Quantum Dynamics in Random Media, QED and Renormalization

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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.