordinates, which can be determined in the principal fiber bundle associated with the problem under the study. These coordinates are similar to those used in the standard approach to quantization of Yang-Mills fields interacting with scalar fields.

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arXiv2019

Path integrals on a manifold that is a product of the total space of the principal fiber bundle and the vector space

Using the path integral measure factorization method based on the nonlinear filtering equation from the stochastic process theory, we consider the reduction procedure in Wiener path integrals for a mechanical system with symmetry that describes the motion of two interacting scalar particles on a special smooth compact Riemannian manifold - the product of total space of the principal fiber bundle and the vector space. The original manifold, the configuration space of this system, is endowed with an isometric free proper action of a compact semisimple unimodular Lie group. The proposed reduction procedure leads to the integral relation between path integrals that represent fundamental solutions of the inverse Kolmogorov equations on the initial and reduced manifolds. For the case of reduction onto the zero-momentum level, the reduction Jacobian is obtained, which is an additional potential term to the Hamiltonian.

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arXiv2019

Transition to the case of "resolved gauge" in the Lagrange-Poincaré equations for a mechanical system with symmetry on the total space of a principal fiber bundle whose base is the bundle space of the associated bundle

This note is a continuation of our earlier articles arXiv:1612.08897 and arXiv:1709.09030, where using the dependent coordinates the local Lagrange-Poincaré equations were obtained for a mechanical system with symmetry describing the motion of two interacting scalar particles on a special Riemannian manifold (the product of the total space of the principal fiber bundle and vector space), on which a free proper and isometric action of a compact semisimple Lie group is given. Assuming the existence of the parametric representations for local sections in the principal bundle, we make the transition to independent coordinates in the obtained Lagrange-Poincaré equations.

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arXiv2018

The Lagrange-Poincaré equations for interacting Yang-Mills and scalar fields

A special case of the Lagrange-Poincaré equations for the gauge field interacting with a scalar field is obtained. For description of the dynamics on the configuration space, the adapted coordinates are used. After neglecting the group variables the obtained equations describe the evolution on the gauge orbit space of the principal fiber bundle which is related to the system under the consideration.

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arXiv2017

Equations of a relative equilibrium in Yang-Mills theory

The equations of a relative equilibrium in a pure Yang--Mills gauge theory with the Coulomb gauge fixing are obtained. They are derived as a direct consequence of the results of our previous work on Wong's equations in gauge theory.The obtained equations are similar to the equations of a relative equilibrium in reducible finite-dimensional dynamical systems with a symmetry. In all these equations, the description of the reduced motion is performed by making use of the dependent coordinates.

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arXiv2017

Coordinate representation of the Lagrange-Poincaré equations for a mechanical system with symmetry on the total space of a principal fiber bundle whose base is the bundle space of the associated bundle

Using the dependent coordinates, the local Lagrange-Poincaré equations and equations for the relative equilibria are obtained for a mechanical system with a symmetry describing the motion of two interacting scalar particles on a special Riemannian manifold (the product of the total space of the principal fiber bundle and the vector space) on which a free proper and isometric action of a compact semi-simple Lie group is given. As in gauge theories, dependent coordinates are implicitly determined by means of equations representing the local sections of the principal fiber bundle.

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arXiv2016

The Lagrange-Poincaré equations for a mechanical system with symmetry on the principal fiber bundle over the base represented by the bundle space of the associated bundle

The Lagrange--Poincaré equations for a mechanical system which describes the interaction of two scalar particles that move on a special Riemannian manifold, consisting of the product of two manifolds, the total space of a principal fiber bundle and the vector space, are obtained. The derivation of equations is performed by using the variational principle developed by Poincaré for the mechanical systems with a symmetry. The obtained equations are written in terms of the dependent variables which, as in gauge theories, are implicitly determined by means of equations representing the local sections of the principal fiber bundle.

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arXiv2014

Dependent coordinates in the Lagrange-Poincaré equations for mechanical systems with symmetry

The Lagrange--Poincaré equations for the mechanical system describing the motion of a scalar particle on a Riemannian manifold with a given free and isometric action of a compact Lie group is obtained. In an arising principle fibre bundle, the total space of which serves as a configuration space of the considered mechanical system, the local description of the reduced motion is done in terms of dependent coordinates. In obtaining of the equations we use the variational principle developed by Poincaré for the mechanical systems with a symmetry.

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arXiv2012

Path integral on a manifold with a non-free group action

The method of the factorization of the path integral measure, based on a nonlinear filtering equation, is extended to the case of a nonfree isometric action of the compact semisimple unimodular Lie group on a smooth compact Riemannian manifold. The method is applied to the path integral which describes the "quantum" motion of the scalar particle on this manifold. The relation between path integral representing the solution of the parabolic equation on initial and reduced manifold is derived. It is shown that reduction reduction leads to the non invariance of the path integral measure.

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arXiv2011

Wong's equations in Yang-Mills theory

We derive Wong's equations for the finite-dimensional dynamical system representing the motion of a scalar particle on a compact Riemannian manifold with a given free isometric smooth action of a compact semisimple Lie group. The obtained equations are written in terms of dependent coordinates which are typically used in an implicit description of the local dynamics given on the orbit space of the principal fiber bundle. Using these equations we obtain Wong's equations in a pure Yang--Mills gauge theory with the Coulomb gauge fixing. This result is based on the existing analogy between the reduction procedures carried out in our finite-dimensional dynamical system and in Yang-Mills gauge fields.

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arXiv2009

Path integral representation of the quantum evolution in dynamical systems with a symmetry for the non-zero momentum level reduction

For the case of reduction onto the non-zero momentum level, 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 semisimle Lie group, the path integral representation of the matrix Green's function, which describes the quantum evolution of the reduced motion, has been obtained. The integral relation between the path integrals representing the fundamental solutions of the parabolic differential equation defined on the total space of the principal fiber bundle and the linear parabolic system of the differential equations on the space of the sections of the associated covector bundle has been derived.

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arXiv2008

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.

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arXiv2007

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.

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