Non-commutative integration of the dirac equation in homogeneous spaces

Alexander Breev, Alexander Shapovalov

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number1867
Pages (from-to)1-30
Number of pages30
Issue number11
Publication statusPublished - Nov 2020


  • 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)

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