Boundary operators in asymptotically flat space-time

In [1] the authors have proposed an interesting framework for studying holography in flat space-time. In this note we explore the relationship between their proposal and the Celestial Holography. In particular, we find that in both the massive and in the massless cases the asymptotic boundary limit of the bulk time-ordered Green’s function G is related to the Celestial amplitudes by an integral transformation. In the massless case the integral transformation reduces to the well known shadow transformation of the celestial amplitude. Now the relation between the asymptotic limit of G and the celestial amplitudes suggests that in asymptotically flat space-time if the scattering states are described by the conformal primary basis then the boundary operators defined by the extrapolate dictionary of [1] are given by the shadow transformation of the conformal primary operators living on the celestial sphere. This result refers to the non-contact part of the extrapolated Green’s function. There are important contact term contributions which we also discuss in the paper.