Derivation of the Vlasov–Maxwell system from the Maxwell–Schrödinger equations with extended charges

Abstract We consider the Maxwell–Schrödinger equations in the Coulomb gauge describing the interaction of extended fermions with their self-generated electromagnetic field. They heuristically emerge as mean-field equations from nonrelativistic quantum electrodynamics in a mean-field limit of many fermions. In the semiclassical regime, we establish the convergence of the Maxwell–Schrödinger equations for extended charges toward the nonrelativistic Vlasov–Maxwell dynamics and provide explicit estimates on the accuracy of the approximation. To this end, we build a well-posedness and regularity theory for the Maxwell–Schrödinger equations and for the Vlasov–Maxwell system for extended charges.