Title: Lecture Notes on Operator Algebras and Quantum Field Theory Show Chinese title

Title: 算子代数与量子场论讲义

Abstract: Lecture notes prepared for the EMS--IAMP Spring School ``Symmetries and Measurement in Quantum Field Theory''. This set of lecture notes covers four lectures: 1. Operator Algebras and Quantum Field Theory, 2. Tomita-Takesaki Modular Theory of von Neumann Algebras and the Bisognano-Wichmann/Borchers Theorem, 3. Local Covariant Quantum Field Theory, 4. Temperature and Entropy-Area Relation of Quantum Fields near Horizons of Dynamical Black Holes. The basic aim is to provide an introduction into some of the contemporary concepts and methods of quantum field theory in the operator-algebraic framework (lectures 1 to 3), and to illustrate (in lecture 4) how they may be applied in a theme of both perpetual and current interest - the ``thermodynamics'' (and also, the ``information content'') of quantum matter in the vicinity of black holes. While the topic of ``measurement'' in quantum field theory is not covered in these lectures (this topic has been presented in the lectures of Chris Fewster [arXiv:2504.17437]), the topic of``symmetry'' in quantum field theory does make an appearance: On one hand, in the form of the geometric action of Tomita-Takesai modular objects associated with certain operator algebras and states as stated in the theorems by Bisognano-Wichmann, and by Borchers, and on the other hand, in the form of local general covariance. The Spring School took place at the University of York, UK, April 7-11, 2025, organized by C.J. Fewster, D.W. Janssen and K. Rejzner, and funded by EPSRC Grant EP/Y000099/1 to the University of York, the European Mathematical Society, the International Association of Mathematical Physics, and COST Action (European Cooperation in Science and Technology) CA23115: Relativistic Quantum Information. The material presented in these notes is an expanded version of the material presented during the lectures by the author. Show Chinese abstract

Abstract: 为EMS-IAMP春季学校“量子场论中的对称性和测量”准备的讲义。 这套讲义涵盖了四次讲座：1. 算子代数与量子场论，2. 诺伊曼代数的Tomita-Takesaki模理论和Bisognano-Wichmann/Borchers定理，3. 局部协变量子场论，4. 动态黑洞视界附近量子场的温度和熵-面积关系。 基本目的是提供一些当代量子场论概念和方法的介绍，在算子代数框架下（讲座1到3），并在讲座4中说明它们如何应用于一个持续且当前感兴趣的主题——黑洞附近的量子物质的“热力学”（以及“信息内容”）。 虽然这些讲座中没有涉及“测量”在量子场论中的主题（该主题已在Chris Fewster的讲座中介绍[arXiv:2504.17437]），但“对称性”在量子场论中的主题有所涉及：一方面，以与某些算子代数和状态相关的Tomita-Takesai模对象的几何作用形式，如Bisognano-Wichmann和Borchers定理所述；另一方面，以局部一般协变性形式。 春季学校于2025年4月7日至11日在英国约克大学举行，由C.J. Fewster、D.W. Janssen和K. Rejzner组织，并由EPSRC资助项目EP/Y000099/1资助约克大学，欧洲数学协会，国际物理数学协会，以及欧洲科学与技术合作计划（COST Action）CA23115：相对论量子信息。 这些笔记中呈现的内容是作者在讲座中所讲内容的扩展版本。