(2+1)D exotic Newton–Hooke symmetry, duality and projective phase

Abstract

A particle system with a (2+1)D exotic Newton-Hooke symmetry is constructed by the method of nonlinear realization. It has three essentially different phases depending on the values of the two central charges. The subcritical and supercritical phases (describing 2D isotropic ordinary and exotic oscillators) are separated by the critical phase (one-mode oscillator), and are related by a duality transformation. In the flat limit, the system transforms into a free Galilean exotic particle on the noncommutative plane. The wave equations carrying projective representations of the exotic Newton-Hooke symmetry are constructed. *

Dynamical sectors for a spinning particle inAdS3

“…The system displays three different phases depending on the values of the parameters [11]. The reduced phase space description reveals a symplectic structure similar to that of Landau problem in the non-commutative plane [12,13].…”

“…In the non-relativistic limit X a ± → X a ± /ω, µ ± → ω 2 µ ± , the Lagrangian takes the NewtonHooke form [11] up to a divergent total derivative, and the equations of motion (18) become those of two harmonic oscillators.…”

Conformal symmetry of an extended Schrödinger equation and its relativistic origin

“…This is due to the fact that the Newton-Hooke group can be obtained from the (A)dS groups as the non-relativistic limit with the velocity of light c going to infinity and the cosmological constant Λ going to zero while keeping c 2 Λ finite, see e.g. [23] and for recent work [24]. The conformal invariance of the relativistic action (23) independently of the power p may be interesting in the search of black hole solutions.…”

The non-commutative oscillator, symmetry and the Landau problem

“…In recent years noncommutative geometry [1,2,3,4,5,6,7,8,9,10] has received much attention due to the fact that spacetime may be noncommutative at very small length scale. For detail study on noncommutative space see the list of Refs.…”