Particle In Noncommutative Space Research Articles

Entropic force for quantum particles

Entropic force has been drawing the attention of theoretical physicists following E Verlinde’s work in 2011 to derive Newton’s second law and Einstein’s field equations of general relativity. In this paper, we extend the idea of entropic force to the distribution of quantum particles. Starting from the definition of Shannon entropy for continuous variables, here we have derived quantum osmotic pressure as well as the consequent entropic forces for bosonic and fermionic particles. The entropic force is computed explicitly for a pair of bosons and fermions. The low temperature limit of this result show that the entropic force for bosons is similar to Hooke’s law of elasticity revealing the importance of this idea in the formation of a Bose–Einstein condensate. For fermions, the low temperature limit boils down to the well known Neumann’s radial force and also reveals the Pauli’s exclusion principle. The classical limit of the entropic force between quantum particles is then discussed. As a further example, the entropic force for quantum particles in noncommutative space is also computed. The result reveals a violation of the Pauli exclusion principle for fermions in noncommutative space.

Noncommutative space–time and Hausdorff dimension

We study the Hausdorff dimension of the path of a quantum particle in noncommutative space–time. We show that the Hausdorff dimension depends on the deformation parameter [Formula: see text] and the resolution [Formula: see text] for both nonrelativistic and relativistic quantum particle. For the nonrelativistic case, it is seen that Hausdorff dimension is always less than 2 in the noncommutative space–time. For relativistic quantum particle, we find the Hausdorff dimension increases with the noncommutative parameter, in contrast to the commutative space–time. We show that noncommutative correction to Dirac equation brings in the spinorial nature of the relativistic wave function into play, unlike in the commutative space–time. By imposing self-similarity condition on the path of nonrelativistic and relativistic quantum particle in noncommutative space–time, we derive the corresponding generalized uncertainty relation.

Two-Particle System in Noncommutative Space with Preserved Rotational Symmetry

We consider a system of two particles in noncommutative space which is rotationally invariant. It is shown that the coordinates of the center-of-mass position and the coordinates of relative motion satisfy noncommutative algebra with corresponding effective tensors of noncommutativity. The hydrogen atom is studied as a two-particle system. We find the corrections to the energy levels of the hydrogen atom up to the second order over the parameter of noncommutativity.

Minimal Length Quantum Mechanics of Dirac Particles in Noncommutative Space

We study the two-dimensional harmonic oscillator in commutative and noncommutative space within the framework of minimal length quantum mechanics for spin-1/2 particles. The energy spectra and the eigenfunction are obtained in both cases. Special cases are also deduced.

Wong's Equations and Charged Relativistic Particles in Non-Commutative Space

In analogy to Wong's equations describing the motion of a charged relativistic point particle in the presence of an external Yang-Mills field, we discuss the motion of such a particle in non-commutative space subject to an external $U_\star(1)$ gauge field. We conclude that the latter equations are only consistent in the case of a constant field strength. This formulation, which is based on an action written in Moyal space, provides a coarser level of description than full QED on non-commutative space. The results are compared with those obtained from the different Hamiltonian approaches. Furthermore, a continuum version for Wong's equations and for the motion of a particle in non-commutative space is derived.

Noncommutative quantum mechanics of a test particle under linearized gravitational waves

We consider the quantum dynamics of a test particle in noncommutative space under the influence of linearized gravitational waves in the long wave-length and low-velocity limit. A prescription for quantizing the classical Hamiltonian for the interaction of gravitational wave with matter in noncommutative space is proposed. The Hamiltonian (and hence the system) is then exactly solved by using standard algebraic methods. The solutions show prominent signatures of the noncommutative nature of space. Computation of the expectation value of the particle's position reveals the inherent quantum nature of spacetime noncommutativity.

COVARIANT FORMULATION OF NOETHER'S THEOREM FOR κ-MINKOWSKI TRANSLATIONS

The problem of finding a formulation of Noether's theorem in noncommutative geometry is very important in order to obtain conserved currents and charges for particles in noncommutative space–times. In this paper, we formulate Noether's theorem for translations of κ-Minkowski noncommutative space–time on the basis of the five-dimensional κ-Poincaré covariant differential calculus. We focus our analysis on the simple case of free scalar theory. We obtain five conserved Noether currents, which give rise to five energy–momentum charges. By applying our result to plane waves it follows that the energy–momentum charges satisfy a special-relativity dispersion relation with a generalized mass given by the fifth charge. In this paper, we provide also a rigorous derivation of the equation of motion from Hamilton's principle in noncommutative space–time, which is necessary for the Noether analysis.