Derivation of the Maxwell-Schr\"odinger Equations: A note on the infrared sector of the radiation field

TL;DR

This paper extends previous derivations of the Maxwell-Schr"odinger equations to include soft photons, demonstrating that the coherence properties of the quantized electromagnetic field remain valid in the infrared sector.

Contribution

It shows that the coherence results for the electromagnetic field also apply to low-energy photons, using an invariance estimate of the photon number domain during evolution.

Findings

Coherence of the quantized electromagnetic field holds for soft photons.

The domain of the photon number operator remains invariant under the Hamiltonian evolution.

Extension of prior results to the infrared sector of the radiation field.

Abstract

We slightly extend prior results about the derivation of the Maxwell-Schr\"odinger equations from the bosonic Pauli-Fierz Hamiltonian. More concretely, we show that the findings from [SIAM J. Math. Anal. 52(5), 4900-4936 (2020)] 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 [arXiv:2202.05015] which proves that the domain of the number of photon operator is invariant during the time evolution generated by the Pauli-Fierz Hamiltonian.