Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

Alexander Breev; Alexander Shapovalov

doi:10.3390/sym12111867

journal article Open Access Nov 13, 2020

Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

Alexander Breev

Alexander Shapovalov

Symmetry Vol. 12 No. 11 pp. 1867 · MDPI AG

View at Publisher Save 10.3390/sym12111867

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.

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Details

Published: Nov 13, 2020
Vol/Issue: 12(11)
Pages: 1867
License: View

Authors

A

Alexander Breev

Department of Theoretical Physics, Tomsk State University, 1 Novosobornaya Sq., 634050 Tomsk, Russia

A

Alexander Shapovalov

Department of Theoretical Physics, Tomsk State University, 1 Novosobornaya Sq., 634050 Tomsk, Russia; Department of Mathematics and Informatics, Tomsk Polytechnic University, 30 Lenin ave., 634034 Tomsk, Russia

Funding

Russian Foundation for Basic Research Award: 20-01-00389 A

Cite This Article

Alexander Breev, Alexander Shapovalov (2020). Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces. Symmetry, 12(11), 1867. https://doi.org/10.3390/sym12111867

Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

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