Algebras of integrals of motion for the Hamilton-Jacobi and Klein-Gordon-Fock equations in spacetime with a four-parameter groups of motions in the presence of an external electromagnetic field

TL;DR

This paper characterizes the algebraic structures of integrals of motion for Hamilton-Jacobi and Klein-Gordon-Fock equations in spacetimes with specific symmetries and external electromagnetic fields, identifying conditions for their existence.

Contribution

It identifies all admissible electromagnetic fields compatible with nontransitive four-parameter motion groups in spacetime and proves that these fields do not deform the symmetry algebra of the equations.

Findings

All admissible electromagnetic fields for the given spacetime symmetries are found.

The algebra of symmetry operators remains undeformed under these admissible fields.

Conditions for the potentials of admissible electromagnetic fields are derived and analyzed.

Abstract

The algebras of the integrals of motion of the Hamilton-Jacobi and Klein-Gordon-Fock equations for a charged test particle moving in an external electromagnetic field in a spacetime manifold are found. The manifold admits a four-parameter groups of motions that act nontransitively on the spacetime. All admissible electromagnetic fields for which such algebras exist are found. In the case of an arbitrary n-dimensional Riemannian space on which the group of motions acts, it is proved that the admissible field does not deform the algebra of symmetry operators of the free Hamilton-Jacobi and Klein-Gordon-Fock equations. In addition, the system of differential equations, which must be satisfied by the potentials of the admissible electromagnetic field, have been investigated for compatibility.