κ -deformed covariant quantum phase spaces as Hopf algebroids

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.

show abstract

“…An important result is that the twist F −1 (19) in the case G = 0 (i.e. χ(p) = 0) is identical [21,22] to …”

Section: Hopf Algebroid Approachmentioning

confidence: 99%

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

Meljanac

Mignemi

et al. 2017

Int. J. Mod. Phys. A

View full text Add to dashboard Cite

show abstract

“…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

Meljanac

Pikutić

2017

Eur. Phys. J. C

View full text Add to dashboard 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.

show abstract

“…This difference in the sign can be traced back to a matter of convention on the signature of the embedding Minkowski space for the AN (n) manifold. 7 For a different approach to deformed phase spaces which makes use of the theory of Hopf algebroids see [34][35][36] sitting at the tip of the cone [2]. This conical metric can be pictured as a wedge cut out from the spatial plane characterized by a deficit angle proportional to the mass of the particle α = 8πGm, where G is Newton's gravitational constant which in two dimensions has units of inverse mass.…”

Section: Deforming Momentum Space To the Group Sl(2 R)mentioning

confidence: 99%

Deformed phase spaces with group valued momenta

Arzano

Nettel

2016

Phys. Rev. D

View full text Add to dashboard Cite

We introduce a general framework for describing deformed phase spaces with group valued momenta. Using techniques from the theory of Poisson-Lie groups and Lie bi-algebras we develop tools for constructing Poisson structures on the deformed phase space starting from the minimal input of the algebraic structure of the generators of the momentum Lie group. The tools developed are used to derive Poisson structures on examples of group momentum space much studied in the literature such as the n-dimensional generalization of the κ-deformed momentum space and the SL(2, Ê) momentum space in three space-time dimensions. We discuss classical momentum observables associated to multi-particle systems and argue that these combine according the usual four-vector addition despite the non-abelian group structure of momentum space.

show abstract

κ -deformed covariant quantum phase spaces as Hopf algebroids

Cited by 34 publications

References 27 publications

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

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

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

Deformed phase spaces with group valued momenta

Contact Info

Product

Resources

About