Conformal basis for flat space amplitudes

We study solutions of the Klein-Gordon, Maxwell, and linearized Einstein equations in R,(1,d11) 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 A 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 Delta is an element of 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 spin, the transition from momentum space to conformal primary wavefunctions is implemented by a Mellin transform. As a consequence of this construction, scattering amplitudes in this basis transform covariantly under SO(1, d+1) as d-dimensional conformal correlators.