Coulomb Potential and Witten Superalgebra

TL;DR

This paper reviews the hidden symmetries in the Coulomb problem, especially in the Dirac equation, revealing that invariance under Witten's superalgebra uniquely identifies the Coulomb potential, with implications for higher dimensions and supersymmetry.

Contribution

It demonstrates that the Coulomb potential is uniquely characterized by invariance under Witten's superalgebra in the Dirac equation, extending the analysis to higher dimensions.

Findings

Hidden symmetries in the Coulomb problem are elucidated.

Invariance under Witten's superalgebra singles out the Coulomb potential.

The traditional view of Coulomb potential is revised in the context of N=2 supersymmetry.

Abstract

The additional hidden symmetry of the Coulomb-Kepler problem is reviewed in classical as well as in quantum mechanics. The main purpose is to elucidate the role of this kind of symmetries in the reduction of physical problems, to show algebraic possibilities of derivation of spectra. The original results are presented also. They are hidden symmetries in the Dirac equation, where it is shown that the requirement of invariance of the Dirac Hamiltonian under some kind of Witten's superalgebra, picks out the Coulomb potential only. The problem in the arbitrary higher dimensions is also considered. It is derived that the traditional view on the Coulomb potential is to be changed in the context of N=2 supersymmetry