 01310 Thomas Chen
 Operatortheoretic infrared renormalization and construction of
dressed 1particle states
(325K, AMSLatex)
Aug 24, 01

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Abstract. We consider the infrared problem in a
model of a freely propagating, nonrelativistic charged particle of
mass $1$ in interaction with the quantized electromagnetic field.
The hamiltonian $H(\ssig)=H_0+g I(\ssig)$ of the system is
regularized by an infrared cutoff $\ssig\ll1$, and an ultraviolet
cutoff $\Lambda\sim1$ in the interaction term, in units of the
mass of the charged particle. Due to translation invariance, it
suffices to study the hamiltonian
$\Hps:=\left.H(\ssig)\right_{\Hp}$, where $\Hp$ denotes the fibre
space of the conserved momentum operator associated to total
momentum $p\in\R^3$. Under the condition that the coupling
constant $g$ is sufficiently small, there exists a constant
$\puppbd\in[\puppbdnum,1)$, such that for all $p$ with
$p\leq\puppbd$, the following statements hold:
(1) For every $\ssig>0$, $\Egrd:=$ inf spec $\Hps$ is an
eigenvalue with corresponding eigenvector $\Omega[p,\ssig]\in\Hp$.
(2) For all $\ssig\geq0$,
$\partial_{p}^\beta\left(\Egrd\frac{p^2}{2} \right)\leq
O(g^{\frac{1}{6}})$ for $\beta=0,1,2$.
(3) $\Omega[p,\ssig]$ is not an element of the Fock space $\Hp$ in
the limit $\ssig\rightarrow0$, if $p>0$.
Our proofs are based on the operatortheoretic renormalization
group of V. Bach, J. Fr\"ohlich, and I.M. Sigal \cite{bfs1,bfs2}.
The key difficulty in the analysis of this system is connected to
the strictly marginal nature of the leading interaction term, and
a main issue in our exposition is to develop analytic tools to
control its renormalization flow.
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