Loop Quantum Gravity, Differential Forms, Quantum Geometry

Exploring the Quanta of Space, Differential Forms, the Tetrad formalism of GR, Canonical Relativity, Ashtekar Variables.

What you'll learn

Grasp the Fundamentals of Loop Quantum Gravity (LQG)

Explore the similarity between Quantum Geometry and Angular Momenta

Master Differential Forms and Their Applications

Familiarize with the ADM formalism of General Relativity, Palatini action, and group theory

Understand Spin-Networks and Quanta of Geometry

Comprehend the Role of Holonomy and Wilson Loops

Explore Properties of the Densitized Triad and Volume Operator

Understand the tetrad formulation of General Relativity and Cartan Equations

Some notions related to the path integral in Loop Quantum Gravity

The importance of the Wheeler DeWitt equation and its relation to loops

Harmonic Analysis over the SU(2) group, key to understanding the basics of Loop Quantum Gravity

Requirements

Quantum physics and Quantum Field Theory (and their maths)

General Relativity (and its math)

Description

Loop Quantum Gravity: A Comprehensive IntroductionFrom the basics to more advanced topics, we will cover angular momenta, holonomy, quantum geometry,  ADM formalism and Palatini action and more (have a look at the syllabus below). There is also an independent section on differential forms, which are important for the final part of the course.Introduction to Loop Quantum Gravity (LQG)Overview of classical gravity and challengesMotivations for Loop Quantum Gravity Discretization of spacetime and fundamental principlesAngular Momenta in LQGProperties of Angular Momentum OperatorsMatrix Representation of Angular MomentumSpin 1/2 Particles in LQGHolonomy and Area OperatorDifferential Equation of the HolonomyConcept of Holonomy in Loop Quantum GravityProperties of the Holonomy, Wilson LoopsDensitized triad in LQGGeneralization of holonomies in LQGQuantum Geometry with Spin-NetworksSpin-Networks and Spin-Network StatesClassical Interpretation of the Densitized TriadVolume Operator in LQGHeisenberg Uncertainty Principle in LQGADM Formalism and TetradsADM FormalismInverse of the Metric Tensor and Projection OperatorFormula for the Determinant of the Metric TensorLie DerivativeAn Introduction to the Tetrads (Generalization of the Triads)Introduction to Differential FormsGeneralization of the Cross Product and Introduction to the Wedge ProductGeometrical Intuition of the Cross and Wedge ProductsCross Product in 2D and 3D Derived from the Wedge ProductWedge Product and Degrees of FormsDifferential Forms and Exterior DerivativeGeneralized Fundamental Theorem of CalculusOverview of the Generalized Fundamental Theorem of CalculusProof of the Generalized Fundamental Theorem of CalculusApplication of the Generalized Theorem of CalculusStokes Theorem in 2D and 3D, Divergence TheoremApplications of Differential FormsTransformation of Volumes in the Language of Differential FormsInvariant Volume Element in D DimensionsSecond Exterior Derivative of a FormApplication of Differential Forms to the Electromagnetic FieldDerivation of Maxwell Equations from Differential FormsHodge Dual and Electromagnetic FormsHodge Dual, Levi Civita Pseudo-TensorExterior Derivative of the Hodge Dual of the Electromagnetic FormDerivation of Remaining Maxwell Equations from Differential FormsExercises with Differential FormsExterior Derivative of a Wedge Product of Differential FormsExercises on Calculating Exterior Derivatives and Hodge DualsSurface Calculation and Hodge Dual ExercisesPalatini action of General Relativity, Path integrals in Loop Quantum GravityPalatini Action of General RelativitySpin Connection, Cartan Equations, Lie Derivatives, and Decomposition of Palatini ActionWheeler DeWitt equation and its relation to loopsBF theoryPath integrals intuition in Loop Quantum GravityHarmonic Analysis over the SU(2) group, Wigner D matricesRepresentation of orbital angular momentum, spherical harmonics, Wigner D matrixOrbital angular momentumSpherical harmonicsLegendre polynomialsWigner D matrices and Spherical HarmonicsAppendix: Some More Mathematical Tools for Advanced UnderstandingTrace of the Logarithm of a Matrix and the DeterminantProof of the Jacobi IdentityNeumann SeriesImportant Properties of Unitary Matrices and Group TheoryMaterial Recommendations for the CourseAdditional resources, readings, and references to enhance understanding (here and there, you will see attachments to the lectures).This course provides a comprehensive introduction to Loop Quantum Gravity, covering fundamental principles, some mathematical tools, and advanced topics to empower learners with a basic but still deep understanding of this intriguing field.

Overview

Section 1: What is Loop Quantum Gravity (LQG) ? Introduction to the main concepts

Lecture 1 The End of Space (and Time): Quantum Fuzziness at the Planck Length

Lecture 2 Intro to the concepts of Loop Quantum Gravity

Lecture 3 Material Recommendations for the Course

Lecture 4 Intro to the section on Angular Momenta

Section 2: Representation of Angular Momentum

Lecture 5 Introduction to Angular Momentum Operators

Lecture 6 Properties of Angular Momentum Operators

Lecture 7 Properties of Angular Momentum Operators part 2

Lecture 8 Matrix Representation of Angular Momentum

Lecture 9 Spin 1/2 Particles

Section 3: Loop Quantum Gravity

Lecture 10 Area Operator

Lecture 11 Differential Equation of the Holonomy

Lecture 12 Concept of Holonomy in Loop Quantum Gravity

Lecture 13 Property of the Holonomy, Wilson Loops

Lecture 14 Densitized triad in Loop Quantum Gravity

Lecture 15 Generalization of holonomies in Loop Quantum Gravity

Lecture 16 Spin-Networks, Spin-Network States, Quanta of Geometry in LQG

Lecture 17 Classical Interpretation of the Densitized Triad in Loop Quantum Gravity

Lecture 18 Volume Operator in Loop Quantum Gravity

Lecture 19 Heisenberg Uncertainty Principle in LQG

Lecture 20 Densitized Triad and Inverse Triad in Loop Quantum Gravity

Section 4: ADM formalism behind Loop Quantum Gravity

Lecture 21 ADM Formalism

Lecture 22 ADM formalism: Inverse of the Metric Tensor and Projection Operator

Lecture 23 Formula for the Determinant of the Metric Tensor

Lecture 24 Lie Derivative

Lecture 25 Generalization of the Triads: Tetrads

Lecture 26 Intro to the subsequent section on differential forms

Section 5: Differential forms

Lecture 27 Intro to Differential Forms

Lecture 28 Generalization of the Cross Product and Introduction to the Wedge Product

Lecture 29 Geometrical Intuition of the Cross and Wedge Products

Lecture 30 Cross Product in 2D Derived from the Wedge Product

Lecture 31 Cross Product in 3D Derived from the Wedge Product

Lecture 32 Wedge Product and Degrees of Forms

Lecture 33 Example with 2-Forms

Lecture 34 Relation between the Wedge Product and the Triple Product in 3D

Lecture 35 Differential Forms

Lecture 36 Exterior Derivative

Lecture 37 Extra: Additional Considerations on the Exterior Derivative

Lecture 38 Examples of Exterior Derivatives

Lecture 39 Overview of the Generalized Fundamental Theorem of Calculus

Lecture 40 Proof of the Generalized Fundamental Theorem of Calculus part 1

Lecture 41 Proof of the Generalized Fundamental Theorem of Calculus part 2

Lecture 42 Example 1: Application of the Generalized Theorem of Calculus

Lecture 43 Example 2: Stokes Theorem in 2D Derived from the Generalized Theorem of Calculus

Lecture 44 Example 3: Divergence Theorem Derived from the Generalized Theorem of Calculus

Lecture 45 Example 4: Stokes Theorem Derived from the Generalized Theorem of Calculus

Lecture 46 Transformation of Volumes in the Language of Differential Forms

Lecture 47 Invariant Volume Element in D Dimensions

Lecture 48 Second Exterior Derivative of a Form

Lecture 49 Application of Differential Forms to the Electromagnetic Field

Lecture 50 First Maxwell Equation

Lecture 51 Second Maxwell Equation

Lecture 52 Hodge Dual, Levi Civita Pseudo-Tensor

Lecture 53 Exterior Derivative of the Hodge Dual of the Electromagnetic Form

Lecture 54 Derivation of the Remaining Maxwell Equations from Differential Forms

Lecture 55 Exterior Derivative of a Wedge Product of Differential Forms

Lecture 56 Exercise 1: Calculation of the Exterior Derivative

Lecture 57 Exercise 2: Calculation of the Exterior Derivative

Lecture 58 Exercise 3: Calculation of the Hodge Dual

Lecture 59 One Observation on the Hodge Duals in 3D

Lecture 60 Calculation of a Surface using Differential Forms

Lecture 61 Exercise with the Hodge Dual in 2 Dimensions

Section 6: General Relativity in terms of Tetrads

Lecture 62 Palatini Action of General Relativity

Lecture 63 Spin Connection for Internal Indices

Lecture 64 Cartan Equations

Lecture 65 Decomposition of Palatini Action part 1

Lecture 66 Decomposition of Palatini Action part 2

Lecture 67 Decomposition of Palatini Action part 3

Lecture 68 Wheeler DeWitt Equation and its relation to Holonomies, Loops

Lecture 69 BF Theory Derived from General Relativity

Lecture 70 Path Integral in Loop Quantum Gravity

Lecture 71 Harmonic Analysis on a Group

Lecture 72 Some more Considerations on Elementary Harmonic Analysis

Section 7: Representation of orbital angular momentum, spherical harmonics, Wigner D matrix

Lecture 73 Orbital Angular Momentum and its Square in Quantum Mechanics

Lecture 74 Laplacian in Spherical Coordinates

Lecture 75 Legendre Differential Equation

Lecture 76 Solution to the Legendre Differential Equation

Lecture 77 Spherical Harmonics

Lecture 78 Eigenfunctions and eigenvalues of Lz and L squared

Lecture 79 Properties of Legendre Polynomials Part 1

Lecture 80 Properties of Legendre Polynomials Part 2: Rodrigues Formula

Lecture 81 Normalization of Legendre Polynomials

Lecture 82 Relation between Beta and Gamma Functions

Lecture 83 Completeness Relation for the Legendre Polynomials

Lecture 84 Properties of the Generalized Legendre Polynomials part 1

Lecture 85 Properties of the Generalized Legendre Polynomials part 2: Orthogonality

Lecture 86 The Full Formula for Spherical Harmonics

Lecture 87 Symmetry Property of Spherical Harmonics

Lecture 88 Completeness of Spherical Harmonics

Lecture 89 Theorem of Addition of Spherical Harmonics

Lecture 90 Final considerations on the Addition Theorem for Spherical Harmonics

Lecture 91 Wigner D Matrices

Section 8: Appendix

Lecture 92 Trace of the Logarithm of a Matrix and the Determinant

Lecture 93 Proof of the Jacobi Identity

Lecture 94 Neumann Series

Lecture 95 Important Properties of Unitary Matrices and Group Theory

Physics Enthusiasts and Students: Undergraduate and graduate students in physics or related fields seeking a deeper understanding of cutting-edge theoretical physics concepts.,Researchers and Academics: Professionals engaged in theoretical physics research, academics, or those working in related fields who want to explore Loop Quantum Gravity as a potential paradigm shift in understanding spacetime.,Science Educators looking to enhance their knowledge of contemporary theoretical physics,Individuals with a genuine interest in the mysteries of the universe, regardless of their academic background, who wish to explore the fascinating realm of Loop Quantum Gravity.,Mathematics Enthusiasts: Learners with a strong mathematical background interested in exploring the mathematical tools and techniques employed in Loop Quantum Gravity, including differential forms, group theory, and advanced mathematical concepts.