Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

Abstract

We develop a non-commutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the non-commutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the non-commutative integration method. In addition, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time AdS3 using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the non-commutative integration method.

Separation of variables in Hamilton–Jacobi equation for a charged test particle in the Stackel spaces of type (2.1)

“…Note that the condition of mutually commutativity of the symmetry operators of the equation (13) is not necessary for the exact integration, since the theory of Stackel spaces can be generalized for a case of non-commutative operators of motion (see, [25] [26]).…”

Exact Solutions of Maxwell Equations in Homogeneous Spaces with the Group of Motions G3(IX)

“…The method was proposed in [27]. The development of the method and its application to gravity theory can be found in [28], [29], [30], [31].…”

Algebra of Symmetry Operators for Klein-Gordon-Fock Equation

“…In particular, in [31][32][33], a complete classification of spaces admitting a simply transitive action of the motions groups G 4 was obtained, provided that the Klein-Gordon-Fock equation is exactly solved by non-commutative integration methods. In [34][35][36][37][38], a similar problem was solved for Dirac-Fock equation.…”