The Stationary Maxwell-Dirac Equations

TL;DR

This paper analyzes the stationary Maxwell-Dirac equations, proving spectral properties and decay behavior of solutions under weak assumptions, with implications for the stability and structure of electronic matter in quantum electrodynamics.

Contribution

It establishes the absence of embedded eigenvalues and characterizes the decay of solutions at spectral edges under minimal regularity and decay conditions.

Findings

No embedded eigenvalues in the essential spectrum.

Exponential decay of solutions for energies within the mass gap.

Asymptotic staticity and decay when energy equals the mass with non-zero charge.

Abstract

The Maxwell-Dirac equations are the equations for electronic matter, the "classical" theory underlying QED. In this article we examine the stationary Maxwell-Dirac equations under weak regularity and decay assumptions. We prove that: There are no embedded eigenvalues in the essential spectrum, $-m\leq E\leq m$. If $|E|< m$ then the Dirac field components (and their derivatives) decay exponentially at spatial infinity. If $E|=m$ then the system is "asymptotically" static and decays exponentially if the total charge is non-zero.