Quantization, complex structures, and spaces of holomorphic functions

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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.