Non-unitary representations of the SU(2) algebra in the Dirac equation with a Coulomb potential

TL;DR

This paper introduces a new non-unitary SU(2) algebra representation for the Dirac hydrogen atom, enabling algebraic solutions and revealing novel properties of the radial eigenfunctions.

Contribution

It presents a novel realization of SU(2) algebra for the Dirac equation with Coulomb potential, incorporating an extra phase variable and enabling algebraic solution methods.

Findings

Radial eigenfunctions form non-unitary SU(2) representations

These representations are not labeled by integer or half-integer quantum numbers

The algebraic approach simplifies solving the Dirac equation with Coulomb potential

Abstract

A novel realization of the classical SU(2) algebra is introduced for the Dirac relativistic hydrogen atom defining a set of operators that, besides, allow the factorization of the problem. An extra phase is needed as a new variable in order to define the algebra. We take advantage of the operators to solve the Dirac equation using algebraic methods. To acomplish this, a similar path to the one used in the angular momentum case is employed; hence, the radial eigenfuntions calculated comprise non unitary representations of the algebra. One of the interesting properties of such non unitary representations is that they are not labeled by integer nor by half-integer numbers as happens in the usual angular momentum representation.