On the Emergence of Time and Space in Closed Quantum Systems (Springer Theses)

✍ Scribed by Tommaso Favalli

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 177
• Edition: 1st ed. 2024
• Category: Library

✦ Synopsis

Time, space and entanglement are the main characters in this book. Their nature is still a great mystery in physics and we study here the possibility that these three phenomena are closely connected, showing how entanglement can be at the basis of the emergence of time and space within closed quantum systems. We revisit and extend the Page and Wootters theory that was originally introduced in order to describe the emergence of time through entanglement between subsystems in a globally static, quantum Universe. In the book, after providing a complete review of the salient aspects of the theory, we establish a connection with recent research on the foundations of statistical mechanics and we provide a new understanding of the thermalization process. Furthermore, we generalize the framework in order describe the spatial degree of freedom and we provide a model of 3+1 dimensional, quantum spacetime emerging from entanglement among different subsystems in a globally "timeless" and "positionless" Universe. Finally, via the Page and Wootters theory, the evolution of quantum clocks within a gravitational field is treated and a time dilation effect is obtained in agreement with the Schwarzschild solution.

✦ Table of Contents

Supervisor’s Foreword
Abstract
=0mmPublications Related to This Thesis
Acknowledgements
Contents
1 Introduction
1.1 What ``Time'' is?
1.2 Inside the Story of a Timeless Universe
References
2 Page and Wootters Theory
2.1 Introduction to the Theory
2.1.1 Two-Particle Universe
2.1.2 Evolution Within a Single State
2.2 Evolution Without Evolution
2.2.1 Conditions and Definitions
2.2.2 Deriving Schrödinger Equation
2.3 Clock as Particle on a Line
2.4 Overcoming Criticisms
2.4.1 The GPPT Proposal
2.4.2 GLM's Multiple Measurements
2.5 Discussion
2.5.1 The Flow of Time: Two Points of View
2.5.2 Experimental Illustration
2.6 Conclusions
References
3 Complement of the Hamiltonian
3.1 The alphaα Quantity
3.2 Unequally-Spaced Energy Levels
3.2.1 General Framework
3.2.2 The s long right arrow normal infinitys -3murightarrowinfty Limit
3.2.3 The Uncertainty Relation
3.3 Equally-Spaced Energy Levels
3.4 Conclusions
3.5 Appendices
References
4 Time Observables within a Timeless Universe
4.1 Equally-Spaced Energy Levels for the Clock Hamiltonian
4.1.1 The Clock Subspace
4.1.2 Dynamics
4.2 The Hermitian Time Operator
4.3 Unequally-Spaced Energy Levels for the Clock Hamiltonian
4.3.1 Discrete Flow of Time
4.3.2 Continuous Flow of Time
4.3.3 Clock Hamiltonian with Generic Spectrum
4.4 On the Quantum Speed Limit Time
4.5 Interacting Clock and System
4.5.1 General Framework
4.5.2 Gravitationally Interacting Clock and System
4.6 Considerations on the Arrow of Time
4.7 Conclusions
4.8 Appendices
References
5 Thermal Equilibrium and Emergence of Time
5.1 Canonical Equilibrium Distribution
5.1.1 Canonical Typicality: An Overview
5.1.2 Derivation of Canonical Typicality
5.2 Environment as a Clock
5.2.1 General Framework
5.2.2 Random Universe and Dynamics
5.2.3 Initial Conditions for upper SS
5.3 A Toy Model
5.4 Non-observable Universe as Clock
5.5 Conclusions
5.6 Appendices
References
6 Quantum Spacetime
6.1 Relative-Position Localization Through Entanglement
6.2 Emergent Space from Entanglement
6.2.1 General Framework
6.2.2 Emergent Relative Distance
6.2.3 On the Position-Momentum Uncertainty Relation
6.3 Spacetime from Entanglement
6.3.1 Introducing the upper CC Subspace
6.3.2 Emergent 1 plus 11+1 Dimensional Spacetime
6.3.3 A Simple Example
6.3.4 On the Quantum Speed Limit Time
6.3.5 Introducing External Potentials in upper RR and upper SS
6.4 Multiple Time Measurements
6.4.1 The GPPT Proposal
6.4.2 GLM's Multiple Measurements
6.5 Generalization to 3 plus 13+1 Dimensional Spacetime
6.5.1 Discrete Values for Space and Time
6.5.2 Continuous Values for Space and Time
6.5.3 Free Particles (with upper M much greater than mMggm) in 3 plus 13+1 Spacetime
6.5.4 System upper SS as a Relativistic Particle
6.6 Conclusions
6.7 Appendices
References
7 Quantum Clocks in a Gravitational Field
7.1 General Framework
7.2 Clocks upper AA and upper BB with Discrete Time Values
7.2.1 Evolution of Free Clocks
7.2.2 A and B Interacting with the Gravitational Field
7.2.3 Bringing upper AA to an Infinite Distance
7.3 Clocks upper AA and upper BB with Continuous Time Values
7.3.1 Evolution of Free Clocks
7.3.2 B and B Interacting with the Gravitational Field
7.3.3 Bringing upper AA to an Infinite Distance
7.4 Gravitational Redshift
7.5 Relativistic Gravitational Potential
7.6 Conclusions
7.7 Appendices
References
8 Conclusions
References
Appendix Biography