### Abstract

Scaling limits of the Hamiltonian H of a system of N charged particles coupled to a quantized radiation field are considered. Ultraviolet cutoffs, λ̂_{1},..., 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, λ̂_{1} = ⋯ = λ̂_{N}, (ii) supports of ultraviolet cutoffs have no intersection, suppλ̂_{i}∩suppλ̂_{j}=∅, i≠j. The Hamiltonian acts on L^{2}(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^{l}B + κ^{2}1 ⊗ 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 L^{2}(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.

Original language | English |
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Pages (from-to) | 1755-1795 |

Number of pages | 41 |

Journal | Journal of Mathematical Physics |

Volume | 43 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 1 2002 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics