Null Infinity and Unitary Representation of The Poincare Group

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.

48 Pages, Latex, References added, Some typos corrected, The section- "Infinite Number of Representations" deleted

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• BMS Symmetry of Celestial OPE
• MHV Graviton Scattering Amplitudes and Current Algebra on the Celestial Sphere
• Celestial Diamonds: Conformal Multiplets in Celestial CFT
• Carrollian Approach to $1+3$D Flat Holography
• The SAGEX Review on Scattering Amplitudes, Chapter 11: Soft Theorems and Celestial Amplitudes
• Carrollian and celestial spaces at infinity
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• MHV Gluon Scattering Amplitudes from Celestial Current Algebras
• Lectures on Celestial Holography
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• Conformal properties of soft-operators - 1 : Use of null-states
• Goldilocks Modes and the Three Scattering Bases
• Symmetries of free massless particles and soft theorems
• On loop celestial amplitudes for gauge theory and gravity
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• Subsubleading soft graviton symmetry and MHV graviton scattering amplitudes
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• Quantizing Carrollian field theories
• (Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes
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• Poincaré Constraints on Celestial Amplitudes
• Celestial insights into the S-matrix bootstrap
• The Holographic Soft $S$-Matrix in QED and Gravity
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• Gauge and Gravity Amplitudes on the Celestial Sphere
• Zwanziger's pairwise little group on the celestial sphere
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• Infrared and Holographic Aspects of the $S$-Matrix in Gauge Theory and Gravity
• A Celestial Description of Planar Super-Yang-Mills Theory
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