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We investigate the impact of the deformed phase space associated with the quantum Snyder space on microphysical systems. The general Fermi–Dirac equation of state and specific corrections to it are derived. We put emphasis on... more
We investigate the impact of the deformed phase space associated with the quantum Snyder space on microphysical systems. The general Fermi–Dirac equation of state and specific corrections to it are derived. We put emphasis on non-relativistic degenerate Fermi gas as well as on the temperature-finite corrections to it. Considering the most general one-parameter family of deformed phase spaces associated with the Snyder model allows us to study whether the modifications arising in physical effects depend on the choice of realization. It turns out that we can distinguish three different cases with radically different physical consequences.
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ABSTRACT We briefly analyze some general questions concerning the twist deformation of the Heisenberg double. We reconsider Heisenberg doubles based on quantized Poincaré (Hopf) algebras as illustrative examples. KeywordsHopf... more
ABSTRACT We briefly analyze some general questions concerning the twist deformation of the Heisenberg double. We reconsider Heisenberg doubles based on quantized Poincaré (Hopf) algebras as illustrative examples. KeywordsHopf algebra–quantum deformation–smash product–Heisenberg double–quantum Poincaré
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In this paper we consider Hopf-algebraic deformations of Lie algebras $iso(g)$ of all inhomogeneous orthogonal groups $ISO(g)$ for a metric tensor $g$ of any dimension with an arbitrary signature. They are known as $\kappa-$deformations... more
In this paper we consider Hopf-algebraic deformations of Lie algebras $iso(g)$ of all inhomogeneous orthogonal groups $ISO(g)$ for a metric tensor $g$ of any dimension with an arbitrary signature. They are known as $\kappa-$deformations and include the well-known $\kappa$-Poincaré Hopf algebra as a special case. Such deformations are determined by a vector $\tau$ which can be taken time-, light- or space-like. We focus on mathematical issues connected with such generalized $\kappa (\tau )$- deformations and we discuss some related problems. Firstly we concentrate on h-adic vs q-analog (polynomial) versions of deformed algebras including specialization of the formal deformation parameter $\kappa$ to some numerical value. Also possible forms of $\kappa$-Minkowski spacetime are reconsidered. The last issue treated in this paper includes some extensions of $\kappa-$deformations by twisting.
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We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations... more
We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator. This general noncommutative geometry construction is then exemplified in the case of k-Minkowski spacetime. The corresponding quantum Poincare'-Weyl Lie algebra of infinitesimal translations, rotations and dilatations is obtained. The d'Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject.
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We review the application of twist deformation formalism and the construction of noncommutative gauge theory on κ-Minkowski space-time. We compare two different types of twists: the Abelian and the Jordanian one. In each case we provide... more
We review the application of twist deformation formalism and the construction of noncommutative gauge theory on κ-Minkowski space-time. We compare two different types of twists: the Abelian and the Jordanian one. In each case we provide the twisted differential calculus and consider U(1) gauge theory. Different methods of obtaining a gauge invariant action and related problems are thoroughly discussed.
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We consider different realizations for the momentum sector of kappa-Poincare Hopf algebra, which is associated with a curved momentum space. We show that the notion of the particle mass as introduced recently by Amelino-Camelia et al. in... more
We consider different realizations for the momentum sector of kappa-Poincare Hopf algebra, which is associated with a curved momentum space. We show that the notion of the particle mass as introduced recently by Amelino-Camelia et al. in the context of relative-locality is realization independent for a wide class of realizations, up to linear order in deformation parameter l. On the other hand, the time delay formula clearly shows a dependence on the choice of realization.
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We discuss a quantum deformation of the Green-Schwarz superstring on flat space, arising as a contraction limit of the corresponding deformation of AdS_5 x S^5. This contraction limit turns out to be equivalent to a previously studied... more
We discuss a quantum deformation of the Green-Schwarz superstring on flat space, arising as a contraction limit of the corresponding deformation of AdS_5 x S^5. This contraction limit turns out to be equivalent to a previously studied limit that yields the so-called mirror model - the model obtained from the light cone gauge fixed AdS_5 x S^5 string by a double Wick rotation. Reversing this logic, the AdS_5 x S^5 superstring is the double Wick rotation of a quantum deformation of the flat space superstring. This quantum deformed flat space string realizes symmetries of timelike kappa-Poincare type, and is T dual to dS_5 x H^5, indicating interesting relations between symmetry algebras under T duality. Our results directly extend to AdS_2 x S^2 x T^6 and AdS_3 x S^3 x T^4, and beyond string theory to many (semi)symmetric space coset sigma models, such as for example a deformation of the four dimensional Minkowski sigma model with timelike kappa-Poincare symmetry. We also discuss poss...
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Twisted deformations of the conformal symmetry in the Hopf algebraic framework are constructed. The first one is obtained by a Jordanian twist built up from dilatation and momenta generators. The second is the light-like κ-deformation of... more
Twisted deformations of the conformal symmetry in the Hopf algebraic framework are constructed. The first one is obtained by a Jordanian twist built up from dilatation and momenta generators. The second is the light-like κ-deformation of the Poincare algebra extended to the conformal algebra, obtained by a twist corresponding to the extended Jordanian r-matrix. The κ-Minkowski spacetime is covariant quantum space under both of these deformations. The extension of the conformal algebra by the noncommutative coordinates is presented in two cases. The differential realizations for κ-Minkowski coordinates, as well as their left-right dual counterparts, are also included.
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We propose a new generalization of the Jordanian twist (building on the previous idea from [J.Phys.A50,26(2017),arXiv:1612.07984]). Obtained this way, the family of the Jordanian twists allows for interpolation between two simple... more
We propose a new generalization of the Jordanian twist (building on the previous idea from [J.Phys.A50,26(2017),arXiv:1612.07984]). Obtained this way, the family of the Jordanian twists allows for interpolation between two simple Jordanian twists. This new version of the twist provides an example of a new type of star product and the realization for noncommutative coordinates. Exponential formulae, used to obtain coproducts and star products, are presented with details.
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We demonstrate that the coproduct of D=2 and D=4 quantum kappa-Poincare algebra in classical algebra basis can not be obtained by the cochain twist depending only on Poincare algebra generators. We also argue that nonexistence of such a... more
We demonstrate that the coproduct of D=2 and D=4 quantum kappa-Poincare algebra in classical algebra basis can not be obtained by the cochain twist depending only on Poincare algebra generators. We also argue that nonexistence of such a twist does not imply the nonexistence of universal R-matrix.
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We extend our previous study of Hopf-algebraic κ-deformations of all inhomogeneous orthogonal Lie algebras iso(g) as written in a tensorial and unified form. Such deformations are determined by a vector τ which for Lorentzian signature... more
We extend our previous study of Hopf-algebraic κ-deformations of all inhomogeneous orthogonal Lie algebras iso(g) as written in a tensorial and unified form. Such deformations are determined by a vector τ which for Lorentzian signature can be taken time-, light- or space-like. We focus on some mathematical aspects related to this subject. Firstly, we describe real forms with connection to the metric's signatures and their compatibility with the reality condition for the corresponding κ-Minkowski (Hopf) module algebras. Secondly, h-adic vs q-analog (polynomial) versions of deformed algebras including specialization of the formal deformation parameter κ to some numerical value are considered. In the latter the general covariance is lost and one deals with an orthogonal decomposition. The last topic treated in this paper concerns twisted extensions of κ-deformations as well as the description of resulting noncommutative spacetime algebras in terms of solvable Lie algebras. We found...
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In this paper we provide universal formulas describing Drinfeld-type quantization of inhomogeneous orthogonal groups determined by a metric tensor of an arbitrary signature living in a spacetime of arbitrary dimension. The metric tensor... more
In this paper we provide universal formulas describing Drinfeld-type quantization of inhomogeneous orthogonal groups determined by a metric tensor of an arbitrary signature living in a spacetime of arbitrary dimension. The metric tensor does not need to be in diagonal form and kappa-deformed coproducts are presented in terms of classical generators. It opens the possibility for future applications in deformed general relativity. The formulas depend on the choice of an additional vector field which parameterizes classical r-matrices. Non-equivalent deformations are then labeled by the corresponding type of stability subgroups. For the Lorentzian signature it covers three (non-equivalent) Hopf-algebraic deformations: time-like, space-like (a.k.a. tachyonic) and light-like (a.k.a. light-cone) quantizations of the Poincare algebra. Finally the existence of the so-called Majid-Ruegg (non-classical) basis is reconsidered.
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We derive an explicit expression for the star product reproducing the κ-Minkowski Lie algebra in any dimension n. The result is obtained by suitably reducing the Wick-Voros star product defined on C^d_θ with n=d+1. It is thus shown that... more
We derive an explicit expression for the star product reproducing the κ-Minkowski Lie algebra in any dimension n. The result is obtained by suitably reducing the Wick-Voros star product defined on C^d_θ with n=d+1. It is thus shown that the new star product can be obtained from a Jordanian twist.
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We study bimodule quantum Riemannian geometries over the field F_2 of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for... more
We study bimodule quantum Riemannian geometries over the field F_2 of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for coordinate algebras up to vector space dimension n< 3, finding a rich moduli of examples for n=3 and top form degree 2, including many that are not flat. Their coordinate algebras are commutative but their differentials are not. We also study the quantum Laplacian Δ=( ,)∇ d on our models and characterise when it has a massive eigenvector.
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Bialgebroids (resp. Hopf algebroids) are bialgebras (Hopf algebras) over noncommutative rings. Drinfeld twist techniques are particularly useful in the (deformation) quantization of Lie algebras as well as underlying module algebras... more
Bialgebroids (resp. Hopf algebroids) are bialgebras (Hopf algebras) over noncommutative rings. Drinfeld twist techniques are particularly useful in the (deformation) quantization of Lie algebras as well as underlying module algebras (=quantum spaces). Smash product construction combines these two into the new algebra which, in fact, does not depend on the twist. However, we can turn it into bialgebroid in the twist dependent way. Alternatively, one can use Drinfeld twist techniques in a category of bialgebroids. We show that both techniques indicated in the title: twisting of a bialgebroid or constructing a bialgebroid from the twisted bialgebra give rise to the same result in the case of normalized cocycle twist. This can be useful for better description of a quantum deformed phase space. We argue that within this bialgebroid framework one can justify the use of deformed coordinates (i.e. spacetime noncommutativity) which are frequently postulated in order to explain quantum gravit...
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The dissertation presents possibilities of applying noncommutative spacetimes description, particularly kappa-deformed Minkowski spacetime and Drinfeld's deformation theory, as a mathematical formalism for Doubly Special Relativity... more
The dissertation presents possibilities of applying noncommutative spacetimes description, particularly kappa-deformed Minkowski spacetime and Drinfeld's deformation theory, as a mathematical formalism for Doubly Special Relativity theories (DSR), which are thought as phenomenological limit of quantum gravity theory. Deformed relativistic symmetries are described within Hopf algebra language. In the case of (quantum) kappa-Minkowski spacetime the symmetry group is described by the (quantum) kappa-Poincare Hopf algebra. Deformed relativistic symmetries were used to construct the DSR algebra, which unifies noncommutative coordinates with generators of the symmetry algebra. It contains the deformed Heisenberg-Weyl subalgebra. It was proved that DSR algebra can be obtained by nonlinear change of generators from undeformed algebra. We show that the possibility of applications in Planck scale physics is connected with certain realizations of quantum spacetime, which in turn leads to d...
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In this paper we perform a parallel analysis to the model proposed in [25]. By considering the central co-tetrad (instead of the central metric) we investigate the modifications in the gravitational metrics coming from the noncommutative... more
In this paper we perform a parallel analysis to the model proposed in [25]. By considering the central co-tetrad (instead of the central metric) we investigate the modifications in the gravitational metrics coming from the noncommutative spacetime of the $\kappa$-Minkowski type in four dimensions. The differential calculus corresponding to a class of Jordanian $ \kappa$-deformations provide metrics which lead either to cosmological constant or spatial-curvature type solutions of non-vacuum Einstein equations. Among vacuum solutions one finds pp-waves.
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We study noncommutative deformations of the wave equation in curved backgrounds and discuss the modification of the dispersion relations due to noncommutativity combined with curvature of spacetime. Our noncommutative differential... more
We study noncommutative deformations of the wave equation in curved backgrounds and discuss the modification of the dispersion relations due to noncommutativity combined with curvature of spacetime. Our noncommutative differential geometry approach is based on Drinfeld twist deformation, and can be implemented for any twist and any curved background. We discuss in detail the Jordanian twist — giving κ-Minkowski spacetime in flat space — in the presence of a Friedman-Lemaître-Robertson-Walker (FLRW) cosmological background. We obtain a new expression for the variation of the speed of light, depending linearly on the ratio E ph/E LV (photon energy/Lorentz violation scale), but also linearly on the cosmological time, the Hubble parameter and inversely proportional to the scale factor.
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In this paper, we perform a parallel analysis to the model proposed in [E. J. Beggs and S. Majid, Gravity induced from quantum spacetime, Class. Quantum Grav. 31 (2014) 035020, arXiv: 1305.2403 [gr-qc]]. By considering the central... more
In this paper, we perform a parallel analysis to the model proposed in [E. J. Beggs and S. Majid, Gravity induced from quantum spacetime, Class. Quantum Grav. 31 (2014) 035020, arXiv: 1305.2403 [gr-qc]]. By considering the central co-tetrad (instead of the central metric), we investigate the modifications in the gravitational metrics coming from the noncommutative spacetime of the [Formula: see text]-Minkowski type in four dimensions. The differential calculus corresponding to a class of Jordanian [Formula: see text]-deformations provides metrics, which lead either to cosmological constant or spatial curvature type solutions of nonvacuum Einstein equations. Among vacuum solutions, we find pp-wave type.
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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... more
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.
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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... more
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.