Algebra of the Symmetry Operators of the Klein–Gordon–Fock Equation for the Case When Groups of Motions <i>G</i>₃ Act Transitively on Null Subsurfaces of Spacetime

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 acts 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 the r-parametric groups of motions of spacetime manifolds if (r≤4).

Keywords

• Klein–Gordon–Fock equation
• algebra of symmetry operators
• theory of symmetry
• separation of variables
• linear partial differential equations