The Group $O(4)$, Separation of Variables and the Hydrogen Atom

The representation theory of the group $O(4)$ is considered systematically in different bases, corresponding to the reduction of $O(4)$ to various continuous or discrete subgroups. The results are applied to the hydrogen atom and we investigate the six different bases corresponding to separation of variables in the Schrödinger equation in momentum space and the four different bases corresponding to separation in coordinate space. It is shown that a classification of different bases (and of complete sets of commuting operators determining the bases) corresponds to a classification of interactions, breaking the original symmetry, while preserving certain aspects of it. Vector and scalar potentials providing such breaking of the $O(4)$ symmetry of the hydrogen atom are constructed explicitly. The relationship between $O(4)$ and separation of variables is used to derive a number of special function identities.