Noncommutative Oscillator Research Articles

Non-commutativity and non-inertial effects on a scalar field in a cosmic string space-time: II. Spin-zero Duffin–Kemmer–Petiau-like oscillator

We study the non-inertial effects of a rotating frame on a spin-zero, Duffin–Kemmer–Petiau-like oscillator in a cosmic string space-time with non-commutative geometry in the momentum space. The spin-zero DKP-like oscillator is obtained from the Klein–Gordon Lagrangian with a non-standard prescription for the oscillator coupling. We find that the solutions of the time-independent radial equation with the non-zero non-commutativity parameter parallel to the string are related to the confluent hypergeometric function. We find the quantized energy eigenvalues of the non-commutative oscillator.

Dirac’s analysis and Ostrogradskii’s theorem for a class of second-order degenerate Lagrangians

This paper analyzes the constraint structure of a class of degenerate second-order particle Lagrangian that includes chiral oscillator, noncommutative oscillator, and two examples from reduced topologically massive gravity. For even-dimensional configuration spaces with maximal nondegeneracy, Dirac bracket is defined solely by coefficient field of highest derivative whereas for odd dimensions almost all fields may contribute. Ostrogradskii’s theorem on energy instability is discussed. Results of Dirac analysis are used to identify ghost degrees of freedom. Translational symmetries are used to construct first-order variational formalisms for oscillator examples, thereby making them ghost-free.

Probing noncommutative theories with quantum optical experiments

One of the major difficulties of modern science underlies at the unification of general relativity and quantum mechanics. Different approaches towards such theory have been proposed. Noncommutative theories serve as the root of almost all such approaches. However, the identification of the appropriate passage to quantum gravity is suffering from the inadequacy of experimental techniques. It is beyond our ability to test the effects of quantum gravity thorough the available scattering experiments, as it is unattainable to probe such high energy scale at which the effects of quantum gravity appear. Here we propose an elegant alternative scheme to test such theories by detecting the deformations emerging from the noncommutative structures. Our protocol relies on the novelty of an opto-mechanical experimental setup where the information of the noncommutative oscillator is exchanged via the interaction with an optical pulse inside an optical cavity. We also demonstrate that our proposal is within the reach of current technology and, thus, it could uncover a feasible route towards the realization of quantum gravitational phenomena thorough a simple table-top experiment.

On the essential spectrum of certain non-commutative oscillators

We show here that the spectrum of the family of non-commutative harmonic oscillators \documentclass[12pt]{minimal}\begin{document}$Q_{(\alpha ,\beta )}^\mathrm{w}(x,D)$\end{document}Q(α,β)w(x,D) for \documentclass[12pt]{minimal}\begin{document}$\alpha ,\beta \in \mathbb {R}_+$\end{document}α,β∈R+ in the range αβ = 1 is [0, +∞) and is entirely essential spectrum. The previous existing results concern the case αβ > 1 (case in which \documentclass[12pt]{minimal}\begin{document}$Q_{(\alpha ,\beta )}^\mathrm{w}(x,D)$\end{document}Q(α,β)w(x,D) is globally elliptic with a discrete spectrum whose qualitative properties are being extensively studied), and ours therefore extend the picture to the range of parameters αβ ⩾ 1.

The Noncommutative Harmonic Oscillator Based on Symplectic Representation of Galilei Group

We study symplectic unitary representations for the Galilei group and derive the Schrödinger equation in phase space. Our formalism is based on the noncommutative structure of the star product. Guided by group theoretical concepts, we construct a physically consistent phase-space theory in which each state is described by a quasi-probability amplitude associated with the Wigner function. As applications, we derive the Wigner functions for the 3D harmonic oscillator and the noncommutative oscillator in phase space.

Drinfel'd twist and Noncommutative oscillators

Using the Drinfel'd twist of Hopf algebras, noncommutative coordinates are derived for quantum mechanics. Two cases are presented, the constant noncommutativity and a Snyder-like one. This approach brings naturally new features, such as non-additivity in the energy and pseudo-hermiticity.

Noncommutative oscillators from a Hopf algebra twist deformation. A first principles derivation

Noncommutative oscillators are first-quantized through an abelian Drinfel'd twist deformation of a Hopf algebra and its action on a module. Several important and subtle issues making possible the quantization are solved. The spectrum of the single-particle Hamiltonians is computed. The multi-particle Hamiltonians are fixed, unambiguously, by the Hopf algebra coproduct. The symmetry under particle exchange is guaranteed. In d=2 dimensions the rotational invariance is preserved, while in d=3 the so(3) rotational invariance is broken down to an so(2) invariance.

Non-commutative oscillator with Kepler-type dynamical symmetry

A 3-dimensional non-commutative oscillator with no mass term but with an appropriate momentum-dependent potential admits a conserved Runge–Lenz vector, derived from the dual description in momentum space. The trajectories lie on ellipses. The dynamical symmetry allows for an algebraic determination of the bound-state spectrum and extends to o ( 4 , 2 ) .

The Dicke-type crossings among eigenvalues of differential operators in a class of non-commutative oscillators

We study the Dicke-type crossings among eigen-values of a type of differential operators describing the non-commutative oscillator in a class. This class consists of some matrix-valued Schrodinger operators. We will show how the crossings occur for the differential operators in detail. In particular, we are interested in the crossing between the ground state energy and an excited state energy.

The non-commutative oscillator, symmetry and the Landau problem

The isotropic oscillator on a plane is discussed where the coordinate and momentum space are both considered to be non-commutative. We also discuss the symmetry properties of the oscillator for three separate cases when the non-commutative parameters Θ and \(\overline{\Theta}\) for x and p-space, respectively, satisfy specific relations. We compare the Landau problem with the isotropic oscillator on non-commutative space and obtain a relation between the two non-commutative parameters and the magnetic field of the Landau problem.

Exact master equation for a noncommutative Brownian particle

We derive the Hu–Paz–Zhang master equation for a Brownian particle linearly coupled to a bath of harmonic oscillators on the plane with spatial noncommutativity. The results obtained are exact to all orders in the noncommutative parameter. As a by-product we derive some miscellaneous results such as the equilibrium Wigner distribution for the reservoir of noncommutative oscillators, the weak coupling limit of the master equation and a set of sufficient conditions for strict purity decrease of the Brownian particle. Finally, we consider a high-temperature Ohmic model and obtain an estimate for the time scale of the transition from noncommutative to ordinary quantum mechanics. This scale is considerably smaller than the decoherence scale.

Noncommutative oscillators and the commutative limit

It is shown in first order perturbation theory that anharmonic oscillators in noncommutative space behave smoothly in the commutative limit just as harmonic oscillators do. The noncommutativity provides a method for converting a problem in degenerate perturbation theory to a nondegenerate problem.