Kontsevich's Deformation Quantization and Quantum Field Theory

This book provides an introduction to deformation quantization and its relation to quantum field theory, with a focus on the constructions of Kontsevich and Cattaneo & Felder.  This subject originated from an attempt to understand the mathematical structure when passing from a commutative classical algebra of observables to a non-commutative quantum algebra of observables. Developing deformation quantization as a semi-classical limit of the expectation value for a certain observable with respect to a special sigma model, the book carefully describes the relationship between the involved algebraic and field-theoretic methods. The connection to quantum field theory leads to the study of important new field theories and to insights in other parts of mathematics such as symplectic and Poisson geometry, and integrable systems. Based on lectures given by the author at the University of Zurich, the book will be of interest to graduate students in mathematics or theoretical physics. Readers will be able to begin the first chapter after a basic course in Analysis, Linear Algebra and Topology, and references are provided for more advanced prerequisites.

本書介紹了變形量子化及其與量子場論的關係，重點介紹了Kontsevich和Cattaneo＆Felder的構造。這個主題起源於試圖理解從可交換的經典可觀察代數到非可交換的量子可觀察代數時的數學結構。通過將變形量子化作為對於特定觀察量的期望值的半經典極限的發展，本書詳細描述了所涉及的代數和場論方法之間的關係。與量子場論的聯繫導致對重要的新場論的研究，並對其他數學領域（如輪廓和泊松幾何以及可積系統）有了深入的了解。本書基於作者在蘇黎世大學的講座，對數學或理論物理學的研究生具有興趣。讀者在完成基礎的分析、線性代數和拓撲課程後即可開始閱讀第一章，並提供了更高級先決條件的參考資料。