Mathematical Sciences: Invariant Variational Problems and Quantization

李群理论是为了纪念挪威数学家索夫斯·李而命名的，它一直是20世纪数学的主要主题之一。作为利用系统中固有对称性的数学工具，李理论对数学本身和理论物理，特别是量子力学、基本粒子物理和相对论产生了深远的影响。特别是，结构和对称性是交织在一起的，正如爱因斯坦和后来的杨-米尔斯的基本工作所说明的那样，在这些工作中，对自然界基本力的深刻成功的非线性理论受到对大量对称性的需要的数学指导。在不同的方向上，数学家和物理学家，通常是独立的，已经开发了大量强大的技术来分类对称的线性实现。这就是“表征理论”的主题。祖克曼教授在李群表示理论的许多方面做了开创性的创造性工作，他现在正在将他的工作应用于基础物理学。十年前，他发明了新的方法来构造李群的线性表示，他的方法借鉴了不同的、看似不相关的数学领域。最近，祖克曼教授的努力都是为了适应物理学中令人兴奋的“弦理论”的新研究。这个理论的目的是对自然界所有的力提供一个统一的描述。在目前的提案中，他将研究弦理论中对称性的线性和非线性表现，以及其他更传统的物理理论。这包括从经典物理理论过渡到量子物理理论的新基础的发展，以及由此产生的两种理论与爱因斯坦的狭义相对论和广义相对论的兼容性。对称性在经典水平上是非线性的，在量子水平上是线性的。对称的双重作用是他作品的一个基本方面。

The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, Lie theory has had a profound impact upon mathematics itself and theoretical physics, especially quantum mechanics, elementary particle physics, and relativity. In particular, the structures and the symmetries are interwoven, as illustrated by the fundamental work of Einstein, and later Yang-Mills, in which profoundly successful nonlinear theories for the fundamental forces in nature have been guided mathematically by the need for a large family of symmetries. In a different direction, mathematicians and physicists, often independently, have developed a vast array of powerful techniques that classify the linear realizations of the symmetries. This is the subject of "representation theory". Professor Zuckerman has done pioneering creative work in many facets of representation theory of Lie groups, and he is now applying his work to fundamental physics. Ten years ago he invented new ways to construct linear representations of Lie groups, his methods drawing from diverse, seemingly unrelated areas of mathematics. Recently, Professor Zuckerman's efforts have been directed toward their adaptation to the exciting new study of "string theory" in physics. The goal of this theory is to provide a unified description of all the forces of nature. In the current proposal he will conduct research on both linear and nonlinear manifestations of symmetry in string theory, as well as other more conventional physical theories. This includes the development of a new basis for the passage from classical to quantum physical theories, and resulting compatibility of both theories with Einstein's special and general relativity. Symmetries are realized nonlinearly at the classical level and linearly at the quantum level. This dual role of symmetry is a fundamental aspect of his work.