APPROACHING THE PLANCK SCALE FROM A GENERALLY RELATIVISTIC POINT OF VIEW: A PHILOSOPHICAL APPRAISAL OF LOOP QUANTUM GRAVITY by Christian W¨uthrich M.Sc., Universit¨atBern, 1999 M.Phil., University of Cambridge, 2000 M.A., University of Pittsburgh, 2005 Submitted to the Graduate Faculty of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2006 UNIVERSITY OF PITTSBURGH FACULTY OF ARTS AND SCIENCES This dissertation was presented by Christian W¨uthrich It was defended on 5 July 2006 and approved by John Earman, Ph.D., History and Philosophy of Science Gordon Belot, Ph.D., Philosophy Jeremy Butterfield, Ph.D., Philosophy (University of Oxford) John Norton, Ph.D., History and Philosophy of Science Laura Ruetsche, Ph.D., Philosophy George Sparling, Ph.D., Mathematics Dissertation Director: John Earman, Ph.D., History and Philosophy of Science ii Copyright c by Christian W¨uthrich 2006 Chapter 2 of the present work is an adapted and extended version of Christian W¨uthrich, “To quantize or not to quantize: fact and folklore in quantum gravity,” Philosophy of Sci- ence, forthcoming. Copyright 2006 by The University of Chicago Press, reproduced with permission. iii APPROACHING THE PLANCK SCALE FROM A GENERALLY RELATIVISTIC POINT OF VIEW: A PHILOSOPHICAL APPRAISAL OF LOOP QUANTUM GRAVITY Christian W¨uthrich, PhD University of Pittsburgh, 2006 My dissertation studies the foundations of loop quantum gravity (LQG), a candidate for a quantum theory of gravity based on classical general relativity. At the outset, I discuss two—and I claim separate—questions: first, do we need a quantum theory of gravity at all; and second, if we do, does it follow that gravity should or even must be quantized? My evaluation of different arguments either way suggests that while no argument can be considered conclusive, there are strong indications that gravity should be quantized. LQG attempts a canonical quantization of general relativity and thereby provokes a foundational interest as it must take a stance on many technical issues tightly linked to the interpretation of general relativity. Most importantly, it codifies general relativity’s main innovation, the so-called background independence, in a formalism suitable for quantization. This codification pulls asunder what has been joined together in general relativity: space and time. It is thus a central issue whether or not general relativity’s four-dimensional structure can be retrieved in the alternative formalism and how it fares through the quantization process. I argue that the rightful four-dimensional spacetime structure can only be partially retrieved at the classical level. What happens at the quantum level is an entirely open issue. Known examples of classically singular behaviour which gets regularized by quantization evoke an admittedly pious hope that the singularities which notoriously plague the classical theory may be washed away by quantization. This work scrutinizes pronouncements claiming that the initial singularity of classical cosmological models vanishes in quantum cosmology based on LQG and concludes that these claims must be severely qualified. In particular, I explicate why casting the quantum cosmological models in terms of a deterministic temporal evolution fails to capture the concepts at work adequately. Finally, a scheme is developed of how the re-emergence of the smooth spacetime from the underlying discrete quantum structure could be understood. iv Keywords: Loop quantum gravity, Hamiltonian general relativity, canonical quantization, general covariance, diffeomorphism invariance, loop quantum cosmology, singularity, spacetime emergence. v TABLE OF CONTENTS PREFACE ......................................... ix 1.0 INTRODUCTION ................................ 1 1.1 Mapping quantum gravity .......................... 3 1.2 String theory .................................. 7 1.3 Synopsis of the chapters ........................... 11 2.0 WHY MUST GRAVITY BE QUANTIZED? ............... 15 2.1 The unificatory reflex in fundamental physics and its disunitist challengers 18 2.2 Why Quantize Gravity? ............................ 23 2.2.1 Early arguments pro quantization ................... 25 2.2.2 Non-commutative Spacetime Operators ................ 27 2.2.3 Inconsistency of Quantum-Classical Dynamics ............ 30 2.3 Why Not Quantize Gravity? ......................... 34 2.3.1 Sakharov’s Induced Gravity ...................... 35 2.3.2 Jacobson’s Gravitational Thermodynamics .............. 36 2.4 Conclusion of this Chapter .......................... 37 3.0 PRINCIPLES OF GENERAL RELATIVITY ............... 39 3.1 The origin of background independence ................... 39 3.2 General covariance .............................. 42 4.0 HAMILTONIAN GENERAL RELATIVITY ................ 55 4.1 GTR as a Hamiltonian system ........................ 55 4.2 Old and new variables ............................. 66 4.2.1 The ADM formalism .......................... 66 4.2.2 The Ashtekar-Barbero formalism ................... 70 4.3 The end of time? ............................... 73 4.4 Implementation of general covariance in canonical formulations ...... 79 5.0 A QUICK GUIDE TO LOOP QUANTUM GRAVITY ......... 87 5.1 The canonical quantization of Hamiltonian GTR .............. 88 5.2 Kinematical loop quantum gravity ...................... 91 5.3 Dynamical loop quantum gravity ....................... 94 5.3.1 Canonical dynamics: Wheeler-DeWitt evolution ........... 94 5.3.2 Covariant dynamics: spinfoams .................... 96 5.3.3 Canonical vs. covariant formulations ................. 100 vi 6.0 CLASSICAL SINGULARITIES AND THEIR FATE IN QUANTUM THEORY ...................................... 103 6.1 Singularities in the classical theory ...................... 103 6.2 Dissolving classical singularities ....................... 109 7.0 INTRODUCING LOOP QUANTUM COSMOLOGY .......... 115 7.1 Foundations of Loop Quantum Cosmology ................. 118 7.2 The disappearance of the curvature singularity ............... 121 7.3 Evolving through the big bang ........................ 128 8.0 EINSTEIN’S NEMESIS CONQUERED AT LAST? ........... 135 8.1 Physical curvature and epicycles of determinism .............. 135 8.2 The infidels: recent critiques of loop quantum cosmology ......... 146 8.2.1 Noui, Perez, and Vandersloot: separable but small physical Hilbert space ................................... 146 8.2.2 Cartin and Khanna: Bianchi I model ................. 148 8.2.3 Green and Unruh: closed model .................... 148 8.2.4 Brunnemann and Thiemann: cosmology or not? ........... 153 9.0 THE EMERGENCE OF SPACETIME ................... 158 9.1 Disappearance of spacetime .......................... 159 9.2 Re-emergence of spacetime .......................... 168 9.2.1 General considerations ......................... 168 9.2.2 Remarks towards a concrete implementation ............. 178 10.0 ALL’S WELL THAT ENDS WELL OR LOVE’S LABOUR’S LOST? 184 APPENDIX A. NOTATION AND CONVENTIONS .............. 186 A.1 General remarks ................................ 186 A.2 Index notation ................................. 186 A.3 Further notations and conventions ...................... 187 A.4 Acronyms ................................... 187 APPENDIX B. CHALLENGING THE SPACETIME STRUCTURALIST 188 APPENDIX C. CLASSICAL COSMOLOGY: FRIEDMANN-LEMAˆıTRE- ROBERTSON-WALKER SPACETIMES .................. 196 APPENDIX D. COMMUTATOR OF THE INVERSE SCALE FACTOR AND HAMILTONIAN CONSTRAINTS OPERATORS ........ 201 BIBLIOGRAPHY .................................... 203 vii LIST OF FIGURES 1 The pullback of a function by a diffeomorphism ................. 43 2 Foliations of spacetimes using a “flow in time” and a “flow in space.” ..... 63 3 Geometric interpretation of Kab. ......................... 67 4 A manifold of genus one. ............................. 82 5 Dependence of pure deformations on the embedding. .............. 84 6 A simple spin network state |si with two trivalent nodes. ........... 93 7 Deforming the (embedded) spin networks as shown does not change the state. 93 8 The basic action of the Hamiltonian on nodes of a spin network. ....... 97 9 Schematic example of how to calculate a transition amplitude W (sf , si). ... 98 −2 10 Kinematic curvature as function of µ (in units of `Pl ). ............. 126 11 Conformal diagram of a FLRW model in LQC. ................. 133 12 Recursive determination of the coefficients ψ(φ, µ). ............... 140 13 Two different ways of determining a third state from two given states. .... 141 viii PREFACE During the last three years, I have had the privilege of working on these exciting foundational problems in quantum gravity with many of our guild’s best people. I am very grateful to all those who contributed to this wonderful and inspiring experience. Having lived and worked in four different countries during that period has created a non-negligible amount of stress and exhaustion, but it has rewarded me in return with academic and personal experiences I would never want to miss. The greatest debt I owe to John Earman, my Doktorvater. He suggested the topic for my dissertation, initiated my visits to Waterloo and Marseille, acted as the chair of my PhD committee and as my mentor, and as such steered me through the whole process of compiling this dissertation with his great competence and experience. This project has greatly benefited from his constant encouragement, thoughtful advice, and insightful criticisms. John has mentored my