This project continues the PI's work related to complex structures and the holomorphic approach to quantization, with the emphasis shifting toward spaces of 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 accessible to 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.
The theory of quantum mechanics governs the behavior of atoms and molecules on the microscopic scale. Quantum theory is an essential tool in chemistry and in the construction of modern computer chips. During the period of the PI's grant, he has developed a number of mathematical techniques related to quantum mechanics and to the theory of quantization, which helps us to understand how quantum mechanics relates to the classical mechanics which governs the behavior of matter at the scale of everyday life. Part of the PI's work concerns "coherent states," which are states of a quantum system in which the particle behaves as classically as possible. Athough coherent states have been part of quantum mechanics from the earliest days, the PI has constructed new families of coherent states describing particles moving on curved surfaces and in the presence of a magnetic field. Another aspect of the PI's research concerns the idea of internal degrees of freedom. In quantum mechanics, a particle can have special types of motion that a classical particle does not have. In the simplest case, these internal degrees of freedom are called spin and can be visualized as tiny, spinning gyroscope attached to the particle. Other internal degrees of freedom are harder to visualize but play an important role in the behavior of particles in experiments. The PI has been studying the behavior of quantum systems as the number of internal degrees of freedom grows large. Surprisingly, the behavior of the system can actually simplify in this situation. The PI has developed new tools for studying the behavior of quantum systems in this so-called "large-N limit." The PI has also been active in promoting interaction between the fields of mathematics and physics. Mathematical results often play an important role in modern theories of physics. In the other direction, ideas from physics often can be inspirational to mathematicians. For this reason, many mathematicians are interested in learning about quantum mechanics, but they often have a difficult time reading books written by (and for) physicists. The PI has written a graduate-level textbook (Quantum Theory for Mathematicians, Springer, 2013) that attempts to bridge this gap. The book translates ideas from physics into language mathematicians feel comfortable with, while also filling in certain mathematical details that sometimes get overlooked in the physics literature.
