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.
- Publication:
-
Journal of High Energy Physics
- Pub Date:
- November 2013
- DOI:
- 10.1007/JHEP11(2013)138
- arXiv:
- arXiv:1306.6540
- Bibcode:
- 2013JHEP...11..138A
- Keywords:
-
- Non-Commutative Geometry;
- Integrable Equations in Physics;
- Statistical Methods;
- High Energy Physics - Theory;
- Mathematical Physics
- E-Print:
- 18 pages. JHEP style. Corrections made