WAVE EQUATIONS ON LORENTZIAN MANIFOLDS AND QUANTIZATION

Preface v
1 Preliminaries 1
1.1 Distributions on manifolds ....................... 1
1.2 Riesz distributions on Minkowski space ................ 9
1.3 Lorentzian geometry .......................... 17
1.4 Riesz distributions on a domain ..................... 29
1.5 Normally hyperbolic operators ..................... 33
2 The local theory 37
2.1 The formal fundamental solution .................... 37
2.2 Uniqueness of the Hadamard coefficients ............... 39
2.3 Existence of the Hadamard coefficients ................. 42
2.4 True fundamental solutions on small domains ............. 44
2.5 The formal fundamental solution is asymptotic ............ 57
2.6 Solving the inhomogeneous equation on small domains ........ 63
3 The global theory 67
3.1 Uniqueness of the fundamental solution ................ 67
3.2 The Cauchy problem .......................... 72
3.3 Fundamental solutions ......................... 86
3.4 Green’s operators ............................ 88
3.5 Non-globally hyperbolic manifolds ................... 92
4 Quantization 102
4.1 C-algebras ............................... 102
4.2 The canonical commutator relations .................. 115
4.3 Quantization functors .......................... 122
4.4 Quasi-local C-algebras ........................ 130
4.5 Haag–Kastler axioms .......................... 136
4.6 Fock space ............................... 140
4.7 The quantum field defined by a Cauchy hypersurface ......... 148
Appendix. Background material 156
A.1 Categories ................................ 156
A.2 Functional analysis ........................... 158
A.3 Differential geometry .......................... 161
A.4 Differential operators .......................... 171
A.5 More on Lorentzian geometry ..................... 173
Bibliography 181
Figures 185
Symbols 187
Index 191

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.