Two and three-fermion 3D equations deduced from Bethe-Salpeter equations

We write a 3D equation for three fermions by combining the three two-body potentials obtained in 3D reductions (based on a series expansion around a relative-energy fixing "approximation" of the free propagators) of the corresponding two-fermion Bethe-Salpeter equations to equivalent 3D equations, putting the third fermion on its positive-energy mass shell. In this way, the cluster-separated limits are 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 below the inelastic threshold and depend only slowly on the total three-fermion energy. This "hand-made" three-fermion 3D equation is also obtained by starting with 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 then transformed into corrections to the 3D equation, in three steps implying each a series expansion. The result is of course complicated, but the lowest-order contributions of these correction terms to the energy spectrum remain manageable.We also present some other 3D reduction procedures and compare them to our's: use of Sazdjian's covariant approximation of the free propagator, 3D reductions performed by a series expansion around instantaneous approximations of the kernels instead of "approximations" of the propagators, Gross' spectator model.