scholarly journals DUAL KAPPA POINCAŔE ALGEBRA

2010 ◽  
Vol 25 (09) ◽  
pp. 1881-1890 ◽  
Author(s):  
JOSE A. MAGPANTAY
Keyword(s):  
Phase Space ◽  
Space Time ◽  
Lorentz Algebra ◽  
Poincaré Algebra ◽  
New Space ◽  
Heisenberg Double

We show a different modification of Poincaré algebra that also preserves Lorentz algebra. The change begins with how boosts affect space–time in a way similar to how they affect the momenta in kappa Poincaré algebra; hence the term "dual kappa Poincaré algebra." Since by construction the new space–time commutes, it follows that the momenta cocommute. Proposing a space–time coalgebra that is similar to the momentum coproduct in the bicrossproduct basis of kappa Poincaré algebra, we derive the phase space algebra using the Heisenberg double construction. The phase space variables of the dual kappa Poincaré algebra are then related to SR phase space variables. From these relations, we complete the dual kappa Poincaré algebra by deriving the action of rotations and boosts on the momenta.

scholarly journals Heisenberg Doubles for Snyder-Type Models

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1055
Author(s):  
Stjepan Meljanac ◽  
Anna Pachoł
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Dual Pair ◽  
One Dimension ◽  
Lorentz Algebra ◽  
Explicit Formulae ◽  
Heisenberg Double

A Snyder model generated by the noncommutative coordinates and Lorentz generators closes a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. This leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given.

scholarly journals NON-COMMUTATIVE SPACE–TIME OF DOUBLY SPECIAL RELATIVITY THEORIES

2003 ◽  
Vol 12 (02) ◽  
pp. 299-315 ◽  
Author(s):  
J. KOWALSKI-GLIKMAN ◽  
S. NOWAK
Keyword(s):  
Phase Space ◽  
Special Relativity ◽  
Space Time ◽  
Kinematic Structure ◽  
Doubly Special Relativity ◽  
Minkowski Space Time ◽  
Space Time Structure ◽  
Commutative Space ◽  
Intrinsic Length ◽  
Heisenberg Double

Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there are infinitely many DSR constructions of the energy–momentum sector, each of whose can be promoted to the κ-Poincaré quantum (Hopf) algebra. Then we use the co-product of this algebra and the Heisenberg double construction of κ-deformed phase space in order to derive the non-commutative space–time structure and the description of the whole of DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space–time of the DSR theory is unique and related to the theory with non-commutative space–time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space–time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space–time, its intrinsic length parameter ℓ becomes observer-independent.

scholarly journals Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1309
Author(s):  
Jerzy Lukierski
Keyword(s):  
Phase Space ◽  
Hopf Algebras ◽  
Deformation Parameter ◽  
Heisenberg Algebra ◽  
Planck Length ◽  
Covariant Quantum ◽  
Drinfeld Twist ◽  
Quantum Deformations ◽  
Heisenberg Double ◽  
Conformal Symmetries

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed D=4 quantum inhomogeneous conformal Hopf algebras Uθ(su(2,2)⋉T4) and Uθ¯(su(2,2)⋉T¯4), where T4 describes complex twistor coordinates and T¯4 the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of D=4 Heisenberg-conformal algebra (HCA) su(2,2)⋉Hℏ4,4 (Hℏ4,4=T¯4⋉ℏT4 is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length λp, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in Hℏ4,4. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra Uθ(su(2,2)⋉T4).

scholarly journals KAPPA-MINKOWSKI SPACETIME, KAPPA-POINCARÉ HOPF ALGEBRA AND REALIZATIONS

2012 ◽  
Vol 09 (06) ◽  
pp. 1261009 ◽  
Author(s):  
DOMAGOJ KOVAČEVIĆ ◽  
STJEPAN MELJANAC
Keyword(s):  
Hopf Algebra ◽  
Star Product ◽  
Minkowski Spacetime ◽  
Inner Product ◽  
Translation Invariance ◽  
Dual Algebra ◽  
Lorentz Algebra ◽  
Heisenberg Algebras ◽  
The Matrix ◽  
Poincaré Algebra

The κ-Minkowski spacetime and Lorentz algebra are unified in unique Lie algebra. Introducing commutative momenta, a family of κ-deformed Heisenberg algebras and κ-deformed Poincaré algebras are defined. They are determined by the matrix depending on momenta. Realizations and star product are defined and analyzed in general. The relation among the coproduct of momenta, realization and the star product is pointed out. Hopf algebra of the Poincaré algebra, related to the covariant realization, is presented in unified covariant form. Left–right dual realizations and dual algebra are introduced and considered. The generalized involution and the star inner product are defined and analyzed. Partial integration and deformed trace property are obtained in general. The translation invariance of the star product is pointed out.

New space-time effects in superradiance: Experiment and theory

2010 ◽  
Vol 74 (7) ◽  
pp. 943-945 ◽  
Author(s):  
A. M. Basharov ◽  
N. V. Znamenskii ◽  
A. Yu. Shashkov
Keyword(s):  
Space Time ◽  
Time Effects ◽  
New Space
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