Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians

by Kasia Rejzner

Overview

Perturbative Algebraic Quantum Field Theory (pAQFT), the subject of this book, is a complete and mathematically rigorous treatment of perturbative quantum field theory (pQFT) that doesn’t require the use of divergent quantities and works on a large class of Lorenzian manifolds.

We discuss in detail the examples of scalar fields, gauge theories and the effective quantum gravity.

pQFT models describe a wide range of physical phenomena and have remarkable agreement with experimental results. Despite this success, the theory suffers from many conceptual problems. pAQFT is a good candidate to solve many, if not all, of these conceptual problems.

Chapters 1-3 provide some background in mathematics and physics. Chapter 4 concerns classical theory of the scalar field, which is subsequently quantized in chapters 5 and 6. Chapter 7 covers gauge theory and chapter 8 discusses effective quantum gravity.

The book aims to be accessible to researchers and graduate students, who are interested in the mathematical foundations of pQFT.

Product Details

ISBN-13:	9783319798578
Publisher:	Springer International Publishing
Publication date:	04/26/2018
Series:	Mathematical Physics Studies
Edition description:	Softcover reprint of the original 1st ed. 2016
Pages:	180
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

Introduction.- Algebraic approach to quantum theory.- Algebraic quantum mechanics.- Causality.- Haag-Kastler axioms.- pAQFT axioms.- LCQFT.- Kinematical structure.- The space of field configurations.- Functionals on the configuration space.- Fermionic field configurations.- Vector fields.- Functorial interpretation.- Classical theory.- Dynamics.- Natural Lagrangians.- Homological characterization of the solution space.- The net of topological Poisson algabras.- Analogy with classical mechanics.- Deformation quantization.- Star products.- The star product on the space of multivector fields.- Kähler structure.- Interaction.- Outline of the approach.- Scatering matrix and time ordered products.- Renormalization group.- Interacting local nets.- Explicit construction.- Gauge theories.- Classical gauge theory.- Gauge-fixing.- BV formalism.- Effective quantum gravity.- From LCQFT to quantum gravity.- Dynamics and symmetries.- Linearized theory.- Quantization.- Relational observables.- Background independence.

Editorial Reviews

The author, who claims to be both a physicist and a mathematician, offers a useful and fascinating book which should be of interest and useful to professional mathematicians and students of both mathematics and physics. It is dedicated to prominent mathematical physicists who passed away recently: Rudolf Haag, Daniel Kastler, Uffe Haagerup, Raymond Stora, and John Roberts ... . The book intends to be pedagogical and self-contained. It establishes the mathematical foundation of perturbation theory in a convincing manner.” (Gert Roepstorff, zbMATH 1347.81011, 2016)

From the Publisher