Current Operators and Majorana Equation for the Hydrogen Atom from Dynamical Groups

It is proved that the dipole operator in the hydrogen atom is the product of an element in the Lie algebra and of a group element of the conformal group $O(4, 2)$. A relativistic wave equation containing the total momentum only is set up which describes the internal structure of the system purely group theoretically and gives the correct mass spectrum. The diagonalization of this equation determines a new basis of states in which the dipole operator is simply an element of the Lie algebra. The angle of transformation to the new basis is evaluated to be $\theta =2\mathrm{}invtan(1-\frac{\epsilon }{n})$, or, $\theta =log\left(\frac{2n}{\epsilon }\right)$, where $n$ is the principal quantum number and $\epsilon$ is essentially the fine-structure constant.