Snyder Space Research Articles

Exact Green’s function for 2D dirac oscillator in constant magnetic field within curved snyder space, and its thermal properties

Following the path integral approach, and in the context of curved Snyder space, we formulate the Green function for a (1+1)-dimensional Dirac oscillator system subject to a homogeneous magnetic field. Using the radial coordinates transformation the Green function and the electron propagator are calculated. Consequently, the exact bound states and their corresponding spectral energies are extracted. Our analysis has revealed that, under specific conditions when mω¯→mωc/2 and c → V F , the behavior of the Dirac oscillator system in the presence of a uniform magnetic field within the SdS algebra closely resembles the dynamics of the monolayer graphene problem in the same algebraic framework. At high temperatures, the thermodynamic properties of the electron gas in the four cases of deformation parameters were extracted. The effect of the deformation parameters on these properties are tested, and also the limit cases for small parameters were inferred.

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.

Symmetric ordering and Weyl realizations for quantum Minkowski spaces

Symmetric ordering and Weyl realizations for non-commutative quantum Minkowski spaces are reviewed. Weyl realizations of Lie deformed spaces and corresponding star products, as well as twist corresponding to Weyl realization and coproduct of momenta, are presented. Drinfeld twists understood in Hopf algebroid sense are also discussed. A few examples of corresponding Weyl realizations are given. We show that for the original Snyder space, there exists symmetric ordering but no Weyl realization. Quadratic deformations of Minkowski space are considered, and it is demonstrated that symmetric ordering is deformed and a generalized Weyl realization can be defined.

Plasma oscillations in a generalized Snyder space

Plasma oscillations in a generalized Snyder space

Plasma oscillations in a generalized Snyder space

С тех пор как Снайдер впервые представил теорию квантованного пространства-времени, были проведены многочисленные исследования, имеющие целью демонстрацию феноменологических аспектов этой теории. В обобщенном пространстве Снайдера исследуется плазменная частота и найдены электронные и ионные плазменные волны. Получены оценки верхней границы масштаба изотропной минимальной длины, которая оказалась близка к $l_0\sim 10^{-9}$ м.

Three-dimensional DKP oscillator in a curved Snyder space

The Snyder–de Sitter model is an extension of the Snyder model to a de Sitter background. It is called triply special relativity (TSR) because it is based on three fundamental parameters: speed of light, Planck mass and cosmological constant. In this paper, we study the three-dimensional DKP oscillator for spin-0 and spin-1 in the framework of Snyder–de Sitter algebra in momentum space. By using the technique of vector spherical harmonics the energy spectrum and the corresponding eigenfunctions are obtained for the both cases.

Unification of κ-Minkowski and extended Snyder spaces

In a recent paper, we have studied associative realizations of the noncommutative extended Snyder model, obtained by including the Lorentz generators (tensorial coordinates) and their conjugated momenta. In this paper, we extend this result to also incorporate a covariant realization of the κ-Poincaré spacetime. We obtain the coproduct, the associative star product and the twist in a Weyl-ordered realization, to first order in the noncommutativity parameters. This could help the construction of a quantum field theory based on this geometry.

Graphene in curved Snyder space

Abstract The Snyder-de Sitter (SdS) model which is invariant under the action of the de Sitter group, is an example of a noncommutative space-time with three fundamental scales. In this paper, we considered the massless Dirac fermions in graphene layer in a curved Snyder space-time which are subjected to an external magnetic field. We employed representation in the momentum space to derive the energy eigenvalues and the eigenfunctions of the system. Then, we used the deduced energy function obtaining the internal energy, heat capacity, and entropy functions. We investigated the role of the fundamental scales on these thermal quantities of the graphene layer. We found that the effect of the SdS model on the thermodynamic properties is significant.

Casimir effect in Snyder space

We show that two indistinguishable aspects of the divergences occurring in the Casimir effect, namely the divergence of the energy of the higher modes and the non-compactness of the momentum space, get disentangled in a given noncommutative setup. To this end, we consider a scalar field between two parallel plates in an anti-Snyder space. Additionally, the large mass decay in this noncommutative setup is not necessarily exponential.

Pair creation in curved Snyder space

We study the problem of pair creation of scalar particles by an electric field in curved Snyder space. We find exact solutions for the Klein–Gordon equation with a constant electric field in terms of hypergeometric functions. Then we calculate the pair creation probability and the number of created pairs of particles through Bogoliubov transformation technique.

Classical electrodynamics on Snyder space

A formulation of classical electrodynamics on an energy-momentum background of constant, non-zero curvature is given. The procedure consists of taking the formulation of standard electrodynamics in the energy-momentum representation, and promoting the energy-momentum vector to belong to a constant (non-zero) curvature space. In particular, special emphasis is given to the definition of integration measure and generalized Dirac's delta function. Finally, simple physical problems as plane waves (solutions outside sources) and point charges are discussed in this context, where the self-energy of a point charge is shown to be finite.

Macroscopic body in the Snyder space and minimal length estimation

We study the problem of the description of a macroscopic body motion in the frame of the nonrelativistic Snyder model. It is found that the motion of the center-of-mass of a body is described by an effective parameter which depends on the parameters of the Snyder algebra for coordinates and momenta of particles forming the body and their masses. We also show that there is a reduction of the effective parameter with respect to the parameters of the Snyder algebra for coordinates and momenta of individual particles. As a result the problem of the extremely small result for the minimal length obtained on the basis of the studies of the Mercury motion in the Snyder space is solved. In addition we find that the relation of the parameter of the Snyder algebra with the mass opens the possibility to preserve the property of independence of the kinetic energy on composition, to recover the weak equivalence principle, to consider coordinates as kinematic variables, to recover the proportionality of momenta to the mass and to consider the Snyder algebra for the coordinates and momenta of the center-of-mass of a body defined in the traditional way.

Twist for Snyder space

We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of momenta is non-coassociative. The twist is constructed using a general definition of the star product in terms of a bi-differential operator in the Hopf algebroid approach. The result is given by a closed analytical expression. We prove that this twist reproduces the correct coproducts of the momenta and the Lorentz generators. The twisted Poincaré symmetry is described by a non-associative Hopf algebra, while the twisted Lorentz symmetry is described by the undeformed Hopf algebra. This new twist might be important in the construction of different types of field theories on Snyder space.

Families of vector-like deformations of relativistic quantum phase spaces, twists and symmetries

Families of vector-like deformed relativistic quantum phase spaces and corresponding realizations are analyzed. A method for a general construction of the star product is presented. The corresponding twist, expressed in terms of phase space coordinates, in the Hopf algebroid sense is presented. General linear realizations are considered and corresponding twists, in terms of momenta and Poincaré–Weyl generators or mathfrak {gl}(n) generators are constructed and R-matrix is discussed. A classification of linear realizations leading to vector-like deformed phase spaces is given. There are three types of spaces: (i) commutative spaces, (ii) kappa -Minkowski spaces and (iii) kappa -Snyder spaces. The corresponding star products are (i) associative and commutative (but non-local), (ii) associative and non-commutative and (iii) non-associative and non-commutative, respectively. Twisted symmetry algebras are considered. Transposed twists and left–right dual algebras are presented. Finally, some physical applications are discussed.

Thermodynamical dimensional reduction in a UV & IR-deformed Snyder space

We study a formulation of statistical mechanics in the context of symplectic structures of the IR and also UV & IR-deformed Snyder phase-spaces. We derive the corresponding invariant Liouville volume and by using it we obtain the deformed partition function. We then study the thermodynamical properties of the 3-dimensional harmonic oscillator in this set-up. By using the equipartition theorem, we show that two of the six degrees of freedom for a 3-dimensional harmonic oscillator will be frozen as the temperature increases. Also, at a constant temperature, whatever is the increase in oscillator length, this reduction of the number of degrees of freedom gets more and more appreciable and it offers an effective dimensional reduction of space from [Formula: see text] to [Formula: see text] when it is close to the IR-length scale.

Quantum field theory in generalised Snyder spaces

We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate perturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT.

The effects of quantum gravity on some thermodynamical quantities

In this paper, using a deformed algebra [Formula: see text] which is originated from various theories of gravity, we study thermodynamical properties of the classical and extreme relativistic gases in canonical ensembles. In this regards, we exactly calculate the modified partition function, Helmholtz free energy, internal energy, entropy, heat capacity and the thermal pressure which conclude to the familiar form of the equation of state for the ideal gas. The advantage of applying this algebra is not only considering all natural cutoffs but also its structure is similar to the other effective quantum gravity models such as polymer, Snyder and noncommutative space–time frameworks. Moreover, after obtaining some thermodynamical quantities including internal energy and entropy, we conclude at high temperature limits due to the decreasing of the number of microstates, these quantities reach to maximal bounds which do not exist in standard cases and it concludes that at the presence of gravity for both micro-canonic and canonic ensembles, the internal energy and the entropy tend to these upper bounds.

Path integral in Snyder space

The definition of path integrals in one- and two-dimensional Snyder space is discussed in detail both in the traditional setting and in the first-order formalism of Faddeev and Jackiw.

Spectrum of the hydrogen atom in Snyder space in a semiclassical approximation

We study the spectrum of the hydrogen atom in Snyder space in a semiclassical approximation based on a generalization of the Born-Sommerfeld quantization rule. While the corrections to the standard quantum mechanical spectrum arise at first order in the Snyder parameter for the $l=0$ states, they are of second order for $l\neq 0$. This can be understood as due to the different topology of the regions of integration in phase space.

Quantum Mechanics on a Curved Snyder Space

We study the representations of the three-dimensional Euclidean Snyder-de Sitter algebra. This algebra generates the symmetries of a model admitting two fundamental scales (Planck mass and cosmological constant) and is invariant under the Born reciprocity for exchange of positions and momenta. Its representations can be obtained starting from those of the Snyder algebra and exploiting the geometrical properties of the phase space that can be identified with a Grassmannian manifold. Both the position and momentum operators turn out to have a discrete spectrum.