arXiv subjects

J. Lukierski

Publications and source records attributed to J. Lukierski.

At least 55 records · Page 3Linked to original sources

arXiv2003

On kappa-deformed D=4 quantum conformal group

This paper is presented on the occasion of 60-th birthday of Jose Adolfo de Azcarraga who in his very rich scientific curriculum vitae has also a chapter devoted to studies of quantum-deformed symmetries, in particular deformations of relativistic and Galilean space-time symmetries [1-4]. In this paper we provide new steps toward describing the $κ$-deformed D=4 conformal group transformations. We consider the quantization of D=4 conformal group with dimensionful deformation parameter $κ$. Firstly we discuss the noncommutativity following from the Lie-Poisson structure described by the light-cone $κ$-Poincaré $r$-matrix. We present complete set of D=4 conformal Lie-Poisson brackets and discuss their quantization. Further we define the light-cone $κ$-Poincaré quantum $R$-matrix in O(4,2) vector representation and discuss the inclusion of noncommutative conformal translations into the framework of $κ$-deformed conformal quantum group. The problem with real structure of $κ$-deformed conformal group is pointed out.

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arXiv2003

Generalized kappa-Deformations and Deformed Relativistic Scalar Fields on Noncommutative Minkowski Space

We describe the generalized kappa-deformations of D=4 relativistic symmetries with finite masslike deformation parameter kappa and an arbitrary direction in kappa-deformed Minkowski space being noncommutative. The corresponding bicovariant differential calculi on kappa-deformed Minkowski spaces are considered. Two distinguished cases are discussed: 5D noncommutative differential calculus (kappa-deformation in time-like or space-like direction), and 4D noncommutative differential calculus having the classical dimension (noncommutative kappa-deformation in light-like direction). We introduce also left and right vector fields acting on functions of noncommutative Minkowski coordinates, and describe the noncommutative differential realizations of kappa-deformed Poincare algebra. The kappa-deformed Klein-Gordon field on noncommutative Minkowski space with noncommutative time (standard kappa-deformation) as well as noncommutative null line (light-like kappa-deformation) are discussed. Following our earlier proposal (see {1,2]) we introduce an equivalent framework replacing the local noncommutative field theory by the nonlocal commutative description with suitable nonlocal star product multiplication rules. The modification of Pauli--Jordan commutator function is described and the kappa-dependence of its light-cone behaviour in coordinate space is explicitly given. The problem with the kappa-deformed energy- momentum conservation law is recalled.

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arXiv2003

κ-deformations of D=4 Weyl and conformal symmetries

We provide first explicite examples of quantum deformations of D=4 conformal algebra with mass-like deformation parameters, in applications to quantum gravity effects related with Planck mass. It is shown that one of the classical $r$-matrices defined on the Borel subalgebra of $sl(4)$ with $o(4,2)$ reality conditions describes the light-cone $κ$-deformation of D=4 Poincaré algebra. We embed this deformation into the three-parameter family of generalized $κ$-deformations, with $r$-matrices depending additionally on the dilatation generator. Using the extended Jordanian twists framework we describe these deformations in the form of noncocommutative Hopf algebra. We describe also another four-parameter class of generalized $κ$-deformations, which is obtained by continuous deformation of distinguished $κ$-deformation of D=4 Weyl algebra, called here the standard $κ$-deformation of Weyl algebra.

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arXiv2003

Basic Twist Quantization of osp(1|2) and kappa-- Deformation of D=1 Superconformal Mechanics

The twisting function describing a nonstandard (super-Jordanian) quantum deformation of $osp(1|2)$ is given in explicite closed form. The quantum coproducts and universal R-matrix are presented. The non-uniqueness of the twisting function as well as two real forms of the deformed $osp(1|2)$ superalgebras are considered. One real quantum $osp(1|2)$ superalgebra is interpreted as describing the $κ$-deformation of D=1, N=1 superconformal algebra, which can be applied as a symmetry algebra of N=1 superconformal mechanics.

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arXiv2003

Doubly Special Relativity versus $κ$-deformation of relativistic kinematics

We argue that recently proposed by Amelino-Camelia et all [1,2] so-called doubly special relativity (DSR), with deformed boost transformations identical with the formulae for $κ$-deformed kinematics in bicrossproduct basis is a classical special relativity in nonlinear disguise. The choice of symmetric composition law for deformed fourmomenta as advocated in [1, 2] implies that DSR is obtained by considering nonlinear fourmomenta basis of classical Poincaré algebra and it does not lead to noncommutative space-time. We also show how to construct large two classes of doubly special relativity theories - generalizing the choice in [1,2] and the one presented by Magueijo and Smolin [3]. The older version of deformed relativistic kinematics, differing essentially from classical theory in the coalgebra sector and leading to noncommutative $κ$-deformed Minkowski space is provided by quantum $κ$-deformation of Poincaré symmetries.

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arXiv2003

$κ$-deformations of D=3 conformal versus deformations of D=4 AdS symmetries

We describe the classical $o(3,2)$ $r$-matrices as generating the quantum deformations of either D=3 conformal algebra with mass-like deformation parameters or D=4 $AdS$ algebra with dimensionless deformation parameters. We describe the quantization of classical $o(3,2)$ $r$-matrices via Drinfeld twist method which locates the deformation in the coalgebra sector. Further we obtain the quantum $o(3,2)$ algebra in a convenient Hopf algebra form by considering suitable deformation maps from classical to deformed $o(3,2)$ algebra basis. It appears that if we pass from $κ$-deformed D =3 conformal algebra basis to the deformed D=4 $AdS$ generators basis the role of dimensionfull parameter is taken over by the $AdS$ radius $R$. We provide also the bilinear $o(3,2)$ Casimir which we express using the deformed D=3 conformal basis.

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arXiv2003

Noncommutative Planar Particle Dynamics with Gauge Interactions

We consider two ways of introducing minimal Abelian gauge interactions into the model presented in [1]. They are different only if the second central charge of the planar Galilei group is nonzero. One way leads to standard gauge transformations and second to a generalized gauge theory with gauge transformations accompanied by time-dependent area-preserving coordinate transformations. Both approaches, however, are related to each other by a classical Seiberg-Witten map supplemented by the noncanonical transformation of the phase space variables for planar particles. We also formulate the two-body problem in the model with a generalized gauge symmetry and consider the case with both CS and background electromagnetic fields, as it is used in the description of fractional quantum Hall effect.

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arXiv2002

Four classes of modified relativistic symmetry transformations

We discuss the nonlinear transformations of standard Poincaré symmetry in the context of recently introduced Doubly Special Relativity (DSR) theories. We introduce four classes of modified relativistic theories with three of them describing various DSR frameworks. We consider four examples of modified relativistic symmetries, which illustrate each of the considered class.

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arXiv2002

Noncommutative planar particles: higher order versus first order formalism and supersymmetrization

We describe the supersymmetrization of two formulations of free noncommutative planar particles -- in coordinate space with higher order Lagrangian [1] and in the framework of Faddeev and Jackiw [2,3], with first order action. In nonsupersymmetric case the first formulation after imposing subsidiary condition eliminating internal degrees of freedom provides the second formulation. In supersymmetric case one can also introduce the split into ``external'' and ``internal'' degrees of freedom both describing supersymmetric models.

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arXiv2002

Nonlinear and Quantum Origin of Doubly Infinite Family of Modified Addition Laws for Fourmomenta

We show that infinite variety of Poincaré bialgebras with nontrivial classical r-matrices generate nonsymmetric nonlinear composition laws for the fourmomenta. We also present the problem of lifting the Poincaré bialgebras to quantum Poincaré groups by using e.g. Drinfeld twist, what permits to provide the nonlinear composition law in any order of dimensionfull deformation parmeter $λ$ (from physical reasons we can put $λ= λ_{p}$ where $λ_{p}$ is the Planck lenght). The second infinite variety of composition laws for fourmomentum is obtained by nonlinear change of basis in Poincaré algebra, which can be performed for any choice of coalgebraic sector, with classical or quantum coproduct. In last Section we propose some modification of Hopf algebra scheme with Casimir-dependent deformation parameter, which can help to resolve the problem of consistent passage to macroscopic classical limit.

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arXiv2002

Generalized Supersymmetries and Composite Structure in M-Theory

We describe generalized D=11 Poincaré and conformal supersymmetries. The corresponding generalization of twistor and supertwistor framework is outlined with $OSp(1|64)$ superspinors describing BPS preons. The $\frac{k}{32}$ BPS states as composed out of $n=32 - k$ preons are introduced, and basic ideas concerning BPS preon dynamics is presented. The lecture is based on results obtained by J.A. de Azcarraga, I. Bandos, J.M. Izquierdo and the author$^1$.

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arXiv2001

Absorption of TeV Photons and $κ$-deformed Poincaré algebra

We consider the process of collision between a hard photon and a soft photon producing an electron-positron pair, under the assumption that the kinematics be described according to the $κ$-deformation of the D=4 Poincaré algebra. We emphasize the relevance of this analysis for the understanding of the puzzling observations of multi-TeV photons from Markarian 501. We find a significant effect of the $κ$-deformation for processes above threshold, while, in agreement with a previous study, we find that there is no leading-order deformation of the threshold condition.

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arXiv2001

$κ$--deformed Wigner construction of relativistic wave functions and free fields on $κ$-Minkowski space

We describe the extension of the Wigner`s infinite-dimensional unitary representations of Poincaré group to the case of $κ$-deformed Poincaré group. We show that the corresponding coordinate wave functions on noncommutative space-time are described by free field equations on $κ$-deformed Minkowski space. The cases of Klein--Gordon, Dirac, Proca and Maxwell fields are considered. Finally some aspects of second quantization are also discussed.

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arXiv2000

Noncommutative Parameters of Quantum Symmetries and Star Products

The star product technique translates the framework of local fields on noncommutative space-time into nonlocal fields on standard space-time. We consider the example of fields on $κ$- deformed Minkowski space, transforming under $κ$-deformed Poincaré group with noncommutative parameters. By extending the star product to the tensor product of functions on $κ$-deformed Minkowski space and $κ$-deformed Poincaré group we represent the algebra of noncommutative parameters of deformed relativistic symmetries by functions on classical Poincaré group.

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arXiv2000

Chern-Simons Particles with Nonstandard Gravitational Interaction

The model of nonrelativistic particles coupled to nonstandard (2+1)-gravity [1] is extended to include Abelian or non-Abelian charges coupled to Chern-Simons gauge fields. Equivalently, the model may be viewed as describing the (Abelian or non-Abelian) anyonic dynamics of Chern-Simons particles coupled, in a reparametrization invariant way, to a translational Chern-Simons action. The quantum two-body problem is described by a nonstandard Schrödinger equation with a noninteger angular momentum depending on energy as well as particle charges. Some numerical results describing the modification of the energy levels by these charges in the confined regime are presented. The modification involves a shift as well as splitting of the levels.

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arXiv2000

Quantum Deformations of Space-Time SUSY and Noncommutative Superfield Theory

We review shortly present status of quantum deformations of Poincaré and conformal supersymmetries. After recalling the $κ$-deformation of $\hbox{D=4}$ Poincaré supersymmetries we describe the corresponding star product multiplication for chiral superfields. In order to describe the deformation of chiral vertices in momentum space the integration formula over $κ$-deformed chiral superspace is proposed.

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arXiv2000

Local Field Theory ON $κ$-Minkowski Space, Star Products and Noncommutative Translations

We consider local field theory on $κ$-deformed Minkowski space which is an example of solvable Lie-algebraic noncommutative structure. Using integration formula over $κ$-Minkowski space and $κ$-deformed Fourier transform we consider for deformed local fields the reality conditions as well as deformation of action functionals in standard Minkowski space. We present explicite formulas for two equivalent star products describing CBH quantization of field theory on $κ$-Minkowski space. We express also via star product technique the noncommutative translations in $κ$-Minkowski space by commutative translations in standard Minkowski space.

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arXiv2000

From Gauging Nonrelativistic Translations to N-Body Dynamics

We consider the gauging of space translations with time-dependent gauge functions. Using fixed time gauge of relativistic theory, we consider the gauge-invariant model describing the motion of nonrelativistic particles. When we use gauge-invariant nonrelativistic velocities as independent variables the translation gauge fields enter the equations through a d\times (d+1) matrix of vielbein fields and their Abelian field strengths, which can be identified with the torsion tensors of teleparallel formulation of relativity theory. We consider the planar case (d=2) in some detail, with the assumption that the action for the dreibein fields is given by the translational Chern-Simons term. We fix the asymptotic transformations in such a way that the space part of the metric becomes asymptotically Euclidean. The residual symmetries are (local in time) translations and rigid rotations. We describe the effective interaction of the d=2 N-particle problem and discuss its classical solution for N=2. The phase space Hamiltonian H describing two-body interactions satisfies a nonlinear equation H={\cal H}(\vec x,\vec p;H) which implies, after quantization, a nonstandard form of the Schrödinger equation with energy dependent fractional angular momentum eigenvalues. Quantum solutions of the two-body problem are discussed. The bound states with discrete energy levels correspond to a confined classical motion (for the planar distance between two particles r\le r_0) and the scattering states with continuum energy correspond to the classical motion for r>r_0. We extend our considerations by introducing an external constant magnetic field and, for N=2, provide the classical and quantum solutions in the confined and unconfined regimes.

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