= Solution
For emission of a soft photon of momentum $q$ from external charged particles, the <Soft photon theorem> gives
$$
\mathcal M^\mu(q)
=\left(\sum_i\eta_i Q_i\frac{p_i^\mu}{p_i\mathbin\cdot q}\right)\mathcal M_0+O(q^0),
$$
where $\eta_i=+1$ for outgoing and $-1$ for incoming particles. Contracting with $q_\mu$ and applying the Ward identity gives
$$
\left(\sum_i\eta_iQ_i\right)\mathcal M_0=0.
$$
For 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.
Back to article page
