Foundations of Quantum Cosmology AAS Editor in Chief Ethan Vishniac, Johns Hopkins University, Maryland, USA About the program: AAS-IOP Astronomy ebooks is the official book program of the American Astronomical Society (AAS), and aims to sharein depththe mostfascinating areas of astronomy, astrophysics, solar physics and planetary science. The program includes publications in the following topics: Books in the program range in level from short introductory texts on fast-moving areas,graduateandupper-levelundergraduatetextbooks,researchmonographsand practical handbooks. For a complete list of published and forthcoming titles, please visit iopscience.org/ books/aas. About the American Astronomical Society The American Astronomical Society (aas.org), established 1899, is the major organization of professional astronomers in North America. The membership (∼7,000) also includes physicists, mathematicians, geologists, engineers and others whose research interests lie within the broad spectrum of subjects now comprising the contemporary astronomical sciences. The mission of the Society is to enhance and share humanity’s scientific understanding of the universe. Editorial Advisory Board Steve Kawaler Bradley M. Peterson Iowa State University, USA The Ohio State University / Space Telescope Science Institute, USA Ethan Vishniac Johns Hopkins University, USA Scott Kenyon Smithsonian Astrophysical Observatory, Dieter Hartmann USA Clemson University, USA Daniel Savin Piet Martens Columbia University, USA Georgia State University, USA Stacy Palen Dawn Gelino Weber State University, USA NASA Exoplanet Science Institute, Caltech, USA Jason Barnes University of Idaho, USA Joan Najita National Optical Astronomy James Cordes Observatory, USA Cornell University, USA Foundations of Quantum Cosmology Martin Bojowald Department of Physics, 104 Davey Lab, University Park, PA 16802, USA IOP Publishing, Bristol, UK ªIOPPublishingLtd2020 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording orotherwise,withoutthepriorpermissionofthepublisher,orasexpresslypermittedbylawor undertermsagreedwiththeappropriaterightsorganization.Multiplecopyingispermittedin accordancewiththetermsoflicencesissuedbytheCopyrightLicensingAgency,theCopyright ClearanceCentreandotherreproductionrightsorganizations. PermissiontomakeuseofIOPPublishingcontentotherthanassetoutabovemaybesought [email protected]. MartinBojowaldhasassertedhisrighttobeidentifiedastheauthorofthisworkinaccordance withsections77and78oftheCopyright,DesignsandPatentsAct1988. ISBN 978-0-7503-2460-1(ebook) ISBN 978-0-7503-2458-8(print) ISBN 978-0-7503-2461-8(myPrint) ISBN 978-0-7503-2459-5(mobi) DOI 10.1088/2514-3433/ab9c98 Version:20200901 AAS–IOPAstronomy ISSN2514-3433(online) ISSN2515-141X(print) BritishLibraryCataloguing-in-PublicationData:Acataloguerecordforthisbookisavailable fromtheBritishLibrary. PublishedbyIOPPublishing,whollyownedbyTheInstituteofPhysics,London IOPPublishing,TempleCircus,TempleWay,Bristol,BS16HG,UK USOffice:IOPPublishing,Inc.,190NorthIndependenceMallWest,Suite601,Philadelphia, PA19106,USA Contents Preamble xi Author biography xvi 1 The Universe on Large and Small Scales 1-1 1.1 Homogeneous and Isotropic Cosmology 1-2 1.2 Hamiltonian 1-7 1.3 Geometry 1-8 1.3.1 The Line Element 1-9 1.3.2 Isotropic Line Elements 1-11 1.3.3 Anisotropic Models 1-14 1.4 Perturbative Inhomogeneity 1-19 1.4.1 Scalar, Vector, and Tensor Perturbations 1-20 1.4.2 Coordinate Dependence 1-21 1.4.3 Evolution Equations 1-24 1.4.4 Inflationary Structure Formation 1-26 1.5 Spherically Symmetric Models 1-33 1.5.1 Line Element and Equations of Motion 1-33 1.5.2 Lemaître–Tolman–Bondi Models 1-36 1.5.3 Canonical Formulation and Static Solutions 1-38 1.5.4 Non-rotating Black Holes 1-41 1.5.5 Horizons 1-44 1.6 Full Inhomogeneity 1-46 References 1-49 2 Covariance 2-1 2.1 Lagrangian and Hamiltonian Formulations 2-1 2.1.1 Equivalence 2-2 2.1.2 Constrained Theories Subject to Gauge Symmetries 2-3 2.1.3 Spacetime Theories 2-4 2.2 Action Principles 2-5 2.2.1 Coordinate Independence 2-6 2.2.2 Higher-curvature Actions 2-17 2.2.3 The Ostrogradsky Problem 2-20 2.2.4 R2 Gravity 2-22 vii FoundationsofQuantumCosmology 2.2.5 Scalar-tensor Theories 2-29 2.2.6 Horǎ va–Lifshitz and Einstein Aether Theories 2-38 2.3 Canonical Formulation 2-39 2.3.1 Spacetime in Special Relativity 2-39 2.3.2 Symmetries 2-41 2.3.3 Spacetime in General Relativity 2-43 2.3.4 Spherical Symmetry 2-47 2.3.5 Perturbative Cosmological Inhomogeneity 2-59 2.3.6 Modified Canonical Gravity 2-70 References 2-73 3 Quantum Corrections 3-1 3.1 Algebra 3-1 3.1.1 Commutator 3-1 3.1.2 Basic Operators 3-3 3.1.3 Representations 3-4 3.1.4 States 3-7 3.2 Measurements 3-10 3.2.1 Moments of a State 3-10 3.2.2 Eigenvalues 3-11 3.2.3 The Divine Dividing Line 3-13 3.2.4 Decoherence 3-14 3.3 Dynamics 3-17 3.3.1 Heisenberg’s and Ehrenfest’s equations 3-17 3.3.2 Harmonic Oscillator 3-20 3.3.3 Effective Potentials 3-23 3.3.4 Low-energy Effective Potential 3-29 3.3.5 States at Non-zero Temperature 3-32 3.3.6 Time-correlation Functions and the Fluctuation–Dissipation 3-36 Theorem 3.4 Transition Amplitude and Path Integrals 3-39 3.4.1 Transition Amplitudes 3-40 3.4.2 Picard–Lefshetz Theory 3-42 3.4.3 Path Integrals 3-47 3.4.4 Relevance 3-52 References 3-53 viii FoundationsofQuantumCosmology 4 Minisuperspace Models 4-1 4.1 Volume Scaling 4-2 4.1.1 Scalar Field 4-3 4.1.2 Quantum Corrections 4-4 4.1.3 Infrared Renormalization 4-5 4.2 Problem of Time 4-7 4.2.1 Hamiltonian Constraints 4-7 4.2.2 Deparameterization 4-10 4.2.3 Examples 4-11 4.3 Algebras 4-14 4.3.1 Positive Volume 4-14 4.3.2 Discrete Volume 4-17 4.3.3 Loop Quantum Cosmology 4-18 4.3.4 Quantum Fluctuations 4-20 4.4 Representations 4-24 4.4.1 Differential and Difference equations 4-24 4.4.2 Initial Conditions 4-26 4.4.3 Instability 4-28 References 4-31 5 Quantum Gravity 5-1 5.1 Classification and Comparison 5-3 5.2 Approaches 5-5 5.2.1 Perturbative Quantum Gravity as an Effective Theory 5-5 5.2.2 Asymptotic Safety 5-6 5.2.3 Causal Dynamical Triangulations 5-8 5.2.4 Non-commutative, Non-associative, 5-28 and Fractional Geometries 5.2.5 String Theory 5-34 5.2.6 Canonical Quantum Gravity 5-57 5.3 Appreciation 5-74 References 5-75 6 Quantum Cosmology of Inhomogeneous Spacetimes 6-1 6.1 Homogeneity in Inhomogeneous Models 6-2 6.1.1 Schwarzschild Interior 6-3 6.1.2 Timelike Homogeneity 6-6 6.1.3 Separate Quantization of Background and Perturbations 6-12 ix FoundationsofQuantumCosmology 6.2 Consistent Spacetime Structures 6-14 6.2.1 Closed Brackets for Spherically Symmetric Spacetimes 6-14 6.2.2 Effective Line Element 6-19 6.2.3 Implications of Signature Change 6-22 6.2.4 Perturbative Inhomogeneity 6-26 6.3 Cosmology at Large Curvature 6-29 6.3.1 Infrared Renormalization 6-30 6.3.2 The Trans-Planckian Problem 6-32 6.3.3 Speculative Proposals 6-33 References 6-38 x