On constant solutions of ${\rm SU}(2)$ Yang-Mills equations with arbitrary current in Euclidean space ${\mathbb R}^n$

TL;DR

This paper classifies all constant solutions of SU(2) Yang-Mills equations with arbitrary currents in Euclidean space, revealing how the number of solutions depends on the current's matrix singular values, with implications for particle physics.

Contribution

It provides a complete classification of constant solutions to SU(2) Yang-Mills equations with arbitrary currents in any dimension, using singular value decomposition and gauge fixing techniques.

Findings

Number of solutions depends on singular values of the current matrix.

Explicit solutions and invariant quantities are expressed via singular values.

Results are relevant for understanding the physical vacuum in gauge theories.

Abstract

In this paper, we present all constant solutions of the Yang-Mills equations with ${\rm SU}(2)$ gauge symmetry for an arbitrary constant non-Abelian current in Euclidean space ${\mathbb R}^n$ 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 $F^2$ 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 ${\rm SU}(2)$. Nonconstant 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.