Algebra of the Symmetry Operators of the Klein–Gordon–Fock Equation for the Case When Groups of Motions G3 Act Transitively on Null Subsurfaces of Spacetime

Valeriy V Obukhov

doi:10.3390/sym14020346

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).

Highlights

The Klein–Gordon–Fock equation describes the dynamics of massive spinless test particles interacting with fields of a gauge nature
The problem of finding the exact basic solutions of the Klein–Gordon–Fock equation in the external intensive fields is of great importance
The basic solution is a common eigenfunction of the complete set of symmetry operators

Summary

Introduction

The Klein–Gordon–Fock equation describes the dynamics of massive spinless test particles interacting with fields of a gauge nature. The noncommutative integration method is based on the complete classification of spacetime manifolds admitting groups of motions, as described in the book [19] The method made it possible to considerably extend the set of fields in which the construction of a complete system of solutions of the classical and quantum equations of a charged test particle motion is reduced to the integration of compatible systems of first-order differential equations. For arbitrary r and n, the following statement is true [24]: If the group of motions (Gr) of the space (Vn) acts transitively on the subspace (Vr,), Equations (6) and (7) form a completely integrable system This system specifies the necessary and sufficient conditions for the existence of symmetry operators that are linear in momenta

Notations and Necessary Information from Petrov Group Classification

Conclutions