Dynamical realization of l-conformal Galilei algebra and oscillators

Abstract

The method of nonlinear realizations is applied to the l-conformal Galilei algebra to construct a dynamical system without higher derivative terms in the equations of motion. A configuration space of the model involves coordinates, which parametrize particles in d spatial dimensions, and a conformal mode, which gives rise to an effective external field. It is shown that trajectories of the system can be mapped into those of a set of decoupled oscillators in d dimensions.

On dynamical realizations of l-conformal Galilei groups

“…Finally, let us compare our results with those of Galajinsky and Masterov [14]. The explicit decoupling of the ρ and x modes depends strongly on the choice of parametrization of coset space.…”

“…This can be explicitly seen by using the parametrization considered in Ref. [14]. Abandoming the SO(3) subgroup one can write the composition law as follows:…”

“…(12) describes the whole family of standard conformal models. To see this we make a substitution suggested by the constraint imposed on the Cartan form related to dilatation generator [14,18] z =ρ ρ .…”

“…The results obtained there were generalized to arbitrary l in Refs. [14] and [15]. The invariant equations appear to describe decoupled SL(2, R) conformal mode as well as the dynamics of additional coordinates related to the generators lying outside the so(3) ⊕ sl(2, R) subalgebra.…”

Chiral and Real N = 2 supersymmetric -conformal Galilei algebras

“…The connection ofconformal Galilei ( > 1) algebras to dynamical systems has been, for a long time, an open problem. Quite recently, however, in the purely bosonic case, systematic methods (based on non-linear realization and Maureer-Cartan equations) to derive dynamical equations from these algebras, have been developed [17,18,19] (see also [20,21,22]). So far, the construction of supersymmetric dynamical systems along these lines has not been addressed in the literature.…”