Conformal basis for flat space amplitudes

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 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.