Microlocal Analysis of Quantum Fields on Curved Spacetimes

Description

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 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