We study solutions of the Klein-Gordon, Maxwell, and linearized Einstein equations in R 1,d+1 that transform as d-dimensional conformal primaries under the Lorentz group SO(1, d + 1). Such solutions, called conformal primary wavefunctions, are labeled by a conformal dimension ∆ and a point in R d , rather than an on-shell (d + 2)-dimensional momentum. We show that the continuum of scalar conformal primary wavefunctions on the principal continuous series ∆ ∈ d 2 + iR of SO(1, d + 1) spans a complete set of normalizable solutions to the wave equation. In the massless case, with or without

It is shown that the tree-level S-matrix for quantum gravity in four-dimensional Minkowski space has a Virasoro symmetry which acts on the conformal sphere at null infinity.

The four-dimensional (4D) Lorentz group SL(2, C) acts as the two-dimensional (2D) global conformal group on the celestial sphere at infinity where asymptotic 4D scattering states are specified. Consequent similarities of 4D flat space amplitudes and 2D correlators on the conformal sphere are obscured by the fact that the former are usually expressed in terms of asymptotic wavefunctions which transform simply under spacetime translations rather than the Lorentz SL(2, C). In this paper we construct on-shell massive scalar wavefunctions in 4D Minkowski space that transform as SL(2, C) conformal primaries. Scattering amplitudes of these wavefunctions are SL(2, C) covariant by construction. For certain mass relations, we show explicitly that their three-point amplitude reduces to the known unique form of a 2D CFT primary three-point function and compute the coefficient. The computation proceeds naturally via Witten-like diagrams on a hyperbolic slicing of Minkowski space and has a holographic flavor.1 The subleading soft theorem has a one-loop exact anomaly [9-12] whose effects remain to be understood but are recently discussed in [13,14].2 One may hope that ultimately 4D quantum gravity scattering amplitudes are found to have a dual holographic representation as some exotic 2D CFT on CS 2 , but at present there are no proposals for such a construction.

The conventional gravitational memory effect is a relative displacement in the position of two detectors induced by radiative energy flux. We find a new type of gravitational 'spin memory' in which beams on clockwise and counterclockwise orbits acquire a relative delay induced by radiative angular momentum flux. It has recently been shown that the displacement memory formula is a Fourier transform in time of Weinberg's soft graviton theorem. Here we see that the spin memory formula is a Fourier transform in time of the recently-discovered subleading soft graviton theorem. Contents

It was shown by Low in the 1950s that the subleading terms of soft-photon S-matrix elements obey a universal linear relation. In this Letter, we give a new interpretation to this old relation, for the case of massless QED, as an infinitesimal symmetry of the S matrix. The symmetry is shown to be locally generated by a vector field on the conformal sphere at null infinity. Explicit expressions are constructed for the associated charges as integrals over null infinity and shown to generate the symmetry. These charges are local generalizations of electric and magnetic dipole charges.