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In this paper a relativistic quantum theory introduced by de Groot and Ruijgrok [Nucl. Phys. 101B, 95(1975)] is reformulated. It is shown that it has the correct nonrelativistic limit and that it gives the same lowest-order relativistic corrections as the Breit equation. In this paper it is applied to bound systems of two charged particles with spin 1/2. For small coupling constants the lowest-order fine structure (up to order alpha superscript 4) for all states and all mass ratios is calculated. No expansion in the mass ratio is made, so that the full recoil effects are obtained. It embraces as special cases the well-established fine structure of the hydrogen atom and positronium and the Fermi splitting of muonium. For the strong-coupling case the critical couplings are calculated. The convergence to the static limit for coupling in the neighborhood of the critical point is extremely slow. The bound-state masses for states with the same total angular momentum J but opposite parity are not equal at the critical point, so there is no parity doubling. The results are compared with a number of other theories.
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