Fumio Hiroshima
Observable effects and parametrized
scaling limits of a model in nonrelativistic quantum electrodynamics
(84K, Latex)
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.