A Conformal Basis for Flat Space Amplitudes

TL;DR

The paper develops a conformal primary basis for flat-space wavefunctions in $\mathbb{R}^{1,d+1}$, showing that the Lorentz group acting as the $d$-dimensional conformal group $SO(1,d+1)$ yields a complete, delta-function-normalizable basis when the conformal dimension lies in the principal continuous series $\Delta=\tfrac{d}{2}+i\mathbb{R}$. It provides explicit constructions for massive and massless scalars, as well as massless spin-1 and spin-2 fields, including closed-form CPWs, shadow transforms, and integral/Mellin transforms that map bulk solutions or scattering amplitudes to $d$-dimensional conformal correlators. The work establishes orthonormality with the Klein-Gordon (and Maxwell/linearized Einstein) inner products and clarifies gauge/diffeomorphism constraints, ultimately enabling a covariant conformal description of flat-space amplitudes and offering a bridge to celestial CFT structures in general dimensions. This framework enhances flat-space holography and provides practical tools for translating scattering data into conformal data on the $d$-dimensional boundary.

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 $Δ$ 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 $Δ\in \frac d2+ i\mathbb{R}$ 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.