  We present a novel form of relativistic quantum mechanics and demonstrate how
to solve it using a recently derived unitary perturbation theory, within
partial wave analysis. The theory is tested on a relativistic problem, with two
spinless, equal mass particles, in which the interaction is entirely given by a
Coulomb potential. As such, it is not meant to reproduce experimental results
for the scattering of two electrons, but is intended as a test of our
calculation methods. We find that this perturbation theory gives finite results
at second order. This is unlike other versions of perturbation theory, which
find divergent results at second and all higher orders. We calculate
differential cross sections in the nonrelativistic regime, where we find
excellent agreement with the Rutherford formula. Then, well into the
relativistic regime, we find differential cross sections with similar shapes to
the M{\o}ller formula and differing from that formula by less than an order of
magnitude.
