Null Infinity and Unitary Representation of The Poincare Group

TL;DR

The paper constructs a massless-particle basis transforming under the unitary principal continuous series of the Lorentz group, labeled by $(h,\bar h, z,\bar z)$ and interpreted as living on null infinity. It develops both 2+1 and 3+1 dimensional realizations, showing how Poincaré symmetry acts naturally on this boundary space and how translations move along a Bondi-like time coordinate $u$. By second quantization, it yields a manifestly unitary, Poincaré-invariant field theory of free massless particles on null infinity, with fields forming ISL(2, C) primaries and a modified Mellin transform of S-matrix elements. The work reveals structure resembling BMS supertranslations at the boundary, connects to conformal primary wave-functions, and points toward a holographic-like description of flat-space scattering at null infinity.

Abstract

Following Pasterski-Shao-Strominger we construct a new basis of states in the single-particle Hilbert space of massless particles as a linear combination of standard Wigner states. Under Lorentz transformation the new basis states transform in the Unitary Principal Continuous Series representation. These states are obtained if we consider the little group of a null momentum \textit{direction} rather than a null momentum. The definition of the states in terms of the Wigner states makes it easier to study the action of space-time translation in this basis. We show by taking into account the effect of space-time translation that \textit{the dynamics of massless particles described by these states takes place completely on the null-infinity of the Minkowski space}. We then second quantize the theory in this basis and obtain a unitary manifestly Poincare invariant (field) theory of free massless particles living on null-infinity. The null-infinity arises in this case purely group-theoretically without any reference to bulk space-time. Action of BMS on massless particles is natural in this picture. As a by-product we generalize the conformal primary wave-functions for massless particles in a way which makes the action of space-time translation simple. Using these wave-functions we write down a modified Mellin(-Fourier) transformation of the S-matrix elements. The resulting amplitude is Poincare covariant. Under Poincare transformation it transforms like products of primaries of inhomogeneous $SL(2,\mathbb{C})$ ($ISL(2,\mathbb{C})$) inserted at various points of null-infinity. $ISL(2,\mathbb{C})$ primaries are defined in the paper.