Abstract:Recently it was conjectured that a certain infinite-dimensional "diagonal" subgroup of BMS supertranslations acting on past and future null infinity (I − and I + ) is an exact symmetry of the quantum gravity S-matrix, and an associated Ward identity was derived. In this paper we show that this supertranslation Ward identity is precisely equivalent to Weinberg's soft graviton theorem. Along the way we construct the canonical generators of supertranslations at I ± , including the relevant soft graviton contributions. Boundary conditions at the past and future of I ± and a correspondingly modified Dirac bracket are required. The soft gravitons enter as boundary modes and are manifestly the Goldstone bosons of spontaneously broken supertranslation invariance.

An infinite number of physically nontrivial symmetries are found for abelian gauge theories with massless charged particles. They are generated by large U (1) gauge transformations that asymptotically approach an arbitrary function ε(z,z) on the conformal sphere at future null infinity (I + ) but are independent of the retarded time. The value of ε at past null infinity (I − ) is determined from that on I + by the condition that it take the same value at either end of any light ray crossing Minkowski space. The ε = constant symmetries are spontaneously broken in the usual vacuum. The associated Goldstone modes are zero-momentum photons and comprise a U (1) boson living on the conformal sphere. The Ward identity associated with this asymptotic symmetry is shown to be the abelian soft photon theorem.Recently a general equivalence relation has emerged between soft theorems and asymptotic symmetries [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Soft theorems are relations between n and n + 1 particle scattering amplitudes, where the extra particle is soft. Any linear relation between scattering amplitudes can be recast as an infinitesimal symmetry of the S-matrix. It is gratifying that in some cases the resulting symmetries have turned out to be known space-time or gauge symmetries. For example Weinberg's soft graviton theorem [20,21] is equivalent to a symmetry of the S-matrix generated by a certain diagonal subgroup [2] of the product of BMS [22] supertranslations acting on past and future null infinity, I + and I − .This equivalence relation is of interest for several reasons. It "explains" why soft theorems exist and are so universal: they arise from a symmetry principle. Moreover, it imparts observational meaning to Minkowskian asymptotic symmetries, which have at times eluded physical interpretation.The framework has proven useful for establishing new symmetries [14] and new soft theorems [4][5][6]. In the quantum gravity case, the symmetries provide the starting point for any attempt at a holographic formulation, see e.g. [23]. In the gauge theory case, they are potentially useful for improving the accuracy of collider predictions, see e.g. [24].The purpose of the present paper is to argue that the soft photon theorem in massless QED [25] [20] [26] can be understood as a new asymptotic symmetry. The symmetry is generated by "large" U(1) gauge transformations which approach an arbitrary function ε(z,z) on the conformal sphere at I but are constant along the null generators, even as they antipodally cross from I − to I + through spatial infinity. Except for the constant transformation, these symmetries are spontaneously broken

Scattering amplitudes of any four-dimensional theory with nonabelian gauge group G may be recast as two-dimensional correlation functions on the asymptotic twosphere at null infinity. The soft gluon theorem is shown, for massless theories at the semiclassical level, to be the Ward identity of a holomorphic two-dimensional G-Kac-Moody symmetry acting on these correlation functions. Holomorphic Kac-Moody current insertions are positive helicity soft gluon insertions. The Kac-Moody transformations are a CP T invariant subgroup of gauge transformations which act nontrivially at null infinity and comprise the four-dimensional asymptotic symmetry group.

No abstract