Some oscillatory representations of fuzzy conformal group SU(2,2) with positive energy
Abstract
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 HF. 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 HA 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 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.