Quantum Dynamics in Random Media, QED and Renormalization

这个项目研究了数学物理中的两个突出问题，对这些问题来说，与量子场论中的非微扰重整化方法密切相关的技术是必不可少的。(1)量子输运。研究随机介质中的电子输运具有重要的技术意义。一些里程碑式的数学定理解释了大无序或极端能量下的绝缘(安德森定域化)。小紊乱时的传导预计在第三维(扩展状态)，但尚未得到证实，并构成了一个关键的数学挑战。这项工作旨在建立随机介质中量子输运问题的流体动力学极限。(2)物质与辐射的数学理论。该项目还研究了非相对论量子电动力学中的红外问题，即非相对论物质与量子化电磁场相互作用的理论。我们主要研究由自由传播的电子与量子化的电磁场相互作用组成的平移不变系统，以及与之相关的红外重整化问题。这项工作的目的是完善和进一步发展算子理论重整化群方法，它是前面分析的基础，专注于无限维Banach空间中的动力系统方面。这个项目研究数学物理中的两个突出问题，它们是基本物理过程的基础。研究随机介质中的电子输运是为了了解半导体等材料的导电和绝缘性质，具有重要的技术意义。量子电动力学的研究旨在解决一个长期存在的关于电子和光子之间相互作用的难题，这个问题对于建立一个关于原子结构和基本粒子相互作用的数学上合理的理论至关重要。

This project investigates two outstanding problems in mathematical physics for which techniques closely related to methods of non-perturbative renormalization in quantum field theory are essential. (1) Quantum transport. The study of electron transport in random media is of key technical importance. Some landmark mathematical theorems explain insulation at large disorders or extreme energies (Anderson localization). Conduction at small disorders is expected in dimension three (extended states), but is unproven, and poses a key mathematical challenge. This work aims to establish the hydrodynamic limit for a quantum transport problem in a random medium. (2) Mathematical theory of matter and radiation. This project also studies problems related to the infrared problem in non-relativistic quantum electrodynamics, the theory of non-relativistic matter interacting with the quantized electromagnetic field. We focus on the translation invariant system consisting of a freely propagating electron that interacts with the quantized electromagnetic field, and the associated problems of infrared renormalization. The work aims to refine and further develop the operator-theoretic renormalization group method that underlies earlier analysis, focusing on dynamical systems aspects in infinite-dimensional Banach spaces. This project investigates two outstanding problems in mathematical physics that underlie fundamental physical processes. The study of electron transport in random media aims at understanding conduction and insulation properties of materials such as semiconductors, and is of key technical importance. The study of quantum electrodynamics aims to solve a longstanding, difficult problem concerning the interaction between electrons and photons that is critical to establishing a mathematically sound theory of atomic structure and elementary particle interaction.