On the three-dimensional reductions of the Bethe - Salpeter equation and their one-body limits (two-boson and boson - fermion cases)

Abstract

We transform the Bethe-Salpeter equation for the two-boson and for the bosonfermion systems into equivalent Salpeter or Sazdjian equations, with a three-dimensional (3D) potential directly computable from a series of reducible and irreducible Feynman graphs. We compute the various one-body limits (in the boson-fermion case we compute the heavy boson and the heavy fermion limits). To be definite, we work in the QED framework, but we also consider the scalar bosons exchanges in the two-boson case.

A cluster-separable Born approximation for the three-dimensional reduction of the three-fermion Bethe-Salpeter equation

“…In [17], we showed that a 3D reduction performed with a propagator ΛG SZ , Λ being a projector, leads to a 3D potential given by…”

“…The 3D reduction of the two-fermion Bethe-Salpeter equation has been performed by many authors [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. All methods are theoretically equivalent at the limit of all correction terms included.…”

A NEW APPROACH TO THE RELATIVISTIC TREATMENT OF THE FERMION–BOSON SYSTEM, BASED ON THE EXTENSION OF THE SL(2, C) GROUP

“…Alternatively, there exist other approaches to the fermion-boson problem that describe the relativistic kinematics exactly. Namely, the phenomenological two-body equation proposed by Królikowski [4], the Barut method [5,6], the reductions of the Bethe-Salpeter equation [7,8] and the relativistic quantum mechanics with constraints [9][10][11][12] are known.…”

3D reduction of the three-fermion Bethe-Salpeter equation

“…The real difficulty appears when only the interactions with fermion 3 (for example) are switched off. If we want a full cluster separability, the resulting equation for the (12) cluster can not refer to the global center of mass frame anymore, as the momentum of fermion 3 enters in the definition of this frame.…”

“…In the Dirac-Coulomb equation, the Coulomb potential is already given by the limit of the Born term, as the higher-order crossed and ladder terms cancel mutually at the one-body limit. In contrast, the one-body limits of the rearrangements contain contributions from all terms [8,9,10,11,12] .…”