Some oscillatory representations of fuzzy conformal group SU(2,2) with
positive energy
- URL: http://arxiv.org/abs/2001.08408v1
- Date: Thu, 23 Jan 2020 08:56:03 GMT
- Title: Some oscillatory representations of fuzzy conformal group SU(2,2) with
positive energy
- Authors: Samuel Bezn\'ak, Peter Pre\v{s}najder
- Abstract summary: We construct the relativistic fuzzy space as a non-commutative algebra of functions with purely structural and abstract coordinates.
We construct two classes of irreducible representations of $su (2,2)$ algebra with textithalf-integer dimension $d$.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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 $\mathcal{H}_F$.
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.
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