Does the Coulomb potential have an algebraic origin?

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.