Quantum Dynamics in Random Media, QED and Renormalization

本计画研究数学物理中两个重要的问题，其中与量子场论中的非微扰重整化方法密切相关的技术是必要的。 (1)量子传输。 随机介质中电子输运的研究具有重要的技术意义。 一些具有里程碑意义的数学定理解释了大混乱或极端能量下的绝缘现象（安德森局域化）。 在三维（扩展状态）中，小障碍的传导是预期的，但未经证实，并提出了一个关键的数学挑战。 本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。 本计画亦研究非相对论量子电动力学中的红外线问题，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的算子理论重整化群方法的基础早期的分析，专注于动力系统方面的无限维Banach空间。 该项目研究了数学物理中两个突出的问题，这些问题是基本物理过程的基础。 随机介质中电子输运的研究旨在了解半导体等材料的导电和绝缘特性，具有关键的技术重要性。 量子电动力学的研究旨在解决一个长期存在的难题，即电子和光子之间的相互作用，这对于建立原子结构和基本粒子相互作用的数学理论至关重要。

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.