Hopf Algebroid Twists for Deformation Quantization of Linear Poisson Structures

Mignemi

et al. 2019

Phys. Rev. D

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

Section: κ-Deformed Relativistic Quantum Phase Spacementioning

confidence: 99%

“…Starting from the right covariant realization, instead of the left one, one arrives at the same results. One can also define a Hopf algebroid structure [14,38,43].…”

Section: Interpolation Between the Left And Right Covariant Realizationmentioning

confidence: 99%

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

Mignemi

et al. 2019

Phys. Rev. D

“…If the star product is associative, the twist operator Eq. (18) satisfies the cocycle condition in the Hopf algebroid sense and vice versa [47][48][49][50][51][52].…”

Section: Star Product and Twist Operatormentioning

confidence: 99%

Families of vector-like deformations of relativistic quantum phase spaces, twists and symmetries

Pikutić

2017

Eur. Phys. J. C

Self Cite

Families of vector-like deformed relativistic quantum phase spaces and corresponding realizations are analyzed. A method for a general construction of the star product is presented. The corresponding twist, expressed in terms of phase space coordinates, in the Hopf algebroid sense is presented. General linear realizations are considered and corresponding twists, in terms of momenta and Poincaré-Weyl generators or gl(n) generators are constructed and R-matrix is discussed. A classification of linear realizations leading to vector-like deformed phase spaces is given. There are three types of spaces: (i) commutative spaces, (ii) κ-Minkowski spaces and (iii) κ-Snyder spaces. The corresponding star products are (i) associative and commutative (but non-local), (ii) associative and noncommutative and (iii) non-associative and non-commutative, respectively. Twisted symmetry algebras are considered. Transposed twists and left-right dual algebras are presented. Finally, some physical applications are discussed.

“…Our results can be rephrased using the formalism of Hopf algebroids [14,15,16,17,18,19,20,21,22], that is for some aspects more suitable for the description of the Snyder models than the usual one based on Hopf algebras, since it deals with the full phase space. We leave however this subject to future investigations.…”

Section: Twist For the Maggiore Realisationmentioning

confidence: 99%

“…[7,8,24,25,26] and generalised in [28], to which we refer for more details. It turns out that the generalised addition of momenta k µ and q µ is given by [7,8,27] …”

Section: Snyder Space and Its Generalisationmentioning

confidence: 99%

Snyder-type space–times, twisted Poincaré algebra and addition of momenta

Mignemi

et al. 2017

Int. J. Mod. Phys. A

We discuss a generalisation of the Snyder model compatible with undeformed Lorentz symmetries, which we describe in terms of a large class of deformations of the Heisenberg algebra. The corresponding deformed addition of momenta, the twist and the R-matrix are calculated to first order in the deformation parameters for all models. In the particular case of the Snyder realisation, an analytic formula for the twist is obtained.