CAREER: Dynamics of complex quantum systems, scaling limits and renormalization

本研究主要包括三个方面的内容：（1）Gross-Pitaevskii（GP）方程组初值问题的适定性理论。Pavlovic（UT Austin）：我们工作的一个主要动机是我们的目标是从非线性色散偏微分方程中推广量子场论（QFT）的分析方法。(2)非相对论量子电动力学（QED）中的问题，特别是与电子的半经典运动，辐射阻尼和QED中的紫外重整化问题有关。(3)从动力学Hartree-Fock理论模拟的相互作用电子气体热动量分布函数的动力学标度极限导出动力学方程。这些项目涉及几个正在进行的合作，包括国际合作。对GP体系的研究，出现在相互作用玻色气体的分析中，是物理学中一些最令人兴奋的最新发展的桥梁（玻色-爱因斯坦凝聚）与一些最令人印象深刻的数学最近的发展非相对论QED是研究非相对论量子力学物质的物理理论（电子、原子、分子）与光的能量量子（光子）相互作用，并描述了技术中广泛的典型领域（化学、电子学、太阳能电池板建模等）的过程。从量子动力学推导动力学方程导致从量子物理学的经典物理学（流体动力学）的精确数学理解。该项目的教育部分涉及在高度多学科的年度主题计划中培养研究生，为他们提供分析，应用和计算数学，非线性偏微分方程和数学物理之间的联系的综合和专业理解。 其目标是为研究生提供对他们的研究领域的非常广泛的理解。

This research project addresses three main topics:(1) The well-posedness theory of the initial value problem for the Gross-Pitaevskii (GP) hierarchy,based on the approach initiated by the PI jointly with N. Pavlovic (UT Austin).A main motivation underlying our work is our aim at generalizing methods from nonlinear dispersive PDE's for the analysis of quantum field theories (QFT's). (2) Problems in non-relativistic Quantum Electrodynamics (QED), in particular relating to the semiclassical motion of electrons, radiation damping, and the problem of ultravioletrenormalization in QED.(3) Derivation of kinetic equations from kinetic scaling limits of the thermal momentum distribution function for interacting electron gases, modeled in dynamical Hartree-Fock theory. These projects involve several ongoing collaborations, including international ones. Study of the GP hierarchy, which emerges in the analysis of interacting Bose gases, bridgessome of the most exciting recent developments in physics (Bose-Einstein condensation)with some of the most impressive recent developments in mathematics (nonlinear dispersive PDE's).Non-relativistic QED is the physical theory of non-relativistic quantum mechanical matter (electrons, atoms, molecules) interacting with the energy quanta (photons) of light, anddescribes processes in a wide spectrum of quintessential areas in technology (chemistry,electronics, modeling of solar panels, etc). The derivation of kinetic equations from quantumdynamics leads to a precise mathematical understanding of classical physics (fluid dynamics) from quantum physics.The educational component of this project involves the training of graduate students in highly multidisciplinary annual thematic programs that provide them with a integrative and specialized understanding oflinks between Analysis, Applied and Computational Mathematics, nonlinear PDE's, and Mathematical Physics. The goal is to provide graduate students with an exceptionally broad understanding of their research fields.