What happens with the relative-time excitations after a three-dimensional reduction of the Bethe-Salpeter equation?

Summary

We perform a three-dimensional reduction of the Bethe-Salpeter equation for two fermions by isolating an instantaneous part of the propagator and treating the rest as a perturbation. Then, in the framework of a static model, we compare the energy spectrum of the Bethe-Salpeter equation with that of the reduced equation. We find that the even «abnormal» solutions (related to excitations in the relative-time degree of freedom) are still present after the reduction. In order to compute their energies, however, one has to sum the perturbation series generated by the reduction, or to re-arrange its terms into Padé approximants. The abnormal solutions appear then as a consequence of an energy dependence of the potential. The computation of the energy of the normal solution is also improved by the use of Padé approximants.