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

经典和量子系统中的动力学、谱分析和重整化

• 批准号: 0704031
• 负责人: Thomas Chen
• 金额: $ 13万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0704031&HistoricalAwards=false
• 关键词: Dynamics; spectral; analysis; renormalization; classical

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.

该项目重点关注数学物理中的分析问题，由三个主要部分组成。在第一部分中，PI 建议开发重整化群 (RG) 方法，用于弱无序安德森模型的谱分析。 作为具体应用，他计划研究态密度的性质。 能够提高当前有关该问题的知识的技术几乎肯定对于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建议继续之前对非相对论量子电动力学（QED）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，并进一步开发用于分析量子场论中光谱问题的等谱重整化群方法。 在第三部分中，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解决了数学物理中的三类问题。安德森模型广泛用于研究随机介质（例如半导体）中电子的量子动力学。 了解其对 2 维或更大维度的小无序的输运特性的预测提出了一个重大的开放问题，并且拟议的项目打算重点关注其某些关键的多尺度方面。 非相对论 QED 描述非相对论量子力学物质（电子、原子、分子）与相对论量子化电磁辐射场（光子）相互作用。它以极高的精度模拟分子物理和化学中的低能过程。由于光子是无质量的，因此电子总是结合无限数量的低能（软）光子，从而形成称为粒子下的束缚态。 由于所谓的红外灾难（微扰计算通常由于软光子云而发散），将后者纳入量子场论的标准框架极其困难。基于 PI 及其合作者之前的工作，拟议的项目旨在进一步发展次粒子的散射理论。 此外，PI 提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。