Quantum Electrodynamics

Quantum electrodynamics couples charged matter to the electromagnetic field. For one Dirac species, it explains emission, absorption, pair production, and electromagnetic scattering with a common coupling. Its perturbative accuracy rests on a small renormalized charge and a physically defined observable, including the treatment of soft photons.

Charged matter and the Maxwell field

In $3+1$ dimensions, the gauge-invariant classical Lagrangian is

$$\mathcal L=-\frac14F_{\mu\nu}F^{\mu\nu} +\bar\psi(i\gamma^\mu D_\mu-m)\psi, \qquad D_\mu=\partial_\mu-iqA_\mu .$$

Here $q$ is the signed charge; for an electron $q=-e$ with $e>0$. Expanding the covariant derivative gives $+qA_\mu\bar\psi\gamma^\mu\psi$. This explicit sign prevents confusing the positive coupling magnitude with the electron’s charge.

The transformation $\psi\mapsto e^{iq\alpha}\psi$, $A_\mu\mapsto A_\mu+\partial_\mu\alpha$ leaves the action invariant. Gauge fixing is needed for many perturbative representations, while gauge-invariant observables and physical predictions must be independent of that choice. Individual propagators and their residues need not be. See Srednicki 2007, §§58–59.

Charge conservation at an external photon

The current matrix element between on-shell fermions contains $\bar u(p')\gamma^\mu u(p)$. If the photon carries $k=p'-p$, then

$$k_\mu\bar u(p')\gamma^\mu u(p) =\bar u(p')(p'\!\!\!/-p\!\!\!/)u(p)=0.$$

The two terms become $m\bar u(p')u(p)$ and cancel. This is the simplest Ward-identity check: replacing an external photon polarization by its momentum makes this current contribution vanish. In a scattering process all diagrams required by the identity must be included.

Running, observables, and infrared limits

For one unit-charge Dirac species at renormalization scales well above its mass, the one-loop running is

$$\mu\frac{\mathrm de}{\mathrm d\mu} =\frac{e^3}{12\pi^2}+O(e^5).$$

The positive sign means the effective coupling grows toward shorter distances. Additional charged species change the coefficient, and threshold matching matters. Extrapolating the one-loop formula to an arbitrarily high scale is not a controlled ultraviolet completion. The calculation is developed in Srednicki 2007, §66.

Cross sections, magnetic moments, and bound-state energy shifts are useful observables. Exclusive charged-particle amplitudes have infrared subtleties because photons are massless. A detector’s energy resolution, an inclusive sum over unresolved radiation, or a specified dressed-state construction is part of a meaningful prediction.

Develop the calculation

The QED Action, Charges, and Observables fixes the detailed structure. Maxwell Theory and Dirac Field recover the uncoupled ingredients. Compare the Schwinger Model: the same gauge/matter pattern in $1+1$ dimensions leads to very different infrared physics.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §§58–59 and 66–68. DOI.