Group conjugations of Dirac operators as an invariant of the Riemannian manifold

摘要

We study a matrix group that. acts on the set of Dirac operators in a four-dimensional Riemannian space with an arbitrary signature. It is proved that the considered group depends neither on the construction of Dirac operators nor on the system of coordinates and in this sense it is some invariant of a Riemannian manifold. The introduced group is a Lie group in a general case. If we know the dimension of Lie group algebra, we can answer the question of how many linearly independent Dirac operators one can construct on the given manifold. As an example, we calculate the considered group in Minkowski space, de Sitter space and in spaces with groups of motion. Some generalizations in higher dimensions are also discussed. One can use the group structure introduced to classify Riemannian spaces and explain the appearance of hidden symmetries of the Dirac equation in a curved spacetime. [References: 19]