##
**Operatoralgebraic methods in quantum field theory. A series of lectures.**
*(English)*
Zbl 0839.46063

Berlin: Akademie Verlag. 228 p. (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.

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.

Reviewer: J.Hamhalter (Praha)

### MSC:

46L60 | Applications of selfadjoint operator algebras to physics |

81T05 | Axiomatic quantum field theory; operator algebras |

46N50 | Applications of functional analysis in quantum physics |