  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.
