Operator-theoretic infrared renormalization and construction of dressed 1-particle states
Operator-theoretic infrared renormalization and construction of dressed 1-particle states
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算子理论红外重整化和修饰 1 粒子态的构造
DOI：
10.3929/ethz-a-004184329
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发表时间：
2001
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影响因子：
--
通讯作者：
Thomas Chen
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文献类型：
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作者：
Thomas Chen
考虑质量为1的自由传播的非相对论带电粒子与量子化电磁场相互作用模型中的红外问题。在相互作用项中，系统的哈密顿量被红外截止值和紫外截止值正则化，单位为带电粒子的质量。由于平移不变性，只需研究哈密顿量对与总动量p有关的守恒动量算子的纤维Hilbert空间的限制就足够了。在耦合常数$g$和守恒动量$p$足够小的条件下，下列命题成立：(1)对于每个$\ssig>；0$，$E_0：=inf规范\hps$是一个本征值，其对应的本征向量为$\omga[p，\ssig]\in\hp$.(2)对于所有$\ssig\geq0$，$(egrd-\frac{|p|^2}{2})$的一阶和二阶导数是$O(g^\Delta)$，对于某些$Delta>；0$。(3)$\Omega[p，\ssig]$不是极限$\ssig\to 0$中Fock空间$\HP$的元素，如果$|p|>；0$。我们的证明是基于V.Bach，J.Fr\“ohlich和I.M.Sigal的算符理论重整化群。分析这个系统的关键困难是与主导相互作用项的严格边际性质有关，而我们论述的一个主要问题是开发分析工具来控制它的重整化流。
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 of the system is regularized by an infrared cutoff $\ssig\ll 1$ and an ultraviolet cutoff $\Lambda\sim 1$ in the interaction term, in units of the mass of the charged particle. Due to translation invariance, it suffices to study $\Hps$, the restriction of the hamiltonian to the fibre Hilbert space of the conserved momentum operator associated to total momentum $p\in\R^3$. Under the condition that the coupling constant $g$ and the conserved momentum $p$ are sufficiently small, the following statements hold: (1) For every $\ssig>0$, $E_0:=inf spec \Hps$ is an eigenvalue with corresponding eigenvector $\Omega[p,\ssig]\in\Hp$. (2) For all $\ssig\geq0$, the first and second derivatives of $(\Egrd-\frac{|p|^2}{2})$ are $O(g^\delta)$ for some $\delta>0$. (3) $\Omega[p,\ssig]$ is not an element of the Fock space $\Hp$ in the limit $\ssig\to0$, if $|p|>0$. Our proofs are based on the operator-theoretic renormalization group of V. Bach, J. Fr\"ohlich, and I.M. Sigal. 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.