A note on asymptotic symmetries and soft-photon theorem

TL;DR

This work reframes Weinberg's soft-photon theorem as a Ward identity for the diagonal subgroup of large gauge transformations acting on radiative data at null infinity, extending the derivation to spacetimes that are close to but not exactly Minkowskian. It develops a systematic treatment of the radiative phase space, showing large-gauge symmetries are genuine symmetries and, upon quantization, identify soft photons as edge states associated with spontaneous symmetry breaking. A key result is the explicit connection between the S-matrix Ward identity and the soft factor, established both in intrinsic $\mathscr{I}$ data and through mode expansions in the $\mathcal{E}$-representation. The approach highlights the role of boundary degrees of freedom in encoding infrared physics and motivates potential extensions to non-Abelian gauge theories and gravity, with speculative links to holography and information aspects of black holes.

Abstract

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.