Covariant Loop Quantum Gravity Carlo Rovelli , Francesca Vidotto Frontmatter More Information

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Cambridge University Press 978-1-107-06962-6 — Covariant Loop Quantum Gravity Carlo Rovelli , Francesca Vidotto Frontmatter More Information

Covariant Loop Quantum Gravity

Quantum gravity is among the most fascinating problems in physics. It modifies our understanding of time, space, and matter. The recent development of the loop approach has allowed us to explore domains ranging from black hole thermodynamics to the early universe. This book provides readers with a simple introduction to loop quantum gravity, centred on its covariant approach. It focuses on the physical and conceptual aspects of the problem and includes the background material needed to enter this lively domain of research, making it ideal for researchers and graduate students. Topics covered include quanta of space, classical and quantum physics without time, tetrad formalism, Holst action, lattice gauge theory, Regge calculus, ADM and Ashtekar variables, Ponzano–Regge and Turaev–Viro amplitudes, kinematics and dynamics of 4d Lorentzian quantum gravity, spectrum of area and volume, coherent states, classical limit, matter couplings, graviton propagator, spinfoam cosmology and black hole thermodynamics.

Carlo Rovelli is Professor of Physics at Aix-Marseille Universite,´ where he directs the gravity research group. He is one of the founders of loop quantum gravity.

Francesca Vidotto is NWO Veni Fellow at the Radboud Universiteit Nijmegen and initiated the spinfoam approach to cosmology.

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-06962-6 — Covariant Loop Quantum Gravity Carlo Rovelli , Francesca Vidotto Frontmatter More Information

“The approach to quantum gravity known as loop quantum gravity has progressed enor- mously in the last decade and this book by Carlo Rovelli and Francesca Vidotto admirably fills the need for an up-to-date textbook in this area. It will serve well to bring beginning students and established researchers alike up-to-date on developments in this fast moving area. It is the only book presenting key results of the theory, including those related to black holes, quantum cosmology and the derivation of general relativity from the funda- mental theory of quantum spacetime. The authors achieve a good balance of big ideas and principles with the technical details.” Lee Smolin, Perimeter Institute for Theoretical Physics “This is an excellent introduction to spinfoams, an area of loop quantum gravity that draws ideas also from Regge calculus, topological field theory and group field theory. It fills an important gap in the literature offering both a pedagogical overview and a platform for further developments in a forefront area of research that is advancing rapidly.” Abhay Ashtekar, The Pennsylvania State University

Covariant Loop Quantum Gravity An Elementary Introduction to Quantum Gravity and Spinfoam Theory

CARLO ROVELLI Universite ’ d’ Aix-Marseille FRANCESCA VIDOTTO Radboud Universiteit Nijmegen

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To our teachers and to all those who teach children to question our knowledge, learn through collaboration, and feel the joy of discovery

Contents

Preface page xi Part I FOUNDATIONS

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 1.3.1 Quanta of area and volume 14 1.4 Physical consequences of the existence of the Planck scale 16 1.4.1 Discreteness: scaling is finite 16 1.4.2 Fuzziness: disappearance of classical space and time 18 1.5 Graphs, loops, and quantum Faraday lines 18 1.6 The landscape 21 1.7 Complements 21 1.7.1 SU(2) representations and spinors 21 1.7.2 Pauli matrices 26 1.7.3 Eigenvalues of the volume 27

2Physicswithouttime 30 2.1 Hamilton function 30 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 39 2.3 General covariant form of mechanics 41 2.3.1 Hamilton function of a general covariant system 44 2.3.2 Partial observables 45 2.3.3 Classical physics without time 46 2.4 Quantum physics without time 47 2.4.1 Observability in quantum gravity 49 2.4.2 Boundary formalism 50 2.4.3 Relational quanta, relational space 52 2.5 Complements 53 2.5.1 Example of a timeless system 53 2.5.2 Symplectic structure and Hamilton function 55

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3Gravity 58 3.1 Einstein’s formulation 58 3.2 Tetrads and fermions 59 3.2.1 An important sign 62 3.2.2 First-order formulation 63 3.3 Holst action and Barbero–Immirzi coupling constant 64 3.3.1 Linear simplicity constraint 65 3.3.2 Boundary term 67 3.4 Hamiltonian general relativity 67 3.4.1 ADM variables 68 3.4.2 What does this mean? Dynamics 70 3.4.3 Ashtekar connection and triads 72 3.5 Euclidean general relativity in three spacetime dimensions 74 3.6 Complements 76 3.6.1 Working with general covariant field theory 76 3.6.2 Problems 79

4 Classical discretization 80 4.1 Lattice QCD 80 4.1.1 Hamiltonian lattice theory 82 4.2 Discretization of covariant systems 83 4.3 Regge calculus 85 4.4 Discretization of general relativity on a two-complex 89 4.5 Complements 95 4.5.1 Holonomy 95 4.5.2 Problems 96

Part II THREE-DIMENSIONAL THEORY

5 Three-dimensional euclidean theory 99 5.1 Quantization strategy 99 5.2 Quantum kinematics: Hilbert space 100 5.2.1 Length quantization 101 5.2.2 Spin networks 102 5.3 Quantum dynamics: transition amplitudes 106 5.3.1 Properties of the amplitude 109 5.3.2 Ponzano–Regge model 110 5.4 Complements 113 5.4.1 Elementary harmonic analysis 113 5.4.2 Alternative form of the transition amplitude 114 5.4.3 Poisson brackets 116 5.4.4 Perimeter of the universe 117

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6 Bubbles and the cosmological constant 118 6.1 Vertex amplitude as gauge-invariant identity 118 6.2 Bubbles and spikes 120 6.3 Turaev–Viro amplitude 123 6.3.1 Cosmological constant 125 6.4 Complements 127 6.4.1 A few notes on SU(2)q 127 6.4.2 Problem 128 Part III THE REAL WORLD

7 The real world: four-dimensional lorentzian theory 131 7.1 Classical discretization 131 7.2 Quantum states of gravity 134 7.2.1 Yγ map 135 7.2.2 Spin networks in the physical theory 137 7.2.3 Quanta of space 140 7.3 Transition amplitudes 141 7.3.1 Continuum limit 143 7.3.2 Relation with QED and QCD 145 7.4 Full theory 146 7.4.1 Face amplitude, wedge amplitude, and the kernel P 147 7.4.2 Cosmological constant and IR finiteness 149 7.4.3 Variants 149 7.5 Complements 151 7.5.1 Summary of the theory 151 7.5.2 Computing with spin networks 152 7.5.3 Spectrum of the volume 155 7.5.4 Unitary representation of the Lorentz group and the Yγ map 159

8 Classical limit 162 8.1 Intrinsic coherent states 162 8.1.1 Tetrahedron geometry and SU(2) coherent states 163 8.1.2 Livine–Speziale coherent intertwiners 167 8.1.3 Thin and thick wedges and time-oriented tetrahedra 168 8.2 Spinors and their magic 169 8.2.1 Spinors, vectors, and bivectors 171 8.2.2 Coherent states and spinors 172 8.2.3 Representations of SU(2) and SL(2,C) on functions of spinors and Yγ map 173 8.3 Classical limit of the vertex amplitude 175 8.3.1 Transition amplitude in terms of coherent states 175 8.3.2 Classical limit versus continuum limit 180 8.4 Extrinsic coherent states 183

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9 Matter 187 9.1 Fermions 187 9.2 Yang–Mills fields 193 Part IV PHYSICAL APPLICATIONS

10 Black holes 197 10.1 Bekenstein–Hawking entropy 197 10.2 Local thermodynamics and Frodden–Ghosh–Perez energy 199 10.3 Kinematical derivation of the entropy 201 10.4 Dynamical derivation of the entropy 204 10.4.1 Entanglement entropy and area fluctuations 208 10.5 Complements 209 10.5.1 Accelerated observers in Minkowski and Schwarzschild metrics 209 10.5.2 Tolman law and thermal time 210 10.5.3 Algebraic quantum theory 210 10.5.4 KMS and thermometers 211 10.5.5 General covariant statistical mechanics and quantum gravity 212

11 Cosmology 215 11.1 Classical cosmology 215 11.2 Canonical loop quantum cosmology 218 11.3 Spinfoam cosmology 220 11.3.1 Homogeneous and isotropic geometry 221 11.3.2 Vertex expansion 222 11.3.3 Large-spin expansion 223 11.4 Maximal acceleration 225 11.5 Physical predictions? 226

12 Scattering 227 12.1 n-Point functions in general covariant theories 227 12.2 Graviton propagator 231

13 Final remarks 235 13.1 Brief historical note 235 13.2 What is missing 236

References 240 Index 252

Preface

This book is an introduction to loop quantum gravity (LQG) focusing on its covariant 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 of 2013. The book is introductory, and assumes only some basic knowledge of general relativity, quantum mechanics, and quantum field theory. It is sim- pler 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 fact focuses on, the momentous 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 introduction Kiefer (2007). At the time of writing, Ashtekar and Petkov are editing a Springer Handbook of Spacetime, with numerous useful contributions, including that of John Engle’s article on spinfoams. A fine book with much useful background material is that of John Baez and Javier Munian (1994). 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 given by Thomas 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. 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 the introduction to LQG by Abhay Ashtekar (Ashtekar 2011), Rovelli’s “Zakopane lectures” (Rovelli 2011), and the introduction by Kristina Giesel and Hanno Sahlmann to the canonical theory, and John Barrett et al.’s review on the semiclassi- cal approximation to the spinfoam dynamics (Barrett et al. 2011b). We also recommend Alejandro Perez’s spinfoam review (Perez 2012), which is complementary to this book in several ways. Finally, we recommend Hal Haggard’s thesis online (Haggard 2011), for a careful and useful introduction to 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

xii Preface

Salminem and, come sempre, Leonard Cottrell, for careful reading of the notes, correc- tions, and clarifications. Several tutorials have been prepared by David Kubiznak and Jonathan Ziprick for the students of the International Perimeter Scholars: Andrzej, Grisha, Lance, Lucas, Mark, Pavel, Brenda, Jacob, Linging, Robert, Rosa; thanks also to them!