Localization of the number of photons of ground states in nonrelativistic QED

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L2(R3) ⊗ F ∼= L2(R3;F), where F is the Boson Fock space over L2(R3 × {1, 2}). It is shown that the ground state, ψg, of H belongs to ∩k=1D(1 ⊗Nk), where N denotes the number operator of F . Moreover it is shown that, for almost every electron position variable x ∈ R3 and for arbitrary k ≥ 0, ‖(1⊗N)ψg(x)‖F ≤ Dke m+1 with some constants m ≥ 0, Dk > 0, and δ > 0 independent of k. In particular ψg ∈ ∩k=1D(e m+1 ⊗ Nk) for 0 < β < δ/2 is obtained.