Conformal Primary Basis for Dirac Spinors

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 $\Delta$. The set of wavefunctions that belong to the principal continuous series, $\Delta =\frac{d}{2}+i\nu$, with $\nu \ge 0$ and $\nu \in \mathbb{R}$ 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.

21 pages, v2: added references, v3: added calculation of the explicit form of the wave function