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

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。泛函积分是由R.P.Feynman引入的，作为理解基本粒子物理量子场理论的概念工具。它们提供了从经典力学和场论到量子对应的桥梁。这些积分已经在物理学中变得无处不在，并为我们目前对一系列学科的理解提供了基础：量子电学和色动力学、基本粒子物理的标准模型、超导理论、相变的统计力学、湍流、金融导数等等。从数学的角度来看，他们提出的主要问题是构造和研究函数或分布空间中的无限维概率测度。在过去的几十年里，这一领域已经取得了令人印象深刻的结果和成功，而这些成功的关键是重整化群。后者可以用以下非技术术语来描述。考虑到有非常多的选民不得不在两个政党之间做出选择。人们可以根据越来越大的地理区域的等级将它们分组，例如县、州、民族等。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人决定。简而言之，重整化群是从一个表示级别的投票赔率配置到下一个更粗的表示级别的配置的转换。这种转换的迭代允许人们从微观行为(个人级别的投票赔率的数学模型)开始预测这个复杂相互作用系统的宏观行为(比方说大多数人在国家级别投票)。以上与投票的类比只是为了说明。在物理学中，人们不是选民，而是考虑原子水平上的自旋或磁矩，并试图使用重整化群机制来推断一块材料(如铁)的整体磁化强度。在物理学中，重整化群的重要性不仅仅在于上面给出的定性描述，而在于基于这一直觉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将特别专注于研究两个固定点之间的完全重整化群轨迹，即需要在从无限小到无限大的全尺度范围内控制系统的情况。PI将在各种模型上研究这样的轨迹，例如具有三维修正传播子的Phi-Four，以及二维的Gross-neveu模型。拟议活动的一个重要组成部分是研究生教育。国际和平研究所将培养研究生参与拟议的研究工作，并将为他们提供从构造量子场论和重整化群论中获得解决该领域研究问题所需的数学工具的培训。

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.