  Starting from ${\cal N}=4$ SYM and using an appropriate Higgs mechanism we
reconsider the construction of a scalar field theory non-minimally coupled to a
Coulomb potential with a relativistic SO(4) symmetry and check for scalar field
consistency conditions. This scalar field theory can also be obtained from a
relativistic particle Lagrangian with a proper implementation of the
non-minimal coupling. We provide the generalization of the non-relativistic
construction of the Runge-Lenz vector to the relativistic case and show
explicitly that this new vector generates the SO(4) algebra. Using the power of
the SO(4) symmetry, we calculate the relativistic hydrogen atom spectrum. We
provide a generalization of the Kustaanheimo-Stiefel transformation to the
relativistic case and relate our results with the corresponding relativistic
oscillator. Finally, in the light of these results, we reconsider the
calculation of the hydrogen atom spectrum from the cusp anomalous dimension
given in [2].
