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Quantum Field Theory in Curved Spacetime by Armen Saharian

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Quantum Field Theory in Curved Spacetime by Armen Saharian QUANTUM FIELD THEORY IN CURVED SPACETIME A. A. Saharian1 Department of Physics, Yerevan State University, 1 Alex Manogian Street, 0025 Yerevan, Armenia 1E-mail: [email protected] 2 Contents 1 Introduction 9 2 Fields in flat spacetime 13 2.1 Lorentz group . 13 2.2 Poincar´egroup . 14 2.3 Group of conformal transformations . 14 2.4 Fields . 16 2.5 Action Principle . 16 2.6 Symmetries of the action and conservation laws . 17 2.6.1 Space-time symmetries . 18 2.6.2 Internal symmetries . 19 2.7 Noether's theorem . 20 2.7.1 Angular momentum tensor and spin tensor . 22 2.8 Scalar field . 23 2.9 Dirac spinor field . 24 2.9.1 Conserved currents . 25 2.10 Gauge fields . 26 2.10.1 Abelian gauge field . 26 2.10.2 Non-abelian gauge fields . 27 3 Quintizing fields in Minkowski spacetime 29 3.1 Methods of quantization . 29 3.2 Klein-Gordon scalar field . 29 3.3 Quantization of a complex scalar field . 33 3.4 Dirac spinor field . 34 3.5 Electromagnetic field . 36 3.5.1 Canonical quantization in the Coulomb gauge . 36 3.5.2 Gupta-Bleuler quantization . 37 3.6 Path integral approach . 38 3.7 Discrete symmetries . 40 3.7.1 Parity transformation . 40 3.7.2 Time reversal . 42 3.7.3 Charge conjugation . 42 4 General Relativity as a classical theory of Gravitation 45 4.1 Gravity as a geometry . 45 4.2 Influence of the gravitational field on non-gravitational matter . 48 4.3 Action for gravitational field . 49 4.4 Total action and Einstein equations . 50 4.5 Tetrad formalism . 51 3 4 CONTENTS 4.6 Cosmological models . 53 4.6.1 Friedmann-Robertson-Walker metric . 53 4.6.2 Friedmann equations and the expansion of the Universe . 54 5 Classical fields in curved spacetime 57 5.1 Scalar field . 57 5.2 Electromagnetic field . 59 5.3 Influence of the gravity on matter: Alternative approach . 59 5.3.1 Covariant derivative in tetrad formalism and the action functional . 59 5.3.2 Dirac spinor field . 63 6 Quantization of fields in curved backgrounds 67 6.1 Canonical quantization . 67 6.2 Bogoliubov transformations . 69 6.3 Notion of particles: Particle detectors . 72 6.3.1 Unruh-DeWitt detector . 72 6.3.2 Inertial detector . 73 6.3.3 Uniformly accelerated detector . 75 6.4 Scalar and fermionic fields in external electromagnetic field . 78 7 Adiabatic expansion of the Green function 81 7.1 Divergences and regularization . 81 7.2 Two-point functions . 82 7.2.1 Two-point functions in Minkowski spacetime . 82 7.3 Adiabatic expansion of Green function in curved spacetime . 86 7.4 Divergences and renormalization on curved backgrounds . 89 7.4.1 Evaluation of the path-integral . 91 8 Renormalization in the effective action 95 8.1 Divergences . 95 8.1.1 Renormalized effective Lagrangian . 97 8.2 Higher spin fields . 98 8.3 Conformal anomalies . 99 8.3.1 Trace anomaly . 99 8.3.2 Energy-momentum tensors in conformally related problems . 103 8.3.3 Examples . 105 8.4 Renormalization of the energy-momentum tensor . 108 8.5 Wald axioms . 109 9 Quantum effects from topology and boundaries 111 9.1 2-dimensional spacetime with compact dimension . 111 9.2 Higher-dimensional spaces with a compact dimension . 115 9.2.1 Hadamard function . 115 9.2.2 Vacuum expectation values of the field squared and energy-momentum tensor 118 9.2.3 Vacuum currents . 120 9.3 Boundary-induced quantum effects: Casimir effect . 121 9.4 Casimir effect for Robin boundary conditions . 122 CONTENTS 5 10 Quantum fields in Rindler spacetime 127 10.1 Worldline for a uniformly accelerated observer . 127 10.2 Rindler coordinates . 128 10.3 Massless scalar field in 2-dimensional Rindler spacetime . 129 11 Quantum fields in de Sitter spacetime 135 11.1 De Sitter spacetime . 135 11.2 Maximally symmetric solutions of Einstein equations with a cosmological constant . 135 11.3 Geometry of de Sitter spacetime and the coordinate systems . 136 11.3.1 Global coordinates . 137 11.3.2 Planar or inflationary coordinates . 138 11.3.3 Static coordinates . 138 11.4 Scalar field mode functions in dS spacetime . 139 11.4.1 Planar coordinates . 139 11.4.2 Global coordinates . 142 11.4.3 Static coordinates . 143 12 Quantum fields in anti-de Sitter spacetime 147 12.1 Introduction . 147 12.2 AdS spacetime: Geometry and coordinate systems . 148 12.2.1 Global coordinates . 148 12.2.2 Poincar´ecoordinates . 149 12.2.3 FRW coordinates . 150 12.3 Scalar field mode functions in anti-de Sitter spacetime . 151 13 Two-point functions in maximally symmetric spaces 155 13.1 Maximally symmetric bitensors . 155 13.2 Scalar two-point function . 156 13.3 Scalar two-point function in dS spacetime . 157 13.4 Scalar two-point function in anti-de Sitter spacetime . 158 13.5 Renormalized energy-momentum tensor in de Sitter spacetime . 158 6 CONTENTS These lectures provide an introduction to quantum field theory on curved backgrounds. The course is divided into three parts. In the first one we give a short introduction to classical and quantum fields in Minkowski spacetime, and to General Relativity as a classical theory of gravitation. In the second part, classical and quantum fields in curved spacetime are considered. The general procedures for the regularization and renormalization are described. In the third part, the applications of general scheme are given to several examples. In particular, we consider quantum fields in de Sitter and anti-de Sitter spacetimes. CONTENTS 7 Notations Spacetime vectors: aµ = (a0; ai). Greek indices correspond to spacetime components and Latin indices are for spatial components. @ Partial derivative: @xµ = @µ =; µ − − Metric tensor: gµν. In flat spacetime inP Galilean coordinates: gµν =diag(1; 1;:::; 1). ν ν Covariant components: aµ = gµνa = ν gµνa Covariant derivative: rµ,; µ µ µ ρ Riemann tensor: R·νρσ = @σΓνρ − · · · , Rµν = R·µρν. Units: ~ = c = 1 8 CONTENTS Chapter 1 Introduction Quantum Field Theory as a theory of elementary particles Quantum Field Theory is a physical theory of elementary particles and their interactions. It has emerged as the most successful physical framework describing the subatomic world. The quantum field theoretical predictions for the interactions between electrons and photons have proved to be correct to within one part in 108. Furthermore, it can adequately explain the interactions of three of the four.
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