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

Abstract

We perform a 3D reduction of the two-fermion homogeneous Bethe-Salpeter equation, by series expansion around a positive-energy instantaneous approximation of the Bethe-Salpeter kernel, followed by another series expansion at the 3D level in order to get a manifestly hermitian 3D potential. It turns out that this potential does not depend on the choice of the starting approximation of the kernel anymore, and can be written in a very compact form. This result can also be obtained directly by starting with an approximation of the free propagator, based on integrals in the relative energies instead of the more usual δ−constraint. Furthermore, the method can be generalized to a system of N particles, consisting in any combination of bosons and fermions. As an example, we write the 3D equation for systems of two or three fermions exchanging photons, in Feynman or Coulomb's gauge.

Relativistic action-at-a-distance interactions: Lagrangian and Hamiltonian to terms of second order

“…The Lagrangian (29) reduces to the Darwin Lagrangian (1) if α = γ = 0 and β = 1. From the Lagrangian (29) one can obtain the equations of motion to terms of second order:…”

“…From the Lagrangian (29) we can obtain the total energy (E), the total linear momentum ( P ) and the total angular momentum ( L), of a system of N interacting relativistic particles, to terms of second order:…”

“…At the quantum level one can describe a system of two spinning particles interacting electromagnetically, including second order terms, by the Breit equation [60]- [62]. An extended Breit equation may be devised in agreement with the Lagrangian (29) to account for the dynamical contributions to the masses and the effect of a linearly rising potential.…”

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

“…Vol. ,pronounced in the scattering of strongly interacting particles but less so for bound-state systems in weak-coupling [13,14,15,16,17] (see ref.[6] for a review). The earliest applications of the BS equation appeared in atomic physics and concerned the proton recoil contribution to the hydrogen fine structure by Salpeter [18] and the positronium energy level structure by Karplus and Klein [4].An important goal for the equation has been the study of strongly interacting particles, which is a fundamental problem in elementary-particle physics.…”

“…pronounced in the scattering of strongly interacting particles but less so for bound-state systems in weak-coupling [13,14,15,16,17] (see ref.…”