# Stability of relativistic quantum electrodynamics in the Coulomb gauge

## TL;DR

This paper proves the stability of relativistic quantum electrodynamics in the Coulomb gauge by establishing a positive lower bound for the renormalized Hamiltonian, ensuring the theory's mathematical consistency.

## Contribution

It demonstrates that QED in the Coulomb gauge is stable by providing a rigorous bound on the renormalized Hamiltonian, a key step in understanding its mathematical foundation.

## Key findings

- Established a positive lower bound for the renormalized Hamiltonian.
- Proved the stability of relativistic QED in the Coulomb gauge.
- Provided a framework for vacuum energy renormalization.

## Abstract

We show that relativistic quantum electrodynamics in the Coulomb gauge satisfies the following bound, which establishes stability: let $H(\Lambda,V)$ denote the Hamiltonian of $QED_{1+3}$ on the three-dimensional torus of volume $V$ and with ultraviolet cutoff $\Lambda$. Then there exists a constant $0<\mu(\Lambda,V)<\infty$ (the vacuum energy renormalization) such that the renormalized Hamiltonian is positive: $H_{ren}(\Lambda,V) \equiv H_{\Lambda,V}+\mu_{\Lambda, V}\cdot \mathbb{1} \ge 0 $.