Quantum electrodynamics

Remarks

Gauge invariance: for a real function $\lambda$, the transformation $\psi \mapsto e^{-ie\lambda}\psi$, $A_\mu \mapsto A_\mu + \partial_\mu \lambda$ leaves $\mathcal{L}$ unchanged. Indeed $D_\mu \psi$ becomes $(\partial_\mu + ieA_\mu + ie\partial_\mu\lambda)(e^{-ie\lambda}\psi) = e^{-ie\lambda} D_\mu\psi$, and $F_{\mu\nu}$ does not change.

Non-example: the Lagrangian $\bar\psi (i\gamma^\mu \partial_\mu - m)\psi - \frac14 F_{\mu\nu}F^{\mu\nu}$ without the coupling term is invariant only when $\lambda$ is constant. The term $-e \bar\psi \gamma^\mu \psi A_\mu$ inside $\bar\psi i\gamma^\mu D_\mu \psi$ is what makes the symmetry local, and it describes the emission and absorption of photons.

The Euler–Lagrange equations are the Dirac equation $(i\gamma^\mu D_\mu - m)\psi = 0$ and Maxwell's equations $\partial_\mu F^{\mu\nu} = e \bar\psi \gamma^\nu \psi$, with the Dirac current as source. For the electron, $m$ is the electron mass and $e^2 / (4\pi) \approx 1/137$ is the fine-structure constant. Sign conventions for $e$ and for the metric vary between books. No construction of QED that satisfies the axioms of quantum field theory is known; in practice the theory is defined order by order in $e$ through renormalization.