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title: The Represented Divergence paper: “Donnay & Herfray, ‘Infrared physics of QED and gravity from representation theory’ (arXiv:2603.06297)” tags: QFT, infrared, representation-theory, BMS, asymptotic-symmetry, gravity, QED, S-matrix

Infrared divergences in quantum electrodynamics and gravity appear when you try to compute scattering amplitudes. Scatter two charged particles, and the amplitude for producing zero soft photons is infinite. The standard treatment regulates the divergence, introduces a cutoff, and argues that physically observable quantities — inclusive cross-sections summed over unresolvable soft radiation — are finite. The divergence is treated as a technical problem with a technical fix.

Donnay and Herfray argue the divergence isn’t a problem. It’s a representation.

The Poincaré group — the symmetry group of flat spacetime — is not the full symmetry of scattering. In QED, large gauge transformations at null infinity extend the symmetry to include angle-dependent U(1) transformations. In gravity, BMS supertranslations extend the Poincaré group to an infinite-dimensional group that acts nontrivially on asymptotic states. These are the asymptotic symmetry groups, and they are larger than the groups textbook scattering theory uses.

The unitary irreducible representations of these extended symmetry groups encode exactly the infrared physics that appears as divergences in the smaller representation. Soft photon theorems, memory effects, charge conservation at every angle — all emerge as properties of the UIRs. The infinite-dimensional symmetry group has representations that naturally carry the information about soft radiation. What looks like a divergence in the Poincaré representation is a structural feature of the BMS representation.

The goal — an infrared-finite S-matrix built from these representations — follows: if you use the right symmetry group, the divergences never appear. They were artifacts of using too small a group, not properties of the physics. The notation was loaded: the Poincaré group carried the implicit claim that scattering doesn’t involve asymptotic symmetry, and that claim manifested as a divergence.