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

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.