LÉVY–LAPLACIAN AND THE GAUGE FIELDS

The following statement is proved for the Lévy–Laplacian defined as the Cesàro mean of second-order directional derivatives: a connection form on a base Riemannian C3-smooth manifold satisfies the Yang–Mills equations if and only if the parallel transport associated with the connection is Lévy harmonic. This statement is an improvement of the well-known result of L. Accardi, P. Gibilisco and I. V. Volovich2 (see also Ref. 1).

Download Full-text

Stochastic Lévy differential operators and Yang–Mills equations

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025717500084 ◽

2017 ◽

Vol 20 (02) ◽

pp. 1750008 ◽

Cited By ~ 8

Author(s):

Boris O. Volkov

Keyword(s):

Differential Operators ◽

Parallel Transport ◽

Directional Derivatives ◽

Chiral Field ◽

Action Functional ◽

Yang Mills ◽

Infinite Dimensional ◽

Lévy Laplacian ◽

Cesaro Averaging ◽

The Relationship

The relationship between the Yang–Mills equations and the stochastic analogue of Lévy differential operators is studied. The value of the stochastic Lévy–Laplacian is found by means of Cèsaro averaging of directional derivatives on the stochastic parallel transport. It is shown that the Yang–Mills equations and the Lévy–Laplace equation for such Laplacian are not equivalent in contrast to the deterministic case. An equation equivalent to the Yang–Mills equations is obtained. The equation contains the Lévy divergence. It is proved that the Yang–Mills action functional can be represented as an infinite-dimensional analogue of the Direchlet functional of a chiral field. This analogue is also derived using Cèsaro averaging.

Download Full-text

Lévy differential operators and Gauge invariant equations for Dirac and Higgs fields

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025719500012 ◽

2019 ◽

Vol 22 (01) ◽

pp. 1950001 ◽

Cited By ~ 2

Author(s):

Boris O. Volkov

Keyword(s):

Differential Operators ◽

Equations Of Motion ◽

Parallel Transport ◽

System Of Differential Equations ◽

Gauge Invariant ◽

Yang Mills ◽

Infinite Dimensional ◽

Dirac Equations ◽

Lévy Laplacian ◽

Higgs Fields

We study the Lévy infinite-dimensional differential operators (differential operators defined by the analogy with the Lévy Laplacian) and their relationship to the Yang–Mills equations. We consider the parallel transport on the space of curves as an infinite-dimensional analogue of chiral fields and show that it is a solution to the system of differential equations if and only if the associated connection is a solution to the Yang–Mills equations. This system is an analogue of the equations of motion of chiral fields and contains the Lévy divergence. The systems of infinite-dimensional equations containing Lévy differential operators, that are equivalent to the Yang–Mills–Higgs equations and the Yang–Mills–Dirac equations (the equations of quantum chromodynamics), are obtained. The equivalence of two ways to define Lévy differential operators is shown.

Download Full-text

CONSISTENT INTERACTIONS BETWEEN GAUGE FIELDS AND LOCAL BRST COHOMOLOGY: THE EXAMPLE OF YANG-MILLS MODELS

International Journal of Modern Physics D ◽

10.1142/s0218271894000149 ◽

1994 ◽

Vol 03 (01) ◽

pp. 139-144 ◽

Cited By ~ 34

Author(s):

G. BARNICH ◽

M. HENNEAUX ◽

R. TATAR

Keyword(s):

Vector Fields ◽

Jacobi Identity ◽

Second Order ◽

Gauge Fields ◽

Coupling Constant ◽

Yang Mills ◽

First Order ◽

Gauge Symmetries ◽

Brst Cohomology ◽

Structure Constants

Recent results on the cohomological reformulation of the problem of consistent interactions between gauge fields are illustrated in the case of the Yang-Mills models. By evaluating the local BRST cohomology through descent equation techniques, it is shown (i) that there is a unique local, Poincaré invariant cubic vertex for free gauge vector fields which preserves the number of gauge symmetries to first order in the coupling constant; and (ii) that consistency to second order in the coupling constant requires the structure constants appearing in the cubic vertex to fulfill the Jacobi identity. The known uniqueness of the Yang-Mills coupling is therefore rederived through cohomological arguments.

Download Full-text

Banach Lie Algebroids

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.2478/v10157-011-0035-y ◽

2011 ◽

Vol 57 (2) ◽

pp. 409-416

Author(s):

Mihai Anastasiei

Keyword(s):

Differential Equations ◽

Smooth Manifold ◽

Vector Bundles ◽

Second Order ◽

Lie Algebroids ◽

Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

Download Full-text

ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

Modern Physics Letters A ◽

10.1142/s0217732392001816 ◽

1992 ◽

Vol 07 (23) ◽

pp. 2077-2085 ◽

Cited By ~ 17

Author(s):

A. D. POPOV

Keyword(s):

Scalar Field ◽

Euclidean Space ◽

Lie Algebra ◽

Linear Equations ◽

Dual Solutions ◽

Gauge Fields ◽

System Of Equations ◽

Semisimple Lie Algebra ◽

Yang Mills ◽

Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

Download Full-text

Some subordination involving polynomials induced by lower triangular matrices

Advances in Difference Equations ◽

10.1186/s13662-020-03002-3 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Saiful R. Mondal ◽

Kottakkaran Sooppy Nisar ◽

Thabet Abdeljawad

Keyword(s):

Unit Disk ◽

Triangular Matrices ◽

Cesàro Mean ◽

Cesaro Mean ◽

Lower Triangular ◽

Lower Triangular Matrices

Abstract The article considers several polynomials induced by admissible lower triangular matrices and studies their subordination properties. The concept generalizes the notion of stable functions in the unit disk. Several illustrative examples, including those related to the Cesàro mean, are discussed, and connections are made with earlier works.

Download Full-text

Gauge × gauge = gravity on homogeneous spaces using tensor convolutions

Journal of High Energy Physics ◽

10.1007/jhep06(2021)117 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

L. Borsten ◽

I. Jubb ◽

V. Makwana ◽

S. Nagy

Keyword(s):

Homogeneous Spaces ◽

Gauge Fields ◽

Convolution Product ◽

Tensor Fields ◽

Einstein Universe ◽

Gauge Transformations ◽

Yang Mills ◽

Static Einstein Universe ◽

Definition Of ◽

Gauge Gravity

Abstract A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of ‘gravity = gauge × gauge’. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the ‘gauge × gauge’ convolution product on, for example, the static Einstein universe, and more generally for ultrastatic spacetimes with compact spatial slices.

Download Full-text

EXTENSION OF THE POINCARÉ GROUP AND NON-ABELIAN TENSOR GAUGE FIELDS

International Journal of Modern Physics A ◽

10.1142/s0217751x10051050 ◽

2010 ◽

Vol 25 (31) ◽

pp. 5765-5785 ◽

Cited By ~ 12

Author(s):

GEORGE SAVVIDY

Keyword(s):

Gauge Group ◽

Gauge Transformation ◽

Space Time ◽

Gauge Fields ◽

Poincaré Group ◽

Matrix Representations ◽

Poincare Group ◽

Yang Mills ◽

The Matrix ◽

Transversal Plane

In the recently proposed generalization of the Yang–Mills theory, the group of gauge transformation gets essentially enlarged. This enlargement involves a mixture of the internal and space–time symmetries. The resulting group is an extension of the Poincaré group with infinitely many generators which carry internal and space–time indices. The matrix representations of the extended Poincaré generators are expressible in terms of Pauli–Lubanski vector in one case and in terms of its invariant derivative in another. In the later case the generators of the gauge group are transversal to the momentum and are projecting the non-Abelian tensor gauge fields into the transversal plane, keeping only their positively definite spacelike components.

Download Full-text