Retardation effects and gauge transformations in the Bethe-Salpeter equations

Abstract

The ladder Bethe-Salpeter equation with Feynman propagators gives rise to spurious corrections of orderv/c to the nonrelativistic limit. The exact cancellation of these retardation effects by the introduction of the crossed-graph interaction kernel is here rigorously proved for the case of scalar bosons. The electrodynamic case is then examined, and the appearance of these spurious terms is related to the lack of exact gauge invariance of the approximate Bethe-Salpeter equation. In particular, the gauge-transformation properties of the wave functions are established, and also the gauge invariance of the crossed-graph equation is shown, up to orderv/c. Finally, for the case of scalar bosons, a more general transformation is defined which leaves theS-matrix invariant, but gives rise to a purely instantaneous interaction in the ladder approximation. Transformations of this kind directly avoid the spurious retardation effects, even for theories which are not gauge invariant.