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Dynamics of Causal Sets Dynamics of Causal Sets by David P. Rideout B.A.E., Georgia Institute of Technology, Atlanta, GA, 1992 M.S., Syracuse University, 1995 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate School of Syracuse University May 2001 Approved Professor Rafael D. Sorkin Date The Causal Set approach to quantum gravity asserts that spacetime, at its smallest length scale, has a discrete structure. This discrete structure takes the form of a locally finite order relation, where the order, corresponding with the macroscopic notion of spacetime causality, is taken to be a fundamental aspect of nature. After an introduction to the Causal Set approach, this thesis considers a simple toy dynamics for causal sets. Numerical simulations of the model provide evidence for the existence of a continuum limit. While studying this toy dynamics, a picture arises of how the dynamics can be generalized in such a way that the theory could hope to produce more physically realistic causal sets. By thinking in terms of a stochastic growth process, and positing some fundamental principles, we are led almost uniquely to a family of dynamical laws (stochastic processes) parameterized by a countable sequence of coupling constants. This result is quite promising in that we now know how to speak of dynamics for a theory with discrete time. In addition, these dynamics can be expressed in terms of state models of Ising spins living on the relations of the causal set, which indicates how non-gravitational matter may arise from the theory without having to be built in at the fundamental level. These results are encouraging in that there exists a natural way to transform this classical theory, which is expressed in terms of a probability measure, to a quantum theory, expressed in terms of a quantum measure. A sketch as to how one might proceed in doing this is provided. Thus there is good reason to expect that Causal Sets are close to providing a background independent theory of quantum gravity. iii Acknowledgements I would like to express deep gratitude for my advisor, Rafael Sorkin, for his patient teaching and support throughout my graduate career. His depth of insight into fundamental issues in physics is extremely helpful and illuminating. I also would like to thank Peter Saulson, for acting as my advisor at a critical stage in my graduate career, providing much needed support and encouragement. Let me also express my appreciation to Fatma Husein for some very illuminating conversations, Scott Klasky for teaching me how to write efficient code, Asif Qamar for introducing me to GNU/Linux at a critical time, and Saul Teukolsky for kind hospitality at Cornell. Thank you to my fellow students who helped me in many ways throughout my time at Syracuse — Bill Kahl, Eric Gregory, Asif Qamar, Rob Salgado, Arshad Momen, Jim Javor, and many others. In addition, let me thank Lawrence Lyon and Nelson Mead, for standing with me in prayer, and especially Lawrence for many discussions both about the physics and the larger perspective of life. Let me also thank Wayne Lytle, for teaching me about C++, and the rest of the people of Covenant Love Community Church/School, especially Kathy Mead, for their support in prayer, meals, babysitting, raw labor, and more, while I was finishing this work. Finally, I would like to thank my family, for their endless patience, and especially my wife, Yvonne Tempel, for her love, exemplary patience, continuous pressure, and reminders of a broader perspective. My daughter Kendra deserves special thanks for her encouraging hugs. This is dedicated to the memory of my father, Donald C. Rideout, whose support in the beginning made all of this possible. Contents 1 Introduction to Causal Sets 1 1.1Quantumgravityingeneral........................... 1 1.2KinematicsofCausalSets............................ 3 1.2.1 MathematicalDefinitions........................ 4 1.2.2 Faithful Embedding — Correspondence with the Continuum . 6 1.2.3 CausalSetDimension.......................... 8 1.2.4 Geometry................................. 13 1.2.5 ClosedTimelikeCurves......................... 16 1.2.6 CoarsegrainingandScaledependenttopology............ 17 1.3DynamicsforCausalSets............................ 19 1.3.1 Generalcovariance............................ 19 1.3.2 “Manifoldness”.............................. 21 1.3.3 Locality.................................. 22 2 Investigation of Transitive Percolation Dynamics 24 2.1Introduction.................................... 24 2.2Features...................................... 29 2.2.1 May resemble continuum spacetime . ............. 29 2.2.2 Homogeneous............................... 29 2.2.3 Timereversalinvariance......................... 30 2.2.4 Existenceofacontinuumlimit..................... 30 2.2.5 Originarytransitivepercolation..................... 30 2.2.6 Suggestivelargescalecosmology.................... 31 2.2.7 Cosmologicalrenormalization...................... 32 2.2.8 Phasetransitionsintheearlyuniverse................. 33 2.2.9 Diffusion-like model . .................... 34 2.2.10Gibbsiandistribution.......................... 35 2.3ContinuumLimit................................. 35 2.3.1 The critical point at p =0,N = ................... 38 ∞ 2.3.2 The large scale effective theory . .................... 39 2.3.3 Evidencefromsimulations........................ 41 2.3.4 ConcludingComments.......................... 60 2.4Scaling....................................... 66 3 Classical Dynamics of Sequential Growth 67 3.1Introduction.................................... 67 3.1.1 Sequentialgrowth............................ 69 iv CONTENTS v 3.1.2 Someexamples.............................. 72 3.2Physicalrequirementsonthedynamics..................... 73 3.2.1 Theconditionofinternaltemporality................. 75 3.2.2 Theconditionofdiscretegeneralcovariance.............. 76 3.2.3 TheBellcausalitycondition....................... 77 3.2.4 TheMarkovsumrule.......................... 79 3.3 The general form of the transition probabilities . ............. 82 3.3.1 Counting the free parameters . .................... 82 3.3.2 The general transition probability in closed form . ...... 85 3.3.3 Inequalities................................ 89 3.3.4 Proofthatthisdynamicsobeysthephysicalrequirements...... 92 3.3.5 Samplecosmologies........................... 95 3.4Originarydynamics................................ 98 3.5Thestochasticgrowthprocessassuch..................... 99 3.6 Two Ising-like state-models . .................... 101 3.7FurtherWork................................... 105 4 Conclusions 108 4.1Summary..................................... 108 4.2QuantumDynamics............................... 110 A Consistency of physical conditions 113 Chapter 1 Introduction to Causal Sets 1.1 Quantum gravity in general The quest for a theory of quantum gravity arises from the fundamental inconsistencies of quantum mechanics and general relativity. Due to the philosophical differences between the two theories, it appears that a successful marriage of the two will occur only with a radical reformulation of both theories. In thinking about how to approach such a reformulation, one must decide which aspects of nature are fundamental, and which arise as “emergent” structures [?]. There are strong indications that nature, at its smallest length scale, has a discrete structure. The ultraviolet divergences of quantum field theory and the singularities of general relativity are two examples. The infinite “entanglement entropy” of a black hole in semi-classical gravity is another indication. (In fact, a continuum is not experimentally verifiable by any finite experiment, even in principle.) In the case of causal sets, two aspects are regarded as fundamental — discreteness and causality. The choice of causality as a fundamental notion is partly aesthetic, and partly due 1 2 CHAPTER 1. INTRODUCTION TO CAUSAL SETS to the great success one has in recovering other aspects of spacetime geometry from a causal order. The Causal Set program postulates that spacetime is a macroscopic approximation to an underlying discrete causal order. The other familiar properties of a spacetime manifold, such as its metrical geometry and Lorentzian signature, arise as “emergent” properties of the underlying discrete order. Mituo Taketani, a Japanese physicist and philosopher of science, has described the process of physical theory construction in terms of three distinct stages [?]. The three stages repeat cyclically, except after each cycle the theory is understood at a deeper level. The three stages, using physics terminology, may be called the phenomenological, kinematical, and dynamical. The phenomenological stage concerns itself with what physical phenomena the theory seeks to address. For what observational results should the theory provide explanation? For the Copernican model of the solar system, an example of the phenomena would be the retrograde motion of the planets with respect to the fixed background stars. In the kinematical stage one decides in what language the theory will be expressed. What are the basic elements of the theory, the “substance”, and how do they interrelate? For the case of general relativity, one chooses a manifold with a Lorentzian metric. This stage is very important, as in it one decides “what really exists” in nature. It determines the mathematical and philosophical construct on which the theory will be based. Once a language is set in place, and the basic constructs
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