Aspects of quantum field theory in curved space-time.

*(English)*Zbl 0677.53081
London Mathematical Society Student Texts, 17. Cambridge etc.: Cambridge University Press. 300 p. £30.00/hbk. £10.95/pbk; $ 49.50/hbk; $ 19.95/pbk (1989).

This is a clearly written introduction to the quantum field theory in curved space-time. Much attention is paid to the physical ideas as well as to the mathematical techniques; in fact the mathematics is here more sophisticated than it is common in such introductory lectures in theoretical physics.

The book consists of nine chapters. The first chapter contains a concise discussion of the basic principles of quantum mechanics. The second one is devoted to the theory of eigenfunction expansions for self-adjoint elliptic differential operators including the problem of self-adjoint extensions, spectral theorems and Weyl-Titchmarsh-Kodaira theory. This is the mathematical basis for the “one loop” physics. In chapter three the scalar field theory is quantized while the next one contains a detailed discussion of two-point functions. In chapter five the stress tensor is calculated and the result applied to the Casimir effect. The general theory of quantum fields on curved space-time is developed in chapter six and applied in the next one to the case of the expanding Universe. The last two chapters are devoted to the problem of renormalization. First, some geometrical tools are introduced which allow to calculate the asymptotic expansions of relevant Green functions. Then the Wald axiomatics defining the renormalized energy-momentum tensor is explained and the proof of Wald’s uniqueness theorem outlined. To calculate effectively the energy-momentum tensor the point-splitting technique is used. Finally, in the Appendix, the Klein paradox is discussed in relation to black holes physics.

The book has all advantages gained from the fact that the author is actively working in the subject.

The book consists of nine chapters. The first chapter contains a concise discussion of the basic principles of quantum mechanics. The second one is devoted to the theory of eigenfunction expansions for self-adjoint elliptic differential operators including the problem of self-adjoint extensions, spectral theorems and Weyl-Titchmarsh-Kodaira theory. This is the mathematical basis for the “one loop” physics. In chapter three the scalar field theory is quantized while the next one contains a detailed discussion of two-point functions. In chapter five the stress tensor is calculated and the result applied to the Casimir effect. The general theory of quantum fields on curved space-time is developed in chapter six and applied in the next one to the case of the expanding Universe. The last two chapters are devoted to the problem of renormalization. First, some geometrical tools are introduced which allow to calculate the asymptotic expansions of relevant Green functions. Then the Wald axiomatics defining the renormalized energy-momentum tensor is explained and the proof of Wald’s uniqueness theorem outlined. To calculate effectively the energy-momentum tensor the point-splitting technique is used. Finally, in the Appendix, the Klein paradox is discussed in relation to black holes physics.

The book has all advantages gained from the fact that the author is actively working in the subject.

Reviewer: P.Maslanka

##### MSC:

53B50 | Applications of local differential geometry to the sciences |

81T20 | Quantum field theory on curved space or space-time backgrounds |

83C47 | Methods of quantum field theory in general relativity and gravitational theory |

47L90 | Applications of operator algebras to the sciences |

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |

83-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to relativity and gravitational theory |