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

TL;DR

This paper develops a simplified three-dimensional reduction method for the N-body Bethe-Salpeter equation, leading to a more manageable and Hermitian potential formulation applicable to complex fermion-boson systems.

Contribution

It introduces a novel series expansion approach that yields a kernel-independent, Hermitian potential for multi-fermion and boson systems from the Bethe-Salpeter equation.

Findings

The reduction produces a simple, kernel-independent potential.

The method generalizes easily to systems with multiple fermions and bosons.

The approach simplifies the computational complexity of the Bethe-Salpeter equation.

Abstract

Starting with the homogeneous Bethe-Salpeter equation for two fermions, we perform a 3D reduction using a series expansion around an unspecified positive-energy instantaneous approximation of the kernel. A second series expansion is made, at the 3D level, in order to get an "hermitian" potential. The combination of both series gives a very simple result, which does not depend of the initial approximation of the kernel anymore, and could be obtained directly by starting with an approximation of the free propagator. The generalisation of this result to a system of f (=0,...N) fermions and N-f bosons is easy.