Yang-Mills gauge fields conserving the symmetry algebra of the Dirac equation in a homogeneous space

TL;DR

This paper investigates Yang-Mills gauge fields that preserve the symmetry algebra of the Dirac equation in homogeneous spaces, providing explicit solutions and eigenvalues in specific geometries like de Sitter space and .

Contribution

It identifies conditions for Yang-Mills fields to maintain Dirac symmetry in homogeneous spaces and demonstrates solution methods in particular geometries.

Findings

Identified Yang-Mills fields preserving Dirac symmetry algebra.

Derived eigenfunctions and eigenvalues in  space.

Provided explicit examples in de Sitter space.

Abstract

We consider the Dirac equation with an external Yang-Mills gauge field in a homogeneous space with an invariant metric. The Yang-Mills fields for which the motion group of the space serves as the symmetry group for the Dirac equation are found by comparison of the Dirac equation with an invariant matrix differential operator of the first order. General constructions are illustrated by the example of de Sitter space. The eigenfunctions and the corresponding eigenvalues for the Dirac equation are obtained in the space $\mathbb{R}^2\times \mathbb{S}^2$ by a noncommutative integration method.