On constant solutions of SU(2) Yang-Mills-Dirac equations

TL;DR

This paper provides a complete classification and explicit forms of all constant solutions to SU(2) Yang-Mills-Dirac equations in Minkowski space, using advanced matrix decomposition methods and group coverings.

Contribution

It introduces a comprehensive classification of constant solutions for SU(2) Yang-Mills-Dirac equations, including explicit forms and new analytical methods.

Findings

Complete classification of constant solutions

Explicit forms of all solutions provided

Methodology involving hyperbolic singular value decomposition

Abstract

For the first time, a complete classification of all constant solutions of the Yang-Mills-Dirac equations with SU(2) gauge symmetry in Minkowski space ${\mathbb R}^{1,3}$ is given. The explicit form of all solutions is presented. We use the method of hyperbolic singular value decomposition of real and complex matrices and the two-sheeted covering of the group SO(3) by the group SU(2). In the degenerate case of zero potential, we use the pseudo-unitary symmetry of the Dirac equation. Nonconstant solutions can be considered in the form of series of perturbation theory using constant solutions as a zeroth approximation; the equations for the first approximation in the expansion are written.