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。潜在的统一是，不同问题领域的解决方案相互启发。该提案的智力价值包含在以下特定研究目标的部分列表中。1. 低密度玻色气体现在处于实验和理论研究的前沿，它打算继续以前在基态能量方面的成功工作，以及玻色-爱因斯坦凝聚问题及其与超流动性的关系。具体问题是基态能量膨胀中的bogolubov第二项的验证和旋转气体的最后线的计算。2. 关于相关电子的哈伯德模型的磁化和长程序的猜想将被研究。凝聚态物理学的其他目标是了解极化子的结合，并建立可以严格证明具有条纹状态的模型。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。PI打算锐化常数，并将不等式扩展到一个具有背景粒子密度而不是真空的新领域。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。大自然似乎在告诉我们一个关于费米子的普遍事实，我们希望从第一性原理来理解它。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。我们将研究穆勒理论的一些悬而未决的问题，以及对交换相关能的利布-牛津界的可能改进。7. 建立在以前的专业知识在该领域，主题在量子信息理论日益重要的领域将继续进行。更广泛的影响：学生和博士后将参与这些项目，这些项目将把指导与研究结合起来，帮助培养下一代具有数学知识的物理学家。它还将进一步加强数学家和物理学家之间的跨学科联系，并建立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. The underlying unity is that solutions in the various problem areas shed light on each other. The intellectual merit of this proposal is contained in the following partial list of specific research goals. 1. Low density Bose gases are now in the forefront of research experimentally and theoretically and it is intended to continue the previous successful work on the ground state energy, as well as the question of Bose-Einstein condensation and its relation to superfluidity. Specific questions are the validation of Bogolubovs second term in the expansion of the ground state energy and the calculation of the yrast line of a rotating gas. 2. Conjectures concerning magnetization and long-range order in the Hubbard model of correlated electrons will be investigated. Other goals 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. The PI intends sharpening the constants and to extend the inequalities to a new domain with a background density of particles instead of a vacuum. 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. 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. 7. Building on previous expertise in the area, topics in the increasingly important field of quantum information theory will be pursued. Broader impact: 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 establish the relevance of modern concepts of mathematical analysis to problems of condensed matter physics.