Lecture Notes on Operator Algebras and Quantum Field Theory

TL;DR

This paper provides an introduction to operator algebraic methods in quantum field theory, illustrating their application to black hole thermodynamics and quantum information near horizons.

Contribution

It offers a comprehensive overview of operator algebra techniques in quantum field theory and their application to black hole thermodynamics and quantum information.

Findings

Application of Tomita-Takesaki modular theory to quantum field algebras

Connection between quantum field entropy and black hole horizons

Illustration of symmetry and covariance in quantum field frameworks

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.