Positive energy representations of the conformal quantum algebra

TL;DR

This paper classifies positive energy unitary representations of the q-deformed conformal algebra, highlighting finite-dimensional cases at roots of unity and analyzing massless representations and their dimensions.

Contribution

It provides a detailed construction of positive-energy unitary representations of the q-deformed conformal algebra, including finite-dimensional cases at roots of unity and massless representation analysis.

Findings

Representations become finite-dimensional at roots of unity.

Massless representations are analyzed within the q-deformed Poincaré subalgebra.

Dimensions of representations vary, with special cases for fundamental representations.

Abstract

The positive-energy unitary irreducible representations of the $q$-deformed conformal algebra ${\cal C}_q = {\cal U}_q(su(2,2))$ 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 Poincar\'e subalgebra of ${\cal C}_q$. Generically, their dimensions are smaller than the corresponding finite-dimensional non-unitary representation of $su(2,2)$, except when $N=2$, $h=0$ and $N = 2 \vert h\vert +1$, where $h$ is the helicity of the representations. The latter cases include the fundamental representations with $h = \pm 1/2$.