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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.

Let $O=\phi ({\mathbf{k}}_1)\phi ({\mathbf{k}}_2)$. Since $Q_S=-2{\omega }_{ij}{\Pi }_{ij}(0)$,For a soft mode ${\gamma }^s=f_q{a}_s+f_q^{\ast }{a}_s^{†}$ and ${\Pi }^s=g_q{a}_s+g_q^{\ast }{a}_s^{†}$, the vacuum obeysThe mode-function Wronskian normalizationimplieswhere $P_{\gamma }(q)$ is the graviton power spectrum per polarization. The equal-time bispectrum is real in this parity-even configuration, and thereforeThis turns the charge insertion into the soft-graviton insertion used in the Soft graviton theorem.