Não-comutatividade via estruturas simpléticas com aplicações em sistemas não lineares, mecânica clássica e cosmologia

Abstract

In this work we have studied how to introduce noncommutativity to some physical systems through the symplectic structures to investigate the properties of the noncommutative (NC) spaces. Initially, we discussed in details one systematic way to introduce noncommutativity, based on Faddeev-Jackiw formalism, called symplectic formalism for induction of noncommutativity (SFINC). This formalism was used to obtain NC versions of the SU(2) Skyrme model and O(3) nonlinear sigma model. After that, the rotational invariant noncommutative Newton’s second law was written in the NC extended classical phase space. Among the main effects of noncommutativity in the dynamics of a harmonic oscillator treated in this space, we note that the noncommutativity induces a stable perturbation in the usual harmonic oscillator and the oscillator may even not be periodic depending on the ratio between the oscillation frequency of the position coordinate and the oscillation frequency of the NC coordinate. Subsequently, considering a phase space with NC symplectic structure and applying the SFINC we obtained the modified Friedmann-Lemaître equations, which have NC corrections. This correction can be interpreted as a NC perfect fluid, which would behave like dust during a period of radiation, preserving the law of conservation of energy. Finally, using the Friedmann-Lemaître equations the NC deceleration parameter q can be determined in terms of the redshift. From the values in literature for the vacuum-energy density parameter Ω0Ʌ and the transition redshift, the range of acceptable values of the matter density parameter of NC fluid Ω0β and the NC parameter β estimated are 0,052+0,03 −0,159 and β = −0,784+2,398−0,453 ×10−36s−2, respectively. This result shows that the NC corrections could be responsible for up to 8.2% of the matter density of the universe, or a third of the dark matter, without violating the literature values of the transition redshift.