The Problem of Time

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.

Table of Contents

Part I: Time in Fundamental Physics

Chapter 1: Introduction: Conceptual Outline of Time

Chapter 2: Time, Space and Laws in Newtonian Mechanics

Chapter 3: Absolute Versus Relational Motion Debate

Chapter 4: Time, Space, Spacetime and Laws in Special Relativity

Chapter 5: Time and Ordinary Quantum Mechanics (QM)

Chapter 6: Quantum Field Theory (QFD

Chapter 7: Time and Spacetime in General Relativity (GR)

Chapter 8: Dynamical Formulations of GR

Chapter 9: Classical-Level Background Independence and the Problem of Time. i. Time and Configuration

Chapter 10: Classical-Level Background Independence and the Problem of Time. ii. Spacetime and Its Interrelation with Space

Chapter 11: Quantum Gravity Programs

Chapter 12: Quantum-Level Background Independence and the Problem of Time

Part II: Classical Problem of Time

Chapter 13: Advanced Nomenclature for Facet Interference

Chapter 14: Configuration Spaces and Their Configurational Relationalism

Chapter 15: Temporal Relationalism (TR)

Chapter 16: Combining Temporal and Configurational Relationalisms

Chapter 17: Temporal Relationalism: More General Geometries

Chapter 18: Configurational Relationalism: Field Theory and GR’s Thin Sandwich

Chapter 19: Relationalism in Various Further Settings

Chapter 20: Other Tempus Ante Quantum Approaches

Chapter 21: Conformal Approach and Its York Time

Chapter 22: Matter Times

Chapter 23: Classical Machian Emergent Time

Chapter 24: Brackets, Constraints and Closure

Chapter 25: Taking Function Spaces Thereover: Beables and Observables

Chapter 26: Fully Timeless Approaches

Chapter 27: Spacetime Relationalism

Chapter 28: Classical Histories Theory

Chapter 29: Classical Machian Combined Approach

Chapter 30: Slightly Inhomogeneous Cosmology (SIC)

Chapter 31: Embeddings, Slices and Foliations

Chapter 32: Applications of Split Spacetime, Foliations and Deformations

Chapter 33: Spacetime Construction and Alternative Emergent Structures

Chapter 34: TRi Foliation (TRiFol)

Chapter 35: Classical-Level Conclusion

Chapter 36: Epilogue II.A. Threading and Null Formulations

Chapter 37: Epilogue 11.B. Global Validity and Global Problems of Time

Chapter 38: Epilogue 11.C. Background Independence and Problem of Time at Deeper Levels of Structure

Part Ill: Quantum Problem of Time

Chapter 39: Geometrical Quantization. i. Kinematical Quantization

Chapter 40: Geometrical Quantization. ii. Dynamical Quantization

Chapter 41: Further Detail of Time and Temporal Relationalism in Quantum Theory

Chapter 42: Geometrical Quantization with Nontrivial g. i. Finite Theories

Chapter 43: Geometrical Quantization with Nontrivial g. ii. Field Theories and GR

Chapter 44: Tempus Ante Quantum

Chapter 45: Tempus Post Quantum. i. Paralleling QFT

Chapter 46: Tempus Post Quantum. ii. Semiclassical Machian Emergent Time

Chapter 47: Tempus Post Quantum. iii. Semiclassical Quantum Cosmological Modelling

Chapter 48: Semiclassicality and Quantum Cosmology: Interpretative Issues

Chapter 49: Quantum Constraint Closure

Chapter 50: Quantum Beables or Observables

Chapter 51: Fully Timeless Approaches at the Quantum Level

Chapter 52: Spacetime Primary Approaches: Path Integrals

Chapter 53: Histories Theory at the Quantum Level

Chapter 54: Combined Histories-Records-Semiclassical Approach

Chapter 55: Quantum Foliation Independence Strategies

Chapter 56: Quantum Spacetime Construction Strategies

Chapter 57: Quantum-Level Conclusion

Chapter 58: Epilogue Ill.A. The Multiple Choice Problem

Chapter 59: Epilogue III.B. Quantum Global Problems of Time

Chapter 60: Epilogue III.C. Deeper Levels’ Quantum Background Independence and Problem of Time

About the Author

Edward Anderson graduated from Cambridge with distinction in Part III Mathematics, and did a PhD in General Relativity at Queen Mary, University of London, before returning to Cambridge as a Research Fellow of Peterhouse and member of DAMTP. E.A. has also occupied positions at the University of Alberta, Universidad Autónoma de Madrid and Université Paris 7 (with a FQXi large grant to study the titular Problem of Time).