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

• Published in: The journal of high energy physics 2013-11, Vol.2013 (11), p.1-17, Article 138
• Main Authors: Abreu, Everton M. C., Marcial, Mateus V., Mendes, Albert C. R., Oliveira, Wilson
• Format: Article
• Language: English
• Subjects: Algebra; Classical and Quantum Gravitation; Elementary Particles; Equations of motion; High energy physics; Hilbert space; Invariants; Mathematical analysis; Mathematical models; Oscillators; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Relativity Theory; Rotational; String Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP11(2013)138

A bstract 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.