We analyze the causal quantum behavior of two charged spinless particles having different masses interacting via an action-at-a-distance relativistic scalar Coulomb potential. Developing de Broglie's idea of splitting the potential energy between the particles, we show that the predictivity constraints of relativistic predictive mechanics established by Droz-Vincent and Sudarshan are satisfied. In this way we develop a relativistic theory of hydrogenlike atoms in the framework of the causal interpretation of quantum mechanics.
An approximate quantum-mechanical two-body equation for spinless particles incorporating relativistic kinematics is derived. The derivation is based on the relativistic energy-momentum relation mc2+ε=m2c4+p2c2+V for each single particle, where mc2 is the particle rest mass energy, p its linear momentum, ε its dynamical energy, and V being the time-like vector interaction potential. The resulting two-body equation assumes rapid wave oscillations in a single, slowly varying potential well. A Bohr-Sommerfeld-type quantization condition is obtained. The approximation is compared to exact results for the harmonic potential.