Derivation of the Maxwell–Schrödinger equations: A note on the infrared sector of the radiation field

We slightly extend prior results about the derivation of the Maxwell-Schrodinger equations from the bosonic Pauli-Fierz Hamiltonian. More concretely, we show that the findings from Leopold and Pickl [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 Ammari et al. [arXiv:2202.05015 (2022)], which proves that the domain of the number of photon operator is invariant during the time evolution generated by the Pauli-Fierz Hamiltonian.