A three-fermion Salpeter equation

Abstract

We write a 3D equation for three fermions by combining the three two-body potentials obtained by the reductions of the corresponding two-fermion Bethe-Salpeter equations to equivalent 3D equations, putting the spectator fermion on the mass shell. In this way, the cluster-separated limits are still exact, and the Lorentz invariance / cluster separability requirement is automatically satisfied, provided no supplementary approximation, like the Born approximation, is made. The use of positive free-energy projectors in the chosen reductions of the two-fermion Bethe-Salpeter equations prevents continuum dissolution in our 3D three-fermion equation. The potentials are hermitian and depend only slowly on the total three-fermion energy. The one high-mass limits are approximately exact. In view of a possible perturbation calculation, correcting the remaining discrepancies with the three-fermion Bethe-Salpeter equation, we succeeded in deriving our 3D equation from an approximation of the three-fermion Bethe-Salpeter equation, in which the three-body kernel is neglected and the two-body kernels approached by positive-energy instantaneous expressions, with the spectator fermion on the mass shell. The neglected terms are transformed into corrections to the 3D equation. A comparison is made with Gross' spectator model.