We focus on free fields and the corresponding quasi-free states and more precisely on Klein–Gordon fields and Dirac fields. The first chapters are devoted to preliminary material on CCR*-algebras, quasi-free states, wave equations on Lorentzian manifolds, microlocal analysis and to the important Hadamard condition, characterizing physically acceptable quantum states on curved spacetimes. In the later chapters more advanced tools of microlocal analysis, like the global pseudo-differential calculus on non-compact manifolds, are used to construct and study Hadamard states for Klein–Gordon fields by various methods, in particular by scattering theory and by Wick rotation arguments. In the last chapter the fermionic theory of free Dirac quantum fields on Lorentzian manifolds is described in some detail.
This monograph is addressed to both mathematicians and mathematical physicists. The first will be able to use it as a rigorous exposition of free quantum fields on curved spacetimes and as an introduction to some interesting and physically important problems arising in this domain. The second may find this text a useful introduction and motivation to the use of more advanced tools of microlocal analysis in this area of research.
Keywords: Quantum Field Theory, curved spacetimes, Hadamard states, microlocal analysis, pseudo-differential calculus
Author(s): Christian Gérard
Series: ESI Lectures in Mathematics and Physics
Publisher: European Mathematical Society
Year: 2019
Language: English
Pages: 230
Introduction......Page 10
Content......Page 11
Notation......Page 14
Minkowski spacetime......Page 16
The Klein–Gordon equation......Page 18
Pre-symplectic space of test functions......Page 21
The complex case......Page 23
Bosonic Fock space......Page 24
Fock quantization of the Klein–Gordon equation......Page 26
Quantum spacetime fields......Page 27
Local algebras......Page 28
Vector spaces......Page 30
Bilinear and sesquilinear forms......Page 31
Algebras......Page 32
States......Page 33
CCR algebras......Page 34
Quasi-free states......Page 35
Covariances of quasi-free states......Page 39
The GNS representation of quasi-free states......Page 42
Pure quasi-free states......Page 46
Examples......Page 50
Background......Page 52
Lorentzian manifolds......Page 56
Globally hyperbolic spacetimes......Page 61
Klein–Gordon equations on Lorentzian manifolds......Page 66
Symplectic spaces......Page 70
Quasi-free states on curved spacetimes......Page 74
Consequences of unique continuation......Page 77
Conformal transformations......Page 78
Wavefront set of distributions......Page 80
Operations on distributions......Page 83
Hörmander's theorem......Page 85
The distinguished parametrices of a Klein–Gordon operator......Page 86
The need for renormalization......Page 90
Old definition of Hadamard states......Page 92
The microlocal definition of Hadamard states......Page 94
The theorems of Radzikowski......Page 95
Equivalence of the two definitions......Page 97
Examples of Hadamard states......Page 99
Existence of Hadamard states......Page 100
Ground states and KMS states......Page 102
Klein–Gordon operators......Page 106
The Klein–Gordon equation on stationary spacetimes......Page 108
Reduction......Page 109
Ground and KMS states for P......Page 110
Hadamard property......Page 111
Pseudodifferential calculus on Rn......Page 112
Pseudodifferential operators on a manifold......Page 115
Riemannian manifolds of bounded geometry......Page 117
The Shubin calculus......Page 120
Seeley's theorem......Page 122
Egorov's theorem......Page 123
Hadamard condition on Cauchy surface covariances......Page 124
Model Klein–Gordon operators......Page 125
Parametrices for the Cauchy problem......Page 126
Spacetime covariances and Feynman inverses......Page 132
Klein–Gordon operators on Lorentzian manifolds of bounded geometry......Page 134
Conformal transformations......Page 135
Hadamard states on general spacetimes......Page 136
Analytic Hadamard states and Wick rotation......Page 138
Boundary values of holomorphic functions......Page 139
The analytic wavefront set......Page 140
Analytic Hadamard states......Page 142
The Reeh–Schlieder property of analytic Hadamard states......Page 143
Existence of analytic Hadamard states......Page 144
Wick rotation on analytic spacetimes......Page 145
The Calderón projectors......Page 146
The Hadamard state associated to Calderón projectors......Page 147
Examples......Page 149
Klein–Gordon fields inside future lightcones......Page 152
The boundary symplectic space......Page 154
The Hadamard condition on the boundary......Page 156
Construction of pure boundary Hadamard states......Page 159
Asymptotically flat spacetimes......Page 160
The canonical symplectic space on I-......Page 163
Klein–Gordon fields on spacetimes with Killing horizons......Page 168
Spacetimes with bifurcate Killing horizons......Page 169
Wick rotation......Page 171
The double -KMS state in M+M-......Page 173
The extended Euclidean metric and the Hawking temperature......Page 174
The Hartle–Hawking–Israel state......Page 175
Klein–Gordon operators on asymptotically static spacetimes......Page 178
The in and out vacuum states......Page 180
Reduction to a model case......Page 182
Feynman propagator on asymptotically Minkowski spacetimes......Page 186
The Feynman inverse of P......Page 187
Proof of Theorem 16.1......Page 189
CAR *-algebras and quasi-free states......Page 196
Clifford algebras......Page 197
Clifford representations......Page 198
Weyl bi-spinors......Page 201
Clifford and spinor bundles......Page 203
Spin structures......Page 205
Spinor connections......Page 206
Dirac operators......Page 207
Dirac equation on globally hyperbolic spacetimes......Page 209
Quantization of the Dirac equation......Page 210
Hadamard states for the Dirac equation......Page 211
Conformal transformations......Page 212
The Weyl equation......Page 213
Relationship between Dirac and Weyl Hadamard states......Page 215
Bibliography......Page 218
General Index......Page 224
Index of Notations......Page 228
