Holomorphic function spaces and quantization

在物理文献中，对涉及大尺寸矩阵的数学模型的研究引起了很大的兴趣。这项工作始于20世纪50年代尤金·维格纳（Eugene Wigner）的开创性成果，在该成果中，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”。维格纳绝妙的结果是当矩阵的大小趋于无穷时能级实际上在极限情况下变得非随机用维格纳著名的半圆分布来描述。这个结果第一次表明，某些复杂的计算实际上随着所涉及的矩阵的大小变大而变得更简单。Brian Hall的这个数学研究项目也是本着同样的精神。霍尔和他的合作者T. Kemp研究了矩阵群上量子力学中的一个重要工具（“西格尔-巴格曼变换”）。他们的第一个结果表明，当矩阵的大小趋于无穷大时，变换确实在极限下大大简化，以至于计算可以在计算机上有效地进行。还有许多美丽的结果有待研究，包括在这种情况下确定维格纳半圆定律的适当类比的问题。这项工作与二维版本的强核力有密切的联系。反过来，这个主题又被诺贝尔奖得主大卫·格罗斯（David Gross）与弦理论联系起来，因为被弦扫出的“世界”是一个二维表面。布赖恩·霍尔的这个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尺度上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，包括固态物理学和计算机芯片的设计。量子力学的思想也对数学产生了深远的影响，1990年物理学家埃德·威滕（Ed Witten）获得菲尔兹奖（也就是所谓的数学诺贝尔奖）就是例证。量子力学的一个关键问题是它与经典力学的联系，经典力学是在宏观尺度上支配物质行为的理论。西格尔-巴格曼变换是一种数学工具，便于在经典力学和量子力学之间进行比较。霍尔的早期工作扩展了西格尔-巴格曼变换的应用范围，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进一步扩大这种转变的范围，将其与强核力和弦理论研究得出的模型联系起来。这项工作在数学本身和物理学两方面都有潜在的应用，前者与令人兴奋的自由概率论的新领域有关，后者可能与弦理论和环量子引力有关。

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.