The Quantum Mechanical Many-Body Problem and Statistical Mechanics

本研究致力于量子力学、凝聚态物理、量子信息论和量子电动力学中多体理论的各个方面，以及一些相关的数学问题。这些不同的主题已经从几十年来的研究中发展出来，潜在的统一性是各个问题领域的解决方案相互启发。智力价值包含在以下具体研究目标的部分列表中。1.在最近成功证明了34年前关于自旋相干态Wehrl熵的猜想的基础上，将研究从SU（2）到其他数学和物理感兴趣的李群的扩展。2.本课程将探讨与关联电子哈伯德模型中的磁化强度和长程有序性有关的各种主题和结构。凝聚态物理学中的其他问题是理解极化子的束缚，并建立可以严格证明存在条纹态的模型。3.物质的一个基本的热力学-力学性质是自由能的热力学极限的存在。到目前为止，当考虑电磁辐射场时，这一点的证明只是部分完成。它旨在完成演示。4. Lieb-Thirring不等式在许多物理和数学问题中是重要的。目标之一是找到更清晰的常数。5.我们将试图验证一个长期存在的猜想，即大原子的最大负电离只有一个电子，即使用计算机也很难确定原子的电离，但大自然似乎告诉我们关于费米子的一个普遍事实，我们希望从第一原理来理解它。6.原子和分子的密度泛函理论具有重要的理论意义和实用价值。一些公开的问题，穆勒理论将进行调查，以及可能的改进，以利布-牛津界的交换相关能，将包括本地动能的概念。7.在该领域以前的专业知识的基础上，将继续研究日益重要的量子信息理论领域。特别是，最近的成功理解的措施，纠缠以及不确定性原则的二分系统将被用来试图解决二分密度矩阵重建问题。这项活动的一个更广泛的影响是，学生和博士后将参与这些项目，这将联合收割机指导与研究相结合，并帮助产生下一代数学知识丰富的物理学家。它还将进一步促进数学家和物理学家社区之间的跨学科联系，并促进现代数学分析概念与凝聚态物理问题的相关性。

This research is devoted to various aspects of many-body theory in quantum mechanics, condensed matter physics, quantum information theory, and quantum electrodynamics, as well as some related mathematical problems. These various topics have grown out of a body of research over several decades and the underlying unity is that solutions in the various problem areas shed light on each other. The intellectual merit is contained in the following partial list of specific research goals. 1. Building on the recent success in proving the 34 year old conjecture about the Wehrl entropy for spin coherent states, the extension from SU(2) to other Lie groups of interest to mathematics and physics will be investigated. 2. Various topics and conjectures concerning magnetization and long-range order in the Hubbard model of correlated electrons will be investigated. Other problems in condensed matter physics are to understand the binding of polarons and to build models that can be rigorously demonstrated to have striped states. 3. A fundamental statistical-mechanical property of matter is the existence of the thermodynamic limit of the free energy. A proof of this has been only partially accomplished so far when the electromagnetic radiation field is taken into account. It is intended to complete the demonstration. 4. The Lieb-Thirring inequalities are important in a variety of physical and mathematical problems. One goal is to find sharper constants. 5. An attempt will be made to verify the long-standing conjecture that the maximum negative ionization of a large atom is only about one electron. It is very difficult, even with computers, to be sure about ionization of atoms, but nature seems to be telling us about a universal fact about fermions that it would be desirable to understand from first principles. 6. Density functional theory for atoms and molecules is important practically and conceptually. Some open questions about the Muller theory will be investigated, as well as possible improvements to the Lieb-Oxford bound for the exchange- correlation energy that will include the concept of local kinetic energy. 7. Building on previous expertise in the area, the increasingly important field of quantum information theory will be pursued. In particular, the recent success in understanding measures of entanglement as well as the the uncertainty principle for bipartite systems will used to try to solve the bipartite density-matrix reconstruction problem. A broader impact of this activity is that students and postdocs will be involved in these projects, which will combine mentoring with research and help produce the next generation of mathematically knowledgeable physicists. It will also further the interdisciplinary bonds between the communities of mathematicians and physicists and promote the relevance of modern concepts of mathematical analysis to problems of condensed matter physics.