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**Quantum field theory in curved spacetime and black hole thermodynamics.**
*(English)*
Zbl 0842.53052

Chicago, IL: Univ. of Chicago Press. xiii, 205 p. (1994).

One of the most difficult subjects in physics is quantum field theory. Physically oriented introductions all seem to use highly questionable mathematical arguments. Too often the authors happily remark that the formulae, while being “formal” (i.e. ill-defined), lead to the “correct value”. Mathematically oriented texts usually do not give enough physical justification, and the uninitiated reader is left wondering what the mathematical structures presented have to do with the physical world. What is needed is a monograph which is honest about the mathematics but does not bypass the physical interpretation. Wald’s lecture notes come very close to striking the perfect balance.

In the first chapter, he gives an overview of the book and its purpose. In chapter 2 he recapitulates the symplectic structure of Hamiltonian mechanics and fundamental aspects of non-relativistic quantum theory, stressing the rôle of the symplectic form. The construction of the associated Hilbert space is not unique; however, any two possible choices lead to unitarily equivalent theories by the Stone-von Neumann theorem. He then illustrates the theory with the familiar example of finitely many harmonic oscillators. In chapter 3, he quantizes the Klein-Gordon field in analogy to the non-relativistic construction. Here, different choices may lead to unitarily inequivalent theories. However, his later discussions show that this problem is not as serious as it first appears to be. Finally, he gives an illuminating explanation of the particle interpretation.

These two chapters are quite easy to read and they provide a much deeper understanding than more traditional texts. This material should be present in any introduction to quantum field theory. The style is informal but the required mathematical theorems are standard knowledge and they are explicitly referred to.

In chapter 4 the Klein-Gordon field is quantized, first for stationary spacetime, then in the general case using the algebraic approach. Wald carefully explains the analogies to the previous constructions and discusses where they fail. In chapter 5 he discusses the Unruh effect, and in chapter 7 the Hawking effect. Chapter 6 is an account of classical black hole thermodynamics and lays the classical groundwork for chapter 7.

In these later chapters, Wald also outlines possible problems and sketches the extent to which they can be solved at present. Further, these discussions are illuminated by many comments comparing them with other parts of the construction. I find this way of presenting the material both honest and also extremely helpful for a real understanding of the subject. However, the material in the later chapters is so complicated that Wald’s informal style leaves the beginner struggling. On the other hand, a book which addresses all the problems in detail, with full proofs, would be much longer than Wald’s lecture notes. Even in this condensed form, his book is the best introduction to the subject available. In my opinion, the second and third chapters are a must for every physicist specializing in quantum field theory, even those with only an interest in ordinary quantum field theory in Minkowski space.

In the first chapter, he gives an overview of the book and its purpose. In chapter 2 he recapitulates the symplectic structure of Hamiltonian mechanics and fundamental aspects of non-relativistic quantum theory, stressing the rôle of the symplectic form. The construction of the associated Hilbert space is not unique; however, any two possible choices lead to unitarily equivalent theories by the Stone-von Neumann theorem. He then illustrates the theory with the familiar example of finitely many harmonic oscillators. In chapter 3, he quantizes the Klein-Gordon field in analogy to the non-relativistic construction. Here, different choices may lead to unitarily inequivalent theories. However, his later discussions show that this problem is not as serious as it first appears to be. Finally, he gives an illuminating explanation of the particle interpretation.

These two chapters are quite easy to read and they provide a much deeper understanding than more traditional texts. This material should be present in any introduction to quantum field theory. The style is informal but the required mathematical theorems are standard knowledge and they are explicitly referred to.

In chapter 4 the Klein-Gordon field is quantized, first for stationary spacetime, then in the general case using the algebraic approach. Wald carefully explains the analogies to the previous constructions and discusses where they fail. In chapter 5 he discusses the Unruh effect, and in chapter 7 the Hawking effect. Chapter 6 is an account of classical black hole thermodynamics and lays the classical groundwork for chapter 7.

In these later chapters, Wald also outlines possible problems and sketches the extent to which they can be solved at present. Further, these discussions are illuminated by many comments comparing them with other parts of the construction. I find this way of presenting the material both honest and also extremely helpful for a real understanding of the subject. However, the material in the later chapters is so complicated that Wald’s informal style leaves the beginner struggling. On the other hand, a book which addresses all the problems in detail, with full proofs, would be much longer than Wald’s lecture notes. Even in this condensed form, his book is the best introduction to the subject available. In my opinion, the second and third chapters are a must for every physicist specializing in quantum field theory, even those with only an interest in ordinary quantum field theory in Minkowski space.

Reviewer: M. Kriele (Berlin)

### MSC:

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

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

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

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

53C80 | Applications of global differential geometry to the sciences |

81T05 | Axiomatic quantum field theory; operator algebras |

83C57 | Black holes |