There are no abnormal solutions of the Bethe$-$Salpeter equation in the static model

TL;DR

This paper demonstrates that in the static QED model, the Bethe-Salpeter equation does not admit abnormal solutions, confirming the stability of the bound state spectrum through gauge analysis.

Contribution

It clarifies that the previously established equation with abnormal solutions is not the Bethe-Salpeter equation, and confirms the absence of abnormal states in the static QED model.

Findings

No abnormal solutions in the Bethe-Salpeter equation for static QED.

The established equation with abnormal solutions differs from the Bethe-Salpeter equation.

Gauge transformations confirm the absence of abnormal states.

Abstract

The four-point Green's function of static QED, where a fermion and an antifermion are located at fixed space positions, is calculated in covariant gauges. The bound state spectrum does not display any abnormal state corresponding to excitations of the relative time. The equation that was established by Mugibayashi in this model and which has abnormal solutions does not coincide with the Bethe$-$Salpeter equation. Gauge transformation from the Coulomb gauge also confirms the absence of abnormal solutions in the Bethe$-$Salpeter equation.