Symmetries of free massless particles and soft theorems

TL;DR

The paper develops a Hilbert-space framework for massless particles in which the Lorentz group acts via the unitary principal series and the null-infinity structure is manifest. It identifies infinite-dimensional symmetries in the free theory—(i) BMS-like supertranslations realized on $(u,z,\bar z)$ and (ii) a global $U(1)$ Kac-Moody sector—then connects these to the leading soft photon and soft graviton theorems through a Mellin-like celestial S-matrix formalism. By introducing the variables $(U,\omega)$ and soft operators $J_{\sigma,\omega}$, the work rewrites the S-matrix such that soft theorems become Ward identities for these symmetry generators, including explicit expressions for $J_{+}$ and $O_{+}$ and a generator $T(g)$ of supertranslations; it also discusses the invariance of the tilde amplitude under global translations. The discussion highlights that while these symmetries are clear at the level of the free theory, extending them to interacting theories (notably with gravity) and to subleading soft theorems, massive cases, and holographic interpretations remain important open directions.

Abstract

In an earlier paper we have constructed a basis of massless single particle quantum states which transform in the unitary principal series representation of the four dimensional Lorentz group. The S-matrix written in this basis gives rise to the Mellin transformed amplitude of Pasterski-Shao-Strominger and its generalization. In this basis the particle can be thought of as living on the null-infinity in the Minkowski space. In this paper we take some preliminary steps to see how the connection between soft theorems and symmetries work out in this picture. For simplicity we consider only the leading soft photon and soft graviton theorems which are related to U(1) Kac-Moody and supertranslations.