Continuous spin representation from contraction of the conformal algebra

Abstract

In this paper, we discuss the Inönü–Winger contraction of the conformal algebra. We start with the light-cone form of the Poincaré 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 to a circle of radius R following the 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, the special conformal translation vector changes and behaves like the four-momentum vector. We also discuss the generalization to d dimensions.