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University of California Santa Cruz a Model of Spacetime UNIVERSITY OF CALIFORNIA SANTA CRUZ A MODEL OF SPACETIME EMERGENCE IN THE EARLY UNIVERSE A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in PHYSICS by Martin W. Tysanner September 2012 The Dissertation of Martin W. Tysanner is approved: Anthony Aguirre, Chair Michael Dine Stefano Profumo Tyrus Miller Vice Provost and Dean of Graduate Studies Copyright c by Martin W. Tysanner 2012 Table of Contents Abstract vi Dedication viii Acknowledgments ix 1 Introduction 1 1.1 Aims and overview . .1 1.2 Inflation and its difficulties . .2 1.2.1 Penrose's entropy argument . .3 1.2.2 Predictivity problem of eternal inflation . .5 1.3 Assessment . .8 1.3.1 Motivation for an emergence picture . 10 1.4 Overview of the Emergence Picture . 12 1.5 Thesis Plan . 19 1.5.1 Thesis outline . 19 1.5.2 How to read this thesis . 22 1.6 Notation and Conventions . 23 I Pre-Emergent Space 28 2 Foundation 29 2.1 Topological Space (Σ; TΣ).......................... 31 2.2 Irreducible Field ' on (Σ; d)......................... 40 2.2.1 Elementary oscillator field,! ~[M].................. 41 2.2.2 Intrinsic stochasticity of ' ...................... 44 2.2.3 Elementary dynamics of ' ...................... 47 2.2.4 Preferred scale . 56 3 Stochastic Processes and Calculus of ' 58 3.1 Stochastic Processes . 59 3.1.1 Stochastic processes and Brownian motion . 59 3.1.2 Spectral analysis of stochastic functions . 66 3.2 Stochastic Calculus . 72 iii 3.2.1 Stochastic differentials . 72 3.2.2 Stochastic integrals . 73 3.3 Differentiation of ' .............................. 76 3.4 Tangent Spaces on (Σ; d)........................... 79 3.5 `Surface' and `Volume' Integrals on (Σ; d)................. 81 3.5.1 Fundamental difficulties of integrals over regions . 82 3.5.2 Obtaining finite covers of balls and spheres on (Σ; d)....... 84 3.5.3 Integrating over uncountably many directions . 85 3.5.4 Volumes and packing measures in M ................ 87 3.5.5 Uncertainty in ' on surfaces and volumes . 88 3.6 Differential Operators on Infinitely Varying ' on M ........... 89 3.6.1 The derived field Φ . 90 3.6.2 Gradient of '(x) on M ........................ 92 3.6.3 Divergence of Φ(x) on M ...................... 92 4 Emergence of Dynamics 94 4.1 The Two Regimes of ' on Mt ........................ 95 4.2 The Stochastic Regime of ' ......................... 97 4.2.1 Characterizing the stochastic regime . 97 4.2.2 Statistics of neighborhoods . 101 4.3 Breaking Scale Invariance . 102 4.3.1 Transient preferred scale . 103 4.3.2 Fluctuations leading to a preferred scale . 107 4.4 Physical Observability of ' Motions . 107 4.4.1 Wave propagation speed . 108 4.4.2 Physical lengths and times . 114 4.4.3 Concretely connecting '[Mt] to physical spacetime . 115 4.4.4 Physical basis for the smoothing intervals ", τ ........... 117 4.5 Field Equation for ' ............................. 119 4.5.1 Generalizing from local dynamics . 120 4.5.2 Field equation in 1+1 dimensions . 122 4.5.3 Generalization to n + 1 dimensions . 127 4.5.4 The need for path integration . 128 II Emergence and Post-Emergent Spacetime 130 5 Composite Field Theory 131 5.1 Composite Theory in 3+1 Dimensions . 132 5.1.1 The quantum and classical-stochastic sectors . 133 5.1.2 Schr¨odinger representation of field theory . 135 5.1.3 Hall-Reginatto ensemble formalism for fields . 138 5.1.4 The composite free field theory with ensembles . 142 5.1.5 Interactions between quantum fields and classical ' ....... 144 5.1.6 Role of the Hall-Reginatto formalism . 146 iv 5.2 Effective Quantization of ' ......................... 147 5.2.1 Origin of effective quantization . 148 5.2.2 ' as a quantum field . 150 5.2.3 Localization of field quanta . 152 5.3 Origin of Quantum Fluctuations of ' .................... 157 5.4 Coupling between ' and Quantum Fields . 160 5.4.1 The model interaction . 161 5.4.2 The quantum picture . 162 5.4.3 The classical picture . 163 5.4.4 Toy model for coupling to ' ..................... 166 6 Emergent Spacetime and Gravity 167 6.1 Manifold emergence . 167 6.1.1 Origin of n spatial dimensions . 167 6.1.2 Equipartition in the ' vacuum . 169 6.1.3 Lorentz symmetry . 172 6.1.4 Metric signature . 176 6.2 Reproducing General Relativity . 178 6.2.1 Computing gµν in general relativity . 178 6.2.2 More general emergent metrics than ηµν .............. 179 6.2.3 Computing gµν in the emergence picture . 181 6.2.4 Manifestation of matter fields . 182 6.3 Cosmogenesis and Inflation-like Era . 183 6.3.1 Initial fluctuation . 183 6.3.2 Initial manifestation of quantum fields . 185 6.3.3 Exponential expansion . 188 A Hamilton-Jacobi Field Theory 190 Bibliography 192 v Abstract A Model of Spacetime Emergence in the Early Universe by Martin W. Tysanner This thesis proposes and develops much of the groundwork for a model of emergent physics, posited to describe the initial condition and early evolution of a universe. Two different considerations motivate the model. First, the spacetime manifold underlying general relativity and quantum theory is a complex object with much structure, but its origin is unexplained by the standard picture. Second, it is argued, the usual assump- tion of the preexistence of this manifold leads to possibly intractable theoretical (not observational) difficulties with the usual cosmological inflation idea. Consistent with both considerations, the assumption of a manifold that precedes a big bang cosmology is dropped; instead, a spacetime manifold with metric, Lorentz symmetry, and man- ifestation of standard quantum fields propagating on the spacetime all emerge in the model from a simpler, statistically scale invariant underlying structure, driven by an inflation-like process. The basic structural components of the model are a stochastic (not quantum or classical) scalar field on a general metric space, plus a collection of quantum fields that supply the matter content once spacetime begins to emerge. Importantly, stan- dard quantum fields cannot be defined on the pre-emergent space; this is addressed by assuming quantum theory exists a priori, and then postulating that quantum fields can begin to manifest once an approximate spacetime has emerged. Atypical fluctuations in the scalar field transiently break the statistical scale invariance in a localized region of the general metric space; a very small subset have field configurations of approximate spacetimes which can potentially evolve into an initial condition for a universe. Space- time structure and geometry then arise from the dependence of propagation speeds and spatial/temporal distances on variations in the scalar field; these variations are seeded by the matter (quantum) fields. The thesis develops the mathematics of the basic components of the model in some detail, outlines a mechanism whereby scale invariance is broken and dimensionality vi is fixed, and develops processes and scenarios wherein variations in the scalar field can lead to spacetime geometry in an inflation-like process. The resulting picture of spacetime is then compared with that of general relativity. vii To Anne, Valerie and Isaac, whose immeasurable patience, sacrifice and support made this endeavor possible viii Acknowledgments I want to express my deep appreciation to my advisor, Anthony Aguirre, for giving me the opportunity to develop and pursue the research program leading to this thesis. Anthony's patience, guidance and willingness to listen to and constructively comment on unfamiliar ideas are just some of his qualities that made working with him a real pleasure and a valuable learning experience. I also thank Michael Hall and Marcel Reginatto for fruitful discussions in better understanding their formalism for mixed quantum/classical systems and offering ideas for how I might apply it to my own work. ix Chapter 1 Introduction 1.1 Aims and overview This thesis will attempt to lay the groundwork for a model of emergent physics, in which a spacetime manifold with metric, as well as standard fields propagating on it, emerge from a simple and general underlying structure. This emergence is viewed both in terms of emergent laws of physics governing a restricted range of length and energy scales, and also as a process of emergence, through an inflation-like phase in the early universe. Indeed, the predictive success and theoretical challenges of inflation form a major motivation for the ideas that will be explored. The Big Bang cosmology plus inflation, when married with general relativity (GR) and the Standard Model of particle physics, form an extremely successful and well- tested description of laboratory physics as well as the physics of the observable universe and its evolution from very early times. However, there are many open questions in pushing beyond these well-described regimes. One of particular interest here is that of cosmological initial conditions, which as described below appear extremely `special' even with the inclusion of inflation. This thesis will approach the problem of unlikely initial conditions by treating the observable universe as a part of a larger system, but one that arises via an emergence process that occurs during an inflation-like era. Because this model will require a significant departure from the typical con- ceptual and mathematical toolkit of high-energy physics and cosmology, this thesis will pay particular attention to motivation of the ideas, both from perceived deficiencies in current models, as well as the mathematical and physical self-consistency and simplicity 1 of the proposed alternatives. This introduction gives, in the next section, a review of the inflationary model of early universe cosmology, and also points out open problems in that view. This motivates the emergence scenario, which the next section sketches in broad form. Following that is an outline of the content of the thesis and the basic ideas it puts forth.
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