Conformal Primary Basis for Dirac Spinors

TL;DR

This paper constructs a basis of solutions to the Dirac equation in Minkowski space that transform as conformal primaries, linking them to momentum space wavefunctions via Mellin transforms, and characterizing their properties across massless and massive cases.

Contribution

It introduces a conformal primary basis for Dirac spinors in Minkowski space, providing a new framework for analyzing solutions with conformal symmetry.

Findings

Principal series solutions form a complete basis

Massless solutions relate to momentum space via Mellin transform

Basis includes delta-function normalizable solutions

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 $\nu\geq 0$ 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.