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Covariant Loop Quantum Gravity An elementary introduction to Quantum Gravity and Spinfoam Theory CARLO ROVELLI AND FRANCESCA VIDOTTO iii . Cover: Martin Klimas, ”Jimi Hendrix, House Burning Down”. Contents Preface page ix Part I Foundations 1 1 Spacetime as a quantum object 3 1.1 The problem 3 1.2 The end of space and time 6 1.3 Geometry quantized 9 1.4 Physical consequences of the existence of the Planck scale 18 1.4.1 Discreteness: scaling is finite 18 1.4.2 Fuzziness: disappearance of classical space and time 19 1.5 Graphs, loops and quantum Faraday lines 20 1.6 The landscape 22 1.7 Complements 24 1.7.1 SU(2) representations and spinors 24 1.7.2 Pauli matrices 27 1.7.3 Eigenvalues of the volume 29 2 Physics without time 31 2.1 Hamilton function 31 2.1.1 Boundary terms 35 2.2 Transition amplitude 36 2.2.1 Transition amplitude as an integral over paths 37 2.2.2 General properties of the transition amplitude 40 2.3 General covariant form of mechanics 42 2.3.1 Hamilton function of a general covariant system 45 2.3.2 Partial observables 46 2.3.3 Classical physics without time 47 2.4 Quantum physics without time 48 2.4.1 Observability in quantum gravity 51 2.4.2 Boundary formalism 52 2.4.3 Relational quanta, relational space 54 2.5 Complements 55 2.5.1 Example of timeless system 55 2.5.2 Symplectic structure and Hamilton function 57 iv v Contents 3 Gravity 59 3.1 Einstein’s formulation 59 3.2 Tetrads and fermions 60 3.2.1 An important sign 63 3.2.2 First-order formulation 64 3.3 Holst action and Barbero-Immirzi coupling constant 65 3.3.1 Linear simplicity constraint 67 3.3.2 Boundary term 68 3.4 Hamiltonian general relativity 69 3.4.1 ADM variables 69 3.4.2 What does this mean? Dynamics 71 3.4.3 Ashtekar connection and triads 73 3.5 Euclidean general relativity in three spacetime dimensions 76 3.6 Complements 78 3.6.1 Working with general covariant field theory 78 3.6.2 Problems 81 4 Classical discretization 82 4.1 Lattice QCD 82 4.1.1 Hamiltonian lattice theory 84 4.2 Discretization of covariant systems 85 4.3 Regge calculus 87 4.4 Discretization of general relativity on a two-complex 91 4.5 Complements 98 4.5.1 Holonomy 98 4.5.2 Problems 99 Part II 3d theory 101 5 3d Euclidean theory 103 5.1 Quantization strategy 103 5.2 Quantum kinematics: Hilbert space 104 5.2.1 Length quantization 105 5.2.2 Spin networks 106 5.3 Quantum dynamics: Transition amplitudes 110 5.3.1 Properties of the amplitude 114 5.3.2 Ponzano-Regge model 115 5.4 Complements 118 5.4.1 Elementary harmonic analysis 118 5.4.2 Alternative form of the transition amplitude 119 5.4.3 Poisson brackets 121 5.4.4 Perimeter of the Universe 122 vi Contents 6 Bubbles and cosmological constant 123 6.1 Vertex amplitude as gauge-invariant identity 123 6.2 Bubbles and spikes 125 6.3 Turaev-Viro amplitude 129 6.3.1 Cosmological constant 131 6.4 Complements 133 6.4.1 Few notes on SU(2)q 133 Part III The real world 135 7 The real world: 4D Lorentzian theory 137 7.1 Classical discretization 137 7.2 Quantum states of gravity 140 7.2.1 Yg map 141 7.2.2 Spin networks in the physical theory 143 7.2.3 Quanta of space 145 7.3 Transition amplitudes 147 7.3.1 Continuum limit 150 7.3.2 Relation with QED and QCD 152 7.4 Full theory 153 7.4.1 Face-amplitude, wedge-amplitude and the kernel P 154 7.4.2 Cosmological constant and IR finiteness 155 7.4.3 Variants 156 7.5 Complements 158 7.5.1 Summary of the theory 158 7.5.2 Computing with spin networks 160 7.5.3 Spectrum of the volume 163 7.5.4 Unitary representation of the Lorentz group and the Yg map 166 8 Classical limit 170 8.1 Intrinsic coherent states 170 8.1.1 Tetrahedron geometry and SU(2) coherent states 171 8.1.2 Livine-Speziale coherent intertwiners 175 8.1.3 Thin and thick wedges and time oriented tetrahedra 176 8.2 Spinors and their magic 177 8.2.1 Spinors, vectors and bivectors 178 8.2.2 Coherent states and spinors 180 8.2.3 Representations of SU(2) and SL(2,C) on functions of spinors and Yg map 181 8.3 Classical limit of the vertex amplitude 183 8.3.1 Transition amplitude in terms of coherent states 184 8.3.2 Classical limit versus continuum limit 188 8.4 Extrinsic coherent states 191 vii Contents 9 Matter 196 9.1 Fermions 196 9.2 Yang-Mills fields 202 Part IV Physical applications 205 10 Black holes 207 10.1 Bekenstein-Hawking entropy 207 10.2 Local thermodynamics and Frodden-Ghosh-Perez energy 209 10.3 Kinematical derivation of the entropy 211 10.4 Dynamical derivation of the entropy 214 10.4.1 Entanglement entropy and area fluctuations 219 10.5 Complements 220 10.5.1 Accelerated observers in Minkowski and Schwarzshild 220 10.5.2 Tolman law and thermal time 220 10.5.3 Algebraic quantum theory 221 10.5.4 KMS and thermometers 222 10.5.5 General covariant statistical mechanics and quantum gravity 223 11 Cosmology 226 11.1 Classical cosmology 226 11.2 Canonical Loop Quantum Cosmology 229 11.3 Spinfoam Cosmology 231 11.3.1 Homogeneus and isotropic geometry 232 11.3.2 Vertex expansion 233 11.3.3 Large-spin expansion 234 11.4 Maximal acceleration 236 11.5 Physical predictions? 237 12 Scattering 238 12.1 n-point functions in general covariant theories 238 12.2 Graviton propagator 242 13 Final remarks 247 13.1 Brief historical note 247 13.2 What is missing 248 Index 252 References 252 Index 265 Preface To ours teachers and all those who teach children to question our knowledge, learn through collaboration and the joy of discovery. This book is an introduction to loop quantum gravity (LQG) focusing on its co- variant formulation. The book has grown from a series of lectures given by Carlo Rovelli and Eugenio Bianchi at Perimeter Institute during April 2012 and a course given by Rovelli in Marseille in the winter 2013. The book is introductory, and as- sumes only some basic knowledge of general relativity, quantum mechanics and quantum field theory. It is simpler and far more readable than the loop quantum gravity text “Quantum Gravity” [Rovelli (2004)], and the advanced and condensed “Zakopane lectures” [Rovelli (2011)], but it covers, and in facts focuses, on the mo- mentous advances in the covariant theory developed in the last few years, which have lead to finite transition amplitudes and were only foreshadowed in [Rovelli (2004)]. There is a rich literature on LQG, to which we refer for all the topics not covered in this book. On quantum gravity in general, Claus Kiefer has a recent general in- troduction [Kiefer (2007)]. Ashtekar and Petkov are editing a ”Springer Handbook of Spacetime” [Ashtekar and Petkov (2013)], with numerous useful contributions, including John Engle article on spinfoams. A fine book with much useful background material is John Baez and Javier Mu- nian’s [Baez and Munian (1994)]. By John Baez, see also [Baez (1994b)], with many ideas and a nice introduction to the subject. An undergraduate level introduction to LQG is provided by Rodolfo Gambini and Jorge Pullin [Gambini and Pullin (2010)]. A punctilious and comprehensive text on the canonical formulation of the theory, rich in mathematical details, is Thomas Thiemann’s [Thiemann (2007)]. The very early form of the theory and the first ideas giving rise to it can be found in the 1991 book by Abhay Ashtekar [Ashtekar (n.d.)]. A good recent reference is the collection of the proceedings of the 3rd Zakopane school on loop quantum gravity, organized by Jerzy Lewandowski [Barrett et al. (2011a)]. It contains an introduction to LQG by Abhay Ashtekar [Ashtekar (2011)], Rovelli’s “Zakopane lectures” [Rovelli (2011)], the introduction by Kristina Giesel and Hanno Sahlmann to the canonical theory, and John Barrett et al review on the semiclassical approximation to the spinfoam dynamics [Barrett et al. (2011b)]. We also recommend Alejandro Perez spinfoam review [Perez (2012)], which is com- plementary to this book in several way. Finally, we recommend the Hall Haggard’s viii ix Preface thesis, online [Haggard (2011)], for a careful and useful introduction and reference for the mathematics of spin networks. We are very grateful to Klaas Landsman, Gabriele Stagno, Marco Finocchiaro, Hal Haggard, Tim Kittel, Thomas Krajewski, Cedrick Miranda Mello, Aldo Riello, Tapio Salminem and, come sempre, Leonard Cottrell, for careful reading the notes, corrections and clarifications. Several tutorials have been prepared by David Ku- biznak and Jonathan Ziprick for the students of the International Perimeter Schol- ars: Andrzej, Grisha, Lance, Lucas, Mark, Pavel, Brenda, Jacob, Linging, Robert, Rosa, thanks also to them! Cassis, November 4th, 2013 Carlo Rovelli and Francesca Vidotto Part I FOUNDATIONS 1 Spacetime as a quantum object This book introduces the reader to a theory of quantum gravity. The theory is covariant loop quantum gravity (covariant LQG). It is a theory that has grown historically via a long indirect path, briefly summarized at the end of this chapter. The book does not follows the historical path.
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