Noncommutative Phase Spaces by Coadjoint Orbits Method

Abstract

Abstract. We introduce noncommutative phase spaces by minimal couplings (usual one, dual one and their mixing). We then realize some of them as coadjoint orbits of the anisotropic Newton-Hooke groups in two-and three-dimensional spaces. Through these constructions the positions and the momenta of the phase spaces do not commute due to the presence of a magnetic field and a dual magnetic field.

Group theoretical construction of planar noncommutative phase spaces

“…These centrally extended anisotropic Lie algebras are new except for the Newton-Hooke groups case [13,14,28].…”

“…Planar noncommutative phase spaces are constructed on both anisotropic kinematical groups by working with their central extensions (subsection 5.1) and on absolute time groups with respect to the isotropy of the twodimensional space by considering their noncentral extensions (subsection 5.2). As it has been said in the introduction, the authors in [12] and [13] have constructed noncommutative phase spaces by coadjoint orbit method starting with the noncentrally extended Galilei and Para-Galilei Lie algebras and the centrally extended anisotropic Newton-Hooke Lie algebras respectively. In this section, we construct noncommutative phase spaces on the other planar kinematical Lie algebras.…”

“…Noncommutative phase spaces on anisotropic kinematical groups 5.1.1 Newton-Hooke noncommutative phase space Nonrelativistic particle models have been constructed following Souriau's method for the two-parameter centrally extended anisotropic Newton-Hooke groups in a two-dimensional space [13] yielding similar results as in [14]. Indeed, by considering the oscillating Newton-Hooke group, the authors in [14] have found a similar symmetry in the so-called Hill problem (the latter is studied in celestial mechanics), which is effectivelly an anisotropic harmonic oscillator in a magnetic field.…”

“…Futhermore, the maximal coadjoint orbit denoted by O (m,h,U ) is equipped with the modified symplectic 2-form: σ = dp i ∧ dq i + ǫ ij mω 0 dp i ∧ dp j (20) which corresponds to the minimal coupling of position with the naturally introduced dual magnetic potential [13] where coordinates are:…”

“…Futhermore, a phase space with noncommutative positions and noncommutative momenta has been realized by Souriau's method on the anisotropic Newton-Hooke groups [13]. In [14], the authors have found a similar symmetry in the so-called Hill problem, which is effectively an anisotropical harmonic oscillator in a magnetic field.…”