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Open Dissertation-Final.Pdf The Pennsylvania State University The Graduate School The Eberly College of Science CORRELATIONS IN QUANTUM GRAVITY AND COSMOLOGY A Dissertation in Physics by Bekir Baytas © 2018 Bekir Baytas Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2018 The dissertation of Bekir Baytas was reviewed and approved∗ by the following: Sarah Shandera Assistant Professor of Physics Dissertation Advisor, Chair of Committee Eugenio Bianchi Assistant Professor of Physics Martin Bojowald Professor of Physics Donghui Jeong Assistant Professor of Astronomy and Astrophysics Nitin Samarth Professor of Physics Head of the Department of Physics ∗Signatures are on file in the Graduate School. ii Abstract We study what kind of implications and inferences one can deduce by studying correlations which are realized in various physical systems. In particular, this thesis focuses on specific correlations in systems that are considered in quantum gravity (loop quantum gravity) and cosmology. In loop quantum gravity, a spin-network basis state, nodes of the graph describe un-entangled quantum regions of space, quantum polyhedra. We introduce Bell- network states and study correlations of quantum polyhedra in a dipole, a pentagram and a generic graph. We find that vector geometries, structures with neighboring polyhedra having adjacent faces glued back-to-back, arise from Bell-network states. The results present show clearly the role that entanglement plays in the gluing of neighboring quantum regions of space. We introduce a discrete quantum spin system in which canonical effective methods for background independent theories of quantum gravity can be tested with promising results. In particular, features of interacting dynamics are analyzed with an emphasis on homogeneous configurations and the dynamical building- up and stability of long-range correlations. We conclude that an analysis of the building-up of long-range correlations in discrete systems requires non-perturbative solutions of the dynamical equations. For the early universe cosmology part of the thesis, we present examples of non-Gaussian statistics that can induce bispectra matching local and non-local (including equilateral) templates in biased sub-volumes. We find cases where the biasing from coupling to long wavelength modes affects only the power spectrum, only the bispectrum or both. iii Table of Contents List of Figures viii List of Tables xii Acknowledgments xiii Chapter 1 Perspectives1 Chapter 2 Fundamentals of correlations in quantum systems4 2.1 Introduction............................... 4 2.2 Building blocks of quantum mechanics ................ 5 2.2.1 States and Dynamics...................... 6 2.2.2 Observables and Measurement................. 9 1 2.2.3 Qubit and spin- 2 ........................ 10 2.3 Quantum bipartite systems and quantum correlations . 11 2.3.1 Quantum entanglement..................... 12 2.3.2 Quantum mutual information and correlations . 14 2.3.3 Two-qubit system and entanglement ............. 15 2.4 Quantum spin chains.......................... 17 2.4.1 Heisenberg (XXX) spin chain ................. 17 2.5 Quantum geometry of space...................... 20 2.5.1 Quantum tetrahedron ..................... 21 2.6 Quantum fields in Minkowski spacetime................ 25 2.6.1 Free scalar fields in Minkowski vacuum............ 25 Chapter 3 Gluing polyhedra with entanglement in loop quantum gravity 29 3.1 Introduction............................... 30 iv 3.2 Classical geometries on a graph Γ ................... 35 3.2.1 Hierarchy of discrete geometries................ 35 3.2.2 Twisted geometries....................... 37 3.2.3 Vector geometries........................ 38 3.2.4 Regge geometries........................ 40 3.3 Quantum polyhedron and Heisenberg uncertainty relations . 41 3.4 Quantum geometries on a graph Γ ................... 44 3.4.1 Quantum twisted geometries.................. 45 3.4.2 Quantum vector geometries .................. 47 3.4.3 Quantum shape-matching conditions ............. 54 3.4.4 Bell states on the dipole graph................. 57 3.4.4.1 Bell states on Γ2 ................... 58 3.4.4.2 Bell states at fixed spins j` on Γ2 . 61 3.4.5 Bell states on the pentagram.................. 63 Chapter 4 Minisuperspace models of quantum spin systems 75 4.1 Introduction............................... 76 4.2 Canonical effective methods ...................... 77 4.3 The quantum spin chain model .................... 79 4.4 Minisuperspace models......................... 84 4.4.1 A minimal minisuperspace model............... 84 4.4.2 A condensate model ...................... 85 4.4.3 Two interacting minisuperspace models............ 86 4.4.4 Effective equations and potentials............... 90 4.4.4.1 Minimal minisuperspace model........... 91 4.4.4.2 Interacting minisuperspace models ......... 93 4.4.4.3 Condensate model .................. 94 4.5 Continuum theories........................... 95 4.5.1 Continuum models ....................... 96 4.5.1.1 Minimal model.................... 96 4.5.1.2 Interacting minisuperspace models ......... 97 4.6 Effective analysis of the discrete theory................ 99 4.6.1 Effective dynamics ....................... 99 4.6.2 Dynamical stability for small number of nodes . 102 4.6.2.1 Single-node graph ..................103 4.6.2.2 Two-node graph and beyond . 106 4.7 Possible implications for quantum cosmology . 109 v Chapter 5 Super cosmic variance from non-Gaussian statistics 111 5.1 Introduction...............................111 5.2 Super cosmic variance .........................113 5.3 Primordial non-Gaussianity ......................114 5.4 Super cosmic variance from general type non-Gaussian fields . 116 5.4.1 The model............................117 5.4.2 Quadratic Terms ........................120 5.4.2.1 Recovering the standard bispectral templates . 122 5.4.2.2 The long-short wavelength split . 124 5.4.2.3 The field observed in biased sub-volumes . 125 5.4.3 Cubic Terms...........................129 5.4.3.1 Generation of cubic terms . 129 5.4.3.2 Constraints on cubic terms from their contributions to the power spectrum . 130 5.4.3.3 The field observed in biased sub-volumes . 131 5.4.3.4 Families of cubic terms from the super cosmic vari- ance point of view ..................134 5.5 Space of non-Gaussian fields with single-clock bispectra . 138 5.5.1 Introduction...........................138 5.5.2 Generating super cosmic variance free models . 140 5.5.3 Constructing a non-Gaussian field with an equilateral bis- pectrum.............................140 5.5.4 Obtaining the single-clock orthogonal template . 143 5.5.5 Comparison with previous work . 148 Chapter 6 Summary and Outlook 150 6.1 Entanglement and quantum spacetimes . 150 6.2 Discrete spin systems and canonical quantum gravity . 154 6.3 Non-local bispectra from super cosmic variance . 156 6.4 Shapes for super cosmic variance free from non-Gaussianities . 158 Appendix A Vector geometry for equal spins jab = j 160 Appendix B Entanglement entropy at fixed total area J 162 vi Appendix C Derivation of the Bell states on a pentagram graph 164 Appendix D Hessian of the action for 15j-symbol 167 Appendix E Shape-matched configurations for a 4-simplex in R3 171 E.1 Pachner move: 1-4 . 172 E.2 Pachner move: 2-3 . 174 Appendix F Cubic terms and the cubic kernel 176 F.1 Quadratic terms induced by a cubic term in biased sub-volumes . 178 F.2 Power spectrum from the cubic field Φ3 . 179 F.3 Special cubic kernels and its corresponding trispectra . 180 Appendix G Basis functions for bispectra 182 Bibliography 185 vii List of Figures 2.1 Graphical representation of a one dimensional isotropic Heisenberg (XXX) chain of spin-1/2 particles si with N sites connected by N-1 links.................................... 18 2.2 A picture for a spin-network graph consisting of 31 nodes [1]. 20 2.3 A description for a 3-valent node and its dual two-dimensional surface. 21 1 2.4 Tetrahedron formed by the edge-vectors ei. Thenormal n3 = e1 × 2A3 e2 is the normal, outward-pointing vector normalized to the area of the associated face............................ 23 3.1 Dipole graph. A pair of nodes connected by l links coloured by spins j`. .................................... 34 3.2 A local patch of a vector geometry. The glued faces have the same areas: Aa = Aa0 , Ab = Ab0 , Ac = Ac0 , and their normals are anti- parallel na = −na0 , nb = −nb0 , nc = −nc0 . The lengths of edges in juxtaposed triangles are not necessarily matched in a vector geometry. 39 3.3 Geometric meaning of the shape-matching conditions (3.12): The a internal angle αbc in the shaded triangle can be expressed in terms of the dihedral angles associated with the thick edges of either tetrahedra. ............................... 41 3.4 The set of L links. Each oriented link corresponds to an ordered pair ` = (s, t), where s(`) and t(`) are the source and target nodes at link l, respectively. The spin at the link ` is j`........... 49 3.5 Dipole graph Γ2. The source (s) and target (t) nodes are connected by four links ` = 1,..., 4. The graph is dual to the triangulation of the 3-sphere formed by two tetrahedra glued along their boundaries. A spin-network state is labelled by spins j` and intertwiners is, it. 58 viii 3.6 Pentagram graph Γ5. The graph is dual to a triangulation of the three-sphere with five tetrahedra. A spin-network state on Γ5 is labeled by five intertwiners ikn and ten spins j` attached to the nodes and links of the graph, respectively. The links are oriented according to t(`) < s(`). ........................ 63 3.7 Pachner move 1-4. A single tetrahedron is divided into a gluing of four tetrahedra by the introduction of a new vertex i in its interior. 69 3.8 Graphical representation of a 1-4 vector geometry in a 3d box gener- ated by using maximal tree method. Each colored point labels differ- ent node of the pentagram. The set of normals at each node satisfy the closure condition (3.96). Dashed black arrows stand for the anti- aligned normals which connect neighbor nodes: na(a+1) = −n(a+1)a, a = 1, ··· , 4.
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