01-310 Thomas Chen

Operator-theoretic infrared renormalization and construction of dressed 1-particle states (325K, AMS-Latex) 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 operator-theoretic 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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