Quantum Electrodynamics

QED is the quantum field theory of electrons and photons. Its central principle is local U(1) gauge invariance: demanding that the Dirac Lagrangian be invariant under local phase rotations ψ → e^{iα(x)}ψ forces the introduction of the photon field Aμ and completely determines the form of the electron-photon interaction. The resulting theory is the most precisely tested in physics — agreement to 12 significant figures.

Key Concepts

Local U(1) gauge invariance: ψ → e^{iα(x)}ψ and Aμ → Aμ − (1/e)∂μα forces the coupling
Covariant derivative: Dμ = ∂μ + ieAμ replaces ∂μ everywhere
QED Lagrangian: ℒ = ψ̄(iD̸−m)ψ − ¼FμνFμν with Fμν = ∂μAν − ∂νAμ
Gauge fixing: Lorenz gauge ∂μAμ = 0 needed to invert photon kinetic operator
Photon propagator (Feynman gauge): Dμν(k) = −igμν/(k²+iε)
QED vertex: −ieγμ — derived from the ψ̄γμψ Aμ term in the Lagrangian

Key Equations

QED Lagrangian

\mathcal{L}_{\rm QED}=\bar{\psi}(i\not\!D-m)\psi-\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu},\quad D_\mu=\partial_\mu+ieA_\mu

Photon propagator

D_{\mu\nu}(k)=\frac{-ig_{\mu\nu}}{k^2+i\varepsilon}\quad(\text{Feynman gauge})

QED vertex factor

-ie\gamma^\mu

Maxwell equations from ℒ

\partial_\mu F^{\mu\nu}=ej^\nu,\quad j^\nu=\bar{\psi}\gamma^\nu\psi

Worked Example

Example Problem

Problem

Show the QED Lagrangian is invariant under ψ → e^{iα(x)}ψ, Aμ → Aμ − (1/e)∂μα.

Solution

Under the transformation: Dμψ = (∂μ+ieAμ)ψ → (∂μ+ie(Aμ−∂μα/e))e^{iα}ψ = e^{iα}(∂μ+ieAμ)ψ = e^{iα}Dμψ. So ψ̄D̸ψ → ψ̄e^{−iα}·e^{iα}D̸ψ = ψ̄D̸ψ. Fμν is trivially invariant since Fμν = ∂μAν−∂νAμ → Fμν−(∂μ∂ν−∂ν∂μ)α/e = Fμν. QED is gauge-invariant. ✓

Key Takeaways

The photon field is not postulated — it emerges from demanding local U(1) invariance of the Dirac Lagrangian
The covariant derivative Dμ = ∂μ+ieAμ automatically generates the minimal coupling −eψ̄γμψAμ
Gauge invariance is preserved quantum-mechanically by Ward identities, ensuring physical predictions are gauge-independent
QED is one of the most successful theories in science: the electron g−2 agrees with experiment to 12 significant figures