Separation of variables in the Dirac equation in Stackel spaces

Abstract

The subspaces of Riemannian space of signature (+---) that admit separation of the Dirac equation have been found in the case of Riemannian space admitting the separation of the Hamilton-Jacobi equation. For the separation of variables in the Hamilton-Jacobi equation it is necessary for the complete set of Killing vectors and tensors to be of a special kind. Every complete set defines its own type of metric of Riemannian space which is called Stackel space. The Dirac equation does not permit the separation of variables in general cases of Stackel space. The main idea of the paper is in the construction, in Stackel space, of a complete set of another kind. This complete set consists of three matrix first-order differential symmetry operators of the Dirac equation. The operators are pairwise commuting and linearly independent. The complete set structure is in agreement with the structure of the Killing vectors and tensors of Stackel space. The separation of variables in the Dirac equation has been carried out in the explicit form in Stackel spaces which admit complete sets of symmetry operators. These operators have been used essentially in the process of separation that differs from Chandrasekhar method.