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Wave equations on Lorentzian manifolds and quantization. (English) Zbl 1118.58016
ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society Publishing House (ISBN 978-3-03719-037-1/pbk). viii, 194 p. (2007).
The book under review deals with the theory of linear wave equations on Lorentzian manifolds which basically are mathematical structures for space-times in the theory of general relativity. Due to the fact that physical fields (e.g., electromagnetic fields) have to satisfy a wave equation on the underlying space-time, the authors focus on global solutions of wave equations on Lorentzian manifolds and consider the concept of quantization from a formal mathematical point of view.
The local theory of wave equations and related concepts have been studied in [F. G. Friedlander, The wave equation on a curved space-time. Cambridge Monographs on Mathematical Physics. 2. London: Cambridge University Press. (1975; Zbl 0316.53021), P. Günther, Huygens’ principle and hyperbolic equations. Perspectives in Mathematics, Academic Press Inc., Boston, MA (1988; Zbl 0655.35003), Y. Choquet-Bruhat, Lect. Math. Phys. 84–106 (1968; Zbl 0169.43202), J. Leray, Hyperbolic differential equations. Princeton (1953), Russian translation (1984; Zbl 0588.35002)] . However, the aim of the authors is to study existence and uniqueness of solutions defined on the entire Lorentzian manifold.
In Chapter 1, the authors give fundamentals about distributions on manifolds, Lorentzian geometry and normally hyperbolic operators as well as the Riesz distributions.
In Chapter 2, by using Riesz distributions and a procedure similar to the construction of the heat kernel for a Laplace type operator on a compact Riemannian manifold, they provide local solution for wave equations on Lorentzian manifolds.
In Chapter 3, they study and construct global fundamental solutions, Green’s operator and solutions to the Cauchy problem by the usage of the local solution developed in the previous chapter. To achieve their results, they assume the underlying Lorentzian manifold is globally hyperbolic. Besides, they consider some non globally hyperbolic Lorentzian manifolds (e.g., anti-de Sitter space-time) on which they can have the existence but not necessarily uniqueness of global solutions.
In Chapter 4, they provide a basic explanation of quantization and describe the theory of \(C^\star\)-algebras as well as CCR-representation. They give a mathematical foundation of quantum field theory. Although, they discuss the quantization functor and check that Haag-Kestler axioms of a local quantum theory are satisfied, they recommend the reader should refer to other sources for \(n\)-point functions, states renormalization, nonlinear fields and physical applications such as Hawking radiation.
They also provide some introductory material on category theory, functional analysis, differential geometry and differential operators in the appendix.

58J45 Hyperbolic equations on manifolds
81T20 Quantum field theory on curved space or space-time backgrounds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
35L05 Wave equation
35L15 Initial value problems for second-order hyperbolic equations
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
81T50 Anomalies in quantum field theory
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