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Jerzy Lukierski

Publications and source records attributed to Jerzy Lukierski.

At least 55 records · Page 3Linked to original sources
arXiv2011

Braided algebras and the kappa-deformed oscillators

Recently there were presented several proposals how to formulate the binary relations describing kappa-deformed oscillator algebras. In this paper we shall consider multilinear products of kappa-deformed oscillators consistent with the axioms of braided algebras. In general case the braided triple products are quasi-associative and satisfy the hexagon condition depending on the coassociator $Phi \in A\otimes A\otimes A$. We shall consider only the products of kappa-oscillators consistent with co-associative braided algebra, with Phi =1. We shall consider three explicite examples of binary kappa-deformed oscillator algebra relations and describe briefly their multilinear coassociative extensions satisfying the postulates of braided algebras. The third example, describing kappa-deformed oscillators in group manifold approach to kappa-deformed fourmomenta, is a new result.

hep-th↗
arXiv2011

Galilean Conformal and Superconformal Symmetries

Firstly we discuss briefly three different algebras named as nonrelativistic (NR) conformal: Schroedinger, Galilean conformal and infinite algebra of local NR conformal isometries. Further we shall consider in some detail Galilean conformal algebra (GCA) obtained in the limit c equal to infinity from relativistic conformal algebra O(d+1,2) (d - number of space dimensions). Two different contraction limits providing GCA and some recently considered realizations will be briefly discussed. Finally by considering NR contraction of D=4 superconformal algebra the Galilei conformal superalgebra (GCSA) is obtained, in the formulation using complex Weyl supercharges.

hep-th↗
arXiv2010

From Quantum Deformations of Relativistic Symmetries to Modified Kinematics and Dynamics

We present a short review describing the use of noncommutative space-time in quantum-deformed dynamical theories: classical and quantum mechanics as well as classical and quantum field theory. We expose the role of Hopf algebras and their realizations (noncommutative modules) as important mathematical tool describing quantum-deformed symmetries: quantum Lie groups and quantum Lie algebras. We consider in some detail the most studied examples of noncommutative space-time geometry: the canonical and $κ$-deformed cases. Finally we briefly describe the modifications of Einstein gravity obtained by introduction of noncommutative space-time coordinates.

hep-th↗
arXiv2010

D=4 Extended Galilei Superalgebras with Central Charges

We perform a nonrelativistic contraction of N-extended Poincare superalgebra with internal symmetry U(N) and general set of N(N-1) real central charges. We show that for even N=2k and particular choice of the dependence of Z_{ij} on light velocity c one gets the N-extended Galilei superalgebra with unchanged number of central charges and compact internal symmetry algebra U(k;H)=USp(2k). The Hamiltonian positivity condition is modified only by one central charge. If we put all the central charges equal to zero one gets the 2k-extended Galilei superalgebra as the subalgebra of recently introduced extended Galilei conformal superalgebra [1,2].

hep-th↗
arXiv2010

Generalized Wigner-Inonu Contractions and Maxwell (Super)Algebras

We consider a class of generalized Inonu-Wigner contraction for semidirect product of two particularly related semisimple Lie (super)algebras. The special class of such contractions provides D=4 Maxwell algebra and recently introduced simple D=4 Maxwell superalgebra. Further we present two types of D=4 N-extended Maxwell superalgebras, the nonstandard one for any N with 1/2 N(N-1) central charges and the standard one, for even N=2k, with k(2k-1) internal symmetry generators.

hep-th↗
arXiv2010

Deformations of Maxwell Superalgebras and Their Applications

We describe the Lie algebra deformations of D=4 Maxwell superalgebra that was recently introduced as the symmetry algebra of a kappa-symmetric massless superparticle in a supersymmetric constant electromagnetic background. Further we introduce the D=3 Maxwell superalgebra and present all its possible deformations. Finally the deformed superalgebras are used to derive via a contraction procedure the complete set of Casimir operators for D=4 and D=3 Maxwell superalgebras.

hep-th↗
arXiv2010

From quantum deformations of relativistic symmetries to modified kinematics and dynamics

Starting from noncommutative generalization of Minkowski space we consider quantum deformed relativistic symmetries which lead to the modification of kinematics of special relativity. The noncommutative field theory framework described by means of the star product formalism is briefly described. We briefly present the quantum modifications of Einstein gravity

hep-th↗
arXiv2010

Maxwell Superalgebra and Superparticle in Constant Gauge Backgrounds

We present SuperMaxwell algebra: an N=1, D=4 algebra with two Majorana supercharges, obtained as the minimal enlargement of superPoincare containing the Maxwell algebra as a subalgebra. The new superalgebra describes the supersymmetries of generalized N=1, D=4 superspace in the presence of a constant Abelian SUSY field strength background. Applying the techniques of non-linear coset realization to the Maxwell supergroup we propose a new kappa-invariant massless superparticle model providing a dynamical realization of the SuperMaxwell algebra.

hep-th↗
arXiv2009

Deformed Maxwell Algebras and their Realizations

We study all possible deformations of the Maxwell algebra. In D=d+1\neq 3 dimensions there is only one-parameter deformation. The deformed algebra is isomorphic to so(d+1,1)\oplus so(d,1) or to so(d,2)\oplus so(d,1) depending on the signs of the deformation parameter. We construct in the dS (AdS) space a model of massive particle interacting with Abelian vector field via non-local Lorentz force. In D=2+1 the deformations depend on two parameters b and k. We construct a phase diagram, with two parts of the (b,k) plane with so(3,1)\oplus so(2,1) and so(2,2)\oplus so(2,1) algebras separated by a critical curve along which the algebra is isomorphic to Iso(2,1)\oplus so(2,1). We introduce in D=2+1 the Volkov-Akulov type model for a Abelian Goldstone-Nambu vector field described by a non-linear action containing as its bilinear term the free Chern-Simons Lagrangean.

hep-th↗
arXiv2009

Kappa-deformed oscillators, the choice of star product and free kappa-deformed quantum fields

In order to obtain free kappa-deformed quantum fields (with c-number commutators) we proposed new concept of kappa-deformed oscillator algebra [1] and the modification of kappa-star product [2], implementing in the product of two quantum fields the change of standard kappa-deformed mass-shell conditions. We recall here that the kappa-deformed oscillators recently introduced in [3]-[5] lie on standard kappa-deformed mass-shell. Firstly, we study kappa-deformed fields with the standard kappa-star product, what implies that in the oscillator algebra the corresponding kappa-deformed oscillators lie on standard kappa-deformed mass-shell. We argue that for the kappa-deformed algebra of such field oscillators which carry fourmomenta on kappa-deformed mass-shell it is not possible to obtain the free quantum kappa-deformed fields with the c-number commutators. Further, we study kappa-deformed quantum fields with the modified kappa-star product which implies the modification of kappa-deformed mass-shell. We obtain large class of kappa-deformed statistics depending on six arbitrary functions which provides the c-number field commutator functions. Such general class of kappa-oscillators can be described as the kappa-deformation of standard oscillator algebra obtained by composing general kappa-deformed multiplication with the deformation of the flip operator.

hep-th↗
arXiv2009

Deformations of Maxwell algebra and their Dynamical Realizations

We study all possible deformations of the Maxwell algebra. In D=d+1\neq 3 dimensions there is only one-parameter deformation. The deformed algebra is isomorphic to so(d+1,1)\oplus so(d,1) or to so(d,2)\oplus so(d,1) depending on the signs of the deformation parameter. We construct in the dS(AdS) space a model of massive particle interacting with Abelian vector field via non-local Lorentz force. In D=2+1 the deformations depend on two parameters b and k. We construct a phase diagram, with two parts of the (b,k) plane with so(3,1)\oplus so(2,1) and so(2,2)\oplus so(2,1) algebras separated by a critical curve along which the algebra is isomorphic to Iso(2,1)\oplus so(2,1). We introduce in D=2+1 the Volkov-Akulov type model for a Abelian Goldstone-Nambu vector field described by a non-linear action containing as its bilinear term the free Chern-Simons Lagrangean.

hep-th↗
arXiv2009

Purely twistorial string with canonical twistor field quantization

We introduce new purely twistorial scale-invariant action describing the composite bosonic D=4 Nambu-Goto string with target space parametrized by the pair of D=4 twistors. We show that by suitable gauge fixing of local scaling one gets the bilinear twistorial action and canonical quantization rules for the two-dimensional twistor-string fields. We consider the Poisson brackets of all constraints characterizing our model and we obtain four first class constraints describing two Virasoro constraints and two U(1)xU(1) Kac-Moody (KM) local phase transformations.

hep-th↗
arXiv2009

kappa-Deformed Oscillators: Deformed Multiplication Versus Deformed Flip Operator and Multiparticle Clusters

We transform the oscillator algebra with kappa-deformed multiplication rule, proposed in [1],[2], into the oscillator algebra with kappa-deformed flip operator and standard multiplication. We recall that the kappa-multiplication of the kappa-oscillators puts them off-shell. We study the explicit forms of modified mass-shell conditions in both formulations: with kappa-multiplication and with kappa-flip operation. On the example of kappa-deformed 2-particle states we study the clustered nonfactorizable form of the kappa-deformed multiparticle states. We argue that the kappa-deformed star product of two free fields leads in similar way to a nonfactorizable kappa-deformed bilocal field. We conclude with general remarks concerning the kappa-deformed n-particle clusters and kappa-deformed star product of n fields.

math-ph↗
arXiv2008

Enlarged NH symmetries: Particle Dynamics and Gauge Symmetries

We show how the Newton-Hooke (NH) symmetries, representing a nonrelativistic version of de-Sitter symmetries, can be enlarged by a pair of translation vectors describing in Galilean limit the class of accelerations linear in time. We study the Cartan-Maurer one-forms corresponding to such enlarged NH symmetry group and by using cohomological methods we determine the general 2-parameter (in D=2+1 4-parameter)central extension of the corresponding Lie algebra. We derive by using nonlinear realizations method the most general group - invariant particle dynamics depending on two (in D=2+1 on four) central charges occurring as the Lagrangean parameters. Due to the presence of gauge invariances we show that for the enlarged NH symmetries quasicovariant dynamics reduces to the one following from standard NH symmetries, with one central charge in arbitrary dimension D and with second exotic central charge in D=2+1.

hep-th↗
arXiv2007

Two-twistor Description of Membrane

We describe D=4 twistorial membrane in terms of two twistorial three-dimensional world volume fields. We start with the D-dimensional p-brane generalizations of two phase space string formulations: the one with $p+1$ vectorial fourmomenta, and the second with tensorial momenta of $(p+1)$-th rank. Further we consider tensionful membrane case in D=4. By using the membrane generalization of Cartan-Penrose formula we express the fourmomenta by spinorial fields and obtain the intermediate spinor-space-time formulation. Further by expressing the worldvolume dreibein and the membrane space-time coordinate fields in terms of two twistor fields one obtains the purely twistorial formulation. It appears that the action is generated by a geometric three-form on two-twistor space. Finally we comment on higher-dimensional (D>4) twistorial p-brane models and their superextensions.

hep-th↗
arXiv2007

Noncommutative Translations and $\star$-Product Formalism

We consider the noncommutative space-times with Lie-algebraic noncommutativity (e.g. $κ$-deformed Minkowski space). In the framework with classical fields we extend the $\star$-product in order to represent the noncommutative translations in terms of commutative ones. We show the translational invariance of noncommutative bilinear action with local product of noncommutative fields. The quadratic noncommutativity is also briefly discussed.

hep-th↗
arXiv2007

Massive Particle Model with Spin from a Hybrid (spacetime-twistorial) Phase Space Geometry and Its Quantization

We extend the Shirafuji model for massless particles with primary spacetime coordinates and composite four-momenta to a model for massive particles with spin and electric charge. The primary variables in the model are the spacetime four-vector, four scalars describing spin and charge degrees of freedom as well as a pair of Weyl spinors. The geometric description proposed in this paper provides an intermediate step between the free purely twistorial model in two-twistor space in which both spacetime and four-momenta vectors are composite, and the standard particle model, where both spacetime and four-momenta vectors are elementary. We quantize the model and find explicitly the first-quantized wavefunctions describing relativistic particles with mass, spin and electric charge. The spacetime coordinates in the model are not commutative; this leads to a wavefunction that depends only on one covariant projection of the spacetime four-vector defining plane wave solutions.

hep-th↗