QED Observables

The set of experimentally measured quantities that test quantum electrodynamics. This page is the inventory — the parallel for QED of the electroweak and Standard Model observable pages, filling the gap flagged in the observables map §4. The machinery that computes each entry lives in cross-sections.md, decay-rates.md, and the general observables map, specialized to the $U (1)_{EM}$ field content of QED.

QED is the most precisely tested theory in physics: its cleanest prediction, the electron $g - 2$ , agrees with experiment to better than one part in $1 0^{12}$ . The observables below are organized by the three structural classes of the observables map: (A) squared amplitudes, (B) correlator poles, (C) static vertex form factors.

1. Precision static properties (Class C)

The headline QED tests come from static electromagnetic form factors $F_{1} (q^{2}), F_{2} (q^{2})$ (the vertex-function parameterization) evaluated at $q^{2} \to 0$ .

1.1 The anomalous magnetic moment $a = (g - 2) /2 = F_{2} (0)$

Observable	Value	Where measured
Electron $a_{e}$	$1159652 180.59 (13) \times 1 0^{- 12}$	Harvard/Northwestern single-electron Penning trap
Muon $a_{μ}$	$116592059 (22) \times 1 0^{- 11}$	Fermilab / BNL storage-ring $g - 2$

The electron $a_{e}$ is the most precise confrontation of theory and experiment in all of science. Its QED prediction is a power series in $α / π$ computed to five loops (12,672 diagrams at tenth order):

$a_{e} = \frac{α}{2 π} - 0.328478 \dots (\frac{α}{π})^{2} + 1.181 \dots (\frac{α}{π})^{3} - \dots,$

the leading Schwinger term $α /2 π \approx 0.00116$ being the first-ever loop calculation (1948). The agreement with experiment is now good enough that $a_{e}$ is used to define $α$ , competitive with atom-interferometry determinations.

The muon $a_{μ}$ is $\sim (m_{μ} / m_{e})^{2} \approx 43000$ times more sensitive to heavy virtual particles, so it also receives sizeable electroweak and hadronic contributions and is a leading BSM probe. The status of the $\sim 4 σ$ "muon $g - 2$ anomaly" hinges on the hadronic vacuum-polarization input (data-driven vs. lattice QCD), an active controversy as of 2026.

1.2 Other form-factor observables

Observable	Form factor	Example
Electric charge	$F_{1} (0) = - e$	universal; charge quantization
Charge radius $⟨ r^{2} ⟩ = - 6 F_{1}^{'} (0)$	slope of $F_{1}$	the proton-radius puzzle (muonic vs. electronic hydrogen)
Electron EDM	$CP$ -odd form factor	SM value $\sim 1 0^{- 40} e cm$ (unobservably tiny); a nonzero measurement = new physics

2. Bound-state spectroscopy (Class B)

Bound states are poles of the off-shell propagator, not S-matrix entries. QED predicts atomic spectra to extraordinary precision via NRQED / Bethe–Salpeter.

Observable	Value / precision	Comment
Hydrogen Lamb shift ( $2 S_{1/2} - 2 P_{1/2}$ )	$1057.8 MHz$	the loop effect that launched QED (1947); self-energy + vacuum polarization
Hydrogen 1S–2S transition	measured to $\sim 1 0^{- 15}$	one of the most precise measurements ever; tests QED + fixes the Rydberg
Hyperfine splitting (21 cm line)	$1420.4 MHz$	electron–proton spin–spin; limited by proton structure
Positronium ( $e^{+} e^{-}$ ) spectrum & decay	1S hyperfine, $para-Ps \to 2 γ$ , $ortho-Ps \to 3 γ$	pure-QED bound state, no hadronic uncertainty
Muonium ( $μ^{+} e^{-}$ )	1S hyperfine to $\sim$ ppb	cleanest test of bound-state QED + muon properties

Positronium and muonium are prized because they contain no hadrons — every correction is pure QED, so they isolate the theory from the QCD uncertainties that limit hydrogen. The full hierarchy of approximations (Schrödinger → Dirac fine structure → Lamb shift → hyperfine) is worked out in QED/hydrogen.md.

3. Scattering cross sections (Class A)

The tree-level QED cross sections are the textbook worked examples; higher orders test the loop structure.

Process	Result	Notes
Compton $e γ \to e γ$	Klein–Nishina formula	worked end-to-end in QED/compton.md
Møller $e^{-} e^{-} \to e^{-} e^{-}$	$t, u$ -channel	polarized Møller measures $sin^{2} θ_{W}$ at low $Q^{2}$
Bhabha $e^{+} e^{-} \to e^{+} e^{-}$	$s, t$ -channel	the luminosity monitor at $e^{+} e^{-}$ colliders
Pair annihilation $e^{+} e^{-} \to γγ$	crossing of Compton
Pair production / $γγ$ scattering	light-by-light via a loop	$γγ \to γγ$ is a pure quantum effect (no tree diagram)

All follow the cross-section master formula with $\overline{∣ M ∣^{2}}$ built from the QED Feynman rules; the general tree-level pipeline is in tree-level.md.

4. The running coupling (Class B + RG)

The fine-structure constant runs with energy scale via the renormalization group:

$α (0) = \frac{1}{137.035999 \dots} running α (m_{Z}) \approx \frac{1}{128.9} .$

The low-energy value $α (0) = F_{2} (0)$ -anchored is the most accurately known fundamental constant; the growth toward the $Z$ scale is the positive QED $β$ -function (charge screening) measured directly, and is a required input to every electroweak precision fit. Extrapolated far beyond, it signals the Landau pole / triviality that marks QED as an effective theory.

5. Summary: what QED observables test

Class	Observable	Precision	Tests
C	electron $a_{e}$	$1 0^{- 12}$	loop structure to 5 loops; defines $α$
C	muon $a_{μ}$	$1 0^{- 9}$	heavy virtual particles (BSM probe)
B	Lamb shift, 1S–2S	$1 0^{- 6}$ – $1 0^{- 15}$	bound-state QED, self-energy + vacuum polarization
B	positronium/muonium	ppb	pure-QED bound states (no hadrons)
A	Compton, Bhabha, Møller	percent–permille	tree + loop amplitudes
RG	$α (Q^{2})$	$1 0^{- 9}$ at $q = 0$	charge screening / running

QED is the template against which the harder theories are calibrated: QCD observables inherit its cross-section machinery but must contend with confinement and IR safety, while electroweak observables add the massive gauge bosons and parity violation.

Pointers

QED from postulates — the theory and its successes list.
QED/compton.md — worked cross-section calculation.
QED/hydrogen.md — bound-state spectroscopy.
cross-sections.md / decay-rates.md — master formulas.
Electroweak observables / QCD observables — the sibling inventories.

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