Derivation of the Maxwell-Schr\"odinger and Vlasov-Maxwell Equations from Non-Relativistic QED

TL;DR

This paper derives classical electromagnetic equations from quantum many-body dynamics of fermions interacting with a quantized radiation field, showing convergence to Maxwell-Schr"odinger and Vlasov-Maxwell systems in certain limits.

Contribution

It establishes the rigorous derivation of Maxwell-Schr"odinger and Vlasov-Maxwell equations from non-relativistic QED for many fermions, connecting quantum and classical descriptions.

Findings

Quantum many-body states converge to classical field equations.

Derivation of Maxwell-Schr"odinger equations from quantum dynamics.

Approximate description of large particle systems by Vlasov-Maxwell system.

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-Schr\"odinger 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.