No book on Green's functions can omit a discussion, however brief, on the subject of Infrared (IR)/no-recoil/Block–Nordsieck models, which have played a central role in the description of those processes for which there is a clear, physical separation between large and small frequencies. The most famous example is probably the removal of all IR divergences in QED, and the way in which this may most simply be illustrated clearly exhibits the power of functional methods. A second example, which has hardly received the attention it merits, is the propensity of virtual soft-photons to damp processes involving large momentum transfers, a subject which will be briefly described in Section 7.2.

In modern language, the true role of soft photons, or very low-frequency photon fluctuations, was anticipated by Bloch and Nordsieck in their seminal paper of 1937; but not until almost two decades had passed was a proof given of the cancellation of all IR divergences for any scattering process in QED. The latter constructions, by Yennie, Frautschi, and Suura, were performed by identifying and extracting the IR divergent terms in every order of perturbation theory, and then summing over them all to obtain a final |amplitude| explicitly free of all IR divergences. Shortly afterwards, Schwinger and Mahanthappa invented a functional method for the direct calculation of probabilities, in which the IR difficulties never appear.