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Anderson E. The Problem of Time: Quantum Mechanics Versus General Relativity

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Anderson E. The Problem of Time: Quantum Mechanics Versus General Relativity
Springer International Publishing AG, 2017. — 917 p. — (Fundamental Theories of Physics 190) — ISBN 978-3-319-58846-9.
This book is a treatise on time and on background independence in physics. It first considers how time is conceived of in each accepted paradigm of physics: Newtonian, special relativity, quantum mechanics (QM) and general relativity (GR). Substantial differences are moreover uncovered between what is meant by time in QM and in GR. These differences jointly source the Problem of Time: Nine interlinked facets which arise upon attempting concurrent treatment of the QM and GR paradigms, as is required in particular for a background independent theory of quantum gravity. A sizeable proportion of current quantum gravity programs - e.g. geometrodynamical and loop quantum gravity approaches to quantum GR, quantum cosmology, supergravity and M-theory - are background independent in this sense. This book's foundational topic is thus furthermore of practical relevance in the ongoing development of quantum gravity programs.
This book shows moreover that eight of the nine facets of the Problem of Time already occur upon entertaining background independence in classical (rather than quantum) physics. By this development, and interpreting shape theory as modelling background independence, this book further establishes background independence as a field of study. Background independent mechanics, as well as minisuperspace (spatially homogeneous) models of GR and perturbations thereabout are used to illustrate these points. As hitherto formulated, the different facets of the Problem of Time greatly interfere with each others' attempted resolutions. This book explains how, none the less, a local resolution of the Problem of Time can be arrived at after various reconceptualizations of the facets and reformulations of their mathematical implementation. Self-contained appendices on mathematical methods for basic and foundational quantum gravity are included. Finally, this book outlines how supergravity is refreshingly different from GR as a realization of background independence, and what background independence entails at the topological level and beyond.
Contents
Time in Fundamental Physics
-Introduction: Conceptual Outline of Time
- Time, Space and Laws in Newtonian Mechanics
- Absolute Versus Relational Motion Debate
-Time, Space, Spacetime and Laws in Special Relativity
-Time and Ordinary Quantum Mechanics (QM)
-Quantum Field Theory (QFT)
-Time and Spacetime in General Relativity (GR)
-Dynamical Formulations of GR
-Classical-Level Background Independence and the Problem of Time. i. Time and Configuration
-Classical-Level Background Independence and the Problem of Time. ii. Spacetime and Its Interrelation with Space
- Quantum Gravity Programs
- Quantum-Level Background Independence and the Problem of Time
Classical Problem of Time
-Advanced Nomenclature for Facet Interference
-Configuration Spaces and Their Configurational Relationalism
-Temporal Relationalism (TR)
-Combining Temporal and Configurational Relationalisms
-Temporal Relationalism: More General Geometries
-Configurational Relationalism: Field Theory and GR’s Thin Sandwich
-Relationalism in Various Further Settings
-Other Tempus Ante Quantum Approaches
-Conformal Approach and Its York Time
- Matter Times
-Classical Machian Emergent Time
- Brackets, Constraints and Closure
-Taking Function Spaces Thereover: Beables and Observables
-Fully Timeless Approaches
-Spacetime Relationalism
- Classical Histories Theory
- Classical Machian Combined Approach
-Slightly Inhomogeneous Cosmology (SIC)
-Embeddings, Slices and Foliations
-Applications of Split Spacetime, Foliations and Deformations
-Spacetime Construction and Alternative Emergent Structures
-TRi Foliation (TRiFol)
-Epilogue II.A. Threading and Null Formulations
-Epilogue II.B. Global Validity and Global Problems of Time
-Epilogue II.C. Background Independence and Problem of Time at Deeper Levels of Structure
Quantum Problem of Time
-Geometrical Quantization. i. Kinematical Quantization
- Geometrical Quantization. ii. Dynamical Quantization
-Further Detail of Time and Temporal Relationalism in Quantum Theory
- Geometrical Quantization with Nontrivial g. i. Finite Theories
- Geometrical Quantization with Nontrivial g. ii. Field Theories and GR
- Tempus Ante Quantum
-Tempus Post Quantum. i. Paralleling QFT
-Tempus Post Quantum. ii. Semiclassical Machian Emergent Time
-Tempus Post Quantum. iii. Semiclassical Quantum Cosmological Modelling
-Semiclassicality and Quantum Cosmology: Interpretative Issues
-Quantum Constraint Closure
-Quantum Beables or Observables
-Fully Timeless Approaches at the Quantum Level
-Spacetime Primary Approaches: Path Integrals
-Histories Theory at the Quantum Level
-Combined Histories-Records-Semiclassical Approach
-Quantum Foliation Independence Strategies
- Quantum Spacetime Construction Strategies
-Quantum-Level Conclusion
-Epilogue III.A. The Multiple Choice Problem
-Epilogue III.B. Quantum Global Problems of Time
- Epilogue III.C. Deeper Levels’ Quantum Background Independence and Problem of Time
Appendices. Mathematical Methods for Basic and Foundational Quantum Gravity
- Basic Algebra and Discrete Mathematics
- Flat Geometry
- Basic Analysis
- Manifold Geometry
- Lie Groups and Lie Algebras
- More Advanced Topology and Geometry
- Configuration Space Geometry: Mechanics
- Field Theory and GR: Unreduced Configuration Space Geometry
- The Standard Principles of Dynamics. ii. Field Theory
- Temporal Relationalism Implementing Principles of Dynamics (TRiPoD)
- Reduced Configuration Spaces for Field Theory and GR
- DE Theorems for Geometrodynamics’ Problem of Time
- Function Spaces, Measures and Probabilities
- Statistical Mechanics (SM), Information, Correlation
- Stochastic Geometry
- Deeper Levels. i. Generalized Configuration Spaces
- Deeper Levels. ii. Grainings, Information, Stochastics and Statistics
- Quantum SM, Information and Correlation
- Further Algebraic Structures
-Alternative Foundations for Mathematics
- Outline of Notation
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