The book is based on the author's PhD thesis, which deals with the concept of time in quantum gravity and its relevance for the physics of the early Universe. It presents a consistent and complete new relational formulation of quantum gravity (more specifically, of quantum mechanics models with diffeomorphism invariance), which is applied to potentially observable cosmological effects. The work provides answers to the following questions: How can the dynamics of quantum states of matter and geometry be defined in a diffeomorphism-invariant way? What is the relevant space of physical states and which operators act on it? How are the quantum states related to probabilities in the absence of a preferred time? The answers can provide a further part of the route to constructing a fundamental theory of quantum gravity. The book is well-suited to graduate students as well as professional researchers in the fields of general relativity and gravitation, cosmology, and quantum foundations.
Author(s): Leonardo Chataignier
Series: Springer Theses
Publisher: Springer
Year: 2022
Language: English
Pages: 253
City: Cham
Supervisor’s Foreword
Abstract
Publications related to this Thesis
Acknowledgments
Contents
Symbols, Conventions, Terminology, and Abbreviations
Mathematical Symbols
Conventions
Terminology
Abbreviations
1 Introduction
1.1 What Is General Relativity About?
1.1.1 Background Independence and Diffeomorphism Invariance
1.1.2 Intrinsic Coordinates
1.1.3 Relational Observables
1.1.4 Physical Events
1.2 What Quantum Gravity Ought to Be About?
1.3 What Is This Thesis About?
1.3.1 Outline of the Thesis
References
2 Classical Diffeomorphism Invariance on the Worldline
2.1 The Abstract Worldline
2.2 Dynamics and Gauge Symmetry
2.3 The Total Hamiltonian
2.3.1 General Case and Gauge Indeterminism
2.3.2 A Particular Case
2.4 Parametrization of Noninvariant Models and Jacobi's Principle
2.5 The Gauge Generator
2.6 Gauge Fixing, Intrinsic Coordinates, and Generalized Reference Frames
2.7 Observables and Invariant Extensions
2.8 Integral Representations of Relational Observables
2.9 Dynamics of Relational Observables
2.9.1 Gauge-Fixed Evolution
2.9.2 The Reduced Phase Space, the Physical Hamiltonian, and the Physical Worldline
2.10 Hamilton–Jacobi Formalism
References
3 Quantum Diffeomorphism Invariance on the Worldline
3.1 The Auxiliary Hilbert Space
3.2 To Constrain or Not to Constrain? Stückelberg's Approach
3.3 The Physical Hilbert Space
3.4 On-Shell and Invariant Operators
3.5 Quantum Relational Observables
3.5.1 Proper-Time Evolution
3.5.2 Evolution in Other Gauges
3.5.3 A Useful Particular Case
3.5.4 An Alternative Factor Ordering
3.5.5 A Strategy
3.5.6 A Perturbative Procedure
3.6 A Tentative Set of Postulates
3.6.1 Proper-Time Quantum Mechanics
3.6.2 Quantum Diffeomorphisms and Changes of Quantum Reference Frames
3.7 Conditional Probabilities
3.7.1 Invariant Extensions of States
3.7.2 Recovering the Page–Wootters Formalism
References
4 The Relativistic Particle as an Archetypical Example
4.1 Classical Theory
4.1.1 Observables
4.1.2 On-Shell Action
4.2 Quantum Theory
4.2.1 The Nonrelativistic Limit of the Induced Inner Product
4.2.2 Quantum Relational Observables and Their Relation to the Classical Expressions I
4.2.3 Dynamics and Nonrelativistic Limit I
4.2.4 Quantum Relational Observables and Their Relation to the Classical Expressions II
4.2.5 Dynamics and Nonrelativistic Limit II
References
5 Homogeneous Classical and Quantum Cosmology
5.1 Singularity Avoidance
5.1.1 The Wave Function of the Universe as Relative Initial Data
5.2 Closed FLRW Model
5.2.1 Classical Theory
5.2.2 Quantum Theory I. The Physical Hilbert Space
5.2.3 Quantum Theory II. Relational Observables
5.2.4 Quantum Theory III. Relational Quantum Dynamics
5.3 The Kasner Model
5.3.1 Classical Theory
5.3.2 Quantum Theory
References
6 Weak-Coupling Expansion
6.1 Classical Theory
6.1.1 The Background Dynamics
6.1.2 WKB Time as a Classical Choice of Gauge
6.1.3 Perturbation Theory
6.2 Quantum Theory
6.2.1 The Auxiliary and Physical Hilbert Spaces. Conditional Probabilities
6.2.2 The Phase-Transformed Constraint Equation
6.2.3 Time-Dependent Measures and Unitarity
6.2.4 Perturbation Theory I
6.2.5 Light-Sector Unitarity. Propagation in a Fixed Background
6.2.6 Perturbation Theory II
6.2.7 WKB Time as a Quantum Choice of Gauge
References
7 Quantum-Gravitational Effects in the Early Universe
7.1 Cosmological Perturbations
7.1.1 The Classical Background
7.1.2 Classical Perturbations
7.1.3 The Master Wheeler-DeWitt Equation
7.2 Weak-Coupling Expansion. Unitarity
7.3 Corrections to Primordial Power Spectra
7.3.1 Restriction to a Single Mode
7.3.2 Relative Initial Data
7.3.3 Unitarity
7.3.4 Power Spectra I. Definitions
7.3.5 Power Spectra II. The Lowest Order
7.3.6 Power Spectra III. Corrections
7.3.7 Power Spectra IV. Discussion
References
8 Conclusions and Outlook
8.1 Conclusions
8.2 Outlook
8.2.1 Relative Initial Data in Field Theory
8.2.2 Whence Probabilities?
References
Appendix A Review of Gauge Systems and Constrained Dynamics
A.1 Gauge Symmetries and Singular Lagrangians
A.1.1 Noether Theorems
A.1.2 Gauge Systems are Singular
A.2 Constrained Dynamics
A.2.1 Primary Constraints
A.2.2 The Total Hamiltonian
A.2.3 The Rosenfeld-Dirac-Bergmann Algorithm
A.2.4 First-Class and Second-Class Functions. The Initial Value Problem
A.2.5 Reference Frames and the Gauge Generator
A.2.6 Gauge Orbits and Gauge Invariance
A.2.7 Gauge Fixing and Invariant Extensions
A.3 The Reduced Phase Space and Its Quantization
A.3.1 Dynamics of Reference Frames in the Reduced Phase Space
A.3.2 Limitations of the Reduced Phase-Space Description
A.3.3 Hamilton-Jacobi Formalism
A.3.4 Quantum Theory
Appendix B The Traditional Born-Oppenheimer Approach to the Problem of Time
B.1 What is the Traditional BO Approach?
B.2 The Semiclassical Derivation of Time
B.3 Backreaction
B.4 The Ambiguity of Backreaction and the Issue of Unitarity
B.4.1 Factorization Ambiguity and Its Physical Meaning
B.4.2 The Traditional and Minimal BO Factorizations are Equivalent
B.4.3 Unitarity and Conditional Probabilities
References
