Infrared physics of QED and gravity from representation theory

The infrared structure of QED and gravity is known to be governed by an infinite-dimensional symmetry group which extends the Poincaré group to include, respectively, large $U(1)$ transformations and BMS supertranslations. We describe how the unitary irreducible representations (UIRs) of these asymptotic symmetry groups encode universal infrared features of a scattering process. Motivated by the goal of defining an infrared-finite $S$-matrix based on these UIRs, we also study supermomentum eigenstates and contrast our construction with the dressed-state approach for infrared-safe amplitudes.