$O(2,1)$ Algebra and the hydrogen atom

It is shown that both the nonrelativistic and the relativistic problems for the hydrogen atom can be formulated in terms of the generators of the noncompact algebra $O(2,1)$. The possible values of the energy (the Balmer and Sommerfeld formulas, respectively), can be determined by an appropriate choice of the basis (tilt) for the generators in the algebra $O(2,1)$.