Does the Coulomb potential have an algebraic origin?

TL;DR

This paper demonstrates that the Coulomb potential uniquely emerges from a Witten N=2 superalgebra structure in the Dirac Hamiltonian with central potentials, providing an algebraic derivation of the energy spectrum.

Contribution

It reveals that the Coulomb potential is the only central potential compatible with a Witten superalgebra in the Dirac equation, offering an algebraic approach to its energy spectrum.

Findings

Witten N=2 superalgebra arises in Dirac Hamiltonian with Coulomb potential.

Only Coulomb-like potential maintains invariance under this superalgebra.

Energy spectrum can be derived algebraically without solving the Dirac equation.

Abstract

It is shown that in case of central potentials, both the fourth component of Lorentz vector as well as Lorentz scalar in the Dirac Hamiltonian, owing to the conserved Dirac spin-orbital matrix, there arises Wittens N=2 superalgebra. The generators of this algebra are constructed and their commutativity with Dirac Hamiltonian is studied. Under the requirement of in-variance relative to this super algebra it follows that only Coulomb like potential obeys the corresponding constraints. This fact allows us to suppose the Wittens superalgebra as an alternative source for emerging of the Coulomb potential. As a byproduct, we obtain energy spectrum without solving the Dirac equation, i.e. by pure algebraically.