Bethe-Salpeter equation: 3D reductions, heavy mass limits and abnormal solutions

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We show that the 3D reductions of the Bethe-Salpeter equation have the same bound state spectrum as the original equation, with the possible exception of solutions for which the 3D reduction vanishes identically. The abnormal solutions of the Bethe-Salpeter equation (corresponding to excitations in the relative time-energy degree of freedom), when they exist, are recovered in the 3D reductions via a complicated dependence of the final potential on the total energy. We know however that the one-body (or one high mass) limit of some 3D reductions of the exact Bethe-Salpeter equation leads to a compact 3D equation (by a mutual cancellation of the ladder and crossed graph contributions), which does not exhibit this kind of dependence on the total energy anymore, and admits thus no abnormal solution. This is in contrast with the results of the ladder approximation, where no such cancellation occurs. We draw the same conclusions for the static model, which we obtain by letting the mass of the lighter particle go also to infinity. These results enforce Wick's conjecture that the abnormal solutions are a spurious consequence of the ladder approximation.