Conformal primary basis for Dirac spinors

Lorenzo Iacobacci,Wolfgang Mück

doi:10.1103/physrevd.102.106025

Lorenzo Iacobacci, Wolfgang Mück

Open Access

https://doi.org/10.1103/physrevd.102.106025

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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.

Highlights

AND SUMMARYThe classification of fields according to the irreducible representations of a symmetry group has been, since the work of Wigner [1], an essential ingredient in quantum mechanics and quantum field theory
We study solutions to the Dirac equation in Minkowski space R1;dþ1 that transform as d-dimensional conformal primary spinors under the Lorentz group SOð1; d þ 1Þ
In the case of fields in Minkowski space-time, which enjoys symmetry under the Poincaregroup, Wigner’s 1939 paper [2] still provides the foundation of our description of particles of any spin. This description is based on wave functions in momentum space, where translation symmetry is manifest

Summary

INTRODUCTION

The classification of fields according to the irreducible representations of a symmetry group has been, since the work of Wigner [1], an essential ingredient in quantum mechanics and quantum field theory. In the case of fields in Minkowski space-time, which enjoys symmetry under the Poincaregroup, Wigner’s 1939 paper [2] still provides the foundation of our description of particles of any spin This description is based on wave functions in momentum space, where translation symmetry is manifest. We generalize the work of Pasterski and Shao to Dirac spinors.1 This is motivated partly by possible applications of flat space holography in quantum electrodynamics and for the explicit construction of the conformal partial wave decomposition [20] in conformal field theories (CFT). For massive Dirac spinors, the basis wave functions are parametrized by a vector w⃗ ∈ Rd representing a boundary point in the momentum space of the particle, and a conformal dimension Δ, which belongs to one-half of the principal continuous series, iR≥0 ðm > 0Þ: ð1Þ. These sections are kept reasonably concise, with all long calculations included in the Appendixes

EMBEDDING SPACE

Solutions of the massive Dirac equation

Dirac inner product

Shadow transform

Conformal primary basis

CONFORMAL PRIMARY BASIS FOR MASSLESS SPINOR FIELDS