|Local Quantum Physics: Fields, Particles, Algebras
||Quantum field theory is the theory that describes all the known forces apart from gravity. However, the standard approach to quantum field theory via path-integrals is fraught with mathematical difficulties. One mathematical approach is to remove the fields as the primary objects and consider the algebras of observables as the fundamental objects of interest. Local Quantum Physics by Rudolf Haag introduces the reader to the ideas of constructive field theory and algebraic field theory.
Haag is a major player in algebraic field theory and this book gives his perspectives on the subject. The main mathematical tools employed in algebraic field theory are nets of c*-algebras. The book does not assume the reader is an expert in operator algebra, though some familiarity with quantum mechanics, quantum field theory and special relativity is a prerequisite. The book is not aimed at experts in constructive or algebraic field theory and so serves as a great introduction to the subject. The reader will be exposed to the main concepts and main theorems of algebraic field theory.
The readership is graduate students and researchers in mathematical physics interested in rigorous approaches to quantum field theory. The book may also be of interest to mathematicians working in operator algebra who would like to understand how to make contact with advanced physics.
In the following I will be referring to the second revised and enlarged edition of the book.
The book is divided into 8 chapters each containing between 3 and 5 sections.
Chapter I lays down the background of quantum field theory. Topics here include: basic concepts of quantum mechanics, locality in relativistic theories, Poincare invariant quantum field theory, the action principle and basic quantum field theory like canonical quantisation, free fields and gauge invariance.
General quantum field theory is the topic of Chapter II. Here one encounters the ideas of constructive field theory via the Wightman axioms. Topics discussed here include: the Wightman axioms, generating functionals, time ordered functions, covariant perturbation theory via Feynman diagrams, asymptotic configurations and particles, the S-matrix and the LSZ-formulation. The CPT theorem, spin-statistics theorem and analytical properties of the S-matrix are also discussed.
Chapter III moves on to algebraic field theory and the algebra of local observables. Topics here include: operator algebras (von Neumann, c* and w*), factors, positive linear forms and states, the GNS construction, nets of algebras of local observables and vacuum states. This chapter discusses the “guts” of algebraic field theory.
The next two chapters discuss some of more advanced aspects of algebraic field theory. That is superselction rules and the KMS-states.
Superselection sectors and symmetry is the topic of Chapter IV. Topic discussed include: charge superselction sectors, the DHR-anaysis, the Buchholz-Fredenhagen analysis, low dimensional space-time and braid statistics.
Chapter V moves on to thermal states and modular automorphisms. Topics here include Gibbs ensembles, the KMS condition, the Tomita-Takesaki theorem, equilibrium states, modular automorphisms of local algebras and nuclearity.
Chapter VI discusses the particle picture of quantum field theory. Topics here include: asymptotic particle cofigurations, particles & infraparticles and the physical state space of QED.
The interpretations and mathematical formalism of quantum physics is the topic of Chapter VII. Topics here include: the Copenhagen interpretation, the classical approximation, “quantum logic” and the EPR-effect.
The final part of the book, Chapter VIII is a concluding chapter which presents a retrospective look at algebraic field theory and presents some future challenges. Topics discussed here include: a comparison with Euclidean quantum field theory, supersymmetry and general relativity. The book presents the challenges faced by gravity, for instance QFT on curved space-times, Hawking radiation and the possibility of quantum gravity.
The book contains a healthy bibliography as well as an author index with references. This allows the reader to chase up the original literature, if desired.
Paperback: 390 pages
Publisher: Springer; 2nd. rev. and enlarged ed. edition (5 Aug 1996)