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The Spin-Foam Approach to Quantum Gravity Living Rev. Relativity, 16, (2013), 3 LIVINGREVIEWS http://www.livingreviews.org/lrr-2013-3 in relativity The Spin-Foam Approach to Quantum Gravity Alejandro Perez Centre de Physique Th´eorique Unit´eMixte de Recherche (UMR 6207) du CNRS et des Universit´esAix-Marseille I, Aix-Marseille II, et du Sud Toulon-Var; laboratoire afili´e`ala FRUMAM (FR 2291) email: [email protected] Accepted on 11 June 2012 Published on 14 February 2013 Abstract This article reviews the present status of the spin-foam approach to the quantization of gravity. Special attention is payed to the pedagogical presentation of the recently-introduced new models for four-dimensional quantum gravity. The models are motivated by a suitable implementation of the path integral quantization of the Plebanski formulation of gravity on a simplicial regularization. The article also includes a self-contained treatment of 2+1 gravity. The simple nature of the latter provides the basis and a perspective for the analysis of both conceptual and technical issues that remain open in four dimensions. This review is licensed under a Creative Commons Attribution-Non-Commercial 3.0 Germany License. http://creativecommons.org/licenses/by-nc/3.0/de/ Imprint / Terms of Use Living Reviews in Relativity is a peer reviewed open access journal published by the Max Planck Institute for Gravitational Physics, Am M¨uhlenberg 1, 14476 Potsdam, Germany. ISSN 1433-8351. This review is licensed under a Creative Commons Attribution-Non-Commercial 3.0 Germany License: http://creativecommons.org/licenses/by-nc/3.0/de/. Figures that have been pre- viously published elsewhere may not be reproduced without consent of the original copyright holders. Because a Living Reviews article can evolve over time, we recommend to cite the article as follows: Alejandro Perez, \The Spin-Foam Approach to Quantum Gravity", Living Rev. Relativity, 16, (2013), 3. URL (accessed <date>): http://www.livingreviews.org/lrr-2013-3 The date given as <date> then uniquely identifies the version of the article you are referring to. Article Revisions Living Reviews supports two ways of keeping its articles up-to-date: Fast-track revision A fast-track revision provides the author with the opportunity to add short notices of current research results, trends and developments, or important publications to the article. A fast-track revision is refereed by the responsible subject editor. If an article has undergone a fast-track revision, a summary of changes will be listed here. Major update A major update will include substantial changes and additions and is subject to full external refereeing. It is published with a new publication number. For detailed documentation of an article's evolution, please refer to the history document of the article's online version at http://www.livingreviews.org/lrr-2013-3. Contents I Introduction7 1 Quantum Gravity: A Unifying Framework for Fundamental Physics 8 1.1 Why non-perturbative quantum gravity? ........................ 9 1.2 Non-renormalizability of general relativity ....................... 10 2 Why Spin Foams? 12 2.1 Loop quantum gravity .................................. 12 2.2 The path integral for generally covariant systems ................... 14 2.3 A brief history of spin foams in four dimensions .................... 17 2.3.1 The Reisenberger model ............................. 17 2.3.2 The Freidel{Krasnov prescription ........................ 18 2.3.3 The Iwasaki model ................................ 18 2.3.4 The Barrett{Crane model ............................ 19 2.3.5 Markopoulou{Smolin causal spin networks ................... 19 2.3.6 Gambini{Pullin model .............................. 20 2.3.7 Capovilla{Dell{Jacobson theory on the lattice . 20 2.3.8 The Engle{Pereira{Rovelli{Livine (EPRL) ................... 20 2.3.9 Freidel{Krasnov (FK) .............................. 21 II Preliminaries: LQG and the canonical quantization of four-dimensional gravity 22 3 Classical General Relativity in Connection Variables 23 3.1 Gravity as constrained BF theory ............................ 23 3.2 Canonical analysis .................................... 24 3.2.1 Constraints algebra ................................ 28 3.3 Geometric interpretation of the new variables ..................... 28 4 Loop Quantum Gravity and Quantum Geometry in a Nutshell 29 4.1 Quantum geometry .................................... 31 4.2 Quantum dynamics .................................... 32 5 Spin Foams and the Path Integral for Gravity in a Nutshell 33 III The new spin-foam models for four-dimensional gravity 36 6 Spin-foam Quantization of BF Theory 37 6.1 Special cases ....................................... 41 6.1.1 SU (2) BF theory in 2D: the simplest topological model . 41 6.1.2 SU (2) BF theory and 3D Riemannian gravity . 42 6.1.3 SU (2) BF theory: The Ooguri model ..................... 44 6.1.4 SU (2) SU (2) BF theory: a starting point for 4D Riemannian gravity . 45 × 6.2 The coherent-states representation ........................... 46 6.2.1 Coherent states .................................. 47 6.2.2 Spin(4) BF theory: amplitudes in the coherent state basis . 48 7 The Riemannian EPRL Model 49 7.1 Representation theory of Spin(4) and the canonical basis . 49 7.2 The linear simplicity constraints ............................ 51 7.2.1 Riemannian model: The Gupta{Bleuler criterion . 52 7.2.2 Riemannian model: The Master-constraint criterion . 53 7.2.3 Riemannian model: Restrictions on the Immirzi parameter . 54 7.2.4 Riemannian model: Overview of the solutions of the linear simplicity con- straints ...................................... 54 7.3 Presentation of the EPRL amplitude .......................... 55 7.3.1 The spin-foam representation of the EPRL amplitude . 56 7.4 Proof of validity of the Plebanski constraints ..................... 58 7.4.1 The dual version of the constraints ....................... 59 7.4.2 The path integral expectation value of the Plebanski constraints . 59 7.5 Modifications of the EPRL model ............................ 62 7.5.1 PEPRL is not a projector ............................. 62 7.5.2 The Warsaw proposal .............................. 63 7.6 The coherent states representation ........................... 65 7.7 Some additional remarks ................................. 68 8 The Lorentzian EPRL Model: Representation Theory and Simplicity Constraints 68 8.1 Representation theory of SL(2; C) and the canonical basis . 68 8.2 Lorentzian model: the linear simplicity constraints . 70 8.2.1 Lorentzian model: The Gupta{Bleuler criterion . 70 8.2.2 Lorentzian model: The Master-constraint criterion . 71 8.2.3 Lorentzian model: Restrictions on the Immirzi parameter . 72 8.2.4 Lorentzian model: Overview of the solutions of the linear simplicity constraints 72 8.3 Presentation of the EPRL Lorentzian model ...................... 72 8.4 The coherent state representation ............................ 73 9 The Freidel{Krasnov Model 74 10 The Barrett{Crane Model 76 10.1 The coherent states representation of the BC amplitude . 77 11 Boundary Data for the New Models and Relation with the Canonical Theory 77 12 Further Developments and Related Models 78 12.1 The measure ....................................... 78 12.2 Relation with the canonical formulation: the scalar constraint . 78 12.3 Spin-foam models with timelike faces .......................... 79 12.4 Coupling to matter .................................... 79 12.5 Cosmological constant .................................. 79 12.6 Spin-foam cosmology ................................... 79 12.7 Constraints are imposed semiclassically ........................ 79 12.8 Group field theories associated to the new spin foams . 80 13 The Semiclassical Limit 80 13.1 Vertex amplitudes asymptotics ............................. 81 13.1.1 SU (2) 15j -symbol asymptotics ......................... 81 13.1.2 The Riemannian EPRL vertex asymptotics . 82 13.1.3 Lorentzian EPRL model ............................. 82 13.2 Full spin-foam amplitudes asymptotics ......................... 82 13.3 Fluctuations: two point functions ............................ 83 IV Three-dimensional gravity 84 14 Spin Foams for Three-Dimensional Gravity 85 14.1 Spin foams in 3D quantum gravity ........................... 85 14.2 The classical theory ................................... 85 14.3 Spin foams from the Hamiltonian formulation ..................... 86 14.4 The spin-foam representation .............................. 87 14.5 Spin-foam quantization of 3D gravity .......................... 90 14.5.1 Discretization independence ........................... 92 14.5.2 Transition amplitudes .............................. 93 14.5.3 The generalized projector ............................ 93 14.5.4 The continuum limit ............................... 94 14.6 Conclusion ........................................ 95 14.7 Further results in 3D quantum gravity ......................... 95 V Conceptual issues and open problems 96 15 Quantum Spacetime and Gauge Histories 97 16 Anomalies and Gauge Fixing 99 17 Discretization Dependence 102 18 Lorentz Invariance in the Effective Low-Energy Regime 105 19 Acknowledgements 106 References 107 The Spin-Foam Approach to Quantum Gravity 7 Part I Introduction Living Reviews in Relativity http://www.livingreviews.org/lrr-2013-3 8 Alejandro Perez 1 Quantum Gravity: A Unifying Framework for Fundamental Physics The revolution brought about by Einstein's theory of gravity lies more in the discovery of the principle of general covariance than in the form
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