Thermodynamics of a Charged Particle in a Noncommutative Plane in a Background Magnetic Field

Landau system in noncommutative space has been considered. To take into account the issue of gauge invariance in noncommutative space, we incorporate the Seiberg-Witten map in our analysis. Generalised Bopp-shift transformation is then used to map the noncommutative system to its commutative equivalent system. In particular we have computed the partition function of the system and from this we obtained the susceptibility of the Landau system and found that the result gets modified by the spatial noncommutative parameter θ. We also investigate the de Hass–van Alphen effect in noncommutative space and observe that the oscillation of the magnetization and the susceptibility gets noncommutative corrections. Interestingly, the susceptibility in the noncommutative scenario is non-zero in the range of the magnetic field greater than the threshold value which is in contrast to its commutative counterpart. The results obtained are valid upto all orders in the noncommutative parameter θ.     

Constraints in Noncommutative Chern-Simons Mechanics and Hall Effect

We study noncommutative Chern-Simons mechanics and noncommutative Hall effect by Dirac theory in this paper. The magnetic field is introduced by means of minimal coupling. We show that the constraint set will enlarge when a dimensionless parameter takes zero value. In order to illustrate our idea, we study two specific models. One is noncommutative Chern-Simons mechanics which describes a charged particle on a noncommutative plane interacting with a perpendicular uniform magnetic field. The other is a charged particle on a noncommutative plane with a background uniform electromagnetic field. We show that when the dimensionless parameter tends to zero, the particle will live in a lower dimensional space in both models. Noncommutative Chern-Simons mechanics will reduce to a harmonic oscillator and the classical equations of motion of a charged particle in the background of a uniform electromagnetic field are governed by classical Hall law when the dimensionless parameter tends to zero.     

Shadow of a noncommutative geometry inspired Ayón Beato García black hole

We introduce the noncommutative geometry inspired Ayón Beato García black hole metric and study various properties of this metric by which we try to probe the allowed values of the noncommutative parameter \(\vartheta \) under certain conditions. We then construct the shadow (apparent shape) cast by this black hole. We derive the corresponding photon orbits and explore the effects of noncommutative spacetime on them. We then study the effects of noncommutative parameter \(\vartheta \), smeared mass m(r), smeared charge q(r) on the silhouette of the shadow analytically and present the results graphically. We then discuss the deformation which arises in the shape of the shadow under various conditions. Finally, we introduce a plasma background and observe how the shadow behaves in this scenario.     

Noncommutativity and Lorentz violation in relativistic heavy ion collisions

We show that relativistic heavy ion collisions at LHC energies could be used as an experimental probe to detect fundamental properties of spacetime long speculated about. Our results rely on the recent proposal that magnetic fields of intensity much larger than that of magnetars should be produced at the beginning of the collisions and this could have an important impact on the experimental manifestation of a noncommutative spacetime. Indeed, in the noncommutative generalization of electrodynamics the interplay between a nonzero noncommutative parameter and an external magnetic field leads us to predict the production of lepton pairs of low invariant mass by free photons (an event forbidden by Lorentz invariant electrodynamics) in relativistic heavy ion collisions at present and future available energies. This unique channel can be clearly considered as a signature of noncommutativity. On the other hand, the search for such decays is worth anyway because their absence would ameliorate of three orders of magnitude the current bound on the noncommutative parameter.     

Star Product Formula of Theta Functions

As a noncommutative generalization of the addition formula of theta functions, we construct a class of theta functions which are closed with respect to the Moyal star product of a fixed noncommutative parameter. These theta functions can be regarded as bases of the space of holomorphic homomorphisms between holomorphic line bundles over noncommutative complex tori.     

Thermodynamics Properties of Confined Particles on Noncommutative Plane

We consider a system of $N$ particles living on the noncommutative plane in the presence of a confining potential and study its thermodynamics properties.Indeed, after calculating the partition function, we determine the corresponding internal energy and heat capacity where different corrections are obtained. In analogy with the magnetic field case, we define an effective magnetization and study its susceptibility in terms of the noncommutative parameter $\theta$. By introducing the chemical potential, we investigate the Bose-Einstein condensation for the present system. Different limiting cases related to the temperature and $\theta$ will be analyzed as well as some numerical illustration will be presented.     

Scalar Field in the Bianchi I: Noncommutative Classical and Quantum Cosmology

Using the ADM formalism in the minisuperspace, we obtain the commutative and noncommutative exact classical solutions and exact wave function to the Wheeler-DeWitt equation with an arbitrary factor ordering, for the anisotropic Bianchi type I cosmological model, coupled to a scalar field, cosmological term and barotropic perfect fluid. We introduce noncommutative scale factors, considering that all minisuperspace variables q ⁱ do not commute, so the symplectic structure was modified. In the classical regime, it is shown that the anisotropic parameter β _±nc and the field φ, for some value in the λ _eff cosmological term and noncommutative θ parameter, present a dynamical isotropization up to a critical cosmic time t _c ; after this time, the effects of isotropization in the noncommutative minisuperspace seems to disappear. In the quantum regimen, the probability density presents a new structure that corresponds to the value of the noncommutativity parameter.     

Noncommutative space from dynamical noncommutative geometries

We present noncommutative topology as a basis for noncommutative geometry phrased completely in terms of partially ordered sets with operations. In this note we introduce a noncommutative space-time starting from a dynamical system of noncommutative topologies based on the notion of temporal points. At every moment a commutative topological space is constructed and it is shown to approximate the noncommutative space in sheaf theoretical terms; this so called moment space should be the space where observed phenomena should be described, the commutative shadow of the noncommutative space is to be thought of as the usual space-time.     

A new limit for the noncommutative space–time parameter

The spectrum of the hydrogen atom in the framework of noncommutative quantum mechanics is studied and the related phenomenology is presented. We find that the noncommutative effects are similar to those obtained by considering the extended charged nature of the proton in the atom. To the first order in the noncommutative parameter, it is equivalent to an electron in the fields of a Coulomb potential and an electric dipole and this allows us to get a bound for this parameter. In a second step, we compute noncommutative corrections of the energy levels and find that they are at the second order in the parameter of noncommutativity. By comparing our results to those obtained from experimental spectroscopy, we get another limit for this parameter.     

On non-commutative geodesic motion

In this work we study the geodesic motion on a noncommutative space–time. As a result we find a non-commutative geodesic equation and then we derive corrections of the deviation angle per revolution in terms of the non-commutative parameter when we specify the problem of Mercury’s perihelion. In this way, we estimate the noncommutative parameter based in experimental data.     

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.     

Tunneling of massive particles from noncommutative inspired Schwarzschild black hole

We apply the generalization of the Parikh–Wilczek method to the tunneling of massive particles from noncommutative inspired Schwarzschild black holes. By deriving the equation of radial motion of the tunneling particle directly, we calculate the emission rate which is shown to be dependent on the noncommutative parameter besides the energy and mass of the tunneling particle. After equating the emission rate to the Boltzmann factor, we obtain the modified Hawking temperature which relates to the noncommutativity and recovers the standard Hawking temperature in the commutative limit. We also discuss the entropy of the noncommutative inspired Schwarzschild black hole and its difference after and before a massive particle’s emission.     

Construction of non-Abelian gauge theories on noncommutative spaces

We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with a constant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories. Received: 13 June 2001 / Published online: 19 July 2001     

Solution of the spin-one DKP oscillator under an external magnetic field in noncommutative space with minimal length

The spin-one Duffin-Kemmer-Petiau(DKP) oscillator under a magnetic field in the presence of the minimal length in the noncommutative coordinate space is studied by using the momentum space representation. The explicit form of energy eigenvalues is found, and the eigenfunctions are obtained in terms of the Jacobi polynomials. It shows that for the same azimuthal quantum number, the energy E increases monotonically with respect to the noncommutative parameter and the minimal length parameter. Additionally, we also report some special cases aiming to discuss the effect of the noncommutative coordinate space and the minimal length in the energy spectrum.     

2+1 Dimensional Noncommutative Dirac Oscillator and (Anti)-Jaynes-Cummings Models

We study Dirac oscillator in 2+1 dimensional noncommutative space. The model is solved exactly and the relationship with Jaynes-Cummings (JC) or anti-Jaynes-Cummings (AJC) models are investigated. We find that for a positive noncommutative parameter, there is an exact map from the 2+1 dimensional noncommutative Dirac oscillator to AJC model. However, for a negative noncommutative parameter, the noncommutative planar Dirac oscillator contains both AJC and JC terms simultaneously. Our investigation may afford a new way to study relativistic quantum mechanics models in noncommutative space by means of quantum optics method, and vice verse.     

Charged thin-shell gravastars in noncommutative geometry

In this paper we construct a charged thin-shell gravastar model within the context of noncommutative geometry. To do so, we choose the interior of the nonsingular de Sitter spacetime with an exterior charged noncommutative solution by cut-and-paste technique and apply the generalized junction conditions. We then investigate the stability of a charged thin-shell gravastar under linear perturbations around the static equilibrium solutions as well as the thermodynamical stability of the charged gravastar. We find the stability regions, by choosing appropriate parameter values, located sufficiently close to the event horizon.     

New bounds for Tsallis parameter in a noncommutative phase–space entropic gravity and nonextensive Friedmann equations

In this paper, we have analyzed the nonextensive Tsallis statistical mechanics in the light of Verlinde’s formalism. We have obtained, with the aid of a noncommutative phase–space entropic gravity, a new bound for Tsallis nonextensive (NE) parameter (TNP) that is clearly different from the ones present in the current literature. We derived the Friedmann equations in a NE scenario. We also obtained here a relation between the gravitational constant and the TNP.     

Composite system in noncommutative space and the equivalence principle

The motion of a composite system made of N particles is examined in a space with a canonical noncommutative algebra of coordinates. It is found that the coordinates of the center-of-mass position satisfy noncommutative algebra with effective parameter. Therefore, the upper bound of the parameter of noncommutativity is re-examined. We conclude that the weak equivalence principle is violated in the case of a non-uniform gravitational field and propose the condition for the recovery of this principle in noncommutative space. Furthermore, the same condition is derived from the independence of kinetic energy on the composition.     

Distortion of the Poisson Bracket by the Noncommutative Planck Constants

In this paper we introduce a kind of “noncommutative neighbourhood” of a semiclassical parameter corresponding to the Planck constant. This construction is defined as a certain filtered and graded algebra with an infinite number of generators indexed by planar binary leaf-labelled trees. The associated graded algebra (the classical shadow) is interpreted as a “distortion” of the algebra of classical observables of a physical system. It is proven that there exists a q-analogue of the Weyl quantization, where q is a matrix of formal variables, which induces a nontrivial noncommutative analogue of a Poisson bracket on the classical shadow.     

Higher order theories and their relationship with noncommutativity

We present a relationship between noncommutativity and higher order time derivative theories using a perturbation method. We make a generalization of the Chern–Simons quantum mechanics for higher order time derivatives. This model presents noncommutativity in a natural way when we project to low-energy physical states without the necessity of taking the strong field limit. We quantize the theory using a Bopp's shift of the noncommutative variables and we obtain a spectrum without negative energies, under the perturbation limits. In addition, we extent the model to high order time derivatives and noncommutativity with variable dependent parameter.