Dynamical Properties of Quantum Systems with Infinitely Many Degrees of Freedom

无限多自由度量子系统的动力学特性

• 批准号: 0905988
• 负责人: Alessandro Pizzo
• 金额: $ 20.9万
• 依托单位: University of California-Davis
• 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: 2009-07-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905988&HistoricalAwards=false
• 关键词: Dynamical; Properties; Quantum; Systems; Infinitely

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The goal of this research program is to obtain mathematical results concerning the dynamical properties of quantum systems with infinitely many degrees of freedom. More specifically, our interest will be focused on two main areas.1) Spectroscopy of systems of non-relativistic, quantum mechanical point-like nuclei and electrons interacting with the quantized radiation field. Because of the growing applications of quantum optics, physicists have become increasingly interested in experimental situations where the theoretical explanation must go beyond regular QED (quantum electrodynamics) perturbation theory. This calls for a more refined analysis of the mathematical theory of non-relativistic QED that describes the interaction of charged quantum matter and the quantized radiation at low energy scales related to those phenomena. Our main goal within this project is to improve the mathematical control on the dynamics of metastable states where a non-perturbative analysis is necessary. To this end, we plan to push forward the analysis of spectral, scattering, and expansion methods developed in recent years also with the contribution of the PI.2) Irreversible processes and transport equations in open quantum systems. Transport theory is intimately related to the study of many-body systems, for which a detailed control of the dynamics is beyond both analytic and numerical methods. Therefore we are forced to deal with effective equations for coarse-grained physical quantities, but one one would like to derive them from the underlying quantum dynamics. This is a mathematically challenging program that links some basic physical concepts to rigorous results expressed in mathematical proofs which hold, at least, in simple models with a non-trivial micro-dynamics. We will extend some first results that we have recently attained regarding quantum diffusion.The concepts and techniques involved in this research project connect different branches of analysis and mathematical physics. Our general strategy is to address mathematical physics problems from different perspectives, because we are confident that solid physical intuition can make links across different mathematical contexts. Conversely we want to gain more insight in the mathematical structures and techniques starting from the solution of physics problems. These problems will provide doctoral research projects for students in mathematics and in physics. The projects represent a valuable training for both students who will pursue academic research in mathematical physics as well as students who will apply tools from analysis and probability to model complex systems in physical applications.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该研究计划的目标是获得有关无限多自由度量子系统动力学性质的数学结果。更具体地说，我们的兴趣将集中在两个主要领域。1）非相对论性，量子力学点状核和电子与量子化辐射场相互作用系统的光谱学。由于量子光学的应用日益广泛，物理学家对实验情况越来越感兴趣，这些实验情况的理论解释必须超越常规的QED（量子电动力学）微扰理论。这就要求对非相对论QED的数学理论进行更精确的分析，该理论描述了带电量子物质和与这些现象相关的低能尺度下的量子化辐射之间的相互作用。我们在这个项目中的主要目标是提高数学控制的动态亚稳态的非微扰分析是必要的。为此，我们计划推进近年来开发的光谱，散射和扩展方法的分析，也包括PI的贡献。2）开放量子系统中的不可逆过程和输运方程。输运理论与多体系统的研究密切相关，对多体系统的动力学的详细控制超出了分析和数值方法。因此，我们不得不处理粗粒度物理量的有效方程，但有人想从底层的量子动力学中推导出它们。这是一个数学上具有挑战性的程序，它将一些基本的物理概念与数学证明中表达的严格结果联系起来，这些数学证明至少在具有非平凡微观动力学的简单模型中成立。我们将扩展我们最近获得的关于量子扩散的一些初步结果。这个研究项目涉及的概念和技术连接了分析和数学物理的不同分支。我们的总体策略是从不同的角度解决数学物理问题，因为我们相信坚实的物理直觉可以在不同的数学背景下建立联系。相反，我们希望从解决物理问题开始，对数学结构和技术有更多的了解。 这些问题将为数学和物理学生提供博士研究项目。 这些项目代表了一个有价值的培训，为学生谁将追求学术研究的数学物理，以及学生谁将应用工具，从分析和概率，以模拟复杂系统的物理应用。