(In-)Consistencies in the relativistic description of excited states in the Bethe-Salpeter equation

TL;DR

This paper investigates the challenges in interpreting solutions of the Bethe-Salpeter equation for relativistic bound states, focusing on complex eigenvalues and their physical significance, especially in scalar and QED systems.

Contribution

It analyzes the origin of complex eigenvalues in the Bethe-Salpeter equation and proposes a consistent interpretation for the bound state spectrum in relativistic quantum field theory.

Findings

Complex eigenvalues arise from crossing of normal and abnormal states.

These crossings occur beyond the ladder approximation's validity.

A consistent interpretation of QED bound states is proposed.

Abstract

The Bethe-Salpeter equation provides the most widely used technique to extract bound states and resonances in a relativistic Quantum Field Theory. Nevertheless a thorough discussion how to identify its solutions with physical states is still missing. The occurrence of complex eigenvalues of the homogeneous Bethe-Salpeter equation complicates this issue further. Using a perturbative expansion in the mass difference of the constituents we demonstrate for scalar fields bound by a scalar exchange that the underlying mechanism which results in complex eigenvalues is the crossing of a normal (or abnormal) with an abnormal state. Based on an investigation of the renormalization of one-particle properties we argue that these crossings happen beyond the applicability region of the ladder Bethe-Salpeter equation. The implications for a fermion-antifermion bound state in QED are discussed, and a consistent interpretation of the bound state spectrum of QED is proposed.