Quantum pre-geometry models for Quantum Gravity by Francesco Caravelli A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2012 © Francesco Caravelli 2012 AUTHOR’S DECLARATION I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii ABSTRACT In this thesis we review the status of an approach to Quantum Gravity through lattice toy models, Quantum Graphity. In particular, we describe the two toy models introduced in the literature and describe with a certain level of details the results obtained so far. We emphasize the connection between Quantum Graphity and emergent gravity, and the relation with Variable Speed of Light theories. iii ACKNOWLEDGEMENTS A long list of sincere thanks has to be given here. First of all, to my supervisors, Fotini Markopoulou and Lee Smolin, for the continuous and discrete support during the PhD. Then, a sincere thanks goes to the people I have closely collaborated with (and friends), Leonardo Modesto, Alioscia Hamma and Simone Severini, Arnau Riera, Dario Benedetti, Daniele Oriti, Lorenzo Sindoni, Razvan Gurau, Olaf Dreyer, Emanuele Alesci, Isabeau Premont-Schwarz,´ Sylvain Carrozza and Cosimo Bambi. A particular thanks goes also to my family (Mamma, Papa´ e le sorelline), for continuous support, moral and economi- cal. Then a long list of friends goes here. Having shared a double life between Waterloo (Canada) and Berlin (Germany), a list of people from all over the world follows! In Wa- terloo, I have to thank my buddy Cozmin(/Cozster Cosniverse Cosferatu) Ududec and my dear friend Mitzy Herrera, Owen Cherry (on top), Cohl Fur(e)y, Coco Escobedo, John Daly and Jessica Phillips, Jonathan Hackett and Sonia Markes . In Berlin I have to thank my russian friend Sasha Latnikova, die kleine Marina Zygelman, Anja Rogacheva, Sara Pernbrunner, Cosimo Restuccia, Ivano Pallante, to my not-girlfriend Elena Bruno, to the si-culo Marco Scalisi for many discussions about all-but-physics, to the puzzling and mis- terious friend Carlos Guedes and to la francesina Estelle de Tournadre in the good and the iv bad (and the bad and the sad). I thank in particular Isabeau, who kindly gave me his thesis in order to use his Latex style. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & In- novation. Part of this research has been also funded by the Templeton Foundation. Part of the numerical work was made possible by the facilities of the Shared Hierarchical Aca- demic Research Computing Network (SHARCNET:www.sharcnet.ca). Also, hospitality at the Santa Fe Institute and the Albert Einstein Institute of Berlin have been much appre- ciated. v DEDICATION Ancora alla mia famiglia e ai miei amici, a qualcun altro che dimentico, e a qualcun altro che non voglio ricordare (Again to my family and my friends, to somebody else I forgot, and to somebody I do not want to remember) vi CONTENTS Author’s Declaration ii Abstract iii Acknowledgements iv Dedication vi 1 Introduction 1 I General Relativity: a troubled son 9 2 General Relativity and its problems 10 2.1 General Relativity . 10 2.1.1 Conceptual problems of GR . 15 2.1.2 Breakdown of General Relativity: singularities . 15 2.2 Why quantizing gravity? . 17 2.3 First attempt of quantization . 20 vii 2.3.1 QFT: linearized theory and divergences . 20 3 Motivations for an analogue description of General Relativity 29 3.1 Condensed Matter, Quantum Mechanics and Analogue models . 30 3.2 Variable Speed of Light theories and the major cosmological problems . 32 II Graph models 38 4 Quantum Graphs without matter 39 4.1 Graph Models . 40 4.2 A brief review of the String-Net condensation . 41 4.3 Graph theory preliminaries . 45 4.4 Quantum mechanics preliminaries . 46 4.5 The Hamiltonian . 49 4.5.1 Valence term . 50 4.5.2 Closed Paths . 51 4.5.3 Interaction Terms . 52 4.6 Extensions with More Degrees of Freedom and String-Net condensation . 52 4.7 Lieb-Robinson Bounds and (non-)locality . 57 4.7.1 Locality and non-locality . 57 4.7.2 Lieb-Robinson Bounds and the emergent speed of sound . 59 4.7.3 Lieb-Robinson Bound for the emergent U(1) model . 60 4.8 Mean field theory approximation at Low Temperature . 63 viii 4.8.1 Graphs and Line Graphs . 63 4.8.2 The line graph representation . 65 4.8.3 Mean field theory approximation and low temperature expansion . 75 4.9 The cosmological horizon problem and the speed of propagation of signals 86 4.10 Problems with this model and the external bath . 87 5 Quantum Graphs with matter 88 5.1 Bose-Hubbard model on Dynamical Graphs . 88 5.1.1 The Bose-Hubbard model . 91 5.1.2 Promoting the edges of the lattice to a quantum degree of freedom . 96 5.1.3 Discussion of the model . 104 5.1.4 Setting of the model . 111 5.1.5 Numerical analysis . 115 5.2 Markov chains analysis of the model . 121 III Emergence 135 6 Geometry and Emergence 136 6.1 Foliated and rotationally invariant graphs . 138 6.1.1 Foliated graphs . 138 6.1.2 Rotationally invariant graphs . 143 6.2 Regions of high connectivity as trapped surfaces . 145 6.2.1 Classical trapping . 146 ix 6.2.2 Quantum case: the KN configuration . 147 6.3 Correspondence between graph connectivity and curved geometry . 156 6.3.1 Restriction of the time-dependent Schrodinger¨ equation to the set of classical states . 156 6.3.2 Dispersion relation and continuum limit . 161 6.3.3 Lieb-Robinson bound for hopping bosons . 163 6.4 Model with parametrized fall-off of connectivity . 166 6.4.1 The Model . 166 6.4.2 Trapped surface . 167 6.5 Emergence of the mass for the scalar field . 169 6.5.1 Emergence of the mass from disordered locality . 169 6.5.2 Emergence of the mass from multi-particles interactions . 191 Appendix A Random Walk 208 Bibliography 216 x ”You can know the name of a bird in all the languages of the world, but when you’re finished, you’ll know absolutely nothing whatever about the bird. So let’s look at the bird and see what it’s doing - that’s what counts. I learned very early the difference between knowing the name of something and knowing something.” Richard Feynman ”Phantasie / Vorstellungskraft ist wichtiger als Wissen, denn Wissen ist begrenzt.” (Imagination is more important than knowledge, because knowledge is limited.) Albert Einstein ”Ero giovanissimo, avevo l’illusione che l’intelligenza umana potesse arrivare a tutto. E percio´ m’ero ingolfato negli studi oltremisura.” (I was young, I had the illusion that human intelligence could get anywhere. For this reason I focused on studies beyond measure.) Enrico Fermi Thanks. xi Chapter 1 Introduction General Relativity is one of the most intriguing physical theories, since it deals with the nature of space and time, and our perception of reality. Using the Planck constant ~, the Newton constant G and the speed of light c, we can build a constant with the dimension of an energy. This is the Planck energy, given by: r 5 ~c 19 Ep = 10 GeV: G ≈ When a gravitational process has an energy of the order of the Planck energy, using the intuition we gained by quantizing the other three known forces, we expect another theory, a Quantum Theory of Gravity, to replace General Relativity. As we will discuss later in detail, this process of quantization is very troublesome. So far, no fully satisfactory quantization of General Relativity exists. What is a Quantum Theory of Gravity? A quantum theory of gravity is a theory which describes gravitational phenomena up to the Planck scale and such that in the 1 limit ~ 0, reduces to a gravitational theory, and that at energies much smaller then the ! Planck energy, reduces to Einstein’s General Relativity. Given this, it is rather compli- cated to describe a theory which do not exist yet. It is true that there are many proposals, but many of these are yet not fully convincing, or cannot predict a valuable and testable phenomenon[45]. Quantum mechanics is one of the milestones of the physics of the last century. It describes the physics of elementary particles and of atoms and molecules and has been tested extensively. General relativity is the other milestone of the past century. It describes the physics of spacetime and it works very well in describing the motion of celestial bod- ies in our solar system, in galaxies and in our Universe. However, when we put the two together and when we try to quantize gravity as it has been done for other forces, many obstructions appear. In spite of progresses in several directions, finding this new theory has proven a chal- lenging problem for several decades. In particular, there are several approaches propos- ing a quantization of gravity [33, 1, 120, 148]. In particular, Loop Quantum Gravity[33] (LQG) is a promising quantization of Gen- eral Relativity. It establishes that spacetime is a foam, in which atoms of space are con- structed on Wilson gauge lines, or spin networks. Many results have been obtained, as the spectrum of the area and volume operators [47] and the Bekenstein-Hawking Area- Entropy law[46].