Deformed relativistic and nonrelativistic symmetries on canonical noncommutative spaces

Abstract

We study the general deformed conformal-Poincaré (Galilean) symmetries consistent with relativistic (nonrelativistic) canonical noncommutative spaces. In either case we obtain deformed generators, containing arbitrary free parameters, which close to yield new algebraic structures. We show that a particular choice of these parameters reproduces the undeformed algebra. The modified coproduct rules and the associated Hopf algebra are also obtained. Finally, we show that for the choice of parameters leading to the undeformed algebra, the deformations are represented by twist functions. [Journal reference: Phys. Rev. D 75 (2007) 045008]

Remarks on the noncommutative gravitational quantum well

“…This shows that rotational symmetry is valid in d = 2 noncommutative space. For higher dimensions deformed transformations have to be appropriately defined which restore this symmetry [18,19,20].…”

Topics in Noncommutative Geometry Inspired Physics

“…Subsequently many peculiar nonperturbative features of QFT in NC spacetime (2), (these are essentially nonlocal effects), were revealed, such as NC solitons without a smooth commutative limit [16][17][18], formal similarity of Abelian NC gauge theories with nonAbelian gauge theories [1,2], (an example being the quantization of ChernSimons coupling in Abelian NC gauge theory [19,20]), structural similarity of NC gauge theories with gravity [21][22][23][24][25][26] to name a few. On the other hand, following a prescription (the Seiberg-Witten map) given in [6] on how to obtain NC corrections in perturbative QFT framework order by order in the NC parameter θ a large amount of work has appeared that studied NC extensions of QFT anomalies [27][28][29][30][31][32][33][34][35][36], NC-solitons having smooth commutative limit [37][38][39][40][41], Chern-Simons gauge theories [42][43][44][45][46][47][48][49][50], (for more recent result valid to all orders of θ see [51] that follow the general method proposed by [52]) etc. NC generalization in Quantum Mechanical models such as Harmonic Oscillator …”

Symmetries of Snyder-de Sitter space and relativistic particle dynamics

“…This is the first such study in this model. Contrary to the analysis done for canonical (constant) noncommutativity [12], here the results are valid in the leading order in the parameters κ (related to the noncommutativity (Snyder) parameter) and α (related to the cosmological constant). A relativistic particle model invariant under these deformed symmetries was given.…”