κ\kappa-Minkowski Spacetimes and DSR Algebras: Fresh Look and Old Problems

Some classes of Deformed Special Relativity (DSR) theories are reconsidered within the Hopf algebraic formulation. For this purpose we shall explore a minimal framework of deformed Weyl-Heisenberg algebras provided by a smash product construction of DSR algebra. It is proved that this DSR algebra, which uniquely unifies $\kappa$-Minkowski spacetime coordinates with Poincar\'e generators, can be obtained by nonlinear change of generators from undeformed one. Its various realizations in terms of the standard (undeformed) Weyl-Heisenberg algebra opens the way for quantum mechanical interpretation of DSR theories in terms of relativistic (St\"uckelberg version) Quantum Mechanics. On this basis we review some recent results concerning twist realization of $\kappa$-Minkowski spacetime described as a quantum covariant algebra determining a deformation quantization of the corresponding linear Poisson structure. Formal and conceptual issues concerning quantum $\kappa$-Poincar\'e and $\kappa$-Minkowski algebras as well as DSR theories are discussed. Particularly, the so-called "$q$-analog" version of DSR algebra is introduced. Is deformed special relativity quantization of doubly special relativity remains an open question. Finally, possible physical applications of DSR algebra to description of some aspects of Planck scale physics are shortly recalled

Heisenberg doubles for Snyder type models

A Snyder model generated by the noncommutative coordinates and Lorentz generators close a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. It 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.Comment: 19 pages, no figures, 1 Appendix; version accepted for publicatio

Fermi equation of state with finite temperature corrections in quantum space-times approach: Snyder model vs GUP case

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.Comment: 15 pages; version accepted for publication in CQ

Constraining Snyder and GUP models with low-mass stars

We investigate the application of an equation of state that incorporates corrections derived from the Snyder model (and the Generalized Uncertainty Principle) to describe the behavior of matter in a low-mass star. Remarkably, the resulting equations exhibit striking similarities to those arising from modified Einstein gravity theories. By modeling matter with realistic considerations, we are able to more effectively constrain the theory parameters, surpassing the limitations of existing astrophysical bounds. The bound we obtain is $\beta_0 \leq 1.36 \times 10^{48}$. We underline the significance of realistic matter modeling in order to enhance our understanding of effects arising in quantum gravity phenomenology and implications of quantum gravitational corrections in astrophysical systems.Comment: 15 pages, 2 figures, 1 tabl

ℏ\hbar-perturbative solutions of quantum Snyder and Yang models with parameters describing spontaneous symmetry breaking

We introduce the perturbative $\hbar$-power series ($\hbar$ = Planck constant) providing the algebraic solutions of $D=4$ quantum Snyder and Yang models which describe relativistic quantum space-times and Lorentz-covariant quantum phase spaces. We argue that if in these series the zero order ($\hbar$-independent) terms are non-vanishing they describe the spontaneous symmetry breaking (SSB) parameters of Lie-algebraic symmetries which characterize the considered models ($D=4$ dS symmetry in Snyder and $D=5$ dS symmetry in Yang cases). The consecutive terms in $\hbar$-power series can be calculated explicitly if we supplement the SSB order parameters (Nambu-Goldstone or NG modes) by dual set of commutative momenta, which together define the canonical tensorial Heisenberg algebra.Comment: Contribution to PoS Corfu 202

Dispersion Relations in κ\kappa-Noncommutative Cosmology

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 $\kappa$-Minkowski spacetime in flat space$-$ in the presence of a Friedman-Lema\^{i}tre-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.Comment: 20 pages. New version: 23 pages, added 4-dim. dispersion relations and numerical estimate

Quantum perturbative solutions of extended Snyder and Yang models with spontaneous symmetry breaking

We propose $\hbar$-expansions as perturbative solutions of quantum extended Snyder and Yang models, with $\hbar$-independent classical zero-th order terms responsible for the spontaneous breaking of $D=4$ and $D=5$ de Sitter symmetries. In such models, with algebraic basis spanned by $\hat o(D,1)$ Lie algebra generators, we relate the vacuum expectation values (VEV) of the spontaneously broken generators with the Abelian set of ten (Snyder, $D=4$) or fifteen (Yang, $D=5$) antisymmetric tensorial generalized coordinates, which are also used as zero order input for obtaining the perturbative solutions of quantum extended Snyder and Yang models. In such a way we will attribute to these Abelian generalized coordinates the physical meaning of the order parameters describing spontaneous symmetry breaking (SSB). It appears that the consecutive terms in $\hbar$-power series can be calculated explicitly if we supplement the SSB order parameters by the dual set of tensorial commutative momenta.Comment: 11 pages; extended version, primary introduction divided into two sections, with a new subsection 2b; substantially extended list of reference

From Snyder space-times to doubly κ\kappa-deformed Yang quantum phase spaces and their generalizations

We propose the double $\kappa$-deformation of Yang quantum phase space which is described by the generalization of $D=4$ Yang model. We postulate that the algebra of such a model is covariant under the generalized Born map, what permits to derive our model from the $\kappa$-deformed Snyder model. Our generalized quantum phase space depends on five deformation parameters defining two Born map-related dimension-full pairs: $(M, R)$ specifying the Yang model and $(\kappa, \tilde{\kappa})$ characterizing the Born-dual $\kappa$-deformations of quantum space-time and quantum fourmomenta sectors; fifth dimensionless parameter $\rho$ is Born-selfdual. Finally, we define the Kaluza-Klein generalization of the Yang model with Lorentz covariance supplemented by internal $\hat{o}(2N)$ symmetries.Comment: 9 page

Remarks on simple interpolation between Jordanian twists

In this paper, we propose a simple generalization of the locally r-symmetric Jordanian twist, resulting in the one-parameter family of Jordanian twists. All the proposed twists differ by the coboundary twists and produce the same Jordanian deformation of the corresponding Lie algebra. They all provide the $\kappa$-Minkowski spacetime commutation relations. Constructions from noncommutative coordinates to the star product and coproduct, and from the star product to the coproduct and the twist are presented. The corresponding twist in the Hopf algebroid approach is given. Our results are presented symbolically by a diagram relating all of the possible constructions.Comment: 12 page