Derivation of the Maxwell-Schrödinger and Vlasov-Maxwell Equations from Non-Relativistic QED

Abstract

We study the spinless Pauli-Fierz Hamiltonian in a semiclassical mean-field limit of many fermions. For appropriate initial conditions, we prove, in the trace norm topology of reduced density matrices, that the many-body quantum state converges to a tensor product of a semiclassically structured Slater determinant and a coherent photon state. These evolve according to a fermionic variant of the Maxwell-Schrödinger equations. By combining this result with [arXiv:2308.16074] through a suitable regularization of the initial data, we further show that, in the limit of large particle number, the dynamics of the Pauli-Fierz Hamiltonian can be approximately described by the non-relativistic Vlasov--Maxwell system for extended charges.

An error in the proof of the Vlasov-Maxwell derivation in Theorem II.8 of the first version has been corrected. Compared to the first version, the rate of convergence in Theorem II.8 is weaker, but the assumption on the support of the velocity variable of the Wigner transform has been relaxed. Moreover, the presentation of Remarks II.5 and II.6 has been improved