Renormalization of QED

Core Idea

Renormalization of QED absorbs ultraviolet divergences into redefinitions of the electron mass, the electric charge, and the field normalizations. Only three types of divergent diagrams exist in QED (self-energy, vacuum polarization, vertex correction), making the theory renormalizable: all divergences at every order are absorbed by a finite number of counterterms.

Explainer

The three divergent diagrams of QED -- the electron self-energy, the vacuum polarization, and the vertex correction -- each modify one of the basic elements of the theory. The self-energy shifts the electron mass and normalizes the electron field. The vacuum polarization modifies the photon propagator and renormalizes the electric charge. The vertex correction modifies the electron-photon coupling. Renormalization absorbs these divergences into redefinitions of the bare parameters.

The procedure is systematic. Start with the bare Lagrangian L = psi_0-bar(i gamma^mu partial_mu - m_0)psi_0 - e_0 psi_0-bar gamma^mu psi_0 A_0_mu - (1/4)F_0². Introduce renormalized fields and parameters: psi_0 = sqrt(Z_2) psi_R, A_0 = sqrt(Z_3) A_R, m_0 = m_R + delta_m, e_0 = Z_1 Z_2^-1 Z_3^-1/2 e_R. Rewrite the Lagrangian in terms of renormalized quantities; the leftover pieces are counterterms that exactly cancel the divergences from loop diagrams. The counterterms delta_m, delta_Z2 = Z_2 - 1, delta_Z3 = Z_3 - 1, and delta_Z1 = Z_1 - 1 are fixed by renormalization conditions that specify the physical mass, charge, and field normalization.

The Ward identity Z_1 = Z_2, a consequence of gauge invariance, is crucial. It ensures that the charge renormalization comes entirely from the vacuum polarization (Z_3), so the renormalized charge is e_R = e_0 Z_3^1/2. This means the electric charge of every particle is renormalized by the same factor, regardless of the particle's mass or spin -- the universality of charge that we observe experimentally. Without the Ward identity, different particles could have different charge renormalizations, and the equality of the proton and electron charges would be an unexplained coincidence.

The triumph of renormalized QED is its predictive power. Once three quantities are measured (the electron mass, the fine structure constant, and the field normalization convention), every other prediction of QED is determined. The anomalous magnetic moment of the electron, the Lamb shift, the hyperfine splitting of hydrogen, photon-photon scattering -- all are computed as power series in alpha with no free parameters. The agreement with experiment (to 12 significant figures for the electron g-2) is the most stringent test of any physical theory ever performed.