Conformal Primary Basis for Dirac Spinors

TL;DR

The paper extends the conformal primary basis program to Dirac spinors in Minkowski space by constructing massive and massless spinor solutions that transform as conformal primaries under the induced conformal group on the boundary. Using the embedding space formalism, it provides explicit integral representations, derives the Dirac inner product and shadow transform, and proves delta-function completeness, including a plane-wave expansion in terms of the conformal primary basis. The massless case is obtained as a $m\to0$ limit and connected to the plane-wave basis via a Mellin transform, with the conformal dimensions restricted to the principal continuous series. These results furnish a fermionic analogue of the scalar conformal primary basis, enabling celestial CFT analyses and potential flat-space holography applications in QED and related theories.

Abstract

We study solutions to the Dirac equation in Minkowski space $\mathbb{R}^{1,d+1}$ that transform as $d$-dimensional conformal primary spinors under the Lorentz group $SO(1,d+1)$. Such solutions are parameterized by a point in $\mathbb{R}^d$ and a conformal dimension $Δ$. The set of wavefunctions that belong to the principal continuous series, $Δ=\frac{d}2 + iν$, with $ν\geq 0$ and $ν\in \mathbb{R}$ in the massive and massless cases, respectively, form a complete basis of delta-function normalizable solutions of the Dirac equation. In the massless case, the conformal primary wavefunctions are related to the wavefunctions in momentum space by a Mellin transform.