Lecture Notes on Quantum Electrodynamics

Abstract

These notes present a self-contained, first-principles derivation of Quantum Electrodynamics as the inevitable consequence of demanding local U(1) gauge invariance of the Dirac Lagrangian. Beginning with the Dirac equation and its zero-momentum solutions, we identify the spinor structure that encodes spin and antiparticle degrees of freedom. We then demonstrate that the Dirac Lagrangian is invariant under global phase transformations but fails under local ones. Insisting on local invariance forces us to introduce a compensating vector field Aµ, whose transformation properties and self-interactions are dictated entirely by symmetry. The antisymmetric combination ∂µAν−∂νAµ furnishes the only gauge-invariant kinetic term, and its components are precisely the electric and magnetic fields. Applying the Euler-Lagrange equations to the resulting QED Lagrangian yields Maxwell's inhomogeneous equations, while the minimal-coupling prescription for a charged particle yields the Lorentz force law. Finally, we prove that the antisymmetry of the field strength tensor guarantees conservation of the Noether current, completing the unification of electromagnetism under the banner of gauge symmetry.