Integrating Klein-Gordon-Fock equations in an external electromagnetic field on Lie groups

TL;DR

This paper explores the symmetry algebra of the Klein-Gordon-Fock equation in external electromagnetic fields on Lie groups, proposing a method for integration using harmonic analysis and illustrating it with a specific four-dimensional example.

Contribution

It introduces a novel approach to integrate the Klein-Gordon-Fock equation in external fields on Lie groups using coadjoint orbit methods and harmonic analysis.

Findings

Symmetry algebra is a one-dimensional central extension in invariant fields.

A new integration method for Klein-Gordon-Fock equations on Lie groups.

Detailed example on the four-dimensional group $E(2) imes \\mathbb{R}$.

Abstract

We investigate the structure of the Klein-Gordon-Fock equation symmetry algebra on pseudo-Riemannian manifolds with motions in the presence of an external electromagnetic field. We show that in the case of an invariant electromagnetic field tensor, this algebra is a one-dimensional central extension of the Lie algebra of the group of motions. Based on the coadjoint orbit method and harmonic analysis on Lie groups, we propose a method for integrating the Klein-Gordon-Fock equation in an external field on manifolds with simply transitive group actions. We consider a nontrivial example on the four-dimensional group $E(2) \times \mathbb{R}$ in detail.