Quantum Electrodynamics

Quantum Electrodynamics (QED) is the relativistic quantum field theory of electrons, positrons, and photons. It is obtained by specializing the general postulates of Quantum Field Theory with three additional inputs: a choice of field content, a gauge symmetry, and the renormalizable Lagrangian uniquely determined by these together with Lorentz and discrete-symmetry invariance.

Three routes to QED. This document takes the gauge-theory viewpoint, in which local $U(1)$ invariance is postulated and minimal coupling is derived. Two companion presentations cover the same theory from different starting points:

• Modern Foundations — Wigner–Weinberg derivation shows (§5.2) that gauge invariance is itself a theorem of Lorentz consistency for any massless spin-1 species. This document picks up where that one ends, taking the QFT postulates as given and adding the QED-specific ingredients. See §Equivalent framings in Step 2 for the relation.
• QED — Historical (Dirac-Equation) Route starts from Dirac's single-particle equation; minimal coupling is postulated there and gauge invariance is derived as a consequence.

Following the tag convention of foundations-modern.md, each step below is labelled as (Empirical input), (Postulate), (Theorem), or (Standard machinery) so the assumption budget on top of the QFT postulates is transparent.

Derivation of QED from QFT

QED inherits all postulates of QFT (relativistic state space, spectrum condition, unique Poincaré-invariant vacuum, fields as operator-valued distributions, microcausality, spin–statistics, cyclicity of the vacuum, dynamics from a local action, and asymptotic completeness). What follows is the chain of choices that turns the generic QFT framework into QED.

Step 1 — Specify the Field Content (Empirical input; specializes QFT Postulate 4)

QED postulates two fundamental fields:

• A Dirac spinor field $\psi (x)$, the electron/positron field, transforming in the $(\frac{1}{2},0)\oplus (0,\frac{1}{2})$ representation of $SL(2,\mathbb{C})$.
• A vector field $A_{\mu }(x)$, the photon field, transforming in the $(\frac{1}{2},\frac{1}{2})$ representation.

By the spin–statistics theorem (QFT Postulate 7), $\psi$ is quantized with anticommutators (fermion) and $A_{\mu }$ with commutators (boson).

Step 2 — Postulate a Local $U(1)$ Gauge Symmetry (Postulate)

The genuinely new ingredient — generic QFTs have no such requirement — is the assumption of an internal local $U(1)$ symmetry:

$$\psi (x)\to e^{i\alpha (x)}\psi (x),A_{\mu }(x)\to A_{\mu }(x)-\frac{1}{e}{\partial }_{\mu }\alpha (x),$$

with $\alpha (x)$ an arbitrary real function. This single postulate has two immediate consequences:

• The photon must be massless: a term $m_{\gamma }^2{A}_{\mu }{A}^{\mu }$ is not gauge-invariant.
• The way electrons couple to photons is uniquely fixed by the requirement that derivatives of $\psi$ appear only through the covariant derivative $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$, which transforms as $D_{\mu }\psi \to e^{i\alpha }{D}_{\mu }\psi$.

Equivalent framings. Postulating gauge invariance and deriving photon masslessness (this doc) is logically equivalent to postulating a massless spin-1 species and deriving gauge invariance — the route taken in foundations-modern.md §5.2. The two are the same theory viewed from opposite ends: foundations-modern starts from Wigner classification (massless spin-1 exists as an irrep) and shows gauge invariance is forced by Lorentz consistency of the non-tensorial polarization vectors; this document starts from gauge invariance and shows the photon must be massless. Either choice fixes the same ${\mathcal{L}}_{\text{QED}}$. The gauge-symmetry-first framing is taken here because it generalizes more cleanly to the non-abelian case (see QCD).

Step 3 — Build the Lagrangian (Theorem; specializes QFT Postulate 9)

The most general Lagrangian density that is

• Lorentz-invariant,
• gauge-invariant under the $U(1)$ above,
• $C$-, $P$-, and $T$-invariant (separately),
• renormalizable (operators of mass dimension $\le 4$ in $d=4$ spacetime dimensions),
• built only from $\psi$, $\bar{\psi }$, $A_{\mu }$, and their derivatives,

is uniquely

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

where

• $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$ is the covariant derivative,
• $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ is the electromagnetic field strength,
• ${\gamma }^{\mu }$ are the Dirac gamma matrices satisfying $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$,
• $e$ is the electric charge and $m$ the electron mass — the only two free parameters.

Renormalizability rules out higher-dimension operators such as $(\bar{\psi }\psi {)}^2$ (dim 6), $\bar{\psi }{\sigma }^{\mu \nu }\psi F_{\mu \nu }$ (dim 5, the Pauli term), and $F^3$-type interactions. The hypothetical $CP$-violating term $\theta {ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }{F}_{\rho \sigma }$ is a total derivative in QED and so does not contribute to perturbation theory.

The classical equations of motion that follow from $\delta S=0$ are the Dirac equation in an external field and Maxwell's equations with a Dirac source:

$$(i{\gamma }^{\mu }{D}_{\mu }-m)\psi =0,{\partial }_{\mu }{F}^{\mu \nu }=e\bar{\psi }{\gamma }^{\nu }\psi =j^{\nu }.$$

Step 4 — Quantize (Standard machinery)

The fields are promoted to operators following the standard QFT machinery (QFT Postulate 9). Two equivalent routes:

Canonical quantization. Compute conjugate momenta:

$${\pi }_{\psi }=\frac{\partial \mathcal{L}}{\partial ({\partial }_0\psi )}=i{\psi }^{†},{\pi }^i=\frac{\partial \mathcal{L}}{\partial ({\partial }_0{A}_i)}=-F^{0i}=E^i,{\pi }^0=0.$$

The vanishing of ${\pi }^0$ is a primary constraint, signalling gauge redundancy in $A_{\mu }$. Resolving it requires a gauge-fixing prescription — Lorenz gauge ${\partial }_{\mu }{A}^{\mu }=0$ (Gupta–Bleuler), Coulomb gauge $\nabla \cdot \mathbf{A}=0$, axial gauge, or (most systematically) BRST quantization. After gauge fixing, impose canonical (anti)commutation relations on the physical degrees of freedom.

Path-integral quantization. Define the generating functional

$$Z[J^{\mu },\bar{\eta },\eta ]=\int \mathcal{D}A\mathcal{D}\bar{\psi }\mathcal{D}\psi \exp \left(i\int d^{4}x[{\mathcal{L}}_{\text{QED}}+{\mathcal{L}}_{\text{gf}}+{\mathcal{L}}_{\text{ghost}}+J^{\mu }{A}_{\mu }+\bar{\eta }\psi +\bar{\psi }\eta ]\right),$$

with a covariant gauge-fixing term, e.g.

$${\mathcal{L}}_{\text{gf}}=-\frac{1}{2\xi }({\partial }_{\mu }{A}^{\mu }{)}^2,$$

and Faddeev–Popov ghosts ${\mathcal{L}}_{\text{ghost}}$. In QED the ghosts decouple (they only matter in non-abelian gauge theories), so they can be ignored for practical computations.

Step 5 — Renormalize (Standard machinery)

Naive perturbation theory in $e$ produces UV-divergent loop integrals. Renormalizability of ${\mathcal{L}}_{\text{QED}}$ guarantees that all divergences can be absorbed into a finite number of multiplicative redefinitions:

$${\psi }_0=Z_2^{1/2}\psi ,A_{0\mu }=Z_3^{1/2}{A}_{\mu },m_0=Z_{m}m,e_0=Z_{e}e.$$

The Ward–Takahashi identities, which are exact consequences of gauge invariance, imply

$$Z_1=Z_2\text{and hence}{Z}_e=Z_3^{-1/2}.$$

Thus the renormalization of the electric charge is governed entirely by the photon self-energy — a non-trivial check that gauge invariance survives quantization.

Step 6 — Predictions via the S-Matrix (Standard machinery; specializes QFT Postulate 10)

Time-ordered correlators $⟨0|T\psi (x_1)\cdots A_{\mu }(y_1)\cdots |0⟩$ are computed perturbatively in $e$ using Feynman rules read off from ${\mathcal{L}}_{\text{QED}}$:

The LSZ reduction formula extracts $S$-matrix elements from amputated time-ordered correlators; cross sections follow from $|\mathcal{M}{|}^2$ via standard kinematics (master formulas and conventions in QFT/cross-sections.md, with the parallel decay-rate treatment in decay-rates.md).

Summary: What QED Adds to QFT

So the "derivation" of QED from QFT amounts to three choices: pick the matter and gauge fields, demand local $U(1)$, and require renormalizability. The dynamics — including the precise form of the electron–photon coupling and all of QED's celebrated predictions ($g-2$, Lamb shift, Compton scattering, ...) — are then uniquely fixed.

Successes and Tested Predictions

QED is the most precisely tested theory in physics. A few highlights:

• Anomalous magnetic moment of the electron: $a_e=(g-2)/2$ agrees with experiment to better than one part in $10^{12}$ when combined with the most precise measurement of the fine-structure constant.
• Lamb shift in hydrogen: the splitting between $2S_{1/2}$ and $2P_{1/2}$ levels, predicted by QED loop corrections.
• Compton, Bhabha, and Møller scattering: tree-level cross sections and their loop corrections.
• Hyperfine structure of hydrogen and positronium.

Caveats and Open Issues

• Landau pole / triviality. The QED coupling $e(\mu )$ grows with energy scale $\mu$ via the renormalization-group equation; extrapolated naively, $e(\mu )$ diverges at an enormous (and unphysical) Landau pole. This suggests pure QED is only an effective theory, valid below some cutoff — and indeed it is embedded in the electroweak theory above $\sim 100\mathrm{GeV}$.
• Haag's theorem prevents the interaction picture from existing rigorously; perturbative QED is best understood as a formal expansion justified by renormalization rather than as a strict consequence of the Wightman axioms.
• No rigorous construction in $d=4$. As with all interacting QFTs in four spacetime dimensions, a fully rigorous mathematical construction of QED satisfying the Wightman/Haag–Kastler axioms is an open problem.