Measures on function spaces, statistical mechanics and the rigorous renormalization group

函数空间、统计力学和严格重正化群的测度

• 批准号: 0907198
• 负责人: Abdelmalek Abdesselam
• 金额: $ 13.99万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907198&HistoricalAwards=false
• 关键词: Measures; function; spaces; statistical; mechanics

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Functional integrals were introduced by R. P. Feynman as a conceptual tool for the understanding of the quantum field theory of elementary particle physics. They provide a bridge from classical mechanics and field theory to their quantum counterparts.These integrals have become ubiquitous in physics and provide a foundation for our current understanding of a wide array of subjects: quantum electro and chromodynamics, the standard model of elementary particle physics, the theory of superconductivity, the statistical mechanics of phase transitions, turbulence, financial derivatives, and much more. From a mathematical standpoint the main question they pose is that of the construction and study of infinite dimensional probability measures in spaces of functions or distributions.The last decades have seen impressive results and successes in this area, and key to these successes is the renormalization group. The later can be described in nontechnical terms as follows. Consider a very large population of voters which have to opt between say two political parties. One can group them according to a hierarchy of larger and larger geographical regions, for instance county, state, nation, etc. One can also assign a `super elector' for each such region whose vote is determined by the majority at the next more detailed level of representation. In a nutshell the renormalization group is the transformation from the configurations of voting odds at one representation level to the next coarser one.The iteration of this transformation allows one to make predictions about the macroscopic behavior of this complex interacting system (the odds on say the majority vote at the national level) starting from the microscopic behavior (a mathematical model for the odds at the level of individuals). The above analogy with voting is given simply for the sake of exposition. In physics, one would, instead of voters, consider for instance spins or magnetic moments at the atomic level, and try to use the renormalization group machinery in order to infer the overall magnetization of a piece of material such as iron.The importance of the renormalization group in physics lies not in the mere qualitative description given above, but rather in the quantitative approximation scheme which has been developed on the basis of this intuition. The main goal of the proposed activity is to develop mathematical tools which allow a rigorous control of the error terms in this approximation scheme. The PI will in particular focus on the study of complete renormalization group trajectories between two fixed points, i.e., situations where one needs to control a system over the full range of scales from the infinitely small to the infinitely large. The PI will study such trajectories on a variety of models such as phi-four with a modified propagator in three dimensions, and the Gross-Neveu model in two dimensions.An important component of the proposed activity is graduate education. The PI will foster the involvementof graduate students in the proposed research work and will provide them with training in the mathematical tools from constructive quantum field theory and renormalization group theory needed in order to tackle research problems in the area.

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。泛函积分由 R. P. Feynman 引入，作为理解基本粒子物理量子场论的概念工具。它们提供了从经典力学和场论到量子对应物的桥梁。这些积分在物理学中已经无处不在，并为我们当前理解一系列广泛的学科奠定了基础：量子电学和色动力学、基本粒子物理学的标准模型、超导理论、相变统计力学、湍流、金融衍生品等等。从数学的角度来看，他们提出的主要问题是在函数或分布空间中构造和研究无限维概率测度。过去几十年在这一领域取得了令人印象深刻的成果和成功，而这些成功的关键是重正化群。后者可以用非技术术语描述如下。考虑一下有大量选民必须在两个政党之间进行选择。人们可以根据越来越大的地理区域的层次结构对它们进行分组，例如县、州、国家等。人们还可以为每个这样的区域分配一个“超级选举人”，其投票由下一个更详细的代表级别的多数决定。简而言之，重正化群是从一个代表级别的投票赔率配置到下一个更粗略的投票赔率配置的转换。这种转换的迭代允许人们从微观行为（个人级别赔率的数学模型）开始对这个复杂的相互作用系统的宏观行为（比如国家层面的多数票的赔率）进行预测。上述与投票的类比只是为了便于说明。在物理学中，人们会代替选民考虑原子水平上的自旋或磁矩，并尝试使用重整化群机制来推断铁等材料的整体磁化强度。重整化群在物理学中的重要性不在于上面给出的单纯定性描述，而在于基于这种直觉而开发的定量近似方案。拟议活动的主要目标是开发数学工具，允许严格控制该近似方案中的误差项。 PI 将特别关注两个固定点之间的完全重正化群轨迹的研究，即需要在从无限小到无限大的整个尺度范围内控制系统的情况。 PI 将在各种模型上研究此类轨迹，例如具有改进的三维传播器的 phi-4 模型和二维的 Gross-Neveu 模型。拟议活动的一个重要组成部分是研究生教育。 PI将促进研究生参与拟议的研究工作，并为他们提供解决该领域研究问题所需的建设性量子场论和重正化群论数学工具的培训。