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 constanth 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

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... ue is a blend of null form bilinear estimates with the machinery of the Wigner transform.