This second edition offers a comprehensive introduction to loop quantum gravity (LQG) in self-dual variables, including the necessary prerequisites. Additionally, it delves into various significant research areas that have emerged in recent years. New content (including an entirely new chapter dedicated to dynamics of quantum spacetime) explores the description of spin networks and spin-foams, their historical development as well as connections to tensor networks, BF theory, and emerging approaches including the spinorial representation of LQG, SU(2) coherent states, and group field theory. Furthermore, the book provides expanded appendices covering essential tools and concepts, such as the connection between information theory and entropy, and overviews of group theory and differential geometry. All topics are presented from a non-expert perspective, ensuring self-containment and accessibility. The primary aim of this second edition remains helping researchers, bewildered by the vast array of topics within this rapidly growing field of quantum gravity, to gain a fundamental understanding of the current developments.
Author(s): Sundance Bilson-Thompson
Edition: 2
Publisher: Springer Nature Switzerland
Year: 2024
Language: English
Pages: 210
Tags: General Relativity, Quantum Field Theory, Quantum Gravity, Spin Networks
Preface
Conventions
Indices
Symbols
Contents
Acronyms
1 Introduction
1.1 Motivation and Some History
1.2 Overview: Loop Quantum Gravity and Friends
2 Classical GR
2.1 Parallel Transport and Curvature
2.2 Einstein's Field Equations
2.3 Coordinates and Diffeomorphism Invariance
3 Quantum Field Theory
3.1 Covariant Derivative and Curvature
3.2 Dual Tensors, Bivectors and k-forms
3.3 Wilson Loops and Holonomies
3.4 Dynamics of Quantum Fields
4 Expanding on Classical GR
4.1 Lagrangian Approach: The Einstein-Hilbert Action
4.2 Hamiltonian Approach: The ADM Splitting
4.2.1 Physical Interpretation of Constraints
4.3 Seeking a Path to Canonical Quantum Gravity
4.3.1 Connection Formulation
4.3.2 Tetrads
4.3.3 Choosing a Gauge Group
4.3.4 Spin Connection
4.3.5 Palatini Action
4.3.6 Palatini Hamiltonian and Constraints
5 First Steps to a Theory of Quantum Gravity
5.1 Ashtekar Formulation: ``New Variables'' for General Relativity
5.2 The Barbero-Immirzi Parameter
5.3 To Be or Not to Be (Real)
5.4 Loop Quantization
5.5 Canonical Quantization
6 Kinematical Hilbert Space
6.1 Kinematics via a Toy Model
6.2 Space of Generalised Connections
6.3 Area Operator
6.4 Volume Operator
6.5 Spin Networks
6.6 Looking Ahead to Spin Foams
7 Dynamics of Spin Networks
7.1 Spin Foams
7.1.1 Transition Amplitudes
7.2 Early Developments
7.2.1 BF Theory
7.2.2 Chern–Simons Theory
7.2.3 The Cosmological Constant
7.3 Some Recent Developments
7.3.1 Tensor Networks
7.3.2 Spinorial LQG and Coherent States
7.3.3 Group Field Theory
8 Applications
8.1 Black Hole Entropy
8.1.1 Rovelli's Counting
8.1.2 Number Theoretical Approach
8.1.3 Chern-Simons Approach
8.1.4 Entropy from Entanglement
8.2 Loop Quantum Cosmology
8.2.1 Isotropy and Homogeneity in the Metric Formulation
8.2.2 FLRW Models
8.2.3 Connection Variables
8.2.4 Holonomy Variables
8.2.5 Quantisation
8.2.6 Triad Eigenstates and Volume Quantization
8.2.7 Regularized FLRW Hamiltonian
8.2.8 Singularity Resolution and Bouncing Cosmologies
9 Discussion
A Groups, Representations, and Algebras
A.1 Lie Groups and Algebras
A.2 Lorentz Lie-Algebra
B Blades, Forms, and Duality
B.1 Blades and Multivectors
B.2 Differential Forms and the Exterior Derivative
B.3 Duality
B.4 Field Strength and the Exterior Derivative
B.5 Spacetime Duality
B.6 Lie-Algebra Duality
B.7 Yang-Mills
B.8 Geometrical Interpretation
B.9 (Anti) Self-dual Connections
C Path Ordered Exponential
D ADM Variables
D.1 Covariant Spatial Derivative
D.2 Extrinsic Curvature
D.3 Canonical Momentum in the ADM Formulation
E Lie Derivative
F 3+1 Decomposition of the Palatini Action
G The Kodama State
H Peter-Weyl Theorem
I Regge Calculus
J Fibre Bundles
K Knots, Links, and the Kauffman Bracket
L Quantum (or q-Deformed) Groups
M Entropy
N Square-Free Numbers
O Brahmagupta-Pell Equation
O.1 Quadratic Integers and the BP Equation
