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UNIVERSITY OF CALIFORNIA SANTA CRUZ QUANTUM GRAVITY AND COSMOLOGY A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in PHYSICS by Lorenzo Mannelli September 2005 The Dissertation of Lorenzo Mannelli is approved: Professor Tom Banks, Chair Professor Michael Dine Professor Anthony Aguirre Lisa C. Sloan Vice Provost and Dean of Graduate Studies °c 2005 Lorenzo Mannelli Contents List of Figures vi Abstract vii Dedication viii Acknowledgments ix I The Holographic Principle 1 1 Introduction 2 2 Entropy Bounds for Black Holes 6 2.1 Black Holes Thermodynamics ........................ 6 2.1.1 Area Theorem ............................ 7 2.1.2 No-hair Theorem ........................... 7 2.2 Bekenstein Entropy and the Generalized Second Law ........... 8 2.2.1 Hawking Radiation .......................... 10 2.2.2 Bekenstein Bound: Geroch Process . 12 2.2.3 Spherical Entropy Bound: Susskind Process . 12 2.2.4 Relation to the Bekenstein Bound . 13 3 Degrees of Freedom and Entropy 15 3.1 Degrees of Freedom .............................. 15 3.1.1 Fundamental System ......................... 16 3.2 Complexity According to Local Field Theory . 16 3.3 Complexity According to the Spherical Entropy Bound . 18 3.4 Why Local Field Theory Gives the Wrong Answer . 19 4 The Covariant Entropy Bound 20 4.1 Light-Sheets .................................. 20 iii 4.1.1 The Raychaudhuri Equation .................... 20 4.1.2 Orthogonal Null Hypersurfaces ................... 24 4.1.3 Light-sheet Selection ......................... 26 4.1.4 Light-sheet Termination ....................... 28 4.2 Entropy on a Light-Sheet .......................... 29 4.3 Formulation of the Covariant Entropy Bound . 30 5 Quantum Field Theory in Curved Spacetime 32 5.1 Scalar Field Quantization .......................... 32 5.2 Bogolubov Coe±cients ............................ 35 5.3 Green Functions ............................... 36 II Publications 39 6 De Sitter Vacua, Renormalization and Locality 40 6.1 Introduction .................................. 40 6.2 Interacting Scalar Field Theory in an ® Vacuum . 44 6.2.1 Notation ................................ 44 6.2.2 Computation ............................. 46 6.3 The Wave Functional in the Antipodal Vacuum . 53 6.4 Conclusion .................................. 56 7 Microscopic Quantum Mechanics of the p = ½ Universe 58 7.1 Introduction .................................. 58 7.2 Local framework for a holographic theory of quantum gravity . 61 7.2.1 The hilbert space of an observer . 65 7.2.2 SUSY and the holoscreens: the degrees of freedom of quantum gravity ................................. 71 7.2.3 Rotation invariance ......................... 74 7.3 Quantum cosmology of a dense black hole ﬂuid . 75 7.3.1 The random operator ansatz .................... 75 7.3.2 Homogeneity, isotropy and ﬂatness . 80 7.3.3 Time dependence - scaling laws ................... 82 7.3.4 Time dependence: a consistency relation, and a failure . 84 7.4 More general space-times .......................... 88 7.5 Discussion ................................... 92 7.6 Appendix ................................... 97 7.6.1 Intersection of causal diamonds ................... 97 7.6.2 Holographic relations in a general FRW cosmology . 102 7.6.3 Computation of ce from geometry and constant in front of HN . 108 7.7 Figures .................................... 111 iv 8 Infrared Divergences in dS/CFT 120 8.1 Introduction ................................ 120 8.2 Review of dS/CFT .............................. 123 8.3 Review of QFT in dS space . 125 8.3.1 Coordinate Systems . 126 8.3.2 Geodesic Distance . 127 8.3.3 The Cut-o® Prescription . 128 8.3.4 Wave Function of the Universe . 130 8.3.5 Representations of the dSd Group . 132 8.4 Scalar Green Functions ............................ 133 8.4.1 Maximally Symmetric Bitensors . 133 8.4.2 Bulk Two-Point Function . 135 8.4.3 Bulk to Boundary Propagators: dS/AdS . 139 8.4.4 Boundary Two-point Function: dS/AdS . 140 8.5 General Structure of the Computation . 140 8.6 Models ..................................... 144 8.7 1-loop Computation: Scalar Fields with Cubic Interaction . 145 8.8 1-loop Computation: Scalar Fields with Derivative Coupling . 148 8.9 1-loop Computation: Spinor Field with Derivative Coupling . 150 8.10 Analysis Divergences ............................. 153 8.10.1 Three Massive Scalar Fields with Cubic Interaction in dSd . 153 8.10.2 Two Massive and One Massless ¯eld in dSd . 154 8.11 The Meaning of the Divergences . 155 8.12 Generalization to a Model with Gravity . 157 8.13 Appendix: Comparison with AdS . 159 8.13.1 Three Scalar Fields AdS . 159 8.14 Appendix: Spinor Green Functions . 161 8.14.1 Spinor Parallel Propagator . 161 8.14.2 Bulk Two-Point Function . 162 8.14.3 Bulk to Boundary Propagators: dS/AdS . 164 Bibliography 166 v List of Figures 4.1 Lightsheets .................................. 21 4.2 Caustic .................................... 25 7.1 Causal diamonds de¯ning one observer . 112 7.2 Causal diamonds de¯ning a pair of observers . 113 7.3 The maximal causal diamond . 114 7.4 Intersection of base spheres in 3D . 115 7.5 Intersection of base spheres in 2D . 116 7.6 Intersection of causal diamonds in 2D . 117 7.7 Detail of the the maximal causal diamond 1 . 118 7.8 Detail of the the maximal causal diamond 2 . 119 8.1 Penrose Diagram: Global coordinates . 129 8.2 Penrose Diagram: Flat coordinates . 129 8.3 Tree diagram ................................. 141 8.4 1-loop diagram ................................ 141 vi Abstract Quantum Gravity and Cosmology by Lorenzo Mannelli The main theme of this Thesis is the connection among Quantum Gravity and Cosmology. In the First Part (Chapters 1 to 5) I give an introduction to the Holographic Principle. The Second Part is a collection of my research work and it is articulated as follows. Chapter 7 is dedicated to analyze the renormalization properties of quantum ¯eld theories in de Sitter space. It is shown that only two of the maximally invariant vacuum states of free ¯elds lead to consistent perturbation expansions. Chapter 8 ¯rst present a complete quantum mechanical description of a ﬂat FRW universe with equation of state p = ½. Then show a detailed correspondence with our heuristic picture of such a universe as a dense black hole ﬂuid. In the end it is explained how features of the geometry are derived from purely quantum input. Chapter 9 study the problem of infrared renormalization of particle masses in de Sitter space. It is shown, in a toy model in which the graviton is replaced with a minimally coupled massless scalar ¯eld, that loop corrections to these masses are infrared (IR) divergent. It is argued that this implies anomalous dependence of masses on the cosmological constant, in a true theory of quantum gravity. To My Family, for their constant support and encouragement. viii Acknowledgments I want to thank my advisor Tom Banks ¯rst of all for showing me a new, more deep and intuitive way to think about physics. For his patience in explaining me the intricate subtleties of Quantum Field Theory and String Theory. And for the strength and dedication with which he guided me through these years of intense research. I would like to thank my other collaborator Willy Fischler for sharing with me his ideas and for the good time spent together. I want to express my gratitude to the other members of the High Energy group in Santa Cruz for all the pleasant discussions we had about science. A special thank goes to David Dorfan for his precious advices about life and physics. To all my friends in Santa Cruz and Santa Barbara where part of this work has been done, I want to thank you just for being my friends and for making the time I spent in California most enjoyable. I'm grateful to Alison Davies for her support and for making my life so special during this last year. Last, I want to thank my family for always encouraging me in what I was doing. My Father for communicating me the love for science, my Mother for always being present in the di±cult moments of my life. My Brother and Sister for their love and support. ix Part I The Holographic Principle 1 Chapter 1 Introduction The main theme of this Thesis is the connection among Quantum Gravity and Cosmology and it mostly contain the research work that I have been doing in this ¯eld. The quantization of gravity is probably the most outstanding unsolved problem in theoretical Physics. It has been studied for more than 70 years, nonetheless we haven't been able yet to ¯nd a complete formulation of a Quantum Theory of Gravity (QTG). The most promising candidate in this direction is String Theory. Even if we don't know what the QTG is going to be there are strong evidence that must be \Holographic". The word Holography in this contest refer to a property of the fundamental degrees of freedom of the theory. In a QTG these are not extensive (growing with volume) as we would expect from a local Field Theory description. They rather scale like the area of a surface. This surprising property has been ¯rst discovered studying Black Holes. In this case the degrees of freedom (dof) describing the black hole grow as the area A of 2 the event horizon. More precisely the dof are counted by the entropy which satisfy the relation A S = 4 in Planck units. The generalization and study of this property in a QTG has initially been carried on by t' Hooft and Susskind. More recently a completely covariant formulation of the \Holographic Principle" has been given by Bousso. String Theory being a QTG give us a description of the fundamental degrees of freedom of Nature. In the String Theory examples where has been explicitly possible to count the dof we have seen that they are Holographic in nature i.e. they scale as an area. The most remarkable among these examples is probably AdS/CFT.