Expressions are obtained for all ${\alpha }^3\mathrm{Ry}$ contributions to the energy levels of the two-fermion system in electrodynamics. These expressions are evaluated from a relativistic two-body equation which takes binding into account in its interaction operator. They are specifically calculated for the $n=2\mathrm{}$ levels of the system. Corrections arise from three sources: (1) improved treatment of pair effects of the Coulomb field and of the exchange of transverse photons, (2) self-energy and vacuum polarization terms, and, in positronium, (3) second-order annihilation processes. The energy shifts resulting from (1) and (2) do not depend on the arbitrary masses through a single parameter like reduced mass. In the limit appropriate to hydrogen, the previously calculated two-body corrections of item (1) are confirmed.

The principal new result is the determination of the $n=2\mathrm{}$ levels of positronium. In contrast to hydrogen, where the self-energy effect is dominant, here all three items yield roughly equal corrections. Together, they amount to about 3 percent of the ${\alpha }^2\mathrm{Ry}$ level splitting.

• Received 11 February 1954

DOI:https://doi.org/10.1103/PhysRev.95.811