Observable effects and parametrized scaling limits of a model in nonrelativistic quantum electrodynamics

Scaling limits of the Hamiltonian H of a system of N charged particles coupled to a quantized radiation field are considered. Ultraviolet cutoffs, λ̂₁,..., λ̂_N, are imposed on the radiation field and the Coulomb gauge is taken. It is the so-called Pauli-Fierz model in nonrelativistic quantum electrodynamics. We mainly consider two cases: (i) all the ultraviolet cutoffs are identical, λ̂₁ = ⋯ = λ̂_N, (ii) supports of ultraviolet cutoffs have no intersection, suppλ̂_i∩suppλ̂_j=∅, i≠j. The Hamiltonian acts on L²(ℝ^dN) ⊗F, where F is a symmetric Fock space, and has the form H=H_el⊗ 1 + B + 1 ⊗ H_quad. Here H_el denotes a particle Hamiltonian, H_quad a quadratic field operator, and B an interaction term. The scaling is introduced as H(κ)=H_el⊗ 1 +κ^lB + κ²1 ⊗ H_quad, where κ is a scaling parameter and l≤2 a parameter of the scaling. Performing a mass renormalization we consider the scaling limit of H(κ) as κ→∞ in the strong resolvent sense. Then effective Hamiltonians H_eff in L²(ℝ^dN) infected with reaction of effect of the radiation field is derived. In particular (1) effective Hamiltonians with an effective potential for l=2, and (2) effective Hamiltonians with an observed mass for l=1, are obtained.