Canonical Quantum Gravity: Fundamentals And Recent Developments by Francesco Cianfrani, Giovanni Montani, Matteo Lulli, and Orchidea Maria Lecian


38578b2c7bd6197.jpg Author Francesco Cianfrani, Giovanni Montani, Matteo Lulli, and Orchidea Maria Lecian
Isbn 9789814556644
File size 2.4 MB
Year 2014
Pages 324
Language English
File format PDF
Category physics


 

CANONICAL QUANTUM GRAVITY Fundamentals and Recent Developments WWW.EBOOK777.COM 8957hc_9789814556644_tp.indd 1 28/4/14 12:30 pm May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory This page intentionally left blank WWW.EBOOK777.COM PST˙ws CANONICAL QUANTUM GRAVITY Fundamentals and Recent Developments Francesco Cianfrani University of Wrocław, Poland Orchidea Maria Lecian “Sapienza”, University of Rome, Italy, Max Planck Institute for Gravitational Physics, Germany & ICRA International Center for Relativistic Astrophysics, Italy Matteo Lulli “Sapienza”, University of Rome, Italy Giovanni Montani ENEA - C.R. Frascati, UTFUS-MAG & “Sapienza”, University of Rome, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I WWW.EBOOK777.COM 8957hc_9789814556644_tp.indd 2 • CHENNAI 28/4/14 12:30 pm Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Use of images from the Millennium Simulation, courtesy of V. Springel & the Virgo Consortium Cover Art by Angel Patricio Susanna, http: //www.behance.net/kunturi CANONICAL  QUANTUM  GRAVITY Fundamentals and Recent Developments Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4556-64-4 Printed in Singapore WWW.EBOOK777.COM April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile Contents Preface xi List of Figures xix List of Notations xxi 1. Introduction to General Relativity 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 Parametric manifold representation . . . . . . . . . Tensor formalism . . . . . . . . . . . . . . . . . . . Affine properties of the manifold . . . . . . . . . . 1.3.1 Ordinary derivative . . . . . . . . . . . . . 1.3.2 Covariant derivative . . . . . . . . . . . . . 1.3.3 Properties of the affine connections . . . . Metric properties of the manifold . . . . . . . . . . 1.4.1 Metric tensor . . . . . . . . . . . . . . . . . 1.4.2 Christoffel symbols . . . . . . . . . . . . . Geodesic equation and parallel transport . . . . . . Levi-Civita tensor . . . . . . . . . . . . . . . . . . Volume element and covariant divergence . . . . . Gauss and Stokes theorems . . . . . . . . . . . . . The Riemann tensor . . . . . . . . . . . . . . . . . 1.9.1 Levi-Civita construction . . . . . . . . . . 1.9.2 Algebraic properties and Bianchi identities Geodesic deviation . . . . . . . . . . . . . . . . . . Einstein’s equations . . . . . . . . . . . . . . . . . 1.11.1 Equivalence Principle . . . . . . . . . . . . 1.11.2 Theory requirements . . . . . . . . . . . . v 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WWW.EBOOK777.COM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 5 5 5 6 7 7 8 9 10 13 14 16 16 19 19 21 21 22 page v April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile CANONICAL QUANTUM GRAVITY vi 1.12 2. Elements of Cosmology 2.1 2.2 2.3 2.4 2.5 3. The Robertson-Walker geometry . . Kinematics of the Universe . . . . . Isotropic Universe dynamics . . . . . Universe thermal history . . . . . . . 2.4.1 Universe critical parameters Inflationary paradigm . . . . . . . . 2.5.1 Standard Model paradoxes . 2.5.2 Inflation mechanism . . . . . 2.5.3 Re-heating phase . . . . . . 3.3 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . Constrained systems . . . . . . . . . . . . . 3.2.1 Primary and secondary constraints 3.2.2 First- and second-class constraints . Canonical transformations . . . . . . . . . . 3.3.1 Strongly canonical transformations 3.3.2 Weakly canonical transformations . 3.3.3 Gauged canonical transformations . Electromagnetic field . . . . . . . . . . . . . 3.4.1 Modified Lagrangian formulation . 3.4.2 Hamiltonian formulation . . . . . . 3.4.3 Gauge transformations . . . . . . . 3.4.4 Gauged canonicity . . . . . . . . . . 4.2 Metric representation . . . . . . . . . . . . 4.1.1 Einstein-Hilbert formulation . . . . 4.1.2 Stress-Energy tensor . . . . . . . . . 4.1.3 ΓΓ formulation . . . . . . . . . . . . 4.1.4 Dirac formulation . . . . . . . . . . 4.1.5 General f (R) Lagrangian densities 4.1.6 Palatini formulation . . . . . . . . . ADM formalism . . . . . . . . . . . . . . . 32 34 37 40 44 46 47 50 54 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lagrangian Formulations 4.1 24 27 31 Constrained Hamiltonian Systems 3.1 3.2 4. 1.11.3 Field equations . . . . . . . . . . . . . . . . . . . Vierbein representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 61 62 64 66 66 68 69 70 70 72 80 83 87 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WWW.EBOOK777.COM . . . . . . . . . 88 . 88 . 90 . 92 . 97 . 100 . 102 . 104 page vi April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile Contents 4.3 5. . . . . . . . . . . . . . . . . . . . . . . . . 105 109 111 113 114 115 Quantization Methods 119 5.1 119 120 121 122 124 124 5.2 5.3 5.4 5.5 6. 4.2.1 Spacetime foliation and extrinsic curvature 4.2.2 Gauss-Codazzi equation . . . . . . . . . . . 4.2.3 ADM Lagrangian density . . . . . . . . . . Boundary terms . . . . . . . . . . . . . . . . . . . 4.3.1 Gibbons-York-Hawking boundary term . . 4.3.2 Comparison among different formulations . vii Classical and quantum dynamics . . . . . . . . . . . . . . 5.1.1 Dirac observables . . . . . . . . . . . . . . . . . . 5.1.2 Poisson brackets and commutators . . . . . . . . . 5.1.3 Schrödinger representation . . . . . . . . . . . . . 5.1.4 Heisenberg representation . . . . . . . . . . . . . . 5.1.5 The Schrödinger equation . . . . . . . . . . . . . . 5.1.6 Quantum to classical correspondence: HamiltonJacobi equation . . . . . . . . . . . . . . . . . . . 5.1.7 Semi-classical states . . . . . . . . . . . . . . . . . Weyl quantization . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Weyl systems . . . . . . . . . . . . . . . . . . . . 5.2.2 The Stone-von Neumann uniqueness theorem . . . GNS construction . . . . . . . . . . . . . . . . . . . . . . . Polymer representation . . . . . . . . . . . . . . . . . . . . 5.4.1 Difference operators versus differential operators . 5.4.2 The polymer representation of Quantum Mechanics 5.4.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Continuum limit . . . . . . . . . . . . . . . . . . . Quantization of Hamiltonian constraints . . . . . . . . . . 5.5.1 Non-relativistic particle . . . . . . . . . . . . . . . 5.5.2 Relativistic particle . . . . . . . . . . . . . . . . . 5.5.3 Scalar field . . . . . . . . . . . . . . . . . . . . . . 5.5.4 The group averaging technique . . . . . . . . . . . Quantum Geometrodynamics 6.1 125 127 129 130 130 131 133 133 134 135 136 137 138 139 140 142 143 145 The Hamiltonian structure of gravity . . . . . . . . . . . . 145 6.1.1 ADM Hamiltonian density . . . . . . . . . . . . . 145 6.1.2 Constraints in the 3+1 representation . . . . . . . 147 WWW.EBOOK777.COM page vii April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile CANONICAL QUANTUM GRAVITY viii 6.1.3 6.2 6.3 6.4 7. 150 151 152 153 153 154 155 155 156 157 Gravity as a Gauge Theory 165 7.1 165 165 168 168 176 176 7.2 7.3 7.4 8. The Hamilton-Jacobi equation for the gravitational field . . . . . . . . . . . . . . . . . . . . . . ADM reduction of the Hamiltonian dynamics . . . . . . . Quantization of the gravitational field . . . . . . . . . . . 6.3.1 Quantization of the primary constraints . . . . . . 6.3.2 Quantization of the supermomentum constraint . 6.3.3 The Wheeler-DeWitt equation . . . . . . . . . . . Shortcomings of the Wheeler-DeWitt approach . . . . . . 6.4.1 The definition of the Hilbert space . . . . . . . . . 6.4.2 The functional nature of the theory . . . . . . . . 6.4.3 The frozen formalism: the problem of time . . . . Gauge theories . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 The Yang-Mills formulation . . . . . . . . . . . . 7.1.2 Hamiltonian formulation . . . . . . . . . . . . . . 7.1.3 Lattice gauge theories . . . . . . . . . . . . . . . . Gravity as a gauge theory of the Lorentz group? . . . . . 7.2.1 Spinors in curved spacetime . . . . . . . . . . . . 7.2.2 Comparison between gravity and Yang-Mills theories . . . . . . . . . . . . . . . . . . . . . . . . Poincaré gauge theory . . . . . . . . . . . . . . . . . . . . Holst action . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Lagrangian formulation . . . . . . . . . . . . . . . 7.4.2 Hamiltonian formulation . . . . . . . . . . . . . . 7.4.3 Ashtekar variables . . . . . . . . . . . . . . . . . . 7.4.4 Removing the time-gauge condition . . . . . . . . 7.4.5 The Kodama functional as a classical solution of the constraints . . . . . . . . . . . . . . . . . . . . 178 179 183 183 184 188 190 197 Loop Quantum Gravity 199 8.1 200 201 203 203 204 208 8.2 8.3 Smeared variables . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Why a reformulation in terms of holonomies? . . . Hilbert space representation of the holonomy-flux algebra 8.2.1 Holonomy-flux algebra . . . . . . . . . . . . . . . 8.2.2 Kinematical Hilbert space . . . . . . . . . . . . . Kinematical constraints . . . . . . . . . . . . . . . . . . . WWW.EBOOK777.COM page viii April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile Contents ix 8.3.1 8.4 8.5 8.6 8.7 8.8 9. Solution of the Gauss constraint: invariant spinnetworks . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Three-diffeomorphisms invariance and s-knots . . Geometrical operators: discrete spectra . . . . . . . . . . 8.4.1 Area operator . . . . . . . . . . . . . . . . . . . . 8.4.2 Volume operator . . . . . . . . . . . . . . . . . . . The scalar constraint operator . . . . . . . . . . . . . . . 8.5.1 Spin foams and the Hamiltonian constraint . . . . Open issues in Loop Quantum Gravity . . . . . . . . . . . 8.6.1 Algebra of the constraints . . . . . . . . . . . . . 8.6.2 Semiclassical limit . . . . . . . . . . . . . . . . . . 8.6.3 Physical scalar product . . . . . . . . . . . . . . . 8.6.4 On the physical meaning of the Immirzi parameter Master Constraint and Algebraic Quantum Gravity . . . . The picture of quantum spacetime . . . . . . . . . . . . . Quantum Cosmology 9.1 9.2 9.3 9.4 9.5 9.6 The minisuperspace model . . . . . . . . . . . . . . . . . General behavior of Bianchi models . . . . . . . . . . . . 9.2.1 Quantum Picture . . . . . . . . . . . . . . . . . 9.2.2 Matter contribution . . . . . . . . . . . . . . . . Bianchi I model . . . . . . . . . . . . . . . . . . . . . . . Bianchi IX model . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Quantum dynamics . . . . . . . . . . . . . . . . 9.4.2 Semiclassical behavior . . . . . . . . . . . . . . . 9.4.3 The quantum behavior of the isotropic Universe BKL conjecture . . . . . . . . . . . . . . . . . . . . . . . Cosmology in LQG . . . . . . . . . . . . . . . . . . . . . 9.6.1 Loop Quantum Cosmology . . . . . . . . . . . . 9.6.2 Other approaches . . . . . . . . . . . . . . . . . 209 210 212 212 214 216 219 221 222 223 224 225 227 228 231 . . . . . . . . . . . . . 234 236 238 240 242 244 247 250 253 256 258 259 270 Bibliography 275 Index 295 WWW.EBOOK777.COM page ix May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory This page intentionally left blank WWW.EBOOK777.COM PST˙ws April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile Preface The request of a coherent quantization for the gravitational field dynamics emerges as a natural consequence of Einstein’s Equations: the energymomentum of a field is source of the spacetime curvature and therefore its microscopic quantum features must be reflected onto microgravity effects. The use of expectation values as sources is well-grounded as a first approximation only, and does not fulfill the requirements of a fundamental theory. However, as is well-known, the achievement of a consistent Quantum Gravity theory remains a complete open task, due to a plethora of different subtleties, which are here summarized in the two main categories: i) General Relativity is a background-independent theory and therefore any analogy with the non-Abelian gauge formalisms must deal with the concept of a dynamic metric field; ii) the implementation in the gravitational sector of standard prescriptions, associated with the quantum mechanics paradigms, appears as a formal procedure, whose real physical content is still elusive. There exists a qualitative consensus on the idea that a convincing solution to the Quantum Gravity problem will not arise before General Relativity and Quantum Mechanics are both deeply revised in view of a converging picture. Indeed, over the last fifteen years, the three most promising approaches to Quantum Gravity (i.e. String Theories, Loop Quantum Gravity and Non-commutative Geometries) revealed common features in defining a “lattice” nature for the microphysics of spacetime. This consideration makes clear that, up to the best of our present understanding, the main effort to improve fundamental formalisms must be the introduction of a “cut-off” physics, able to replace the notion of a spacetime continuum with a consistent discrete scenario. The correctness of such a statement will probably xi WWW.EBOOK777.COM page xi April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile xii CANONICAL QUANTUM GRAVITY be regarded as the main success of the end of the last century reached in Theoretical Physics. A proper task for the present century is now to constrain the morphology of such a discrete microstructure of spacetime, to get, at least, a phenomenological description for Quantum Gravity effects. From a theoretical point of view, an important aim would consist of a unified picture containing common features of the present approaches, but synthesized into a more powerful mathematical language. On the one hand, it would be important to recognize non-commutative properties in the loop representation of spacetime; on the other hand, transporting the background independence of the “spin networks” into the interaction framework characterizing String Theories would constitute a relevant progress. The viability of these two goals is an intuitive perspective, but it contrasts with the rigidity of the corresponding formalisms, which confirms the request for more general investigation tools. The simplest approach to the quantization of General Relativity relies on the implementation of the canonical method on the phase-space structure associated with the gravitational degrees of freedom. This attempt is the one originally pursued by B. DeWitt in 1967 and it immediately revealed all the pathologies contained in the canonical quantum geometrodynamics. It was this clear inconsistence of the canonical quantum procedure (referred to the second order formalism) that attracted the attention of a large number of researchers, active on the last four decades. This strong effort of re-analysis, which could give the feeling of an overestimation of the real chances allowed by the canonical method, found its “merry ending” in the developments in Loop Quantum Gravity and its applications. In fact, the formulation of the Hamiltonian problem for General Relativity provided by A. Ashtekar in 1986, allowed for a new paradigm for the canonical method, in close analogy with an SU (2) gauge theory. The main success of Loop Quantum Gravity is recognizing a discrete structure of spacetime, by starting from continuous variables in the phase space of the theory. The origin of such a valuable result consists in the discrete nature acquired by the spectra of areas and volumes, reflecting to some extent the compactness of the SU (2) group emerging in this formulation. The merit of the loop quantization of gravity can be identified in the possibility to deal with non-local geometric variables, like holonomies and fluxes, instead of the simple metric analysis faced by DeWitt. In fact, using the physical properties associated with the connection and the so-called electric fields, the background independence of the theory naturally emerges, and the quantum structure of spacetime comes out, as far as the Hilbert space is characterized via the WWW.EBOOK777.COM page xii April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile Preface xiii spin-network basis. Quantum Cosmology is a privileged arena in which different Quantum Gravity approaches can be implemented and it is expected to provide a consistent picture of the Universe birth. In this respect, the standard minisuperspace implementations of both the Wheeler-DeWitt formulation and of Loop Quantum Gravity (Loop Quantum Cosmology) are predictive to some extent. In particular the Wheeler-DeWitt equation, as restricted to the homogeneous cosmological spaces, presents some pleasant features, such that the Universe dynamics resembles that of a scalar field interacting with a nontrivial and non-perturbative potential. In this formulation, the cosmological singularity is not removed, but a consistent Universe evolution throughout the Planck era can be clearly traced. For what concerns Loop Quantum Cosmology, a Big-Bounce emerges due to the discreteness of the space volume spectrum, and it offers a viable semiclassical picture of the singularity removal by a phenomenological repulsion of the high curvature gravity. Nonetheless, a critical revision of the standard loop Universe quantization can be given, based on observing how the homogeneity restriction implies a gauge fixing on the SU (2) internal symmetry of the full theory. A recent promising alternative reformulation of the Loop Quantum Gravity dynamics for the primordial Universe is presented together with its basic tools (“in primis” the presence of an intertwiner structure), which makes it closer to the full theory than the Loop Quantum Cosmology paradigm. The analysis of Quantum Cosmology here presented is probably one of the most original contributions offered in this volume to the comprehension of the impact that quantum physics can have on the dynamics of the gravitational field. It can be regarded as the prize the reader gets out of the study of the preceding chapters concerning the successes and the limits of the full Quantum Gravity theory. The aim of this book is to provide a detailed account of the physical content emerging from the canonical approach to Quantum Gravity. All the crucial steps in our presentation have a rather pedagogical character, providing the reader with the necessary tools to become involved in the field. Such a pedagogical aspect is then balanced and completed by subtle discussions on specific topics which we regard as relevant for the physical insight they outline on the treated questions. Our analysis is not aimed at convincing the reader about a pre-constituted point of view instead, our principal goal is to review the picture of Canonical Quantum Gravity on WWW.EBOOK777.COM page xiii April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile xiv CANONICAL QUANTUM GRAVITY the basis of the concrete facts at the ground of its clear successes, and also of its striking shortcomings. In order to focus our attention on the physical questions affecting the consistence of a canonical quantum model, when extended to the gravitational sector, we provide a critical discussion of all the fundamental concepts of quantum physics, by stressing the peculiarity of the gravitational case in their respect. All the key features of Loop Quantum Gravity are described in some details, presenting the specific technicalities, but privileging their physical interpretation. Finally, all the open questions concerning the faced topics are clearly outlined by a precise criticism on the weak points present in the quantum construction. The detailed structure of the book is • the main goal of the first chapter is to provide the basic formalisms and concepts at the ground of Einsteinian theory of gravity. We start by constructing the tensor representation of a manifold differential geometry in terms of the so-called embedding procedure, i.e. we consider a parametric representation for a generic hypersurface. The concepts of spacetime curvature and geodesic deviation are then fully addressed. We discuss the fundamental principles of General Relativity and show how they fix the form of Einstein’s equations. As a last step we provide the vierbein formalism for further applications throughout the book. • in the second chapter we provide a brief presentation of the Universe evolution based on the isotropic Robertson-Walker geometry. We first discuss the morphology of a homogeneous and isotropic Universe and its kinematic implications, able to account for observations such as the photons redshift and the Hubble law. Then, we address the dynamic features of the isotropic Universe as described by Einstein’s equations. The structure of the dynamic equations is outlined by characterizing the matter source via a perfect fluid endowed with a proper equation of state. This allows one to develop a brief sketch of the Universe thermal history, describing the different stages of its evolution. Finally, we analyze the inflationary paradigm, offering a compact review of the main shortcomings of the standard cosmological model, as well as the basic ideas at the ground of the inflationary scenario. WWW.EBOOK777.COM page xiv April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile Preface xv • in the third chapter, we introduce the main concepts of the standard Hamiltonian formulation for constrained systems, namely the distinction between primary and secondary constraints as well as between first-class and second-class constraints. Moreover the definition of canonical transformations is reviewed and broadened introducing the concepts of weak- and gauged-canonicity. In order to give some practical insight, we use an example based on the free electromagnetic field where all the concepts previously faced are sketched in a rather pedagogical way. In this respect we will present the L. Castellani algorithm for constructing the generators of gauge transformations. • in the fourth chapter, the Lagrangian formulation of General Relativity is critically discussed giving different but equivalent (under certain hypothesis) Lagrangian densities for the gravitational field. We start by introducing the commonly used Einstein-Hilbert Lagrangian density followed by the study of the action principle when matter is included. We discuss the non-covariant Lagrangian density approach due to Einstein and Dirac. The study of an extended theory of gravity in the form of an f (R) Lagrangian density is developed followed by description of the Palatini formulation. The ADM formulation is presented using the embedding technique developed in the first chapter and a critical discussion on the boundary terms of some of the formulations is given allowing one to perform the transformation linking some of the presented approaches, namely the ADM to Einstein’s and Dirac’s ones. • in the fifth chapter, we recall the basic features of Quantum Mechanics. In particular, we review the definition of quantum states as elements of a Hilbert space and of observables as linear operators. The classical to quantum correspondence between Poisson brackets and commutators is discussed as a tool for the quantization of a generic mechanical system. Then, we present the Schrödinger and Heisenberg representations. The Hamilton-Jacobi equation and the development of semiclassical coherent states are analyzed in order to bridge the quantum formulation with the classical world. Furthermore, we introduce Weyl quantization and outline the main steps of the GNS construction. The polymer representation is then presented as an application. Finally, the quantization of some physically-relevant constrained systems and its shortcomings are discussed. WWW.EBOOK777.COM page xv April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile xvi CANONICAL QUANTUM GRAVITY • in the sixth chapter, the standard Hamiltonian formulation of General Relativity is reviewed. Then, the two possible scenarios for the canonical quantization of the gravitation field are illustrated. On one hand, we present the reduction to the canonical form and its main issues are briefly outlined. On the other hand, we focus on the Wheeler-De Witt approach. We emphasize how the implementation of the constraints via the Dirac prescription leads to the Wheeler-DeWitt equation. The main technical and interpretative shortcomings of this formulation (which prevents it from being a well-grounded quantum description for the gravitational field) are discussed: the absence of a real Hilbert space structure, the need for a consistent regularization scheme and the problem of time. Some proposals to identify a time variable are presented: the BrownKuchař model, the multitime approach and the Vilenkin proposal. • in the seventh chapter, Yang-Mills gauge theories are presented. The classical Lagrangian and Hamiltonian formulations are analyzed. Then, we introduce some tools which are useful for lattice quantization (spin networks and spin foams). Furthermore, a formal comparison between Yang-Mills models and General Relativity is performed by outlining the similarities but also the unavoidable differences. Poincaré gauge theory is then discussed as an alternative formulation in which gravity behaves as a Yang-Mills interaction. The Holst formulation for gravity is then revised and the emergence of a SU(2) gauge symmetry is emphasized. The analysis of the associated constrained system in vacuum and in the presence of matter fields is carefully discussed in a generic local Lorenz frame. Finally, the Kodama state is presented. • the eighth chapter is devoted to Loop Quantum Gravity. The role of holonomies and fluxes in quantum theories is briefly discussed, while the quantization of the corresponding algebra in Loop Quantum Gravity is presented. The main achievements of this framework are emphasized: the definition of a kinematical Hilbert space, the success in the quantization of kinematical constraints, the achievement of discrete spectra for the area and volume operators, the implementation of the scalar constraint on a quantum level. A careful explanation of the main technical and conceptual issues (the study of the dynamics, the semiclassical limit, the constraint algebra, the role of the Immirzi parameter) is given, while the Master Constraint program and the Algebraic Quantum Grav- WWW.EBOOK777.COM page xvi April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile Preface xvii ity setup are briefly discussed as attempts towards the solution of some shortcomings. Finally, a comparison between the picture of the quantum spacetime coming out from the Loop Quantum Gravity and the Wheeler-DeWitt frameworks is performed. • in the ninth chapter, Quantum Cosmology is presented as an arena in which the predictions of Quantum Gravity can be investigated. The minisuperspace approximation is discussed: we point out the simplifications that the restriction to homogeneous models provides for the quantization issue, such that a proper quantum description of the Bianchi type I and IX models in the Wheeler-DeWitt approach can be given, even though this is not generically enough to obtain a consistent quantum model for the Universe (the presence of singularity and the chaos in Bianchi IX). Then, the paradigm of Loop Quantum Cosmology is introduced and the bouncing solution replacing the Big-Bang singularity is explicitly shown to arise. Finally, a critical discussion on the foundation of Loop Quantum Cosmology is given and a new research line for the cosmological sector of Loop Quantum Gravity, namely Quantum Reduced Loop Gravity, is presented. We would like to thank Abhay Ashtekar for interesting discussions on Loop Quantum Gravity during the First Stueckelberg Workshop (Pescara, June 25-July 1 2006), and which inspired the writing of this book. We would also like to thank Richard Arnowitt, Stanley Deser and Charles W. Misner for their comments and discussion on the Hamiltonian formulation of General Relativity and for having called our attention on the specific relevance of reference [17]. WWW.EBOOK777.COM page xvii April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile xviii CANONICAL QUANTUM GRAVITY The work of FC was supported by funds provided by the National Science Center under the agreement DEC- 2011/02/A/ST2/00294. The work of OML was partially supported by the research grant ’Reflections on the Hyperbolic Plane’ from the Albert Einstein Institute for Gravitational Physics - MPI, Potsdam-Golm, and partially by the research grant ’Classical and Quantum Physics of the Primordial Universe’ from Sapienza University of Rome- Physics Department, Rome. WWW.EBOOK777.COM page xviii April 7, 2014 12:27 BC: 8957 – Canonical Quantum Gravity mainfile List of Figures 1.1 1.2 Manifold embedded in a Minkowskian space . . . . . . . . . . . Riemann Levi-Civita construction . . . . . . . . . . . . . . . . 4.1 ADM foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8.1 8.2 8.3 SU (2) invariant spin network . . . . . . . . . . . . . . . . . . . 210 Tetrahedron adapted to a graph . . . . . . . . . . . . . . . . . 218 Quantum space in LQG . . . . . . . . . . . . . . . . . . . . . . 228 9.1 9.2 9.3 Bianchi IX potential . . . . . . . . . . . . . . . . . . . . . . . . 246 Potential for closed isotropic Universe . . . . . . . . . . . . . . 254 Bounce in LQC . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 xix WWW.EBOOK777.COM 2 17 page xix

Author Francesco Cianfrani, Giovanni Montani, Matteo Lulli, and Orchidea Maria Lecian Isbn 9789814556644 File size 2.4 MB Year 2014 Pages 324 Language English File format PDF Category Physics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare This book aims to present a pedagogical and self-consistent treatment of the canonical approach to Quantum Gravity, starting from its original formulation to the most recent developments in the field. We start with an innovative and enlightening introduction to the formalism and concepts on which General Relativity has been built, giving all the information necessary in the later analysis. A brief sketch of the Standard Cosmological Model describing the Universe evolution is also given alongside the analysis of the inflationary mechanism. After deepening the fundamental properties of constrained dynamic systems, the Lagrangian approach to the Einsteinian Theory is presented in some detail, underlining the parallelism with non-Abelian gauge theories. Then, the basic concepts of the canonical approach to Quantum Mechanics are provided, focusing on all those formulations which are relevant for the Canonical Quantum Gravity problem. The Hamiltonian formulation of General Relativity and its constrained structure is then analyzed by comparing different formulations. The resulting quantum dynamics, described by the Wheeler–DeWitt equation, is fully discussed in order to outline its merits and limits. Afterwards, the reformulation of Canonical Quantum Gravity in terms of the Ashtekar–Barbero–Immirzi variables is faced by a detailed discussion of the resulting Loop Quantum Gravity Theory. Finally, we provide a consistent picture of canonical Quantum Cosmology by facing the main features of the Wheeler–DeWitt equation for the homogeneous Bianchi models and then by a detailed treatment of Loop Quantum Cosmology, including very recent developments. Contents: Introduction to General Relativity Elements of Cosmology Constrained Hamiltonian Systems Lagrangian Formulations Quantization Methods Quantum Geometrodynamics Gravity as a Gauge Theory Loop Quantum Gravity Quantum Cosmology Readership: Researchers in theoretical physics, quantum physics, general relativity and astrophysics.       Download (2.4 MB) Quantum Gravity Recent Developments in Gravitation: Cargèse 1978 Mathematical Physics: Proceedings of the 12th Regional Conference A First Course In Loop Quantum Gravity Advanced Concepts In Particle And Field Theory Load more posts

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