Dynamics, spectral analysis, and renormalization in classical and quantum systems

本项目主要研究数学物理中的分析问题，主要由三个部分组成。在第一部分中，PI建议开发用于弱无序Anderson模型谱分析的重整化群（RG）方法。作为一个具体的应用，他计划研究态密度的性质。能够改进关于这个问题的现有知识的技术几乎肯定会对有关安德森模型的各种其他基本问题很重要。在第二部分中，PI建议继续他之前关于非相对论性量子电动力学（QED）的数学基础的研究。在各种合作中，他计划研究粒子内散射理论的各个方面，并进一步发展一种等光谱重整化群方法，用于分析量子场论中的光谱问题。在第三部分，PI计划研究与高维哈密顿系统动力学相关的稳定性问题。本提案中提出的项目涉及数学物理中的三种类型的问题。安德森模型被广泛用于研究随机介质中电子的量子动力学，如半导体。理解它对2维或更大的小紊乱的输运性质的预测是一个主要的开放问题，拟议的项目打算关注它的某些关键的多尺度方面。非相对论性QED描述了非相对论性量子力学物质（电子、原子、分子）与相对论性量子化电磁辐射场（光子）的相互作用。它以极好的精度模拟分子物理和化学中的低能过程。因为光子是无质量的，电子总是结合无限数量的低能量（软）光子，从而形成一个被称为下粒子的束缚态。将后者纳入量子场论的标准框架是极其困难的，因为所谓的红外突变（微扰计算通常由于软光子云而发散）。在PI和他的合作者先前工作的基础上，提议的项目打算进一步发展下粒子的散射理论。此外，PI建议研究具有大量自由度的哈密顿系统的动力学和稳定性方面，例如，描述经典晶体的晶格振动。

This project focuses on analytical problems in mathematical physics, and consists of three main parts. In the first part, the PI proposes to work on the development of renormalization group (RG) methods for the spectral analysis of the weakly disordered Anderson model. As a concrete application, he plans to study properties of the density of states. Techniques capable of improving the current knowledge about this problem will almost certainly be important for a variety of other essential questions concerning the Anderson model. In the second part, the PI proposes to continue his previous research on the mathematical foundations of non-relativistic Quantum Electrodynamics (QED). In various collaborations, he plans to study aspects of infraparticle scattering theory, and to further develop an isospectral renormalization group method for the analysis of spectral problems in quantum field theory. In the third part, the PI plans to investigate stability questions related to the dynamics of high-dimensional Hamiltonian systems.The projects presented in this proposal address three types of problems in mathematical physics. The Anderson model is widely used for the study of the quantum dynamics of electrons in random media, such as semiconductors. Understanding its predictions on transport properties at small disorders in dimensions 2 or larger poses a major open problem, and the proposed project intends to focus on certain key multiscale aspects of it. Non-relativistic QED describes non-relativistic quantum mechanical matter (electrons, atoms, molecules) interacting with the relativistic quantized electromagnetic radiation field (photons). It models low-energy processes in molecular physics and chemistry with excellent accuracy. Because photons are massless, electrons always bind an infinite number of low energetic (soft) photons, thus forming a bound state referred to as an infraparticle. Accommodating the latter into the standard framework of quantum field theory is extremely difficult, because of the so-called infrared catastrophe (perturbative computations typically diverge due to the soft photon cloud). Building on previous work of the PI and his collaborators, the proposed projects intend to further develop the scattering theory of infraparticles. Moreover, the PI proposes to investigate aspects of the dynamics and stability of Hamiltonian systems with a large number of degrees of freedom, which describe, for instance, the lattice vibrations of a classical crystal.