Loop Quantum Gravity, Differential Forms, Quantum Geometry

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.