We consider a charged particle, spin 1 2, with relativistic kinetic energy and minimally coupled to the quantized Maxwell field. Since the total momentum is conserved, the Hamiltonian admits a fiber decomposition as H(P), P ∈ R3. We study the spectrum of H(P). In particular we prove that, for non-zero photon mass, the ground state is exactly two-fold degenerate and separated by a gap, uniformly in P, from the rest of the spectrum. 1 Introduction and Main Results Let us consider a classical point charge, charge e, mass M, position q, velocity ˙q, coupled to the Maxwell field with electric field E and magnetic field B. The coupling to the field is through a rigid charge distribution ϕ: R 3 → R+ normalized as ∫ dxϕ(x) = 1. Then the equations of motion for the coupled system read, i
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