On the nature of "hidden symmetry" or "accidental degeneracy" in the Kepler problem

TL;DR

This paper investigates a special symmetry in the Dirac equation related to the Coulomb potential, revealing that only the Coulomb potential maintains this symmetry under a specific eigenvalue sign interchange.

Contribution

It identifies a unique symmetry of the Dirac equation that holds exclusively for the Coulomb potential, enhancing understanding of hidden symmetries in quantum systems.

Findings

The symmetry applies to the Dirac equation with Coulomb potential.

Only Coulomb potential satisfies the symmetry condition.

The symmetry involves interchange of eigenvalues of the Dirac's K operator.

Abstract

We propose a symmetry of the Dirac equation under the interchange of signs of eigenvalues of the Dirac's $K$ operator. We show that the only potential which obeys this requirement is the Coulomb one for both vector and scalar cases.