Relativistic bound-state equations for two-fermion systems

TL;DR

This paper develops a relativistic wave equation for two-fermion systems that uses a propagator satisfying the Lehmann spectral representation, leading to new insights into bound states like positronium and deep bound states.

Contribution

It introduces a two-fermion relativistic wave equation based on a propagator that satisfies the Lehmann representation, improving upon previous models.

Findings

Eigenstates for positronium were obtained.

Discovered a deep bound state with binding energy close to twice the fermion mass.

Identified a bound state with an anomalously small mass.

Abstract

We point out that the free single-fermion propagator which is used in the QFT equations for two-fermion systems, has a bosonic structure, transforms to the single-boson propagator for the Klein-Gordon equation in the nonrelativistic limit, and, therefore, does not satisfy the Lehmann spectral representation for fermions. Proceeding from the Lippmann-Schwinger integral equation, we obtain a two-fermion relativistic wave equation in which the free two-fermion equal-time propagator satisfying the Lehmann representation, is used. This wave equation has been applied to the electron-positron system. In addition to the eigenstates for positronium, we find a solution which describes a deep bound state with the binding energy $\simeq 2m$ and anomalously small mass of the bound system.