Manifestations the hidden symmetry of Coulomb problem in the relativistic quantum mechanics - from Pauli to Dirac electron

TL;DR

This paper generalizes a Pauli theorem to the Dirac equation, revealing a hidden symmetry in the relativistic Coulomb problem and connecting it to the Johnson-Lippmann operator and the Laplace-Runge-Lenz vector.

Contribution

It introduces a simple, general proof of the hidden symmetry operator in the Dirac Coulomb problem, linking it to well-known symmetry operators without complex calculations.

Findings

The hidden symmetry operator coincides with the Johnson-Lippmann operator.

The operator is connected to the Laplace-Runge-Lenz vector.

The derivation simplifies understanding of relativistic Coulomb symmetry.

Abstract

The theorem known from Pauli equation about operators that anticommute with Dirac's $K$-operator is generalized to the Dirac equation. By means of this theorem the operator is constructed which governs the hidden symmetry in relativistic Coulomb problem (Dirac equation). It is proved that this operator coincides with the familiar Johnson-Lippmann one and is intimately connected to the famous Laplace-Runge-Lenz (LRL) vector. Our derivation is very simple and informative. It does not require a longtime and tedious calculations, as is offten underlined in most papers.