QED test of a Bethe-Salpeter solution method.

Abstract

A method is described of solving Bethe-Salpeter equations for spin-\textonehalf{} particles using a standard quasipotential approximation with a correction series. A three-dimensional, 16-component bound-state equation is obtained whose form is different from the Breit equation. The equation has two equivalent first-order forms which show each particle obeying a Dirac equation in the presence of the other. The formalism also reconstitutes the four-dimensional bound-state vertex function from the three-dimensional wave function. The method is tested on positronium and the hydrogen atom, using a single-photon-exchange kernel in the Coulomb gauge. In the bound-state equation, in addition to the Coulomb potential and Breit interaction, the formalism gives a subtracted box potential $(\frac{1}{2E})\frac{{\ensuremath{\alpha}}^{2}}{{r}^{2}}$ where $E$ is the mass of the bound state. With this term included, the energy levels (fine structure and hyperfine structure) are correct to order ${\ensuremath{\alpha}}^{4}$. The reconstituted bound-state vertex function, when substituted in Feynman triangle diagrams, gives the lowest-order atomic dipole transition amplitudes correctly.