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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.

For emission of a soft photon of momentum $q$ from external charged particles, the Soft photon theorem giveswhere ${\eta }_i=+1$ for outgoing and $-1$ for incoming particles. Contracting with $q_{\mu }$ and applying the Ward identity givesFor a nonzero underlying amplitude, total outgoing charge equals total incoming charge. Thus photon gauge redundancy enforces conservation of electric charge.

The analogous Soft graviton theorem for helicity two requires universal coupling to energy-momentum and yields conservation of total four-momentum, the seed of the Equivalence principle. For massless helicity greater than two, the corresponding soft consistency conditions demand conserved higher-rank momentum charges that generic interacting scattering cannot satisfy. Under the assumptions of a Lorentz-invariant local S-matrix with long-range interactions, such particles therefore have no nontrivial coupling; this is the soft-theorem obstruction to interacting massless higher-spin particles.