Algebra of symmetry operators for Klein-Gordon-Fock equation

TL;DR

This paper characterizes the external electromagnetic fields allowing first-order symmetry operators for the Klein-Gordon-Fock equation in certain space-times, linking symmetry operators to the underlying group actions.

Contribution

It identifies all external electromagnetic fields in which the Klein-Gordon-Fock equation admits first-order symmetry operators under specific group action conditions.

Findings

Symmetry operators correspond to the algebra of the acting group.

Complete classification of admissible electromagnetic fields for symmetry operators.

Equivalence of symmetry operator algebra to the group algebra in Riemannian spaces.

Abstract

All external electromagnetic fields in which the Klein-Gordon-Fock equation admits the first-order symmetry operators are found, provided that in the space-time $V_4$ a group of motion $G_3$ acts simply transitively on a non-null subspace of transitivity $V_3$. It is shown that in the case of a Riemannian space $V_n$, in which the group $G_r$ acts simply transitively, the algebra of symmetry operators of the $n$-dimensional Klein-Gordon-Fock equation in an external admissible electromagnetic field coincides with the algebra of operators of the group $G_r$.