This book provides a detailed introduction to linear wave equations on Lorentzian manifolds (for vector-bundle valued fields). After a collection of preliminary material in the first chapter, one finds in the second chapter the construction of local fundamental solutions together with their Hadamard expansion. The third chapter establishes the existence and uniqueness of global fundamental solutions on globally hyperbolic spacetimes and discusses Green's operators and well-posedness of the Cauchy problem. The last chapter is devoted to field quantization in the sense of algebraic quantum field theory. The necessary basics on $C^*$-algebras and CCR-representations are developed in full detail. The text provides a self-contained introduction to these topics addressed to graduate students in mathematics and physics. At the same time, it is intended as a reference for researchers in global analysis, general relativity, and quantum field theory. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Author(s): Christian Bar, Nicolas Ginoux, Frank Pfaffle
Series: Esi Lectures in Mathematics and Physics
Edition: 1
Publisher: European Mathematical Society
Year: 2007
Language: English
Pages: 202
Preface......Page 5
Contents......Page 7
Distributions on manifolds......Page 9
Riesz distributions on Minkowski space......Page 17
Lorentzian geometry......Page 25
Riesz distributions on a domain......Page 37
Normally hyperbolic operators......Page 41
The formal fundamental solution......Page 45
Uniqueness of the Hadamard coefficients......Page 47
Existence of the Hadamard coefficients......Page 50
True fundamental solutions on small domains......Page 52
The formal fundamental solution is asymptotic......Page 65
Solving the inhomogeneous equation on small domains......Page 71
Uniqueness of the fundamental solution......Page 75
The Cauchy problem......Page 80
Fundamental solutions......Page 94
Green's operators......Page 96
Non-globally hyperbolic manifolds......Page 100
C*-algebras......Page 110
The canonical commutator relations......Page 123
Quantization functors......Page 130
Quasi-local C*-algebras......Page 138
Haag–Kastler axioms......Page 144
Fock space......Page 148
The quantum field defined by a Cauchy hypersurface......Page 156
Categories......Page 164
Functional analysis......Page 166
Differential geometry......Page 169
Differential operators......Page 179
More on Lorentzian geometry......Page 181
Bibliography......Page 189
Figures......Page 193
Symbols......Page 195
Index......Page 199
