The Quantum Mechanical Many-Body Problem and Statistical Mechanics

这项研究致力于量子力学、凝聚态物理和量子电动力学中的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。这些不同的主题是经过几十年的研究而产生的，其根本的统一是各种问题领域的解决方案相互揭示。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，帮助培养下一代具有数学知识的物理学家。它还将促进数学家和物理学家群体之间的跨学科联系，并促进现代数学分析概念与凝聚态物理问题的相关性。这一建议的学术价值包含在以下具体研究目标的部分清单1中。低密度玻色气体现在处于研究的前沿(从实验和理论上)，它打算继续之前关于基态能量的成功工作，以及玻色-爱因斯坦凝聚问题及其与超流动性的关系。一个问题是关于固体和超流同时发生的。另一种方法是验证博古洛博夫在基态能量展开中的第二个任期。三是考察Lieb-Liniger一维模型的激发谱是否能为实验中使用的真实三维气体建立。我们将研究相关电子的Hubbard模型中有关磁化和长程有序的各种主题和猜想。同时，还将尝试证明二维Heisenberg模型基态的长程有序性。这两个模型目前在凝聚态物理中都很重要。物质的一个基本统计力学性质是存在‘自由能的热力学极限’。到目前为止，如果考虑到电磁辐射，这一点只有部分得到了证明。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，特别是关于多体和束缚态的问题。这延续了在之前的资助下开始的一项成功的非微扰研究，在该研究中，我们发展了一个对束缚态有意义的相对论QED模型。一个完整的理论还没有发明出来，尝试这样做是很重要的，因为QED是我们的基本物理理论之一。我们将尝试验证一个长期存在的猜想，即一个大原子的最大负电离度大约只有一个电子。要确定原子的电离是非常困难的，即使用计算机也是如此，但大自然似乎在告诉我们一个关于费米子的普遍事实，从第一原理来理解这一事实是可取的。量子信息论中的几个主题和密切相关的#领域。特别是，将为Holevo的熵猜想的可加性寻找一个证明(或反证)。Successhere将暗示纠缠态不能提高量子通道的容量。

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.