# Journal of Nonlinear Mathematical Physics

Volume 27, Issue 2, January 2020, Pages 199 - 218

# On constant solutions of SU(2) Yang-Mills equations with arbitrary current in Euclidean space ℝn

Authors
Dmitry Shirokov
National Research University Higher School of Economics, Myasnitskaya str. 20, Moscow, 101000, Russia,dm.shirokov@gmail.com
Institute for Information Transmission Problems of Russian Academy of Sciences, Bolshoy Karetny per. 19, Moscow, 127051, Russia
Received 21 July 2019, Accepted 20 August 2019, Available Online 27 January 2020.
DOI
10.1080/14029251.2020.1700625How to use a DOI?
Keywords
Yang-Mills equations; singular value decomposition; cubic equations; constant solutions; SU(2)
Abstract

In this paper, we present all constant solutions of the Yang-Mills equations with SU(2) gauge symmetry for an arbitrary constant non-Abelian current in Euclidean space ℝn of arbitrary finite dimension n. Using the invariance of the Yang-Mills equations under the orthogonal transformations of coordinates and gauge invariance, we choose a specific system of coordinates and a specific gauge fixing for each constant current and obtain all constant solutions of the Yang-Mills equations in this system of coordinates with this gauge fixing, and then in the original system of coordinates with the original gauge fixing. We use the singular value decomposition method and the method of two-sheeted covering of orthogonal group by spin group to do this. We prove that the number (0, 1, or 2) of constant solutions of the Yang-Mills equations in terms of the strength of the Yang-Mills field depends on the singular values of the matrix of current. The explicit form of all solutions and the invariant F2 can always be written using singular values of this matrix. The relevance of the study is explained by the fact that the Yang-Mills equations describe electroweak interactions in the case of the Lie group SU(2). Non-constant solutions of the Yang-Mills equations can be considered in the form of series of perturbation theory. The results of this paper are new and can be used to solve some problems in particle physics, in particular, to describe physical vacuum and to fully understand a quantum gauge theory.

Open Access

Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
27 - 2
Pages
199 - 218
Publication Date
2020/01/27
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2020.1700625How to use a DOI?
Open Access

TY  - JOUR
AU  - Dmitry Shirokov
PY  - 2020
DA  - 2020/01/27
TI  - On constant solutions of SU(2) Yang-Mills equations with arbitrary current in Euclidean space ℝⁿ
JO  - Journal of Nonlinear Mathematical Physics
SP  - 199
EP  - 218
VL  - 27
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1700625
DO  - 10.1080/14029251.2020.1700625
ID  - Shirokov2020
ER  -