A note on asymptotic symmetries and soft-photon theorem

We use the asymptotic data at conformal null-infinity $\mathscr{I}$ to formulate Weinberg's soft-photon theorem for Abelian gauge theories with massless charged particles. We show that the angle-dependent gauge transformations at $\mathscr{I}$ are not merely a gauge redundancy, instead they are genuine symmetries of the radiative phase space. In the presence of these symmetries, Poisson bracket between the gauge potentials is not well-defined. This does not pose an obstacle for the quantization of the radiative phase space, which proceeds by treating the conjugate electric field as the fundamental variable. Denoting by $\mathcal{G}_+$ and $\mathcal{G}_-$ as the group of gauge transformations at $\mathscr{I}^+$ and $\mathscr{I}^-$ respectively, Strominger has shown that a certain diagonal subgroup $ \mathcal{G}_{diag} \subset \mathcal{G}_+ \times \mathcal{G}_-$ is the symmetry of the S-matrix and Weinberg's soft-photon theorem is the corresponding Ward identity. We give a systematic derivation of this result for Abelian gauge theories with massless charged particles. Our derivation is a slight generalization of the existing derivations since it is applicable even when the bulk spacetime is not exactly flat, but is only "almost" Minkowskian.