Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations

TL;DR

This paper extends Galilean symmetries to include constant accelerations, introduces central charges in the algebra, and constructs classical mechanics models that realize these extended symmetries, including an exotic planar particle in electromagnetic fields.

Contribution

It introduces acceleration-extended Galilean algebras with central charges and provides dynamical models realizing these symmetries, including an exotic planar particle in electromagnetic fields.

Findings

Extended Galilean algebra with central charges in any dimension.

Classical mechanics models with higher derivatives realize the algebra.

Reinterpretation as an exotic planar particle in electromagnetic fields.

Abstract

We add to Galilean symmetries the transformations describing constant accelerations. The corresponding extended Galilean algebra allows, in any dimension $D=d+1$, the introduction of one central charge $c$ while in $D=2+1$ we can have three such charges: c, \theta and \theta'. We present nonrelativistic classical mechanics models, with higher order time derivatives and show that they give dynamical realizations of our algebras. The presence of central charge $c$ requires the acceleration square Lagrangian term. We show that the general Lagrangian with three central charges can be reinterpreted as describing an exotic planar particle coupled to a dynamical electric and a constant magnetic field.