18 research outputs found
-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 -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 -Minkowski spacetime described as a quantum covariant
algebra determining a deformation quantization of the corresponding linear
Poisson structure. Formal and conceptual issues concerning quantum
-Poincar\'e and -Minkowski algebras as well as DSR theories are
discussed. Particularly, the so-called "-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 . 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
-perturbative solutions of quantum Snyder and Yang models with parameters describing spontaneous symmetry breaking
We introduce the perturbative -power series ( = Planck
constant) providing the algebraic solutions of 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 (-independent) terms are non-vanishing they describe the spontaneous symmetry
breaking (SSB) parameters of Lie-algebraic symmetries which characterize the
considered models ( dS symmetry in Snyder and dS symmetry in Yang
cases). The consecutive terms in -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 -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 -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 (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 -expansions as perturbative solutions of quantum extended
Snyder and Yang models, with -independent classical zero-th order terms
responsible for the spontaneous breaking of and de Sitter
symmetries. In such models, with algebraic basis spanned by Lie
algebra generators, we relate the vacuum expectation values (VEV) of the
spontaneously broken generators with the Abelian set of ten (Snyder, ) or
fifteen (Yang, ) 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 -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 -deformed Yang quantum phase spaces and their generalizations
We propose the double -deformation of Yang quantum phase space which
is described by the generalization of 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 -deformed Snyder model. Our
generalized quantum phase space depends on five deformation parameters defining
two Born map-related dimension-full pairs: specifying the Yang model
and characterizing the Born-dual
-deformations of quantum space-time and quantum fourmomenta sectors;
fifth dimensionless parameter is Born-selfdual. Finally, we define the
Kaluza-Klein generalization of the Yang model with Lorentz covariance
supplemented by internal 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
-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