Quantum 101

Table of Contents Introduction Motivation for Loop Quantum Gravity Background Independence Reformulating General Relativity Ashtekar Variables Holonomies and Wilson Loops The Loop Representation Quantum Configuration Space Spin Networks Kinematical Hilbert Space Quantum Geometry: Discrete Spectra Area...

Table of Contents Introduction Motivation for Canonical Quantization 3+1 Decomposition of Spacetime Hamiltonian Formulation of General Relativity The ADM Formalism Canonical Variables and Constraints Dirac's Procedure for Constrained Systems Wheeler–DeWitt Equation Problem of...

Table of Contents Introduction Why Quantum Gravity? Classical Gravity: General Relativity Quantum Field Theory Overview Incompatibility Between GR and QFT Conceptual Challenges in Quantizing Gravity Approaches to Quantum Gravity Perturbative Quantum Gravity Graviton...

Table of Contents Introduction Historical Background The Chern–Simons Action Gauge Invariance and Quantization Topological Nature of the Theory Connection to Topological Quantum Field Theory Classical Equations of Motion Quantization of Chern–Simons Theory Canonical...

Table of Contents Introduction What is a TQFT? Contrast with Conventional Quantum Field Theories Topological Invariance Mathematical Definition of TQFT Cobordism and Categories Atiyah–Segal Axioms Examples of TQFTs BF Theory Chern–Simons Theory Donaldson–Witten Theory Observables in...

Table of Contents Introduction What is a TQFT? Contrast with Conventional Quantum Field Theories Topological Invariance Mathematical Definition of TQFT Cobordism and Categories Atiyah–Segal Axioms Examples of TQFTs BF Theory Chern–Simons Theory Donaldson–Witten Theory Observables in...

Table of Contents Introduction Historical Context and Maldacena's Proposal Anti-de Sitter (AdS) Space Conformal Field Theory (CFT) Basics Duality Statement: AdS/CFT D3-Branes and AdS(_5) × S(^5) ( mathcal{N}=4 ) Super Yang–Mills...

Table of Contents Introduction Compactification and Extra Dimensions T-Duality: Basics and Motivation Closed Strings on a Circle Momentum and Winding Modes T-Duality Transformation Rules Physical Consequences of T-Duality T-Duality and Open Strings D-Branes:...

Table of Contents Introduction Strings and the Worldsheet Worldsheet Coordinates and Notation The Nambu–Goto Action The Polyakov Action Equivalence of Nambu–Goto and Polyakov Formulations Classical Equations of Motion Constraints and Virasoro Conditions Symmetries...

Table of Contents Introduction Motivation for String Theory Particles as One-Dimensional Strings Open vs Closed Strings Vibrational Modes and Mass Spectrum The String Action: Nambu–Goto and Polyakov Quantization of the String Conformal...

Table of Contents Introduction Motivation for Superspace and Superfields Supersymmetry Generators and Algebra Representation of SUSY Algebra Superspace Coordinates Superfields: Definition and Expansion Types of Superfields Chiral Superfields Vector Superfields Supersymmetry Transformations of Superfields Component...

Table of Contents Introduction The Motivation for Supersymmetry Fermions and Bosons in Quantum Field Theory The Hierarchy Problem Basics of Supersymmetry (SUSY) Supersymmetry Algebra Superpartners and Supermultiplets Chiral and Vector Supermultiplets Superspace and...