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

TL;DR

This paper develops a new method for reducing the complex 4D Bethe-Salpeter equation for two-fermion systems into a manageable 3D form, ensuring hermiticity and generalizability to N-particle systems with various interactions.

Contribution

It introduces a series expansion approach around an instantaneous approximation to derive a hermitian 3D potential independent of initial kernel choices, applicable to multi-particle systems.

Findings

Derived a compact, hermitian 3D potential for two-fermion systems.

Showed the method's applicability to systems with multiple particles and different gauge interactions.

Demonstrated the equivalence of the potential with direct integral-based approximations.

Abstract

We perform a 3D reduction of the two-fermion 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 delta-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.