Introduction to algebraic quantum field theory. Transl. from the Russian by K. M. Cook.

*(English)*Zbl 0711.46051
Mathematics and Its Applications: Soviet Series, 19. Dordrecht etc.: Kluwer Academic Publishers. xi, 301 p. Dfl. 260.00; $ 139.00; £84.00 (1990).

This book on the algebraic quantum field theory has three chapters.

Chapter 1 is concerned with the axiomatic formalism. Because the mathematical literature on the theory of algebras is abundant, the first section gives only the basic definitions and brief comments on some less known subjects such as Arveson spectral theory. In Section 2 are given the axioms of the algebraic approach. Section 3 on the structure of the local quantum theory constitutes the most important part of this chapter and presents all the theorems derived from the axioms. The following subjects are discussed:

- nature of the local algebras,

- relations between local algebras of different regions,

- duality,

- statistical (or local) independence,

- symmetry properties of the axiomatic formalism.

The title of Chapter 2 is: From the theory of observables to the theory of quantum fields. In fact there is no mention of a quantum field in Chapter 1 and to transform the axiomatic algebraic approach into a realistic field theory, an analysis of the superselection rules is presented, first the global theory of these rules in Section 1 and their local theory in section 2. The superselection rules appear as certain conditions on the set of all observables, both local and global. And these conditions complete the local structure of the theory. Finally Section 3 gives an introductory treatment of the methods used to reconstruct field theoretical properties but no concrete model is discussed.

In section 1 of Chapter 3 entitled: field algebras and their applications, two topics are discussed: first the theory of unbounded operator algebras which is presented in a systematic way and second the vacuum structure of quantum field theory. In section 2 the properties of von Neumann field algebras are considered in connection with the problem of constructing an algebraic description of a quantum field system.

The space available made it impossible for the author to present the algebraic methods of constructive field theory and to analyze particular kinds of field algebras. But in Section 3 to give an explicit example of local quantum theory and to complete his introduction to this theory the author discusses free and generalized free neutral fields.

There is an Appendix on the problem of constructing algebraic gauge quantum field theory.

[The Russian original (1986) has been reviewed in Zbl 0656.46055.]

Chapter 1 is concerned with the axiomatic formalism. Because the mathematical literature on the theory of algebras is abundant, the first section gives only the basic definitions and brief comments on some less known subjects such as Arveson spectral theory. In Section 2 are given the axioms of the algebraic approach. Section 3 on the structure of the local quantum theory constitutes the most important part of this chapter and presents all the theorems derived from the axioms. The following subjects are discussed:

- nature of the local algebras,

- relations between local algebras of different regions,

- duality,

- statistical (or local) independence,

- symmetry properties of the axiomatic formalism.

The title of Chapter 2 is: From the theory of observables to the theory of quantum fields. In fact there is no mention of a quantum field in Chapter 1 and to transform the axiomatic algebraic approach into a realistic field theory, an analysis of the superselection rules is presented, first the global theory of these rules in Section 1 and their local theory in section 2. The superselection rules appear as certain conditions on the set of all observables, both local and global. And these conditions complete the local structure of the theory. Finally Section 3 gives an introductory treatment of the methods used to reconstruct field theoretical properties but no concrete model is discussed.

In section 1 of Chapter 3 entitled: field algebras and their applications, two topics are discussed: first the theory of unbounded operator algebras which is presented in a systematic way and second the vacuum structure of quantum field theory. In section 2 the properties of von Neumann field algebras are considered in connection with the problem of constructing an algebraic description of a quantum field system.

The space available made it impossible for the author to present the algebraic methods of constructive field theory and to analyze particular kinds of field algebras. But in Section 3 to give an explicit example of local quantum theory and to complete his introduction to this theory the author discusses free and generalized free neutral fields.

There is an Appendix on the problem of constructing algebraic gauge quantum field theory.

[The Russian original (1986) has been reviewed in Zbl 0656.46055.]

Reviewer: P.Hillion

##### MSC:

46L60 | Applications of selfadjoint operator algebras to physics |

47L60 | Algebras of unbounded operators; partial algebras of operators |

81T05 | Axiomatic quantum field theory; operator algebras |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

46N50 | Applications of functional analysis in quantum physics |