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 Maxwell–Schrödinger equations from bosonic Pauli–Fierz Hamiltonian. More concretely, we show that findings Leopold and Pickl [SIAM J. Math. Anal. 52(5), 4900–4936 (2020)] coherence quantized electromagnetic field also hold for soft photons with small energies. This is achieved help an estimate Ammari et al. [arXiv:2202.05015 (2022)], which proves domain number photon operator invariant during time evolution generated by