Conformal basis for flat space amplitudes

We study solutions of the Klein-Gordon, Maxwell, and linearized Einstein equations in ${\mathbb{R}}^{1,d+1}$ 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 $\mathrm{\Delta }$ and a point in ${\mathbb{R}}^d$, rather than an on-shell ($d+2$)-dimensional momentum. We show that the continuum of scalar conformal primary wavefunctions on the principal continuous series $\mathrm{\Delta }\in \frac{d}{2}+i\mathbb{R}$ 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.