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

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