Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces
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-commutati...
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| Veröffentlicht in: | Symmetry (Basel) 2020-11, Vol.12 (11), p.1867 |
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| creator | Breev, Alexander Shapovalov, Alexander |
| description | 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. |
| doi_str_mv | 10.3390/sym12111867 |
| format | Article |
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| source | MDPI - Multidisciplinary Digital Publishing Institute; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Dirac equation Exact solutions Generators Integrals Lie groups Mathematical analysis Ordinary differential equations Partial differential equations Quantum field theory Separation Spacetime Symmetry |
| title | Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces |
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