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Convergence of the Dirac–Maxwell System to the Vlasov–Poisson System

We study the behavior of solutions of the (mixed state) Dirac–Maxwell system in the simultaneous non-relativistic and semi-classical limits, i.e., as the speed of light c tends to infinity and the Planck constant ℏ tends to zero. Under a constraint of the form , where M is a sufficiently large integer, and with suitable conditions on the initial data, we prove that the scalar Wigner transform of the Dirac spinors converges weakly to a solution of the classical Vlasov–Poisson system. Our key technique is a blend of null form bilinear estimates with the machinery of the Wigner transform.