Bound state equation for 4 or more relativistic particles

Abstract

We apply the 3D reduction method we recently proposed for the N-particle Bethe-Salpeter equation to the 4-particle case. We find that the writing of the Bethe-Salpeter equation is not a straightforward task when N is larger or equal to 4, owing to the presence of mutually unconnected interactions, which could lead to an overcounting of some diagrams in the resulting full propagator. We overcome this difficulty in the N=4 case by including three counterterms in the Bethe-Salpeter kernel. The application of our 3D reduction method to the resulting Bethe-Salpeter equation suggests us a modified 3D reduction method, which gives directly the 3D potential, without the need of writing the Bethe-Salpeter kernel explicitly. The modified reduction method is usable for all N.Comment: 16 pages, three figures in one PS file. In this improved version, the writing of the 3D potential is straightforwar

Direct interaction along light cones at the quantum level

“…Both possibilities (a) and (b) are simpler than the BS equation for N particles. The latter contains k-particle interaction terms for k = 2, ..., N [37]. It is therefore hard to even write down the equation explicitly.…”

Many-body-QED perturbation theory: Connection to the two-electron Bethe–Salpeter equation

“…The BS equation has also been applied to three or more particles [26,27,28], although serious problems have been encountered for more than three particles [29].…”

“…In recent years there have been numerous applications in QCD, dealing mainly with the quark-quark, quark-antiquark interactions, quark confinement and related problems [14,19,20,21]. Here, the problems mentioned above are more serious, as recently summarized by Namyslowski [6].There have also been many applications in surface and solid-state physics, ranging from electronhole interactions in ion crystals [22] and studies of the two-dimensional Hubbard model [23] and Cooper pairs [24] to quantum dots [25].The BS equation has also been applied to three or more particles [26,27,28], although serious problems have been encountered for more than three particles [29].Various approximation schemes for treating the BS equation have been developed over the time. The simplest approximation is the "ladder approximation", where all intermediate states evolve only in the forward (positive) time direction.…”

3D reduction of the N-body Bethe–Salpeter equation

“…For N ≥ 4 the writing of K ′ becomes more complicated, because of the presence of commutating kernels like K ′ 12 and K ′ 34 which would lead, without corrections, to an overcounting of some graphs in the expansion of G. We examine this problem elsewhere [32].…”