Building the Full Fermion-Photon Vertex of QED by Imposing Multiplicative Renormalizability of the Schwinger-Dyson Equations for the Fermion and Photon Propagators

Understand

In principle, calculation of a full Green's function in any field theory requires knowledge of the infinite set of multi-point Green's functions, unless one can find some way of truncating the corresponding Schwinger-Dyson equations.

• For the fermion and boson propagators in QED this requires an {\it ansatz} for the full three point vertex.
• Here we illustrate how the properties of gauge invariance, gauge covariance and multiplicative renormalizability impose severe constraints on this fermion-boson interaction, allowing a consistent truncation of the propagator equations.
• We demonstrate how these conditions imply that the 3-point vertex {\bf in the propagator equations} is largely determined by the behaviour of the fermion propagator itself and not by knowledge of the many higher point functions.