## 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 AdS_{3} using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the non-commutative integration method.

Original language | English |
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Article number | 1867 |

Pages (from-to) | 1-30 |

Number of pages | 30 |

Journal | Symmetry |

Volume | 12 |

Issue number | 11 |

DOIs | |

Publication status | Published - Nov 2020 |

## Keywords

- Dirac equation
- Homogeneous spaces
- Induced representations
- Non-commutative integration
- Orbit method

## ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Chemistry (miscellaneous)
- Mathematics(all)
- Physics and Astronomy (miscellaneous)