Quantum Electrodynamics

Quantum electrodynamics is the simplest relativistic gauge theory with charged matter, but its precision comes from a demanding chain of reasoning: define gauge-invariant observables, normalize scattering states, renormalize without violating the Ward–Takahashi identity, combine unresolved radiation with virtual corrections, and separate physical scales. Choose the route below by the output you need. For a first pass, follow the pages in order.

Enter this chapter

If you need to…	Start here	You should leave with…
Fix the model, charges, propagators, vertex, and observable boundary	The QED action, charges, and observables	A convention-complete gauge theory and a list of gauge-invariant measurable quantities
Calculate and validate a leading scattering process	Tree-level QED processes	Normalized amplitudes and cross sections with crossing, gauge, and kinematic checks
Remove ultraviolet divergences in the electron and photon sectors	Electron and photon renormalization	Defined counterterms, pole cancellation, and an on-shell–$\overline{\mathrm{MS}}$ scheme translation
Derive the identity that protects electric charge universality	Ward–Takahashi identity and charge renormalization	The contact-term and momentum-space identities, $Z_1=Z_2$, and failure diagnostics
Turn a charged loop into screening, thresholds, and a running coupling	Vacuum polarization and the running charge	A transverse polarization tensor, its analytic structure, and low-/high-energy limits
Define a finite observable in the presence of arbitrarily soft photons	Soft photons and infrared-finite QED	An explicit real–virtual cancellation tied to energy, angular, and detector resolution
Extract a magnetic moment and organize a precision comparison	Lepton magnetic moments	$F_1(0)=1$, $a_\ell=F_2(0)$, the one-loop anchor, and a covariance-aware sector decomposition
Predict atomic energy levels without mixing hard, potential, and nuclear physics	Bound-state QED and NRQED	An NRQED operator and scale hierarchy plus an evidence-ready spectroscopy workflow

Suggested order. A first pass follows the table from top to bottom.

Hard dependencies. The “Required background” note on each leaf is binding; a “Helpful background” note only improves fluency. Within this chapter, tree-level processes require the action; renormalization requires the action; the Ward–Takahashi derivation and vacuum polarization require renormalization; soft-photon predictions require tree amplitudes plus the external KLN prerequisite; magnetic moments require the Ward identity plus form factors; and bound-state QED requires vacuum polarization plus nonrelativistic EFT. These dependencies are physical, not merely editorial.

The renormalized vertex and propagators must satisfy the Ward–Takahashi identity. Vacuum polarization changes the effective charge, soft radiation changes which cross section is finite, and both effects feed precision observables. Bound states then add nonrelativistic scale separation and nonperturbative Coulomb exchange.

Shared QED conventions

This volume uses the $(+---)$ metric and natural units. For a field of signed charge $q$,

$$D_\mu=\partial_\mu+iqA_\mu, \qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \alpha=\frac{e^2}{4\pi},$$

where $e>0$ and an electron has $q=-e$. The invariant Lagrangian is

$$\mathcal L_{\mathrm{QED}} =-\frac14F_{\mu\nu}F^{\mu\nu} +\bar\psi(i\gamma^\mu D_\mu-m)\psi.$$

Covariant gauge fixing is a calculational choice; physical observables must be independent of its parameter. External-state normalization, charge sign, momentum flow, subtraction scheme, infrared resolution, and scale choices are stated where they first affect a result. Dirac contractions use $p\!\!\!/\equiv\gamma^\mu p_\mu$.

The chapter mostly treats one charged Dirac species and then states how additional lepton masses or a second bound constituent enter. It does not silently identify three distinct quantities:

• a bare parameter in a regulated Lagrangian;
• a renormalized parameter in a named scheme at a named scale;
• a coupling or form factor extracted from a physical observable.

Keeping those layers distinct is essential when translating between low-energy electric charge, $\overline{\mathrm{MS}}$ parameters, and momentum-dependent effective couplings. Standard treatments of this chain are given in Schwartz 2014, Chapters 13–20, pp. 224–380 and Weinberg 1995, Chapters 10–14, pp. 436–589.

The calculation–validation loop

Every worked QED prediction in this chapter follows the same compact loop.

1. Name the observable. Give external states, inclusiveness, polarization treatment, cuts or resolution, and the perturbative order.
2. Fix conventions. State signed charges, momentum flow, state normalization, regulator, subtraction scheme, and scale.
3. Construct the amplitude or EFT. Include every diagram or operator required by the symmetry and power counting.
4. Renormalize and factorize. Cancel ultraviolet poles with defined counterterms and infrared singularities only after forming a sufficiently inclusive or dressed observable.
5. Apply identities. Test current conservation, the Ward–Takahashi relation, photon transversality, and renormalization-scale cancellation.
6. Take diagnostic limits. Use soft, collinear, threshold, static, high-energy, heavy-source, or point-source limits as appropriate.
7. Propagate uncertainty. Separate input, truncation, numerical, model, and experimental components and retain correlations.

A result is not validated merely because it is finite. The following table identifies what different checks can actually establish.

Check	Typical QED test	What a failure means
Gauge invariance	Replace $\varepsilon^\mu$ by $k^\mu$; verify $k_\mu\Pi^{\mu\nu}=0$	Missing diagrams, a routing error, or a symmetry-breaking regulator/counterterm choice
Ultraviolet consistency	Cancel every $1/\bar\epsilon$ pole and verify $Z_1=Z_2$	Incorrect counterterm definitions or an incomplete renormalized amplitude
Infrared consistency	Cancel the common regulator between real and virtual terms	An observable that is not inclusive enough or mismatched phase-space definitions
Analytic structure	Locate the pair-production threshold and branch cut	A wrong continuation, subtraction, or imaginary-part sign
Power counting	Recover Coulomb, heavy-source, and point-source limits	Double counting or an operator assigned to the wrong scale
Dimensions and normalization	Check flux, spin averages, $F_1(0)=1$, and mass dimensions	A convention mismatch that can survive algebraic simplification

Passing one row does not imply the others. A transverse vacuum polarization may still use the wrong subtraction, and an infrared-finite rate may still be normalized with the wrong flux.

Observable and preparation readiness

Before calculating, be able to state the external states or bound levels, the measured quantity, its polarization treatment and inclusiveness, the scheme and scale of every renormalized input, and the target perturbative accuracy. Use these unscored checks to repair the first missing element.

• If you cannot derive the photon and fermion propagators from a gauge-fixed quadratic action or explain Gauss’s law, begin with the action and observables.
• If traces, spin averages, two-body phase space, or crossing are unfamiliar, work through tree-level processes before loops.
• If you can compute a loop integral but cannot state which finite quantity defines the coupling or mass, begin with electron and photon renormalization.
• If ultraviolet renormalization is familiar but a photon mass or dimensional infrared pole remains in a cross section, use the soft-photon method.
• If your target is a magnetic moment or atomic transition, first verify the form-factor and scale prerequisites listed on those pages; precision does not shorten the dependency chain.
• If you cannot say which energy/angular resolution makes two radiative events equivalent, repair the observable on the soft-photon page.
• If a prediction–measurement comparison lacks input versions or covariance, repair it with the magnetic-moment comparison or the spectroscopy comparison, according to the observable.

No score or completion claim attaches to this diagnostic. Its purpose is to expose the first missing capability.

Informal synthesis check

Consider an unpolarized charged-lepton scattering observable measured at a hard scale $Q\gg m_\ell$, inclusive over photons below a declared energy and angular resolution. Give a reproducible prediction through the first radiative correction.

A satisfactory analysis should:

1. derive the Born amplitude with signed charges, spin averages, flux, and phase space;
2. define $Z_2$, $Z_3$, $Z_m$, and the vertex counterterm in one scheme;
3. use the Ward–Takahashi identity to relate the vertex and external-leg ultraviolet structure;
4. include the vacuum-polarization correction with its subtraction and timelike/Euclidean qualification;
5. add unresolved real radiation to the virtual term and show the auxiliary infrared regulator cancels;
6. retain the physical energy, angle, mass, and scale logarithms rather than deleting them;
7. report parametric, missing-order, and numerical uncertainties with any shared covariance.

Use this repair map: step 1 → tree-level processes; step 2 → electron and photon renormalization; step 3 → the Ward–Takahashi identity; step 4 → vacuum polarization; step 5 → the soft-photon construction; step 7 → magnetic-moment uncertainty separation and, for levels, the NRQED comparison. If the fixed-order answer contains large logarithms, exit to the volume’s factorization and resummation prerequisites rather than interpreting the logarithm as a divergence.

Scope and exits

Choose the exit by purpose:

• for non-Abelian self-interactions, color factors, and gauge-boson vertices, continue to Yang–Mills theory;
• for weak corrections and spontaneous symmetry breaking, continue to electroweak theory and the Higgs;
• for hadronic contributions to electromagnetic observables, continue to hadrons and heavy quarks and the relevant QCD route;
• for global combinations, electroweak fits, and dated comparison records, continue to precision Standard Model tests.

The chapter’s boundary is also methodological. Gauge-parameter independence does not remove the need to define an observable; ultraviolet renormalization does not guarantee infrared finiteness; and a small coupling does not make Coulomb binding perturbative term by term.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.