# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Operatoralgebraic methods in quantum field theory. A series of lectures. (English) Zbl 0839.46063
Berlin: Akademie Verlag. 228 p. DM 68.00 (1995).

This book gives a systematic exposition of basic as well as advanced methods of the theory of operator algebras in local quantum field theory. Starting with an approach based on nets of local algebras pioneered by R. Haar and D. Kastler (1964), the author surveys further developments in this area and the connection with deep results in the theory of operator algebras.

The book is organized into six chapters. In the first chapter the basic structure of nets of local algebras is introduced and studied. Classical results on vacuum representation (Reeh-Schlieder theorem, Borchers theorem, etc.) are exhibited. Some results on the classification of local algebras are also proved. The second chapter deals with localizable and transportable endomorphisms of quasi-local ${C}^{*}$-algebras and their equivalent classes (so-called DHR-superselection theory). The permutators of such endomorphisms and their basic properties are studied. The third chapter focuses on DHR-theory for automorphisms. In that case the field algebra and symmetry groups are constructed by means of permutators, monomorphic sections, cohomology theory of Abelian algebras and left modules over an operator algebra. Finally, the Wick-Wightman-Wigner picture is obtained as the Fourier transform of the field algebra. The fourth chapter is devoted to continue the study of DHR-theory (case of endomorphisms). Some important invariants for classes of irreducible endomorphisms are derived. Standard left inverse and conjugation is studied. Connection with Jones projection and canonical endomorphisms arising in the theory of inclusions of von Neumann algebras is shown. The fifth chapter is focused on construction of the field algebra in general case. This construction is based mainly on properties of finite endomorphisms. The regular representations and the Fourier transform of field algebras are studied. In the concluding chapter the unitary representation of the unit sphere is discussed. Among others the interrelation between the split property and the trace class condition is shown. A surprising result saying that the split property implies that all local algebras are injective type ${\text{III}}_{1}$ factors (and thereby isomorphic) is proved. The chapter ends with the Bisognano-Wichmann theorem.

This book clearly demonstrates that deep ideas in operator algebras theory (modular theory, Connes classification of type III factors, etc.) yield the sharpest results in the local quantum field theory. The book is self-contained (only basic knowledge on operator algebras is needed), well-organized and readable without missing depth. It is highly recommended for mathematicians as well as physicists interested in quantum field theory and should become one of the standard references for years to come.

##### MSC:
 46L60 Applications of selfadjoint operator algebras to physics 81T05 Axiomatic quantum field theory; operator algebras 46N50 Applications of functional analysis in quantum physics