Classical and quantum mechanics of the nonrelativistic Snyder model

Abstract

The Snyder model is an example of noncommutative spacetime admitting a fundamental length scale $\beta$ and invariant under Lorentz transformations, that can be interpreted as a realization of the doubly special relativity axioms. Here, we consider its nonrelativistic counterpart, i.e. the Snyder model restricted to three-dimensional Euclidean space. We discuss the classical and the quantum mechanics of a free particle in this framework, and show that they strongly depend on the sign of a coupling constant $\lambda$, appearing in the fundamental commutators and proportional to ${\beta }^2$. For example, if $\lambda$ is negative, momenta are bounded. On the contrary, for positive $\lambda$, positions and areas are quantized. We also give the exact solution of the harmonic oscillator equations both in the classical and the quantum case, and show that its frequency is energy dependent.

21 pages, 5 figures, plain TeX. Some points have been simplified. Version published on PRD

References in corpus (3)

• Deformed Special Relativity and Deformed Symmetries in a Canonical Framework
• Doubly special relativity and translation invariance
• The Area Quantum and Snyder Space

Cited by in corpus (63)

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• High temperature dimensional reduction in Snyder space
• Particle Dynamics on Snyder space
• Kappa-Minkowski spacetime, Kappa-Poincaré Hopf algebra and realizations
• Snyder dynamics in a Schwarzschild spacetime
• Spacetime singularity resolution in Snyder noncommutative space
• Classical dynamics on curved Snyder space
• Natural Cutoffs via Compact Symplectic Manifolds
• Quantum field theory in generalised Snyder spaces
• The Minimal Length and the Quantum Partition Functions
• Bumblebee gravity and particle motion in Snyder noncommutative spacetime structures
• Relative locality in a quantum spacetime and the pregeometry of $\kappa$-Minkowski
• Dynamics of a Dirac Fermion in the presence of spin noncommutativity
• The Constraints and Spectra of a Deformed Quantum Mechanics
• Effect of Snyder-de Sitter model on the Black hole thermodynamics in the context of rainbow gravity
• Geometrizing the Klein-Gordon and Dirac equations in Doubly Special Relativity
• Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry
• Snyder-type spaces, twisted Poincaré algebra and addition of momenta
• On deformations of classical mechanics due to Planck-scale physics
• UV/IR Mixing in Nonassociative Snyder phi^4 Theory
• Bounds on quantum gravity parameter from the $SU(2)$ NJL effective model of QCD
• Spectrum of the hydrogen atom in Snyder space in a semiclassical approximation
• Casimir effect in Snyder Space
• Constraining Snyder and GUP models with low-mass stars
• Noncommutative Yang model and its generalizations
• Polymer quantization versus the Snyder noncommutative space
• Klein-Gordon theory in noncommutative phase space
• Generalized Extended Uncertainty Principles, Liouville theorem and density of states: Snyder-de Sitter and Yang models
• Macroscopic body in Snyder space and minimal length estimation
• Generalized uncertainty principle and thermostatistics: a semiclassical approach
• Deformed Quantum Phase Spaces, Realizations, Star Products and Twists
• Graphene in curved Snyder space
• Late time cosmic acceleration from natural infrared cutoff?
• Vacuum energy from noncommutative models
• Path integrals in Snyder space
• Noncommutative spaces and covariant formulation of statistical mechanics
• Classical electrodynamics on Snyder space
• Exact Solutions of D-dimensional Klein-Gordon Oscillator with Snyder-de Sitter Algebra
• Snyder-de Sitter meets the Grosse-Wulkenhaar model
• Deformed Gazeau-Klauder Schrödinger-Cat States with Modified Commutation Relations
• Quantum phase transitions of the Dirac oscillator in the Anti-Snyder model
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Generalized relativistic kinematics in Poincaré-invariant models
• Twist for Snyder space
• Non-Relativistic Anti-Snyder Model and Some Applications
• Entropy bound for the photon gas in noncommutative spacetime
• Minimal length estimation on the basis of studies of the Sun-Earth-Moon system in deformed space
• Linear-Quadratic GUP and Thermodynamic Dimensional Reduction
• Two-particle system in Coulomb potential for twist-deformed space-time
• $\kappa$-Galilean and $\kappa$-Carrollian noncommutative spaces of worldlines
• Kinetic energy properties and weak equivalence principle in a space with GUP
• Relative-locality geometry for the Snyder model
• Black hole thermodynamics in Snyder phase space
• Propagation of spinors on a noncommutative spacetime: equivalence of the formal and the effective approach
• Covariant dynamics on the momentum space
• Generalized Relativistic Wave Equations with Intrinsic Maximum Momentum
• Supersymmetric Extension of the Snyder Algebra
• Formal developments in curved momentum space: the quantum field theory roadmap
• Symplectic realization of generalized Snyder-Poisson algebra
• Spinorial Snyder and Yang Models From Superalgebras And Noncommutative Quantum Superspaces
• Einstein crystals in Snyder and Snyder-de Sitter noncommutative backgrounds
• Statistical physics on Euclidean Snyder space: connections with the GUP and cosmological implications