Algebra of the symmetry operators of the Klein-Gordon-Fock equation for the case when groups of motions $G_3$ act transitively on null subsurfaces of spacetime

TL;DR

This paper classifies the symmetry algebras of the Klein-Gordon-Fock equation for a charged particle in specific electromagnetic fields within spacetimes admitting a three-parameter group of motions acting transitively on null hypersurfaces.

Contribution

It identifies all admissible electromagnetic fields that preserve the symmetry algebra structure of the Klein-Gordon-Fock equation in these spacetimes.

Findings

All admissible electromagnetic fields preserving symmetry algebras are found.

Such fields do not deform the algebra of symmetry operators for free equations.

Complete classification of admissible fields for symmetry algebras up to four parameters.

Abstract

The algebras of the symmetry operators for the Hamilton-Jacobi and Klein-Gordon-Fock equations are found for a charged test particle moving in an external electromagnetic field in a spacetime manifold, on the isotropic (null) hypersurface of which a three-parameter groups of motions act transitively.on the isotropic (null) hypersurface of which a three-parameter groups of motions act transitively. We have found all admissible electromagnetic fields for which such algebras exist. We have proved that an admissible field does not deform the algebra of symmetry operators for the free Hamilton-Jacobi and Klein-Gordon-Fock equations. The results complete the classification of admissible electromagnetic fields in which the Hamilton-Jacobi and Klein-Gordon-Fock equations admit algebras of motion integrals that are isomorphic to the algebras of operators of $r$-parametric groups of motions of spacetime manifolds if $(r \leq 4)$.