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.

该项目延续了 PI 与复杂结构和全纯量化方法相关的工作，重点转向无限维群上的全纯函数空间。 PI 与 W. Kirwin 合作，开发了一种新方法来理解黎曼流形上的某些所谓的适应复杂结构，使用虚时间测地线流，并期望通过引入磁力来构造一类新的复杂结构。项进入测地线流。该项目的另一部分包括回归 PI 早期研究部分，即无限维群上的全纯函数空间的研究。 当前的目标是更好地理解该主题中的各种基本结构，即使找到正确的定义也是极其困难的。 PI 正在开发的技术预计将为量子物理学做出贡献，尤其是量子场论的基础学科。场论是具有无限多个自由度的系统，并且此类理论的量化非常困难。 事实证明，全纯量化方法在有限维设置中已经取得了丰硕的成果，并且相对于无限维限制具有一定的优势。特别是无限维群经常出现在量子场论中，因此 PI 在此类群上的工作离应用并不遥远。PI 在量子理论方面的工作促使他开始写一本关于量子力学的书。本书的目标是让数学家读者能够理解量子力学。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技术细节，而是用数学家熟悉的语言解释量子理论的主要思想。 PI 希望这本书能够为量子物理和数学之间的长期互利互动做出贡献。 PI 将继续撰写更多的说明性文章和研究文章，并将使用为本书开发的材料教授研究生课程。