New non-unitary representations in a Dirac hydrogen atom

TLDR: In this paper, a non-unitary representation of the SU(2) algebra for the Dirac equation with a Coulomb potential is introduced. But the authors do not define a new set of operators for the relativistic hydrogen atom.

Abstract: New non-unitary representations of the SU(2) algebra are introduced for the case of the Dirac equation with a Coulomb potential; an extra phase, needed to close the algebra, is also introduced. The new representations does not require integer or half-integer labels. The set of operators defined are used to span the complete space of bound-state eigenstates of the problem thus solving it in an essentially algebraic way. Hydrogen-like atoms are some of the most important quantum systems solved. Even for describing stabilization properties and for testing QED and weak interaction theories a great deal can be done at the relativistic atomic physics level (Greiner 1991, Kylstra et al 1997, Quiney et al 1997). It is, therefore, very important to extend our insight into the properties of hydrogen-like systems. An important tool has been the algebraic properties of the set of operators defining the system; these are not only connected with the corresponding group and its symmetry algebra but often offer simplified methods for carrying out some calculations. It is the purpose of this letter to define a new set of operators for the Dirac relativistic hydrogen atom. This comprises a non-unitary representation of the SU(2) algebra and defines ladder operators for the problem. An extra phase is needed to close the algebra but this allows us to solve the Dirac hydrogen atom in a neat algebraic way. The Dirac Hamiltonian for a hydrogen-like atom is