The Stationary Maxwell-dirac Equations

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 = eφ, with φ independent of t. The system is said to be isolated if the independent variables obey quite weak regularity and decay conditions. In this paper we prove the following results 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; • if the system is asymptotically static then, under a certain restriction on the decay, it is electrically neutral. Date: November 13, 2001. 1991 Mathematics Subject Classification. Primary: 35Q40; Secondary: 00A79.