Mariusz Woronowicz

New Lie-algebraic and quadratic deformations of Minkowski space from twisted Poincaré symmetries

Abstract We consider two new classes of twisted D = 4 quantum Poincare symmetries described as the dual pairs of noncocommutative Hopf algebras. Firstly we investigate a two-parameter class of twisted Poincare algebras which provide the examples of Lie-algebraic noncommutativity of the translations. The corresponding associative star-products and new deformed Lie-algebraic Minkowski spaces are introduced. We discuss further the twist deformations of Poincare symmetries generated by the twist with its carrier in Lorentz algebra. We describe corresponding deformed Poincare group which provides the quadratic deformations of translation sector and define the quadratically deformed Minkowski space–time algebra.

Regular black holes in quadratic gravity

The first-order correction of the perturbative solution of the coupled equations of the quadratic gravity and nonlinear electrodynamics is constructed, with the zeroth-order solution coinciding with the ones given by Ayón-Beato and Garcí a and by Bronnikov. It is shown that a simple generalization of the Bronnikovs electromagnetic Lagrangian leads to the solution expressible in terms of the polylogarithm functions. The solution is parametrized by two integration constants and depends on two free parameters. By the boundary conditions the integration constants are related to the charge and total mass of the system as seen by a distant observer, whereas the free parameters are adjusted to make the resultant line element regular at the center. It is argued that various curvature invariants are also regular there that strongly suggests the regularity of the spacetime. Despite the complexity of the problem the obtained solution can be studied analytically. The location of the event horizon of the black hole, its asymptotics and temperature are calculated. Special emphasis is put on the extremal configuration.

κ-DEFORMED STATISTICS AND CLASSICAL FOUR-MOMENTUM ADDITION LAW

We consider κ-deformed relativistic symmetries described algebraically by modified Majid–Ruegg bi-cross-product basis and investigate the quantization of field oscillators for the κ-deformed free scalar fields on κ-Minkowski space. By modification of standard multiplication rule, we postulate the κ-deformed algebra of bosonic creation and annihilation operators. Our algebra permits one to define the n-particle states with classical addition law for the four-momentum in a way which is not in contradiction with the nonsymmetric quantum four-momentum co-product. We introduce κ-deformed Fock space generated by our κ-deformed oscillators which satisfy the standard algebraic relations with modified κ-multiplication rule. We show that such a κ-deformed bosonic Fock space is endowed with the conventional bosonic symmetry properties. Finally we discuss the role of κ-deformed algebra of oscillators in field-theoretic noncommutative framework.

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

The aim of this paper is to study in D = 4 the general framework providing various κ-deformations of field oscillators and consider the commutator function of the corresponding κ-deformed free fields. In order to obtain free κ-deformed quantum fields (with c-number commutators) we proposed earlier a particular model of a κ-deformed oscillator algebra (Daszkiewicz M, Lukierski J and Woronowicz M 2008 Mod. Phys. Lett. A 23 9 (arXiv:hep-th/0703200)) and the modification of κ-star product (Daszkiewicz M, Lukierski J, Woronowicz M 2008 Phys. Rev. D 77 105007 (arXiv:0708.1561 [hep-th])), implementing in the product of two quantum fields the change of standard κ-deformed mass-shell conditions. We recall here that other different models of κ-deformed oscillators recently introduced in Arzano M and Marciano A (2007 Phys. Rev. D 76 125005 (arXiv:0707.1329 [hep-th])), Young C A S and Zegers R (2008 Nucl. Phys. B 797 537 (arXiv: 0711.2206 [hep-th])), Young C A S and Zegers R (2008 arXiv: 0803.2659 [hep-th]) are defined on a standard κ-deformed mass shell. In this paper, we consider the most general κ-deformed field oscillators, parametrized by a set of arbitrary functions in 3-momentum space. First, we study the fields with the κ-deformed oscillators defined on the standard κ-deformed mass shell, and argue that for any such choice of a κ-deformed field oscillators algebra we do not obtain the free quantum κ-deformed fields with the c-number commutators. Further, we study κ-deformed quantum fields with the modified κ-star product and derive a large class of κ-oscillators defined on a suitably modified κ-deformed mass shell. We obtain a large class of κ-deformed statistics depending on six arbitrary functions which all provide the c-number field commutator functions. This general class of κ-oscillators can be described by the composition of suitably defined κ-multiplications and the κ-deformation of the flip operator.

κ-deformed covariant quantum phase spaces as Hopf algebroids

Abstract We consider the general D = 4 ( 10 + 10 ) -dimensional κ -deformed quantum phase space as given by Heisenberg double H of D = 4 κ -deformed Poincare–Hopf algebra H . The standard ( 4 + 4 ) -dimensional κ -deformed covariant quantum phase space spanned by κ -deformed Minkowski coordinates and commuting momenta generators ( x ˆ μ , p ˆ μ ) is obtained as the subalgebra of H . We study further the property that Heisenberg double defines particular quantum spaces with Hopf algebroid structure. We calculate by using purely algebraic methods the explicit Hopf algebroid structure of standard κ -deformed quantum covariant phase space in Majid–Ruegg bicrossproduct basis. The coproducts for Hopf algebroids are not unique, determined modulo the coproduct gauge freedom. Finally we consider the interpretation of the algebraic description of quantum phase spaces as Hopf algebroids.

Braided tensor products and the covariance of quantum noncommutative free fields

We introduce the free quantum noncommutative fields as described by braided tensor products. The multiplication of such fields is decomposed into three operations, describing the multiplication in the algebra of functions on noncommutative spacetime, the product in the algebra of deformed field oscillators, and the braiding by factor between algebras and . For noncommutativity of quantum spacetime generated by the twist factor, we shall employ the ⋆-product realizations of the algebra in terms of functions on the standard Minkowski space. The covariance of single noncommutative quantum fields under deformed Poincare symmetries is described by the algebraic covariance conditions which are equivalent to the deformation of generalized Heisenberg equations on a Poincare group manifold. We shall calculate the braided field commutator covariant under deformed Poincare symmetries, which for free quantum noncommutative fields provides the field quantization condition and is given by the standard Pauli–Jordan function. For the illustration of our new scheme, we present explicit calculations for the well-known case in the literature of canonically deformed free quantum fields.

NONCOMMUTATIVE TRANSLATIONS AND ⋆-PRODUCT FORMALISM

We consider the noncommutative space-times with Lie-algebraic noncommutativity (e.g.

NONCOMMUTATIVE SPACE-TIME FROM QUANTIZED TWISTORS

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On Hopf algebroid structure of κ-deformed Heisenberg algebra

-deformed Minkowski space). In the framework with classical fields we extend the

Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach

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