Analytical and numerical analysis of a rotational invariant D = 2 harmonic oscillator in the light of different noncommutative phase-space configurations

Abstract

In this work we have investigated some properties of classical phase-space with symplectic structures consistent, at the classical level, with two noncommutative (NC) algebras: the Doplicher-Fredenhagen-Roberts algebraic relations and the NC approach which uses an extended Hilbert space with rotational symmetry. This extended Hilbert space includes the operators θ ij and their conjugate momentum π ij operators. In this scenario, the equations of motion for all extended phase-space coordinates with their corresponding solutions were determined and a rotational invariant NC Newton's second law was written. As an application, we treated a NC harmonic oscillator constructed in this extended Hilbert space. We have showed precisely that its solution is still periodic if and only if the ratio between the frequencies of oscillation is a rational number. We investigated, analytically and numerically, the solutions of this NC oscillator in a two-dimensional phase-space. The result led us to conclude that noncommutativity induces a stable perturbation into the commutative standard oscillator and that the rotational symmetry is not broken. Besides, we have demonstrated through the equations of motion that a zero momentum π ij originated a constant NC parameter, namely, θ ij = const., which changes the original variable characteristic of θ ij and reduces the phase-space of the system. This result shows that the momentum π ij is relevant and cannot be neglected when we have that θ ij is a coordinate of the system.

Self-quartic interaction for a scalar field in an extended DFR noncommutative space–time

“…It can be constant, which causes the Lorentz invariance breaking, or as a coordinate of a NC phase-space that is formed by (x, p, θ µν , k µν ) where k µν is the canonical momentum conjugate to θ µν . It can be demonstrated that k µν is also connected to the Lorentz invariance [19].…”

“…It can be shown that this well known standard DFR space is in fact incomplete. As a matter of fact, in [19], one of us showed that the existence of the canonical momentum k µν is intrinsically connected with θ µν and, consequently, with the Lorentz invariance. That is the reason we have called a DFR * system (that will be analyzed here) where the momentum associated to θ µν is zero, a toy model.…”

“…All the corresponding operators belong to the same algebra and have the same hierarchical level. Recently it was demonstrated [19] that the canonical momentum k µν is in fact connected to the Lorentz invariance of the system. Hence, it is possible to construct a filed theory with this extended phase-space [20].…”

“…19) and consequently, both (2.16) and (2.17) are combined into a single equation to give the Klein-Gordon equation in the DFR space for the scalar field φ…”

Harmonic Oscillator Chain in Noncommutative Phase Space with Rotational Symmetry

“…Harmonic oscillator was intensively studied in the frame of noncommutative algebra [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. Recently experiments with micro-and nano-oscillators were implemented for probing minimal length [49].…”