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: 0940145
• 负责人: Thomas Chen
• 金额: $ 8.54万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-01-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0940145&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.

这个项目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建议发展重整化群(RG)方法来分析弱无序Anderson模型的谱。作为一个具体的应用，他计划研究态密度的性质。能够改进当前关于这个问题的知识的技术几乎肯定对于关于安德森模型的各种其他基本问题将是重要的。在第二部分中，PI建议继续他之前对非相对论量子电动力学(QED)的数学基础的研究。在各种合作中，他计划研究粒子下散射理论的各个方面，并进一步发展一种等谱重正化群方法来分析量子场论中的光谱问题。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。该提案中提出的项目涉及数学物理中的三类问题。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维或更大维度小无序的输运性质的预测是一个重大的悬而未决的问题，拟议的项目打算集中在它的某些关键的多尺度方面。非相对论QED描述了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)的相互作用。它以极高的精度模拟了分子物理和化学中的低能过程。由于光子是无质量的，电子总是与无限数量的低能(软)光子结合，从而形成一种被称为亚粒子的束束态。将后者纳入量子场论的标准框架是极其困难的，因为所谓的红外灾难(由于软光子云，微扰计算通常会发散)。在PI和他的合作者以前工作的基础上，拟议的项目旨在进一步发展亚粒子的散射理论。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，例如，描述经典晶体的晶格振动的哈密顿系统。