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.
Highlights
This paper is organized as follows — in section 2 we review the definition of asymptotic flatness and the geometry of null infinity I ; in section 3 we review Weinberg’s soft-photon theorem and express it in terms of the coordinates intrinsic to I ; in section 4 we introduce the radiative phase space of electromagnetism and quantize it; in section 5 we review Strominger’s proposal for new symmetries of the scattering problem in massless quantum electrodynamics; in section 6 we impose the invariance of the S-matrix under large gauge symmetry to derive the Weinberg’s soft-photon theorem
In order to promote the symmetry of the phase space to the symmetry of the S-matrix, Strominger suggested to first identify the null generators of I + and I − by the antipodal mapping of the S2 cross-sections and equate the gauge parameters on I + and I − along the identified generators [5]
In this paper we have derived the Weinberg’s soft-photon theorem for massless Abelian gauge theory as a Ward identity corresponding to the diagonal subgroup of the group of large gauge symmetries acting on the radiative data on I + and I −
Summary
We review the geometry of null infinity I. The vector fields on I that respect this universal structure are the infinitesimal generators of the asymptotic symmetry group of asymptotically flat spacetimes. This group is called the BMS1 group after the founders Bondi, van der Burg, Metzner [25] and Sachs [26]. Where β is a scalar on I such that Lnβ = 0 Those ξa which are of the form ξa = f na, where f is a function such that Lnf = 0, generate the normal subgroup of the BMS group and are called supertranslations (ST).
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