κ -deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra, and dispersion relations

Abstract

We consider κ-deformed relativistic quantum phase space and possible implementations of the Lorentz algebra. There are two ways of performing such implementations. One is a simple extension where the Poincaré algebra is unaltered, while the other is a general extension where the Poincaré algebra is deformed. As an example we fix the Jordanian twist and the corresponding realization of noncommutative coordinates, coproduct of momenta and addition of momenta. An extension with a one-parameter family of realizations of the Lorentz generators, dilatation and momenta closing the Poincaré-Weyl algebra is considered. The corresponding physical interpretation depends on the way the Lorentz algebra is implemented in phase space. We show how the spectrum of the relativistic hydrogen atom depends on the realization of the generators of the Poincaré-Weyl algebra. * Daniel.Meljanac@irb.hr † meljanac@irb.hr ‡ smignemi@unica.it § rina.strajn@unidu.hr

Interpolations between Jordanian Twists Induced by Coboundary Twists

“…Jordanian deformations have been considered in applications in AdS/CFT correspondence [25,26,34,35], as integrable deformations of sigma models in relation to deformation of AdS 5 and supergravity [20,55]. Jordanian twists have been applied in deformation of spacetime metrics [5], Maxwell equations and dispersion relations [1], as well as classical and quantum mechanics [36]. Gauge theories under Jordanian deformation were also investigated in [13].…”

One Parameter Family of Jordanian Twists

“…We note that for the twist F −1 GZ [10], the star product, an explicit form of the twist F GZ and the deformed Hopf algebra structure, were not known in the literature so far. Jordanian twists have been of interest in the recent literature [6,13,22,28,33,36]. We note that our results could be useful in future applications of Jordanian twists.…”

Interpolations between Jordanian twists, the Poincaré–Weyl algebra and dispersion relations

“…The physical interpretation depends on the realization of the generators of the Poincaré-Weyl algebra [14,24,30]. Particularly, the spectrum of the relativistic hydrogen atom depending on the parameter u was investigated in [14]. Differences in realizations of NC coordinates could also lead to different physical predictions, see e.g.…”

“…The Jordanian twist reappeared in the context of the κ-Minkowski NC space time [11,12,13,14]. A relation with the symmetry of the κ-Minkowski space time is established with the introduction of the generators of relativistic symmetries, dilatation D and momenta p α (instead of generators H and E of the sl(2) algebra) satisfying the same commutation relation, [p α , D] = p α .…”

“…Integrable deformations of sigma models in relation to deformations of AdS 5 and supergravity were investigated [21]. Jordanian twists have been applied in the deformation of spacetime metrics [22], dispersion relations [13,14] and gauge theories [23].…”