Quantization, complex structures, and spaces of holomorphic functions

量子化、复数结构和全纯函数空间

• 批准号: 1001328
• 负责人: Brian Hall
• 金额: $ 13.74万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001328&HistoricalAwards=false
• 关键词: Quantization; complex; structures; spaces; holomorphic

This project continues the PI's work related to complex structures and theholomorphic approach to quantization, with the emphasis shifting toward spacesof holomorphic functions on infinite-dimensional groups. The PI, in work with W. Kirwin, has developed a new method of understanding certain so-called adapted complex structures on a Riemannian manifold, using the imaginary-time geodesic flow and expects to construct a new class of complex structures by introducing a magnetic term into the geodesic flow. Another part of this project includes a return to an earlier part of the PI's research, namely the study of holomorphic function spaces over infinite-dimensional groups. The current goal is to develop a better understanding of various basic constructs in the subject where even finding the proper definitions is extremely difficult. The techniques the PI is developing are expected to contribute to quantum physics, especially to the fundamental subject of quantum field theory. Field theories are systems with infinitely many degrees of freedom, and quantization of such theories is notoriously difficult. The holomorphic approach to quantization has proved fruitful already in the finite-dimensional setting, and it has certain advantages with respect to the infinite-dimensional limit. In particular, infinite-dimensional groups show up often in quantum field theory, so the PIs work on such groups is not far from applications.The PI's work in quantum theory has led him to start writing a book on quantum mechanics. The goal of this book is to make quantum mechanics accessibleto an audience of mathematicians. This book will fill in the necessary background from classical mechanics and then explain quantum mechanics using notation that is familiar to mathematicians, and showing respect for the significant technical mathematical issues that are glossed over in the physics literature. The goal, however, is not to emphasize the mathematical technicalities, but rather to explain the main ideas of quantum theory in language that mathematicians feel comfortable with. The PI hopes that this book will contribute to the long and mutually beneficial interaction between quantum physics and mathematics. The PI will continue to write additional expository articles as well as research articles and will teach a graduate course using the materials being developed for the book.

这个项目延续了PI在复杂结构和量子化的全纯方法方面的工作，重点转向无限维群上的全纯函数空间。PI与W. Kirwin，已经开发出一种新的方法来理解某些所谓的适应复杂结构的黎曼流形上，使用时间测地线流，并期望通过引入一个磁项到测地线流构造一类新的复杂结构。该项目的另一部分包括返回到PI研究的早期部分，即研究无限维群上的全纯函数空间。 目前的目标是更好地理解该主题中的各种基本结构，即使找到适当的定义也非常困难。PI正在开发的技术预计将有助于量子物理学，特别是量子场论的基本主题。场论是具有无限多个自由度的系统，量子化这种理论是出了名的困难。 量子化的全纯方法在有限维的情况下已经证明是卓有成效的，而且在无限维的情况下也有一定的优势。特别是量子场论中经常出现无限维群，PI对这类群的研究离应用不远了，PI在量子理论方面的工作使他开始写一本关于量子力学的书。这本书的目标是使量子力学为数学家所理解.本书将填补经典力学的必要背景，然后使用数学家熟悉的符号解释量子力学，并尊重物理文献中掩盖的重要技术数学问题。然而，我们的目标不是强调数学的技术性，而是用数学家感到舒服的语言来解释量子理论的主要思想。PI希望这本书将有助于量子物理学和数学之间的长期互利互动。PI将继续撰写更多的临时文章以及研究文章，并将使用为该书开发的材料教授研究生课程。