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