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Mariusz Woronowicz

Publications and source records attributed to Mariusz Woronowicz.

17 recordsLinked to original sources
arXiv2025

Towards new relativistic doubly $κ$-deformed D=4 quantum phase spaces

We propose new noncommutative models of quantum phase spaces, containing a pair of $κ$-deformed Poincaré algebras, with two independent double ($κ,\tildeκ$)-deformations in space-time and four-momenta sectors. The first such quantum phase space can be obtained by contractions $M,R\to \infty$ of recently introduced doubly $κ$-deformed $(κ,\tildeκ)$-Yang models, with the parameters $M,R$ describing inverse space-time and four-momenta curvatures and constant four-vectors $a_μ, b_μ$ determining nine types of $(κ,\tildeκ)$-deformations. The second considered model is provided by the nonlinear doubly $κ$-deformed TSR algebra spanned by 14 coset $\hat{o}(1,5)/\hat {o}(2)$ generators. The basic algebraic difference between the two models is the following: the first one, described by $\hat{o}(1,5)$ Lie algebra can be supplemented by the Hopf algebra structure, while the second model contains the quantum phase space commutators $[\hat{x}_μ,\hat{q}_ν]$, with the standard numerical $i\hbarη_{μν}$ term; therefore it describes the quantum-deformed Heisenberg algebra relations which cannot be equipped with the Hopf algebra.

hep-th↗
arXiv2024

From Snyder space-times to doubly $κ$-dependent Yang quantum phase spaces and their generalizations

We propose the doubly $κ$-dependent Yang quantum phase space which describes the generalization of $D = 4$ Yang model. We postulate that such model is covariant under the generalized Born map, what permits to derive this new model from the earlier proposed $κ$-Snyder model. Our model of $D=4$ relativistic Yang quantum phase space depends on five deformation parameters which form two Born map-related dimensionful pairs: $(M,R)$ specifying the standard Yang model and $(κ,\tildeκ)$ characterizing the Born-dual $κ$-dependence of quantum space-time and quantum fourmomenta sectors; fifth parameter $ρ$ is dimensionless and Born-selfdual. In the last section, we propose the Kaluza-Klein generalization of $D=4$ Yang model and the new quantum Yang models described algebraically by quantum-deformed $\hat{o}(1,5)$ algebras.

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arXiv2022

Noncommutative spaces and superspaces from Snyder and Yang type models

The relativistic $D=4$ Snyder model is formulated in terms of $D=4$ $dS$ algebra $o(4,1)$ generators, with noncommutative Lorentz-invariant Snyder quantum space-time provided by $\frac{O(4,1)}{O(3,1)}$ coset generators. Analogously, in relativistic $D=4$ Yang models the quantum-deformed relativistic phase space is described by the algebras of coset generators $\frac{O(5,1)}{O(3,1)}$ or $\frac{O(4,2)}{O(3,1)}$. We extend these algebraic considerations by using respective $dS$ superalgebras, which provide Lorentz-covariant quantum superspaces (SUSY Snyder model) as well as relativistic quantum phase super spaces (SUSY Yang model).

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arXiv2021

Spinorial Snyder and Yang Models From Superalgebras And Noncommutative Quantum Superspaces

The relativistic Lorentz-covariant quantum space-times obtained by Snyder can be described by the coset generators of (anti) de-Sitter algebras. Similarly, the Lorentz-covariant quantum phase spaces introduced by Yang, which contain additionally quantum curved fourmomenta and quantum-deformed relativistic Heisenberg algebra, can be defined by suitably chosen coset generators of conformal algebras. We extend such algebraic construction to the respective superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY Snyder model) and indicate also how to obtain the quantum relativistic phase superspaces (SUSY Yang model). In last Section we recall briefly other ways of deriving quantum phase (super)spaces and we compare the spinorial Snyder type models defining bosonic or fermionic quantum-deformed spinors.

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arXiv2020

Two $θ_{μν}$ -deformed covariant relativistic quantum phase spaces as Poincare-Hopf algebroids

We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known $θ_{μν}$-deformed $D=4$ Poincare-Hopf algebra $\mathbb{H}$. The first algebroid describes $θ_{μν}$-deformed relativistic phase space with canonical NC space-time (constant $θ_{μν}$ parameters) and the second one incorporates dual to $\mathbb{H}$ quantum $θ_{μν}$-deformed Poincare-Hopf group algebra $\mathbb{G}$, which contains noncommutative space-time translations given by $Λ$-dependent $Θ_{μν}$ parameters ($% Λ$ $\equiv Λ_{μν}$ parametrize classical Lorentz group). The canonical $θ_{μν}$-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincare transformations described by $\mathbb{G}$. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.

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arXiv2018

Quantum Twist-Deformed D=4 Phase Spaces with Spin Sector and Hopf Algebroid Structures

We consider the generalized (10+10)-dimensional D=4 quantum phase spaces containing translational and Lorentz spin sectors associated with the dual pair of twist-quantized Poincare Hopf algebra $\mathbb{H}$ and quantum Poincare Hopf group $\widehat{\mathbb{G}}$. Two Hopf algebroid structures of generalized phase spaces with spin sector will be investigated: first one $% \mathcal{H}^{(10,10)}$ describing dynamics on quantum group algebra $% \widehat{\mathbb{G}}$ provided by the Heisenberg double algebra $\mathcal{HD=% }\mathbb{H}\rtimes \widehat{\mathbb{G}}$, and second, denoted by $\mathcal{% \tilde{H}}^{(10,10)}$, describing twisted Hopf algebroid with base space containing twisted noncommutative Minkowski space $\hat{x}_{μ}$. We obtain the first explicit example of Hopf algebroid structure of relativistic quantum phase space which contains quantum-deformed Lorentz spin sector.

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arXiv2017

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

We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the considered generalized phase spaces into the ones describing the quantum-deformed Heisenberg algebras.

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arXiv2016

On Hopf algebroid structure of kappa-deformed Heisenberg algebra

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 $\mathbb{H}$ and the standard $(4+4)$-dimensional space is its subalgebra generated by $κ$-Minkowski coordinates $\hat{x}_μ$ and corresponding commuting momenta $\hat{p}_μ$. 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.

math-ph↗
arXiv2015

Deformed Covariant Quantum Phase Spaces as Hopf Algebroids

We consider the general D=4 (10+10)-dimensional kappa-deformed quantum phase space as given by Heisenberg double \mathcal{H} of D=4 kappa-deformed Poincare-Hopf algebra H. The standard (4+4) -dimensional kappa - deformed covariant quantum phase space spanned by kappa - deformed Minkowski coordinates and commuting momenta generators ({x}_{μ},{p}_{μ}) is obtained as the subalgebra of \mathcal{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 explicite Hopf algebroid structure of standard kappa - 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.

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arXiv2013

Noncommutative Space-time from Quantized Twistors

We consider the relativistic phase space coordinates (x_μ,p_μ) as composite, described by functions of the primary pair of twistor coordinates. It appears that if twistor coordinates are canonicaly quantized the composite space-time coordinates are becoming noncommutative. We obtain deformed Heisenberg algebra which in order to be closed should be enlarged by the Pauli-Lubanski four-vector components. We further comment on star-product quantization of derived algebraic structures which permit to introduce spin-extended deformed Heisenberg algebra.

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arXiv2012

Braided Field Quantization from Quantum Poincare Covariance

We demonstrate that the covariance of the algebra of quantum NC fields under quantum-deformed Poincare symmetries implies the appearence of braided algebra of fields and the notion of braided locality in NC QFT. We briefly recall the historical development of NC QFT which was firstly formulated in the framework using classical relativistic symmetries but further it was described as generated by the quantum-deformed symmetries. We argue that consistent covariant quantum-deformed formalism requires "braiding all the way", in particular braided commutator of deformed field oscillators as well as the braid between the field oscillators and noncommutative Fourier exponentials. As example of braided quantum-deformed NC QFT we describe the NC scalar free fields on noncommutative canonical (Moyal-Weyl) space-time with braided c-number field commutator which implies braided locality.

hep-th↗
arXiv2012

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 M of functions on noncommutative space-time, the product in the algebra H of deformed field oscillators, and the braiding by factor Psi_{M,H} between algebras M and H. For noncommutativity generated by the twist factor we shall employ the star-product realizations of the algebra M in terms of functions on 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 Poincare group manifold. We shall calculate the covariant braided field commutator, which for free quantum noncommutative fields provides the field quantization condition and is given by standard Pauli-Jordan function. For ilustration of our new scheme we present explicit calculations for the well-known case in the literature of canonically deformed free quantum fields.

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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↗
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↗
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↗
arXiv2006

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 Garc{\'ı}a and by Bronnikov. It is shown that a simple generalization of the Bronnikov's 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.

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