Positive energy representations of the conformal quantum algebra

Abstract

The positive-energy unitary irreducible representations of the $q$-deformed conformal algebra ${\mathcal{C}}_q={\mathcal{U}}_q(su(2,2))$ are obtained by appropriate deformation of the classical ones. When the deformation parameter $q$ is $N$-th root of unity, all these unitary representations become finite-dimensional. For this case we discuss in some detail the massless representations, which are also irreducible representations of the $q$-deformed Poincaré subalgebra of ${\mathcal{C}}_q$. Generically, their dimensions are smaller than the corresponding finite-dimensional non-unitary representation of $su(2,2)$, except when $N=2$, $h=0$ and $N=2|h|+1$, where $h$ is the helicity of the representations. The latter cases include the fundamental representations with $h=\pm 1/2$.

arXiv comment10 pages, plain-TEX, preprint IC/92/187, SISSA 137/92/FM, UTS-DFT-22-92, (July 1992), Phys. Lett. 302B (1993) 215