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Semiclassical Analysis of Loop Quantum Gravity

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Semiclassical Analysis of Loop Quantum Gravity UNIVERSITA’ DEGLI STUDI ROMA TRE Department of Mathematics Doctoral School in Mathematical and Physical Sciences XXII Cycle A.Y. 2009 Doctorate Thesis Università Roma Tre - Université de la Méditerranée Co-tutorship agreement Semiclassical analysis of Loop Quantum Gravity Dr. Claudio Perini Director of the Ph.D. School: prof. Renato Spigler Roma Tre University, Department of Mathematics, Rome Supervisors: prof. Fabio Martinelli Roma Tre University, Department of Mathematics, Rome prof. Carlo Rovelli Université de la Méditerranée, Centre de Physique Théorique de Luminy, Marseille Dedicated to my wife Elena, to my family, to my friends in Rome and to all quantum gravity people I met in the long road to this PhD thesis on the back: cellular decomposition of the lighthouse off Cassis Contents 1 Introduction 6 2 Hamiltonian General Relativity 10 2.1 Canonical formulation of General Relativity in ADM variables ............... 10 2.2 Lagrangian and ADM Hamiltonian ............................... 11 2.3 Triad formalism ......................................... 13 2.4 Ashtekar-Barbero variables .................................... 13 2.5 Constraint algebra ........................................ 14 2.6 The holonomy ........................................... 16 3 The structure of Loop Quantum Gravity 18 3.1 Kinematical state space ..................................... 18 3.2 Solution to the Gauss and diffeomorphism constraints ..................... 20 3.3 Electric flux operator ....................................... 24 3.4 Holonomy-flux algebra and uniqueness theorem ........................ 26 3.5 Area and volume operators .................................... 27 3.6 Physical interpretation of quantum geometry .......................... 28 3.7 Dynamics ............................................. 28 4 Covariant formulation: Spin Foam 32 4.1 Path integral representation ................................... 32 4.2 Regge discretization ........................................ 35 4.3 Quantum Regge calculus and spinfoam models ......................... 36 4.4 BF theory ............................................. 38 4.5 Barret-Crane model ....................................... 40 5 A new model: EPRL vertex 43 5.1 The goal of the model: imposing weakly the simplicity constraints .............. 43 5.2 Description of the model .................................... 44 6 Asymptotics of LQG fusion coefficients 50 6.1 Analytical expression for the fusion coefficients ........................ 50 6.2 Asymptotic analysis ....................................... 52 6.3 Semiclassical behavior ...................................... 56 6.4 The case γ = 1 .......................................... 58 6.5 Summary of semiclassical properties of fusion coefficients .................. 59 6.6 Properties of 9j-symbols and asymptotics of 3j-symbol .................... 60 7 Numerical investigations on the semiclassical limit 61 7.1 Wave-packets propagation .................................... 61 7.2 The semi-analytic approach ................................... 66 7.3 Physical expectation values .................................... 67 4 7.4 Improved numerical analysis ................................... 68 8 Semiclassical states for quantum gravity 70 8.1 Livine-Speziale coherent intertwiners .............................. 71 8.2 Coherent tetrahedron ....................................... 73 8.3 Coherent spin-networks ..................................... 75 8.4 Resolution of the identity and holomorphic representation .................. 79 9 Graviton propagator in Loop Quantum Gravity 83 9.1 n-point functions in generally covariant field theories ..................... 84 9.2 Compatibility between of radial, Lorenz and harmonic gauges ................. 86 9.2.1 Maxwell theory ...................................... 88 9.2.2 Linearized General Relativity .............................. 90 10 LQG propagator from the new spin foams 93 10.1 Semiclassical boundary state and the metric operator ..................... 94 10.2 The new spin foam dynamics .................................. 99 10.3 LQG propagator: integral formula ................................ 100 10.4 Integral formula for the amplitude of a coherent spin-network ................ 100 10.5 LQG operators as group integral insertions ........................... 101 10.6 LQG propagator: stationary phase approximation ....................... 103 10.7 The total action and the extended integral ........................... 103 10.8 Asymptotic formula for connected two-point functions ..................... 104 10.9 Critical points of the total action ................................ 105 10.10 Hessian of the total action and derivatives of the insertions ................. 106 10.11 Expectation value of metric operators ............................. 108 10.12 LQG propagator: the leading order ............................... 109 10.13 Comparison with perturbative quantum gravity ........................ 112 11 Conclusions and outlook 114 5 1. INTRODUCTION 1 Introduction In the last century, General Relativity (GR) and Quantum Mechanics (QM) revolutionized the physics and demolished two deep-rooted prejudices about Nature: the determinism of physical laws and the absolute, inert character of space and time. At the same time they explained most of the phenomena that we observe. In particular the microscopic observations of nuclear and subnuclear physics have been explained within the Quantum Field Theory (QFT) called Standard Model of particle physics, and, on the other side, the large scale phenomena of the Universe have been explained within General Relativity (GR). QM and GR are the two conceptual pillars on which modern physics is built. GR has modified the notion of space and time; QM the notion of causality, matter and measurements, but these modified notions do not fit together easily. The basic assumptions of each one of the two theories are contradicted by the other. On the one hand QM is formulated in a Newtonian absolute, fixed, non-dynamical space-time, while GR describes spacetime as a dynamical entity: it is no more an external set of clocks and rods, but a physical interacting field, namely the gravitational field. Moreover the electromagnetic, weak and strong interactions, unified in the language of Quantum Field Theory, obeys the laws of quantum mechanics, and on the other side the gravitational interaction is described by the classical deterministic theory of General Relativity. Both theories work extremely well at opposite scales but this picture is clearly incomplete [1] unless we want to accept that Nature has opposite foundations in the quantum and in the cosmological realm. QM describes microscopic phenomena involving fundamental particles, ignoring completely gravity, while GR describes macroscopic systems, whose quantum properties are in general (safely) neglected. This is not only a philosophical problem, but assumes the distinguishing features of a real scientific problem as soon as one considers measurements in which both quantum and gravitational effects cannot be neglected. The search for a theory which merges GR and QM in a whole coherent picture is the search for a theory of Quantum Gravity (QG). If one asks the question: why do we need Quantum Gravity? the answers are many. It is worth recalling that the theory describing the gravitational interaction fails in giving a fully satisfactory description of the observed Universe. General relativity, indeed, leads inevitably to space-time singularities as a number of theorems mainly due to Hawking and Penrose demonstrate. The singularities occur both at the beginning of the expansion of our Universe and in the collapse of gravitating objects to form black holes. Classical GR breaks completely down at these singularities, or rather it results to be an incomplete theory. On the quantum mechanical side, we can ask what would happen if we managed to collide an electron-positron pair of energy (in the center of mass) greater than the Planck energy. We are unable to give an answer to this question. The reason of this failure is related to the fact that, in such an experiment, we cannot neglect the gravitational properties of the involved particles at the moment of the collision. But, we do not have any scientific information on how taking into account such an effect in the framework of QFT. In other words, when the gravitational field is so intense that space-time geometry evolves on a very short time scale, QFT cannot be consistently applied any longer. Or, from another perspective, we can say that when the gravitational effects are so strong to produce the emergence of spacetime singularities, field theory falls into troubles. A possible attempt for quantizing gravity is the one that leads to perturbative Quantum Gravity, the conventional quantum field theory of gravitons (spin 2 massless particles) propagating over Minkowski 6 1. INTRODUCTION spacetime. Here the metric tensor is splitted in gµν = ηµν + hµν and η is treated as a background metric (in general any solution to Einstein equations) and h is a dynamical field representing self-interacting gravitons. This theory gives some insights but cannot be considered a good starting point because it brakes the general covariance of General Relativity, and (more practically) it is non-renormalizable. It is remarkable that it is possible to construct a quantum theory of a system with an infinite number of degrees of freedom, without assuming a fixed background causal structure; this is the framework of modern background
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