The stationary Maxwell–Dirac equations

Abstract

The Maxwell–Dirac equations are the equations for electronic matter, the 'classical' theory underlying QED. The system combines the Dirac equations with the Maxwell equations sourced by the Dirac current. A stationary Maxwell–Dirac system has ψ = e−iEt, with independent of t. The system is said to be isolated if the dependent variables obey quite weak regularity and decay conditions. In this paper, we prove the following strong localization result for isolated, stationary Maxwell–Dirac systems, there are no embedded eigenvalues in the essential spectrum, i.e. −m ≤ E ≤ m; if |E| < m then the Dirac field decays exponentially as |x| → ∞; if |E| = m then the system is 'asymptotically' static and decays exponentially if the total charge is non-zero.