Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 2 / iii

The leading Soft photon theorem for a photon of momentum $q$ iswhere ${\eta }_i=+1$ for outgoing and $-1$ for incoming particles. Replacing $ϵ$ by $q$ and applying the Ward identity gives ${\sum }_i{\eta }_i{Q}_i=0$, which is electric charge conservation.

The analogous Weinberg soft theorem for helicity two forces every particle to couple to a massless spin-two field with the same strength; its Ward identity reduces to momentum conservation and yields the equivalence principle. For helicity strictly greater than two, the corresponding polynomial constraints have no nontrivial solution for a generic interacting S-matrix under the usual assumptions of Lorentz invariance, locality, and finitely many particle species. This is the soft-theorem obstruction to ordinary long-range interactions mediated by massless higher-spin particles.