Derivation of the Maxwell-Schrödinger Equations: A note on the infrared sector of the radiation field

Abstract

We slightly extend prior results about the derivation of the Maxwell-Schrödinger equations from the bosonic Pauli-Fierz Hamiltonian. More concretely, we show that the findings from [25] about the coherence of the quantized electromagnetic field also hold for soft photons with small energies. This is achieved with the help of an estimate from [3] which proves that the domain of the number of photon operator is invariant during the time evolution generated by the Pauli-Fierz Hamiltonian.

Norm approximation for the Fröhlich dynamics in the mean-field regime

“…where pψ t , ϕ t q P L 2 pR 3 q ˆL2 pR 3 q solves the time-dependent Landau hptq " ´∆ `?αΦ ϕt ´µptq (1. 13) and initial datum pψ 0 , ϕ 0 q " pψ, ϕq. The Landau-Pekar equations model the interaction between a single quantum particle and a classical phonon field.…”

“…Here, the appearance of classical radiation rests on the fact that many weakly correlated particles in the same quantum state create radiation. This mechanism has been investigated for the Nelson model with ultraviolet cutoff [1,12,14,29], the Pauli-Fierz Hamiltonian [13,30], the renormalized Nelson model [2] and for the Nelson model with ultraviolet cutoff in a limit of many weakly interacting fermions [28]. We would like to remark that the interaction of the renormalized Nelson model is more singular than the one of the Fröhlich model and that an analysis is more complicated in this case.…”