# A Conformal Basis for Flat Space Amplitudes

## TL;DR

This paper introduces conformal primary wavefunctions for solutions to wave equations in flat spacetime, enabling a conformal basis for scattering amplitudes that transform as $d$-dimensional conformal correlators, thus linking flat space physics with conformal symmetry.

## Contribution

It constructs a complete set of conformal primary wavefunctions for scalar, Maxwell, and Einstein equations, and shows how scattering amplitudes transform as conformal correlators in this basis.

## Key findings

- Conformal primary wavefunctions form a complete basis for solutions.
- Scattering amplitudes transform covariantly as conformal correlators.
- Mellin transform connects momentum space to conformal primary basis.

## Abstract

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 $\Delta$ 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 $\Delta\in \frac d2+ i\mathbb{R}$ 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.