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

Abstract

We classify the various 3D reductions of the Bethe-Salpeter equation for two fermions, according to the constraint chosen. We show that, for a given choice of the constraint, the final 3D equations can all be obtained from a basic one, by letting a simple operator act on the wavefunction and by rearranging the terms of the series giving the 3D potential, without leaving the 3D framework. We examine the relations of the various reductions with each other and with the scattering amplitude and we specialize to a few popular reductions. When the mass of one of the fermions becomes infinite (one-body limit), we find Dirac's equation with a potential (Coulombian in QED) for our basic equation and this same equation or a projection of it (according to the relation existing before taking the limit) in the other cases.

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

“…independent of p 0 ), one would get Salpeter's equation by integration with respect to p 0 [3]. In the realistic case of a non-instantaneous kernel, it is possible to compute the bound state energies at the 4D level by perturbations around an instantaneous approximation of the kernel [10,31,14]. Here, we want to build a 3D reduction around an approximation K 0 of the Bethe-Salpeter kernel.…”

“…where T 12,23 contains neither initial (23) nor final (12) interaction, these interactions being included in G 23 and G 12 respectively. The full (23) propagator can be expanded as [33,14] G 23 (P 23 , p ′ 23 , p 23 ) = Φ 23 ( P 23 , p ′ 23 ) −i P 230 − E 23 + iǫ Φ 23 ( P 23 , p 23 ) + · · · (92)…”

“…This will of course depend on the specific problem studied. The mutual cancellation between the higher-order "modified ladder terms" (generated by the 3D reduction) and the "crossed terms", which occurs at the one-body limit [14,15,16] is an encouraging fact. The starting homogeneous Bethe-Salpeter equation (for the Bethe-Salpeter amplitude) is deduced from the inhomogeneous one (for the full propagator) by postulating the presence of a bound state pole, and is thus a priori valid only for bound states.…”

SU(2) Lattice Gluon Propagator and Potential Models

“…Ref. [2] and [3]). More precisely, one makes an instantaneous approximation to the wave function (which leads to the Salpeter equation), a nonrelativistic approximation to the kinetic term in the Hamiltonian and a local approximation to the potential.…”

Bound state equation for 4 or more relativistic particles

“…The Bethe-Salpeter equation [1,2] is the usual tool for computing relativistic bound states [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The principal difficulty of this equation comes from the presence of N-1 (for N particles) unphysical degrees of freedom: the relative time-energy degrees of freedom.…”