Scaling limits of the Hamiltonian H of a system of N charged particles coupled to a quantized radiation field are considered. Ultraviolet cutoffs, λ̂1,..., λ̂N, 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 = ⋯ = λ̂N, (ii) supports of ultraviolet cutoffs have no intersection, suppλ̂i∩suppλ̂j=∅, i≠j. The Hamiltonian acts on L2(ℝdN) ⊗F, where F is a symmetric Fock space, and has the form H=Hel⊗ 1 + B + 1 ⊗ Hquad. Here Hel denotes a particle Hamiltonian, Hquad a quadratic field operator, and B an interaction term. The scaling is introduced as H(κ)=Hel⊗ 1 +κlB + κ21 ⊗ Hquad, 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 Heff in L2(ℝdN) 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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics