Continuous Spin Representations of the Poincar\'e and Super-Poincar\'e Groups

TL;DR

This paper constructs and analyzes continuous spin representations of the Poincaré and super-Poincaré groups in higher dimensions, revealing their structure, classification, and supersymmetric extensions, including cases with finite dimensionality and central charges.

Contribution

It extends the construction of continuous spin representations to higher dimensions and explores their supersymmetric counterparts, including finite-dimensional cases with nilpotent translations.

Findings

Continuous spin representations are classified by the length of a space-like vector and Dynkin indices.

Supermultiplets are formed by combining bosonic and fermionic representations.

Finite-dimensional representations with nilpotent translations contain zero or negative norm states.

Abstract

We construct Wigner's continuous spin representations of the Poincar\'e algebra for massless particles in higher dimensions. The states are labeled both by the length of a space-like translation vector and the Dynkin indices of the {\it short little group} $SO(d-3)$, where $d$ is the space-time dimension. Continuous spin representations are in one-to-one correspondence with representations of the short little group. We also demonstrate how combinations of the bosonic and fermionic representations form supermultiplets of the super-Poincar\'e algebra. If the light-cone translations are nilpotent, these representations become finite dimensional, but contain zero or negative norm states, and their supersymmetry algebra contains a central charge in four dimensions.