Some oscillatory representations of fuzzy conformal group SU(2,2) with positive energy

We construct the relativistic fuzzy space as a non-commutative algebra of functions with purely structural and abstract coordinates being the creaction and annihilation (C/A) operators acting on a Hilbert space $\mathcal{H}_F$. Using these oscillators, we represent the conformal algebra $su(2,2)$ (containing the operators describing physical observables, that generate boosts, rotations, spatial and conformal translations, and dilatation) by operators acting on such functions and reconstruct an auxiliary Hilbert space $\mathcal{H}_A$ to describe this action. We then analyze states on such space and prove them to be boost-invariant. Eventually, we construct two classes of irreducible representations of $su(2,2)$ algebra with \textit{half-integer} dimension $d$ ([1]): (i) the classical fuzzy massless fields as a doubleton representation of the $su(2,2)$ constructed from one set of C/A operators in fundamental or unitary inequivalent dual representation and (ii) classical fuzzy massive fields as a direct product of two doubleton representations constructed from two sets of C/A operators that are in the fundamental and dual representation of the algebra respectively.