On the three-dimensional reductions of the Bethe-Salpeter equation and their one-body limits (two-fermion case)

We classify the various 3D reductions of the Bethe-Salpeter equation for two fermions, according to the constraint chosen. We show that, for a given choice of the constraint, the final 3D equations can all be obtained from a basic one, by letting a simple operator act on the wavefunction and by rearranging the terms of the series giving the 3D potential, without leaving the 3D framework. We examine the relations of the various reductions with each other and with the scattering amplitude and we specialize to a few popular reductions. When the mass of one of the fermions becomes infinite (one-body limit), we find Dirac's equation with a potential (Coulombian in QED) for our basic equation and this same equation or a projection of it (according to the relation existing before taking the limit) in the other cases. (author)