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
Publisher: American Mathematical Society
Year: 2007
Language: English
Commentary: no
Pages: 204
Cover......Page 1
Editors......Page 3
Title......Page 4
ISBN 978-3-03719-037-1......Page 5
Preface......Page 6
Contents......Page 8
Distributions on manifolds......Page 10
Riesz distributions on Minkowski space......Page 18
Lorentzian geometry......Page 26
Riesz distributions on a domain......Page 38
Normally hyperbolic operators......Page 42
The formal fundamental solution......Page 46
Uniqueness of the Hadamard coefficients......Page 48
Existence of the Hadamard coefficients......Page 51
True fundamental solutions on small domains......Page 53
The formal fundamental solution is asymptotic......Page 66
Solving the inhomogeneous equation on small domains......Page 72
Uniqueness of the fundamental solution......Page 76
The Cauchy problem......Page 81
Fundamental solutions......Page 95
Green's operators......Page 97
Non-globally hyperbolic manifolds......Page 101
C*-algebras......Page 111
The canonical commutator relations......Page 124
Quantization functors......Page 131
Quasi-local C*-algebras......Page 139
Haag–Kastler axioms......Page 145
Fock space......Page 149
The quantum field defined by a Cauchy hypersurface......Page 157
Categories......Page 165
Functional analysis......Page 167
Differential geometry......Page 170
Differential operators......Page 180
More on Lorentzian geometry......Page 182
Bibliography......Page 190
Figures......Page 194
Symbols......Page 196
Index......Page 200
Back Cover......Page 204
