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▶Foundations

1.1The Principle of Least Action 1.2Lagrangian Mechanics 1.3Symmetries & Conservation 1.4Hamiltonian Mechanics 1.5Mathematical Foundations

▶Quantum Mechanics

2.1QM Foundations 2.2Exactly Solvable Problems 2.3Approximation Methods 2.4Entanglement & Measurement 2.5Interpretations 2.6Path Integrals

▼Electromagnetism

3.1Maxwell's Equations 3.2Electromagnetic Waves 3.3Gauge Invariance 3.4Relativistic Formulation

▶Special Relativity

4.1Lorentz Transformations 4.2Spacetime Diagrams 4.34-Vectors & Invariants 4.4Relativistic Mechanics 4.5E = mc²

▶Statistical Physics

5.1Entropy & the Second Law 5.2Ensembles 5.3Quantum Statistics 5.4Phase Transitions 5.5Information & Entropy

▶General Relativity

6.1Differential Geometry 6.2Einstein Field Equations 6.3Schwarzschild Solution 6.4Black Holes 6.5Gravitational Waves 6.6Cosmological Solutions

▶Quantum Field Theory

7.1Free Fields 7.2Interactions 7.3Renormalization 7.4Gauge Theories

▶The Standard Model

8.1Particle Content 8.2Electroweak Unification 8.3QCD 8.4Neutrino Physics 8.5Beyond the Standard Model

▶Cosmology

9.1The Big Bang 9.2Inflation 9.3CMB Physics 9.4Dark Matter 9.5Dark Energy 9.6Structure Formation

▶Frontiers

10.1Quantum Gravity 10.2Foundational Questions 10.3Information & Physics 10.4Cross-Field Connections

▶Research Frontiers

11.1Holography & AdS/CFT 11.2Entanglement & Spacetime 11.3The Island Formula 11.4Holographic Error Correction 11.5The Amplituhedron 11.6The Swampland Program 11.7Complexity & Black Holes 11.8Topological Order & Fractons 11.9Information Geometry

▶Open Problems

12.1Black Hole Information Paradox 12.2The Dark Energy Problem 12.3The Hubble Tension 12.4Cosmic Topology 12.5Novel Directions

Lesson 3.3 · 3. Electromagnetism

Gauge Invariance

← 3.2 Electromagnetic Waves 3.4 Relativistic Formulation →