The Quantum Mechanical Many-Body Problem and Statistical Mechanics

本研究致力于量子力学、凝聚态物理和量子电动力学中多体理论的各个方面，以及量子信息论和纯数学中的一些问题。这些不同的主题是从几十年来的研究中发展出来的，潜在的统一性是各个问题领域的解决方案相互启发。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，这些项目将把指导与研究结合起来，帮助培养下一代数学知识渊博的物理学家。它也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.这一建议的智力价值包含在以下部分具体研究目标清单中.低密度玻色气体现在处于研究的最前沿（实验和理论），它旨在继续以前在基态能量方面的成功工作，以及玻色-爱因斯坦凝聚及其与超流性的关系。一个问题是关于固体性和超流性的同时出现。二是验证Bogolubov在基态能量展开式中的第二项。第三是考察Lieb-Liniger一维模型的激发谱是否可以建立在实验所用的真实三维气体上.本课程将探讨哈伯德关联电子模型中有关磁化强度和长程有序性的各种问题。与此同时，我们将尝试证明二维海森堡模型基态的长程有序性。这两个模型在凝聚态物理学中都是重要的。物质的一个基本物理力学性质是自由能的极限。到目前为止，当考虑到电磁辐射时，这一点的证明只是部分完成。它的目的是完成演示.研究量子电动力学（QED）中的重整化问题，特别是在多体和束缚态方面。这延续了一项成功的非微扰研究，该研究是在先前的资助下开始的，在该研究中，我们开发了一个对束缚态有意义的相对论QED模型。完整的理论尚未发明，尝试这样做很重要，因为QED是我们的基本物理理论之一。5.我们将试图验证一个长期存在的猜想，即一个大原子的最大负电离只有一个电子。即使使用计算机，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，我们希望从第一性原理来理解这个事实。量子信息论及其密切相关领域的若干论题。特别是，证明（或反证）将寻求Holevo的可加性熵猜想。这意味着纠缠态不能提高量子信道的容量。

This research is devoted to various aspects of many-body theory in quantummechanics, condensed matter physics and quantum electrodynamics, as well as someproblems in quantum information theory and pure mathematics. These various topicshave grown out of a body of research over several decades and the underlying unity isthat solutions in the various problem areas shed light on each other. A broader impactof this activity is that students and postdocs will be involved in these projects, which willcombine mentoring with research and help produce the next generation ofmathematically knowledgeable physicists. It will also further the interdisciplinary bondsbetween the communities of mathematicians and physicists and promote the relevanceof modern concepts of mathematical analysis to problems of condensed matter physics.The intellectual merit of this proposal is contained in the following partial list of specificresearch goals.1. Low density Bose gases are now in the forefront of research {experimentally andtheoretically) and it is intended to continue the previous successful work on the groundstate energy, as well as the question of Bose-Einstein condensation and its relation tosuperfluidity. One question concerns the simultaneous occurrence of solidity andsuperfluidity. Another is to validateBogolubov's second term in the expansion of the ground state energy. A third is to see ifthe excitation spectrum of the Lieb-Liniger one-dimensional model can be establishedfor realistic three-dimensional gases used in experiments.2. Various topics and conjectures concerning magnetization and long-range order in theHubbard model of correlated electrons will be investigated. In parallel, an attempt will bemade to prove long-range order in the ground states of two-dimensional Heisenbergmodels. Both models are presently important in condensed matter physics.3. A fundamental statistical-mechanical property of matter is the existence of the`thermodynamical limit of the free energy. A proof of this has been only partiallyaccomplished so far when the electromagnetic radiation #eld is taken into account. It isintended to complete the demonstration.4. To study the renormalization problem in quantum electrodynamics (QED), especiallywith regard to the many-body and bound state aspects. This continues a successfulnon-perturbative study begun under the previous grant in which we developed a modelof relativistic QED that makes sense for bound states. A complete theory has not yetbeen invented, and it is important to try to do so because QED is one of ourfundamental physical theories.5. An attempt will be made to verify the long-standing conjecture that the maximumnegative ionization of a large atom is only about one electron. It is very di#cult, evenwith computers, to be sure about ionization of atoms, but nature seems to be telling us auniversal fact about fermions that it would be desirable to understand from firstprinciples.6. Several topics in quantum information theory and closely related #elds. In particular,a proof (or disproof) will be sought for Holevo's additivity of entropy conjecture. Successhere will imply that entangled states cannot improve the capacity of quantum channels.