Null Infinity and Unitary Representation of The Poincare Group

TL;DR

This paper constructs a new basis for massless particles transforming under a unitary principal series representation, revealing that their dynamics occur at null-infinity and enabling a Poincare covariant S-matrix formulation.

Contribution

It introduces a novel basis for massless particles linked to null-infinity, facilitating a group-theoretic approach to their dynamics and Poincare invariance in quantum field theory.

Findings

States transform under Unitary Principal Continuous Series

Massless particle dynamics are confined to null-infinity

Poincare covariant S-matrix with modified Mellin transform

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.