Complete reduction of fermion-antifermion bethe-salpeter equation with static kernel

The Bethe-Salpeter (BS) equation for a $\text{spin}\text{-}\text{1}\text{2}$ fermion-antifermion bound system is considered for the case in which the kernel is static and is the fourth component (i.e., three-scalar part) of a vector potential. Relative time (or relative energy) dependence can be eliminated easily. The 16 BS bispinor amplitudes are reexpressed in the usual way in terms of corresponding tensor amplitudes which satisfy 16 coupled integrodifferential equations. If Lorentz-, parity-, and charge-conjugation invariance are used, these equations can be reduced through a sequence of transformations to single eigenvalue equations, involving scalar and three-vector wavefunctions for singlet and triplet states, respectively. The effective Hamiltonians obtained in these equations are correct to all orders in the coupling constant and have a simple structure, consisting in general of a scalar, a spin-orbit, and a tensor part, which are explicitly exhibited.Although the equations could well be used for consideration of a general particleantiparticle system (e.g., quark-antiquark), for the present only positronium with a Coulomb interaction kernel is treated as an illustrative example. There exists a singlettriplet splitting in leading order mα⁶ ln α even though no spin-spin forces are directly introduced in the kernel. The splitting is calculated in detail in perturbation theory to order mα⁶ ln α and mα⁶.