Instantaneous Bethe-Salpeter Equation: (Semi-)Analytical Solution

TL;DR

This paper presents a method to convert the Bethe-Salpeter equation for fermion-antifermion bound states into a matrix eigenvalue problem using an instantaneous approximation, enabling more straightforward solutions.

Contribution

It provides a semi-analytical approach to solving the Bethe-Salpeter equation by explicitly constructing matrices in the instantaneous approximation.

Findings

Matrix eigenvalue formulation simplifies solving the Bethe-Salpeter equation.

Explicit matrices enable algebraic solutions for bound state problems.

Approach improves analytical understanding of fermion-antifermion interactions.

Abstract

The Bethe-Salpeter equation for bound states of a fermion-antifermion pair in the instantaneous approximation for the involved interaction kernel is converted into an equivalent matrix eigenvalue problem with explicitly (algebraically) given matrices.