The Quantum Mechanical Many-Body Problem and Statistical Mechanics

本研究涉及量子力学、凝聚态物理、量子信息论、量子电动力学等多体理论的各个方面，以及一些相关的数学问题。这些不同的主题是在几十年的研究中发展起来的，其潜在的统一性是，不同问题领域的解决方案相互阐明。智力价值包含在以下特定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)到其他对数学和物理感兴趣的李群的扩展。2. 在相关电子的哈伯德模型有关磁化和长程顺序的各种主题和猜想将被调查。凝聚态物理中的其他问题是理解极化子的结合，并建立能够严格证明具有条纹态的模型。3. 物质的一个基本统计力学性质是自由能的热力学极限的存在。到目前为止，当考虑到电磁辐射场时，对这一点的证明只是部分完成。它的目的是完成演示。4. Lieb-Thirring不等式在许多物理和数学问题中都很重要。一个目标是找到更清晰的常数。5. 人们将尝试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左右。即使用计算机，也很难确定原子的电离，但大自然似乎在告诉我们一个关于费米子的普遍事实，从第一原理来理解它是可取的。6. 原子和分子的密度泛函理论在实践和概念上都是重要的。一些关于穆勒理论的悬而未决的问题将被研究，以及可能的改进交换相关能的利布-牛津界，将包括局部动能的概念。7. 在该领域以前的专业知识的基础上，将追求日益重要的量子信息理论领域。特别是，最近在理解纠缠度量以及二部系统的不确定性原理方面的成功将用于尝试解决二部密度-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到这些项目中来，这将把指导与研究结合起来，帮助培养下一代精通数学的物理学家。它还将进一步加强数学家和物理学家之间的跨学科联系，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.