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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.

The Weinberg soft theorem constrains interactions of a massless particle by requiring its soft-emission amplitude to be invariant under shifts of an unphysical polarization. For helicity one it gives charge conservation, for helicity two it gives universal gravitational coupling, and for higher helicity it obstructs generic long-range interactions.