Quantization, complex structures, and spaces of holomorphic functions

该项目延续了PI在复杂结构和全纯方法量化方面的工作，重点转移到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，利用虚时测地线流来理解黎曼流形上某些所谓的适应性复杂结构，并期望通过在测地线流中引入磁项来构建一类新的复杂结构。这个项目的另一部分包括对PI早期研究的回归，即对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主题中的各种基本结构，即使找到适当的定义也是极其困难的。PI正在开发的技术有望对量子物理学，特别是量子场论的基础学科做出贡献。场论是具有无限多个自由度的系统，这种理论的量子化是出了名的困难。全纯量化方法在有限维情况下已经被证明是卓有成效的，并且相对于无限维极限具有一定的优势。特别是，无限维群经常出现在量子场论中，所以pi研究这些群离应用不远了。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。这本书的目的是使量子力学对数学家的读者来说是容易理解的。这本书将填补必要的背景，从经典力学，然后解释量子力学使用符号，是数学家熟悉的，并表示对重要的技术数学问题，在物理文献中掩盖的尊重。然而，我们的目标不是强调数学技术，而是用数学家们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长期互利的相互作用。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。

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.