Holomorphic function spaces and quantization

全纯函数空间和量化

• 批准号: 1301534
• 负责人: Brian Hall
• 金额: $ 15.29万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301534&HistoricalAwards=false
• 关键词: Holomorphic; function; spaces; quantization

There has been much interest in the physics literature in the study of mathematical models involving matrices of large size. This work began with the pioneering results Eugene Wigner in the 1950's, in which the energy levels of certain models in nuclear physics were modeled as the "eigenvalues" of large random matrices. The beautiful result of Wigner is that the energy levels actually become nonrandom in the limit as the size of the matrices goes to infinity and are described by Wigner's famous semicircular distribution. This result was the first indication that certain complicated calculations actually become simpler as the size of the matrices involved gets larger. This mathematics research project by Brian Hall is in the same spirit. Hall and his collaborator T. Kemp study an important tool in quantum mechanics (the "Segal--Bargmann transform") on groups of matrices. Their first results indicate that the transform does indeed simplify substantially in the limit as the size of the matrices goes to infinity, to the point that the calculations can be carried out efficiently on a computer. There are many beautiful results still to investigate, including the problem of determining the appropriate analog of Wigner's semicircular law in this setting. This work has close connections to the two-dimensional version of strong nuclear force. That subject, in turn, has been connected to string theory by the Nobel Prize winner David Gross, since the "worldsheet" swept out by a string is a two-dimensional surface. This mathematics research project by Brian Hall concerns the mathematical theory of quantum mechanics. Quantum mechanics is the fundamental physical theory describing the behavior of matter at the atomic scale. Quantum mechanics is foundational to many areas of science and engineering, including solid state physics and the design of computer chips. Ideas from quantum mechanics also have had a profound impact in mathematics, as exemplified by the awarding of the Fields Prize (the so-called Nobel Prize for mathematics) to a physicist, Ed Witten, in 1990. A key issue in quantum mechanics concerns its connection with classical mechanics, the theory that governs the behavior of matter on the macroscopic scale. The Segal--Bargmann transform is a mathematical tool that facilitates a comparison between classical and quantum mechanics. Hall's earlier work has extended the range of application of the Segal--Bargmann transform, and has been cited extensively in both the physics and mathematics literature. Hall's ongoing research will expand the scope of the transform still farther, by connecting it to models derived from study of the strong nuclear force and string theory. This work has potential applications in both mathematics itself, with connections to the exciting new field of free probability theory, and in physics, with possible connections to both string theory and loop quantum gravity.

在涉及大尺寸矩阵的数学模型的研究中，对物理文献引起了极大的兴趣。这项工作始于1950年代的开创性结果Eugene Wigner，其中核物理学中某些模型的能级被建模为大型随机矩阵的“特征值”。 Wigner的美丽结果是，随着矩阵的大小输入无限，能量水平实际上变成了极限，并由Wigner著名的半圆形分布描述。该结果是第一个迹象表明，随着涉及的矩阵的大小变大，某些复杂的计算实际上变得更加简单。布莱恩·霍尔（Brian Hall）的这项数学研究项目也具有同样的精神。霍尔和他的合作者T. Kemp研究了量子力学（“ Segal-Bargmann Transform”）对矩阵组的重要工具。他们的第一个结果表明，随着矩阵的大小输入无穷大，转换确实确实在极限上进行了基本简化，以至于可以在计算机上有效地进行计算。仍有许多美丽的结果要进行调查，包括在这种情况下确定Wigner的半圆形定律的适当类似物的问题。这项工作与强核力量的二维版本有着密切的联系。反过来，该主题与诺贝尔奖获得者大卫·格罗斯（David Gross）联系在一起，因为“世界表”被弦扫除是二维表面。 Brian Hall的这项数学研究项目涉及量子力学的数学理论。量子力学是描述物质在原子量表的行为的基本物理理论。量子力学是许多科学和工程领域的基础，包括固态物理和计算机芯片的设计。量子力学的想法也对数学产生了深远的影响，这是1990年的物理学家埃德·维滕（Ed Witten）授予现场奖（所谓的诺贝尔奖）的例子。与经典力学的联系，该理论在宏观规模上控制物质的行为。 Segal-Bargmann变换是一种数学工具，可促进经典和量子力学之间的比较。霍尔的早期工作扩展了Segal-Bargmann Transform的应用范围，并在物理和数学文献中被广泛引用。霍尔正在进行的研究将通过将其连接到从强大的核力量和弦理论中得出的模型来扩大转换的范围。这项工作在两种数学本身中都有潜在的应用，并与令人兴奋的自由概率理论的新领域以及物理学的联系以及与字符串理论和循环量子重力的可能性都有可能的联系。