01-408 Fumio Hiroshima

Observable effects and parametrized scaling limits of a model in nonrelativistic quantum electrodynamics (84K, Latex) Nov 5, 01

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Abstract. Scaling limits of the Hamiltonian $H$ of a system of $N$ charged particles coupled to a quantized radiation field are considered. Ultraviolet cutoffs, $\la_1,....,\la_N$, are imposed on the radiation field and the Coulomb gauge is taken. It is so called the Pauli-Fierz model in nonrelativistic quantum electrodynamics. We mainly consider two cases: (i) all the ultraviolet cutoffs are identical, $\la_1=\cdots=\la_N$, (ii) supports of ultraviolet cutoffs have no intersection, ${\rm supp}\la_i\cap{\rm supp}\laj=\emptyset$, $i\not= j$. The Hamiltonian acts on $\LR\otimes\fff$, where $\fff$ is a symmetric Fock space and has the form $H=\ele\otimes\I+B+\I\otimes\fq$. Here $\ele$ denotes a particle Hamiltonian, $\fq$ a quadratic field operator, and $B$ an interaction term. The scaling is introduced as $H(\k)=\ele\otimes\I+\k^l B+\k^2\I\otimes\fq$, where $\k$ is a scaling parameter and $l\leq 2$ a parameter of the scaling. Performing a mass renormalization we consider the scaling limit of $H(\k)$ as $\k\rightarrow \infty$ in the strong resolvent sense. Then effective Hamiltonians $\eff$ in $\LR$ 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.

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