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We prove the existence of a $L^2$-normalized solitary wave solution for theMaxwell-Dirac equations in (3+1)-Minkowski space. In addition, for theCoulomb-Dirac model, describing fermions with attractive Coulomb interactionsin the mean-field limit, we prove the existence of the (positive) energyminimizer.

A normalized solitary wave solution of the Maxwell-Dirac equations