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New Dynamics for Canonical Loop Quantum Gravity

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New Dynamics for Canonical Loop Quantum Gravity NEW DYNAMICS FOR CANONICAL LOOP QUANTUM GRAVITY Mehdi Assanioussi Faculty of Physics University of Warsaw Dissertation submitted for the degree of Doctor of Philosophy Under the supervision of Prof. dr hab. Jerzy Lewandowski December 2016 Declaration I hereby declare that this thesis presents my original research work. Wherever contributions of others are involved, every effort is made to indicate this clearly, with due reference to the literature, and acknowledgements of collaborative research and discussions. I also declare that the submitted thesis has not been the subject of proceedings resulting in the award of any other academic degree or diploma. The work was done under the supervision of professor dr hab. Jerzy Lewandowski at the Faculty of Physics, University of Warsaw. Mehdi Assanioussi December 2016 Acknowledgements I would like to express my deepest gratitude to my supervisor prof. Jerzy Lewandowski, for the freedom he gave me in my work, for his constant support to attend any conference I liked, and for his guidance and encouragement without which I would not have accomplished the work presented in this thesis. This journey would not have been as fruitful and exciting without multi- ple passionate and enlightening discussions I had with Emanuele Alesci, Ilkka Mäkinen, Andrea Dapor, Norbert Bodendorfer, Jędrzej Świeżewski and Wo- jciech Kamiński, which often lasted for many late evening hours and which helped enormously to shape my ideas the way they are now, and for that I am forever grateful. I also thank all my colleagues in the Institute of Theoretical Physics at the faculty of Physics, University of Warsaw, for the stimulating and convivial environment we shared. I would like to give special thanks to Simone Speziale who set me on the path of quantum gravity. I also thank prof. J. Fernando Barbero G. and prof. Hanno Sahlmann for assessing this dissertation. I would like to thank my parents for their unconditional support, unfading encouragement and all they sacrificed so that I could be where I am today. To both of my sisters, Ikhlas and Kenza, I want to address special thanks for their warm feelings and the immense joy they bring to my life. To my friends Taha, Faissal, Ghassane, Yasser, Saad, Kevin and Antoine, I am extremely fortunate to have met you and very grateful for your friendship and all the laughing with me and at me. A friendship which will always make me push forward through life. Last but not least, I would like to express my deepest gratitude and af- fection to Basia for her support, patience and all the wonderful moments we enjoyed together which helped me keep up the hard work. « إن من الواجب على من يحقق في كتابات العلماء، إذا كان البحث عن الحقيقة هدفه، هو ٔان يستنكر جميع ما ٔيقراه، [:::] وعليه ٔان يتشكك في نتائج دراسته ٔايضا حتى يتجنب الوقوع في ٔاي تحيز ٔاو تساهل. » – ابن الهيثم، كتاب المناظر، 1021 – « The duty of those who investigate the writ- ings of scientists, if learning the truth is their goal, is to disapprove all that they read, [:::] They should also suspect their own critical ex- amination of it, so that they may avoid falling into either prejudice or leniency. » – Alhazen, Book of Optics, 1021 – Abstract Canonical loop quantum gravity (LQG) is a canonical quantization of general relativity in its Hamiltonian formulation, which has successfully completed the construction of a kinematical Hilbert space and the implementation of the Gauss constraints and the spatial diffeomorphism constraints. In the first part of this thesis I give a general introduction to the Ashtekar– Barbero Hamiltonian formulation of general relativity and a detailed overview of the LQG program. In the second part, I present a new approach for quantizing the Hamiltoni- ans in various LQG models. The result allows to make a step forward toward improving the status of the dynamics in the theory. The construction is based on novel regularization procedures of the classical functionals, starting with the construction of a new geometrical operator, the curvature operator, related to the three–dimensional Ricci curvature. This new approach eventually leads to a diffeomorphism covariant scalar constraint operator in vacuum LQG with an anomaly–free constraints algebra. Moreover, thanks to the new prescription, it is possible to construct eligible symmetric extensions with the perspective to obtain self–adjoint extensions. In the third part, I expose how the new regularization is used to imple- ment symmetric Hamiltonian operators in the context of two particular LQG deparametrized models, completing the quantization of the two models with a full quantum gravity sector. Finally, I present an approximation method based on time–independent perturbation theory, which was applied to the Hamilto- nian operators in those deparametrized models and where the perturbation parameter depends on the Barbero–Immirzi parameter. Table of contents I A centenary quest: Quantum gravity! 1 II On Dirac’s footsteps: canonical loop quantum gravity 9 II.1 Hamiltonian formulation of GR and Ashtekar-Barbero variables 9 II.1.1 ADM formulation of general relativity . 10 II.1.2 General relativity in Ashtekar-Barbero variables . 14 II.2 Canonical quantization: LQG . 18 II.2.1 Holonomy-flux algebra and kinematical Hilbert space . 19 II.2.2 Implementation of the Gauss and spatial diffeomorphism constraints . 25 II.2.3 On the dynamics in loop quantum gravity . 29 III Alternative dynamics in vacuum LQG 43 III.1 Vertex Hilbert space . 44 III.2 Curvature operator . 45 III.2.1 Regge calculus . 46 III.2.2 Construction of the curvature operator . 51 III.2.3 Properties of the curvature operator . 61 III.2.4 The Lorentzian part of the scalar constraint operator . 66 III.3 Euclidean operator . 67 III.3.1 Construction of the Euclidean operator . 68 III.3.2 Properties of the Euclidean operator . 73 III.4 Scalar constraint operator, quantum algebra and physical states 74 III.4.1 Quantum constraints algebra . 75 III.4.2 Physical states . 76 IV Towards new dynamics in LQG deparametrized models 79 IV.1 Emergent time in examples . 80 IV.1.1 Gravity coupled to a free Klein–Gordon field . 80 IV.1.2 Gravity coupled to non–rotational dust . 83 xii Table of contents IV.2 LQG quantization . 84 IV.2.1 Physical Hilbert spaces . 85 IV.2.2 Quantum dynamics . 85 IV.3 An approximation method for LQG dynamics . 95 IV.3.1 Perturbation theory with the Barbero–Immirzi parameter 96 IV.3.2 Numerical analysis of simple examples . 98 V Summary and outlook 105 References 109 Appendix A Key publications 117 Chapter I A centenary quest: Quantum gravity! When the theory of general relativity [1, 2] was proposed by Albert Einstein in 1915 as a relativistic theory of gravity, it was welcomed in the scientific community with less enthusiasm than it later proved to deserve. Despite its early successful predictions such as Mercury anomalous perihelion (1915) and the deflection of starlight by the Sun during the solar eclipse (1919), general relativity was for several years viewed as an intriguing theory with an unusual mathematical complexity, which stands aside from the generally accepted fields of theoretical physics at that time. Nevertheless, with the derivation of the first non–trivial solutions to the field equations of the theory and the emergence of Cosmology, many physicists and mathematicians of the time took interest in investigating the structure and implications of the theory. After almost two decades, general relativity started imposing itself as a revolutionizing theory and a cornerstone of modern theoretical physics. On one of his lecture in 1939, Paul Dirac adequately summarized the common perception about the theory then: “What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating. [...] The restricted theory changed our ideas of space and time in a way that may be summarized by stating that the group of transformations to which the space-time continuum is subject must be changed from the Galilean group to the Lorentz group. [...] The general theory 2 A centenary quest: Quantum gravity! of relativity involved another step of a rather similar character, although the increase in beauty this time is usually considered to be not quite so great as with the restricted theory, which results in the general theory being not quite so firmly believed in as the restricted theory.” Few decades later will see the theory of relativity flourish in several aspects with new exact solutions, new mathematical formulations of the theory and better understanding of the dynamical nature of spacetime. Ironically, these achievements also exposed certain issues in the theory, namely the problem of observables and the problem of singularities present in cosmological and black hole solutions. In the meantime, another revolution was running: the rise of quantum mechanics. The base of quantum mechanics arose gradually from various ex- perimental observations and theoretical descriptions of different phenomena thought originally to be independent. Some of the essential concepts were already developed in the nineteenth century with the wave theory of light, followed later by Max Planck’s hypothesis of exchange of energy in discrete amounts, the quanta. It is until the late 1920’s that the theory was consistently formulated and accepted as the standard theory of atoms and photons, with a new understanding of once trivial concepts, such as measurement and locality. Quantum mechanics was since then extended and successfully applied in many domains, setting itself at the base of uncountable theoretical and technological developments. The greatest consequent theoretical achievement is undeniably the extension of the principles of quantum mechanics to field theories, giving birth to the realm of quantum field theories (QFTs).
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