Electrodynamic Displacement of Atomic Energy Levels. III. The Hyperfine Structure of Positronium

A functional integro-differential equation for the electron-positron Green's function is derived from a consideration of the effect of sources of the Dirac field. This equation contains an electron-positron interaction operator from which functional derivatives may be eliminated by an iteration procedure. The operator is evaluated so as to include the effects of one and two virtual quanta. It contains an interaction resulting from quantum exchange as well as one resulting from virtual annihilation of the pair. The wave functions of the electron-positron system are the solutions of the homogeneous equation related to the Green's function equation. The eigenvalues of the total energy of the system may be found by a four-dimensional perturbation technique. The system bound by the Coulomb interaction is here treated as the unperturbed situation. Numerical values for the spin-dependent change of the energy from the Coulomb value in the ground state are finally obtained accurate to order $\alpha$ relative to the hyperfine structure ${\alpha }^2$ Ry. The result for the singlet-triplet energy difference is $\Delta W_{\mathrm{ts}}=\frac{1}{2}{\alpha }^2{\mathrm{Ry}}_{\infty }[\frac{7}{3}-(\frac{32}{9}+2\mathrm{}\ln 2\left)\frac{\alpha }{\pi }\right]=2.0337\mathrm{}\times {10}^5\frac{\mathrm{Mc}}{sec}.$ Theory and experiment are in agreement.