Part IV, Chapter 3 | Page 1 of 3

Quantum Electrodynamics: The QED Lagrangian

Deriving the most precisely tested theory in physics from gauge invariance

3.1 The Free Dirac and Maxwell Lagrangians

We begin with the two free-field Lagrangians that QED unifies. The free Dirac Lagrangian for a spin-1/2 fermion of mass m is:

$$\mathcal{L}_{\text{Dirac}} = \bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi$$

where $\psi$ is a four-component Dirac spinor and $\bar{\psi} = \psi^\dagger\gamma^0$is the Dirac adjoint. The gamma matrices satisfy the Clifford algebra $\{\gamma^\mu,\gamma^\nu\} = 2g^{\mu\nu}$.

This Lagrangian has a global U(1) symmetry: it is invariant under

$$\psi(x) \to e^{i\alpha}\psi(x), \qquad \bar{\psi}(x) \to \bar{\psi}(x)e^{-i\alpha}$$

where $\alpha$ is a constant. By Noether's theorem, this symmetry implies a conserved current:

$$J^\mu = \bar{\psi}\gamma^\mu\psi, \qquad \partial_\mu J^\mu = 0$$

The free electromagnetic field is described by the Maxwell Lagrangian:

$$\mathcal{L}_{\text{Maxwell}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}, \qquad F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$

This is invariant under gauge transformations $A_\mu \to A_\mu - \partial_\mu\alpha(x)$ for any function $\alpha(x)$. The question is: can we couple these two sectors in a way that respects both symmetries?

3.2 Gauging the U(1) Symmetry

The central idea of gauge theory is to promote the global symmetry to a local one. We demand invariance under:

$$\psi(x) \to e^{ie\alpha(x)}\psi(x)$$

where $\alpha(x)$ is now an arbitrary function of spacetime. The free Dirac Lagrangian is not invariant under this local transformation because the derivative acts on $\alpha(x)$:

$$\partial_\mu\psi \to e^{ie\alpha(x)}\big(\partial_\mu\psi + ie(\partial_\mu\alpha)\psi\big)$$

The unwanted $\partial_\mu\alpha$ term breaks the invariance. To restore it, we introduce the gauge covariant derivative:

$$\boxed{D_\mu \equiv \partial_\mu + ieA_\mu}$$

If the gauge field simultaneously transforms as $A_\mu \to A_\mu - \partial_\mu\alpha$, then:

$$D_\mu\psi \to (\partial_\mu + ie(A_\mu - \partial_\mu\alpha))\,e^{ie\alpha}\psi = e^{ie\alpha}(\partial_\mu + ieA_\mu)\psi = e^{ie\alpha}D_\mu\psi$$

The covariant derivative transforms homogeneously, just like $\psi$ itself! Therefore, replacing$\partial_\mu \to D_\mu$ in the Dirac Lagrangian restores local gauge invariance.

💡The Gauge Principle

The gauge principle is remarkably powerful: demanding local U(1) invariance forces the existence of the photon field and uniquely determines how it couples to matter. We did not put in the interaction by hand — it emerged from a symmetry requirement. This is the paradigm for all gauge theories in the Standard Model: the form of the interaction is dictated by the gauge group (U(1) for QED, SU(2) for the weak force, SU(3) for the strong force).

3.3 The Full QED Lagrangian

Combining the gauged Dirac Lagrangian with the Maxwell kinetic term:

$$\boxed{\mathcal{L}_{\text{QED}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \bar{\psi}(i\gamma^\mu D_\mu - m)\psi}$$

Expanding the covariant derivative $D_\mu = \partial_\mu + ieA_\mu$:

$$\mathcal{L}_{\text{QED}} = \underbrace{-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}_{\text{photon kinetic}} + \underbrace{\bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi}_{\text{electron kinetic}} \underbrace{- e\bar{\psi}\gamma^\mu\psi\,A_\mu}_{\text{interaction}}$$

The three terms have clear physical interpretations:

Photon kinetic term: Describes free photon propagation (from Maxwell's equations)
Electron kinetic term: Describes free electron/positron propagation (from the Dirac equation)
Interaction term: $\mathcal{L}_{\text{int}} = -eJ^\mu A_\mu$ couples the Dirac current to the photon field

Note that a photon mass term $\frac{1}{2}m_\gamma^2 A_\mu A^\mu$ is forbidden by gauge invariance, since it would transform as $m_\gamma^2(A_\mu - \partial_\mu\alpha)(A^\mu - \partial^\mu\alpha) \neq m_\gamma^2 A_\mu A^\mu$. Gauge invariance thus predicts that the photon is massless — confirmed by experiment to extraordinary precision ($m_\gamma < 10^{-18}$ eV).

3.4 Summary of Gauge Transformations

Under a local U(1) gauge transformation parametrized by $\alpha(x)$:

\begin{align*} \psi(x) &\to e^{ie\alpha(x)}\psi(x) \\ \bar{\psi}(x) &\to \bar{\psi}(x)\,e^{-ie\alpha(x)} \\ A_\mu(x) &\to A_\mu(x) - \partial_\mu\alpha(x) \\ F_{\mu\nu}(x) &\to F_{\mu\nu}(x) \quad \text{(invariant)} \\ D_\mu\psi(x) &\to e^{ie\alpha(x)}D_\mu\psi(x) \end{align*}

The invariance of the full Lagrangian follows because every term is built from gauge-covariant building blocks: $\bar{\psi}\psi$, $\bar{\psi}D_\mu\psi$, and $F_{\mu\nu}$ are all gauge-invariant (or transform covariantly so that their combinations in $\mathcal{L}$ are invariant).

For quantization, we must fix a gauge. Common choices include:

Lorenz/Feynman Gauge

Add $-\frac{1}{2\xi}(\partial_\mu A^\mu)^2$ to the Lagrangian

$\xi = 1$: Feynman gauge (simplest propagator)

Coulomb Gauge

Impose $\nabla \cdot \mathbf{A} = 0$

Physical (only transverse photons), but not manifestly Lorentz-invariant

3.5 The Fine Structure Constant

The coupling strength of QED is characterized by the fine structure constant:

$$\boxed{\alpha = \frac{e^2}{4\pi} \approx \frac{1}{137.036}}$$

Because $\alpha \ll 1$, the perturbative expansion in powers of $\alpha$ converges rapidly. Each additional vertex in a Feynman diagram contributes a factor of $e \sim \sqrt{\alpha}$, so an n-vertex diagram is suppressed by $\alpha^{n/2}$ relative to the leading order.

This is why QED is the most precisely tested theory in physics: perturbation theory works spectacularly well. The anomalous magnetic moment of the electron has been computed to order $\alpha^5$ (five loops, involving 12,672 Feynman diagrams) and agrees with experiment to better than one part in $10^{12}$.

💡Running of the Coupling

The coupling $\alpha$ is not truly constant — it "runs" with energy scale due to vacuum polarization (virtual electron-positron pairs screen the bare charge). At $q^2 = 0$, $\alpha \approx 1/137$. At the Z-boson mass scale ($q \approx 91$ GeV),$\alpha \approx 1/128$. The coupling grows at higher energies, but remains perturbatively small throughout the experimentally accessible range.

Key Concepts

The free Dirac Lagrangian has a global U(1) symmetry; Noether's theorem gives the conserved current $J^\mu = \bar{\psi}\gamma^\mu\psi$
Promoting U(1) to a local symmetry requires the covariant derivative $D_\mu = \partial_\mu + ieA_\mu$
The gauge principle uniquely determines the interaction: $\mathcal{L}_{\text{int}} = -e\bar{\psi}\gamma^\mu\psi A_\mu$
Gauge invariance forbids a photon mass term, predicting $m_\gamma = 0$
The fine structure constant $\alpha \approx 1/137$ ensures perturbation theory converges rapidly
QED is the most precisely tested theory in all of physics

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