Snyder Algebra Research Articles

Realizations and star-product of doubly κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-deformed Yang models

The Yang algebra was proposed a long time ago as a generalization of the Snyder algebra to the case of curved background spacetime. It includes as subalgebras both the Snyder and the de Sitter algebras and can therefore be viewed as a model of noncommutative curved spacetime. A peculiarity with respect to standard models of noncommutative geometry is that it includes translation and Lorentz generators, so that the definition of a Hopf algebra and the physical interpretation of the variables conjugated to the primary ones is not trivial. In this paper we consider the realizations of the Yang algebra and its κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-deformed generalization on an extended phase space and in this way we are able to define a Hopf structure and a twist.

Generalizations of Snyder model to curved spaces

We consider generalizations of the Snyder algebra to a curved spacetime background with de Sitter symmetry. As special cases, we obtain the algebras of the Yang model and of triply special relativity. We discuss the realizations of these algebras in terms of canonical phase space coordinates, up to fourth order in the deformation parameters. In the case of triply special relativity we also find exact realization, exploiting its algebraic relation with the Snyder model.

Translation in momentum space and minimal length

We show that in the presence of the Snyder algebra the notion of translation in momentum space is modified to a formula similar to the relativistic addition of velocities. These results confirm the strict connection between Snyder algebra and the Lorentz group.

Heisenberg Doubles for Snyder-Type Models

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.

Bumblebee gravity and particle motion in Snyder noncommutative spacetime structures

A metric with a Lorentz violating parameter is associated with thebumblebee gravity model. We study the motion of a particle in thisbumblebee background where the dynamical variables satisfynon-canonical Snyder algebra along with some critical survey onthe classical observations in the bumblebee background to see howthese would likely differ from Schwarzschild background. It hasbeen found that the perihelion shift acquires a generalizedexpression with two independent parameters. One of these two isconnected with the Lorentz violating factor and the other isinvolved in the Snyder algebraic formulation. We also observe thatthe time period of revolution, in general, acquires a Lorentzviolating factor in the bumblebee background, however, for thecircular orbit, it remains unchanged even in the presence of theLorentz violating factor in the bumblebee background. Theparameters used here can be constrained with the same type ofconjecture used earlier.

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.

Dirac oscillator and minimal length

We show how to solve the Dirac oscillator with a minimal length by using previous results on the harmonic oscillator in the Snyder algebra.

Generalized higher-order Snyder model

In this paper we consider the higher-order generalization of the Snyder algebra. We find the modified inner product for this algebra. We also discuss the one-dimensional hydrogen atom eigenvalue problem. We find the invariant phase space volume for the generalized Snyder algebra. Finally we use the density of states in three dimension to discuss the Maxwellian distribution and cosmological constant.

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.

Conformal invariance in noncommutative geometry and mutually interacting Snyder particles

A system of relativistic Snyder particles with mutual two-body interaction that lives in a Non-Commutative Snyder geometry is studied. The underlying novel symplectic structure is a coupled and extended version of (single particle) Snyder algebra. In a recent work by Casalbuoni and Gomis, Phys.Rev. D90, 026001 (2014), a system of interacting conventional particles (in commutative spacetime) was studied with special emphasis on it's Conformal Invariance. Proceeding along the same lines we have shown that our interacting Snyder particle model is also conformally invariant. Moreover, the conformal Killing vectors have been constructed. Our main emphasis is on the Hamiltonian analysis of the conformal symmetry generators. We demonstrate that the Lorentz algebra remains undeformed but validity of the full conformal algebra requires further restrictions.

GUP-BASED AND SNYDER NONCOMMUTATIVE ALGEBRAS, RELATIVISTIC PARTICLE MODELS, DEFORMED SYMMETRIES AND INTERACTION: A UNIFIED APPROACH

We have developed a unified scheme for studying noncommutative algebras based on generalized uncertainty principle (GUP) and Snyder form in a relativistically covariant point particle Lagrangian (or symplectic) framework. Even though the GUP-based algebra and Snyder algebra are very distinct, the more involved latter algebra emerges from an approximation of the Lagrangian model of the former algebra. Deformed Poincaré generators for the systems that keep space–time symmetries of the relativistic particle models have been studied thoroughly. From a purely constrained dynamical analysis perspective the models studied here are very rich and provide insights on how to consistently construct approximate models from the exact ones when nonlinear constraints are present in the system. We also study dynamics of the GUP particle in presence of external electromagnetic field.

PROPERTIES OF SNYDER SPACE

We argue that Snyder's algebra has two distinct representations and they yield a lattice description of space. We also argue that particle dynamics can be defined on such a lattice in a manner which is consistent with Poincaré invariance.

Supersymmetric extension of the Snyder algebra

We obtain a minimal supersymmetric extension of the Snyder algebra and study its representations. The construction differs from the general approach given in Hatsuda and Siegel ( arXiv:hep-th/0311002) and does not utilize super-de Sitter groups. The spectra of the position operators are discrete, implying a lattice description of space, and the lattice is compatible with supersymmetry transformations.

Snyder space revisited

We examine basis functions on momentum space for the three-dimensional Euclidean Snyder algebra. We argue that the momentum space is isomorphic to the SO ( 3 ) group manifold, and that the basis functions span either one of two Hilbert spaces. This implies the existence of two distinct lattice structures of space. Continuous rotations and translations are unitarily implementable on these lattices.

Emergent geometry from quantized spacetime

We examine the picture of emergent geometry arising from a mass-deformed matrix model. Because of the mass-deformation, a vacuum geometry turns out to be a constant curvature spacetime such as d-dimensional sphere and (anti-)de Sitter spaces. We show that the mass-deformed matrix model giving rise to the constant curvature spacetime can be derived from the d-dimensional Snyder algebra. The emergent geometry beautifully confirms all the rationale inferred from the algebraic point of view that the d-dimensional Snyder algebra is equivalent to the Lorentz algebra in (d+1)-dimensional {\it flat} spacetime. For example, a vacuum geometry of the mass-deformed matrix model is completely described by a G-invariant metric of coset manifolds G/H defined by the Snyder algebra. We also discuss a nonlinear deformation of the Snyder algebra.

Deformed general relativity and torsion

We argue that the natural framework for embedding the ideas of deformed, or doubly, special relativity (DSR) into a curved spacetime is a generalization of Einstein–Cartan theory, considered by Stelle and West. Instead of interpreting the noncommuting ‘spacetime coordinates’ of the Snyder algebra as endowing spacetime with a fundamentally noncommutative structure, we are led to consider a connection with torsion in this framework. This may lead to the usual ambiguities in minimal coupling. We note that observable violations of charge conservation induced by torsion should happen on a timescale of 103 s, which seems to rule out these modifications as a serious theory. Our considerations show, however, that the noncommutativity of translations in the Snyder algebra need not correspond to noncommutative spacetime in the usual sense.

Lorentz-covariant deformed algebra with minimal length

TheD-dimensional two-parameter deformed algebra with minimal length introduced by Kempf is generalized to a Lorentz-covariant algebra describing a (D + 1)-dimensional quantized space-time. ForD=3, it includes Snyder algebra as a special case. The deformed Poincare transformations leaving the algebra invariant are identified. Uncertainty relations are studied. In the case ofD=1 and one nonvanishing parameter, the bound-state energy spectrum and wavefunctions of the Dirac oscillator are exactly obtained.

Lorentz-covariant deformed algebra with minimal length and application to the (1 + 1)-dimensional Dirac oscillator

The $D$-dimensional $(\beta, \beta')$-two-parameter deformed algebra introduced by Kempf is generalized to a Lorentz-covariant algebra describing a ($D+1$)-dimensional quantized space-time. In the D=3 and $\beta=0$ case, the latter reproduces Snyder algebra. The deformed Poincar\'e transformations leaving the algebra invariant are identified. It is shown that there exists a nonzero minimal uncertainty in position (minimal length). The Dirac oscillator in a 1+1-dimensional space-time described by such an algebra is studied in the case where $\beta'=0$. Extending supersymmetric quantum mechanical and shape-invariance methods to energy-dependent Hamiltonians provides exact bound-state energies and wavefunctions. Physically acceptable states exist for $\beta < 1/(m^2 c^2)$. A new interesting outcome is that, in contrast with the conventional Dirac oscillator, the energy spectrum is bounded.

Deformed symmetry in Snyder space and relativistic particle dynamics

We describe the deformed Poincare-conformal symmetries implying the covariance of the noncommutative space obeying Snyder's algebra. Relativistic particle models invariant under these deformed symmetries are presented. A gauge (reparametrisation) independent derivation of Snyder's algebra from such models is given. The algebraic transformations relating the deformed symmetries with the usual (undeformed) ones are provided. Finally, an alternative form of an action yielding Snyder's algebra is discussed where the mass of a relativistic particle gets identified with the inverse of the noncommutativity parameter.

The AdS particle

In this Letter we have considered a relativistic Nambu–Goto model for a particle in AdS metric. With appropriate gauge choice to fix the reparameterization invariance, we recover the previously discussed [S. Ghosh, P. Pal, Phys. Lett. B 618 (2005) 243, hep-th/0502192] “exotic oscillator”. The Snyder algebra and subsequently the κ-Minkowski spacetime are also derived. Lastly we comment on the impossibility of constructing a non-commutative spacetime in the context of open string where only a curved target space is introduced.