Quantum Point Charges Interacting with Quasi-classical Electromagnetic Fields
S. Breteaux, M. Correggi, M. Falconi, J. Faupin
TL;DR
This work analyzes quantum particles interacting with quantized fields in the quasi-classical regime where field excitations are abundant. By tracing out the field on quasi-classical product states, it derives reduced quadratic forms that converge, in the Gamma-sense, to effective particle Hamiltonians determined by a classical field described by a Wigner measure. For the Nelson and Pauli–Fierz models, the authors establish strong (and norm, under confinement) resolvent convergence to limit operators that incorporate classical potentials $V_{\mu}$ and, in the PF case, vector and scalar components $\mathbf{A}_{\mu}$ and $W_{\mu}$ (and $\mathbf{B}_{\mu}$). They also show that ultraviolet renormalization either commutes with the quasi-classical limit (Nelson) or can be handled consistently in the PF setting, yielding well-posed reduced dynamics without ultraviolet cut-offs. The results provide a rigorous bridge between fully quantum field interactions and effective quasi-classical descriptions relevant for non-relativistic QED-like systems.
Abstract
We study effective models describing systems of quantum particles interacting with quantized (electromagnetic) fields in the quasi-classical regime, i.e., when the field's state shows a large average number of excitations. Once the field's degrees of freedom are traced out on factorized states, the reduced dynamics of the particles' system is described by an effective Schrödinger operator keeping track of the field's state. We prove that, under suitable assumptions on the latter, such effective models are well-posed even if the particles are point-like, that is no ultraviolet cut-off is imposed on the interaction with quantum fields.