Continuous Spin Representation from Contraction of the Conformal Algebra

TL;DR

This paper demonstrates how contracting the conformal algebra via Inönü-Wigner contraction leads to the continuous spin representation in four dimensions, revealing new insights into the structure of conformal symmetries.

Contribution

It introduces a novel contraction method of the conformal algebra that results in continuous spin representations, extending the understanding of conformal symmetry in various dimensions.

Findings

Continuous spin representation obtained from conformal algebra contraction.

Scaling symmetry persists after contraction, but special conformal vectors transform like momenta.

Generalization of the contraction process to arbitrary dimensions discussed.

Abstract

In this paper, we discuss the In\"on\"u-Winger contraction of the conformal algebra. We start with the light-cone form of the Poincar\'e algebra and extend it to write down the conformal algebra in $d$ dimensions. To contract the conformal algebra, we choose five dimensions for simplicity and compactify the third transverse direction in to a circle of radius $R$ following Kaluza-Klein dimensional reduction method. We identify the inverse radius, $1/R$, as the contraction parameter. After the contraction, the resulting representation is found to be the continuous spin representation in four dimensions. Even though the scaling symmetry survives the contraction, but the special conformal translation vector changes and behaves like the four-momentum vector. We also discussed the generalization to $d$ dimensions.