Null infinity and unitary representation of the Poincare group

Abstract

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 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 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 manifestly Poincare invariant (field) theory of free massless particles living on null-infinity. This theory has unitary time evolution. The null-infinity arises in this case purely group-theoretically without any reference to bulk space-time. Action of BMS is particularly 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, ℂ) (ISL(2, ℂ)) inserted at various points of null-infinity. ISL(2, ℂ) primaries are defined in the paper.