Twists, realizations and Hopf algebroid structure of κ-deformed phase space

Jurić, Tajron; Meljanac, Stjepan; Štrajn, Rina

doi:10.1142/s0217751x14500225

Cited by 37 publications

(80 citation statements)

References 143 publications

(263 reference statements)

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“…In [40], a phenomenological analysis related to vector-like deformations of the relativistic quantum phase space and relativistic kinematics was elaborated up to first order in the deformation, particularly on particle propagation in spacetime. Note that if NC coordinatesx μ close a Lie algebra inx μ , then the corresponding deformed quantum phase space has a Hopf algebroid structure [46][47][48][49]. Otherwise, the coproduct is non-coassociative and the corresponding structure should be quasi-bialgebroid.…”

Section: Outlook and Discussionmentioning

confidence: 99%

“…For functions f (x) and g(x) which can be Fourier transformed, the relation between star product and twist operator is given by [46,47] …”

Section: Star Product and Twist Operatormentioning

confidence: 99%

“…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%

“…Twist (39) is written in Hopf algebroid approach [46,47]. The main point is that it can be written in the standard form (41), where x α p β is identified with the gl(n) generators L αβ , satisfying (28).…”

Section: Twist and R-matrixmentioning

confidence: 99%

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

Meljanac

Pikutić

2017

Eur. Phys. J. C

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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.

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Section: Outlook and Discussionmentioning

confidence: 99%

“…For functions f (x) and g(x) which can be Fourier transformed, the relation between star product and twist operator is given by [46,47] …”

Section: Star Product and Twist Operatormentioning

confidence: 99%

“…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%

Section: Twist and R-matrixmentioning

confidence: 99%

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

Meljanac

Pikutić

2017

Eur. Phys. J. C

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“…Noncommutative κ-deformed Minkowski space has been complemented with appropriate momenta into a noncommutative phase space in a number of works. It has been recently argued that this noncommutative phase space has a structure of a (topological) Hopf algebroid [1,2] and the same for general Lie algebra type phase spaces [4]; the covariant structure is however not included in those works. It is known that both the standard and general κ-phase spaces have a structure of a Heisenberg double [5], a Hopf algebraic generalization of a Heisenberg algebra prominently used in quantum group theory [6,7].…”

Section: Introductionmentioning

confidence: 99%

On Hopf algebroid structure of κ-deformed Heisenberg algebra

2017

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The (4 + 4)-dimensional κ-deformed quantum phase space as well as its (10 + 10)-dimensional covariant extension by the Lorentz sector can be described as Heisenberg doubles: the (10 + 10)-dimensional quantum phase space is the double of D = 4 κ-deformed Poincaré Hopf algebra H and the standard (4 + 4)-dimensional space is its subalgebra generated by κ-Minkowski coordinates xµ and corresponding commuting momenta pµ. Every Heisenberg double appears as the total algebra of a Hopf algebroid over a base algebra which is in our case the coordinate sector. We exhibit the details of this structure, namely the corresponding right bialgebroid and the antipode map. We rely on algebraic methods of calculation in Majid-Ruegg bicrossproduct basis. The target map is derived from a formula by J-H. Lu. The coproduct takes values in the bimodule tensor product over a base, what is expressed as the presence of coproduct gauge freedom.

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Some features of scattering problem in a κ‐Deformed minkowski spacetime

Khodadi

2016

Annalen der Physik

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The doubly special relativity (DSR) theories are suggested in order to incorporate an observerindependent length scale in special theory of relativity. The Magueijo-Smolin proposal of DSR is realizable through a particular form of the noncommutative (NC) spacetime (known as κ-Minkowski spacetime) in which the Lorentz symmetry is preserved. In this framework, the NC parameter κ provides the origin of natural cutoff energy scale. Using a nonlinear deformed relativistic dispersion relation along with the Lorentz transformations, we investigate some phenomenological facets of two-body collision problem (without creation of new particles) in a κ-Minkowski spacetime. By treating an elastic scattering problem, we study effects of the Planck scale energy cutoff on some relativistic kinematical properties of this scattering problem. The results are challenging in the sense that as soon as one turns on the κ-spacetime extension, the nature of the two-body collision alters from elastic to inelastic one. It is shown also that a significant kinematical variable involving in heavy ion collisions, the rapidity, is not essentially an additive quantity under a sequence of the nonlinear representation of the Lorentz transformations. 4 In the framework of Newtonian kinematics and for the case of two equal masses m1 = m2, the ratio K 1,a K 1,b in terms of ψ results in the simple relation K 1,a K 1,b = cos 2 ψ which the laboratory scattering angle ψ is limited to the interval 0 ≤ ψ ≤ π 2 . For more details one can see "Classical Mechanics" text book such as Ref. [86]. If ψ = 0 then K 1,a K 1,b = 1, namely, no collision takes place since the speed remains constant before and after the collision and no momentum is transferred to particle-2. However, if ψ → π

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Twists, realizations and Hopf algebroid structure of κ-deformed phase space

Cited by 37 publications

References 143 publications

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

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

On Hopf algebroid structure of κ-deformed Heisenberg algebra

Some features of scattering problem in a κ‐Deformed minkowski spacetime

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