Challenging Routes In Quantum Cosmology

"Quantum cosmology has gradually emerged as the focus of devoted research, mostly within the second half of last century. As we entered the 21st century, the subject is still very much alive. The outcome of results and templates for investigation have been enlarged, some very recent and fascinating. Hence this book, where the authors bequeath some of their views, as they believe this current century is the one where quantum cosmology will be fully accomplished. Though some aspects are not discussed (namely, supersymmetry or loop structures), there are perhaps a set of challenges that in the authors' opinion remain, some since the dawn of quantum mechanics and applications to cosmology. Others could have been selected, at the readers' discretion and opinion. The authors put herewith a chart and directions to explore, some of which they have worked on or aimed to work more, in the twilight of their current efforts. Their confidence is that someone will follow in their trails, venturing in discovering the proper answer, by being able to formulate the right questions beforehand. The authors' shared foresight is that such discoveries, from those formulations, will be attained upon endorsing the routes within the challenges herewith indicated"-- Contents Preface Acknowledgments List of Figures List of Tables List of Symbols A Chart 1. Introduction 2. A Readers' Digest Settings 3. Foliation of Space-Time and the Geometry of Closed 3-Manifolds 3.1 Hypersurfaces 3.1.1 Extrinsic curvature 3.1.2 Gauss and Codazzi–Mainardi equations 3.2 Spacelike Foliations 3.3 Spacelike Hypersurfaces and the Gauss Theorem 3.4 Geometry of Closed 3-Manifolds 3.4.1 Geometric structures 4. Hamiltonian Formulation of General Relativity 4.1 3+1 Analysis of Einstein–Hilbert Action Functional 4.2 Matter Coupling 4.3 General Relativity as a Gauge Theory 4.4 Dirac Observables 4.5 Superspace 4.6 Minisuperspace 4.6.1 Jacobi's action and the geodesic motion 4.6.2 Conformal diagrams in minisuperspace 5. Minisuperspace Quantization and the Time Problem 5.1 Superspace Canonical Quantization 5.2 Dirac Quantization and the Wheeler–DeWitt Equation 5.3 Minisuperspace Dirac Quantization and the Wheeler–DeWitt Equation 5.4 Unitary Evolution in Quantum Cosmology 5.4.1 First category: Time before quantization 5.4.1.1 The internal Schrödinger interpretation 5.4.1.2 Matter clocks and reference uids 5.4.1.3 Unimodular gravity 5.4.2 Second category 5.4.2.1 The Klein–Gordon interpretation 5.4.2.2 Third quantization 5.4.2.3 The semi-classical interpretation 5.4.3 Third category 5.4.3.1 The naŁ ve Schrödinger interpretation 5.4.3.2 The conditional probability interpretation 5.4.3.3 Consistent (decoherent) histories approach 5.4.3.4 The frozen time formalism (evolving constants of motion) 5.5 Deterministic Time Evolution 5.5.0.1 The de Broglie–Bohm (pilot wave) interpretation 5.5.0.2 The complex de Broglie–Bohm interpretation Challenges 6. Solutions: Multiplicity and Degeneracy 6.1 Context 6.2 Hamiltonian Framework 6.2.1 WKB approach 6.2.2 The "Tunneling" condition 6.3 Path Integral Framework 6.3.1 The "no-boundary" condition 6.3.2 Contour difficulties 6.3.3 The "Tunneling" condition 6.3.4 Picard–Lefschetz methodology 6.4 Probing (Beyond) Minisuperspace 6.4.1 S-expansion 6.4.2 (Classical) breakdown 6.5 Outlook 6.5.1 Predictions 6.5.2 Supersymmetry and Picard–Lefschetz 7. Fractional Quantum Cosmology 7.1 Framework 7.1.1 Historical (Introduction) 7.1.2 Fractional Schrödinger equation 7.1.2.1 The case of Hα = 0 7.1.2.2 Harmonic oscillator and beyond 7.1.2.3 Tunneling 7.2 Fractional Quantum Cosmology 7.2.1 Feynman's path integral 7.2.2 Fractional Wheeler–DeWitt equation 7.2.3 Space-fractional quantum cosmology 7.3 Time-Fractional Quantum Cosmology 7.4 Fractional Quantization of a Schwarzchild Black Hole 7.4.1 Minisuperspace quantization of Schwarzschild black hole 7.4.2 Fractional modifications and Schwarzschild thermodynamics 7.5 Fractal Quantum Cosmology 7.6 Quantum Cosmology with Variable Spatial Dimensions 7.7 Outlook 8. Quantum Chaos and Quantum Cosmology 8.1 Context 8.2 Classical Framework 8.2.1 Dynamical billiards 8.2.2 Chaos indicators 8.3 Quantum Framework 8.3.1 WKB and EBK method(s) 8.3.2 Quantum billiards 8.3.3 Trace formula 8.3.4 Quantum chaology 8.4 Quantum Cosmology Framework 8.4.1 Minisuperspace case study 8.4.2 WKB and quantum cosmology 8.5 Outlook 9. Boundary Conditions from Algebraic Criteria 9.1 Context 9.2 Quantum Solutions for Three FLRW Case Studies 9.2.1 Quantum FLRW with uid: A simple case study 9.2.2 Quantum FLRW with conformal scalar eld 9.2.3 Quantum FLRW with a non-minimal scalar eld 9.3 Boundary Conditions from Algebraic Criteria 9.3.1 FLRW quantum cosmology with a perfect uid 9.3.2 Conformal FLRW quantum cosmology 9.3.3 FLRW non-minimally coupled to a scalar eld 9.4 Outlook 10. Quantum Groups and Quantum Cosmology 10.1 Essential Tools 10.2 An Application: The Heisenberg–Weyl q-Algebra 10.2.1 Heisenberg–Weyl q-algebra at the root of unity 10.2.2 Realization of Uq(su(2)) 10.2.3 Realization of Uq(su(1; 1)) 10.3 Cosmological Framework 10.3.1 Quantum groups and minisuperspace states 10.3.2 Quantum deformation of the gravitational sector 10.3.3 Another application: The nature of Λ 10.3.4 Holography and quantum deformation 10.4 Outlook Bibliography Index Aims to convey a perspective of what quantum cosmology is and how it can be a relevant field of research in the 21st century. This book provides a series of challenges as the means of forwarding some of the main elements of quantum cosmology. It presents reasonable amount of technical features in a compact appendix.