Supercharge Operator of Hidden Symmetry in the Dirac Equation

TL;DR

This paper explores a new operator that reveals hidden supersymmetry in the Dirac equation, specifically selecting the Coulomb potential and connecting to Johnson and Lippmann's earlier work.

Contribution

It constructs a general operator that anticommutes with the Dirac K-operator and identifies conditions under which it reveals supersymmetry, uniquely selecting the Coulomb potential.

Findings

Operator coincides with Johnson and Lippmann's operator for Coulomb potential

Unveils hidden supersymmetry in the Dirac equation

Provides a method to select Coulomb potential via symmetry

Abstract

As is known, the so-called Dirac $K$-operator commutes with the Dirac Hamiltonian for arbitrary central potential $V(r)$. Therefore the spectrum is degenerate with respect to two signs of its eigenvalues. This degeneracy may be described by some operator, which anticommutes with $K$. If this operator commutes with the Dirac Hamiltonian at the same time, then it establishes new symmetry, which is Witten's supersymmetry. We construct the general anticommuting with $K$ operator, which under the requirement of this symmetry unambiguously select the Coulomb potential. In this particular case our operator coincides with that, introduced by Johnson and Lippmann many years ago.