Galilean Conformal Mechanics from Nonlinear Realizations

TL;DR

This paper constructs new Galilean conformal mechanics models using nonlinear realizations, deriving invariant actions and extending standard conformal mechanics to higher dimensions with or without central charges.

Contribution

It introduces a systematic method to build Galilean conformal models from the algebra's nonlinear realizations, including new actions in arbitrary dimensions.

Findings

Derived new Galilean conformally invariant actions in various dimensions.

Extended standard conformal mechanics to include spatial dimensions.

Provided geometric constraints that lead to dynamical equations.

Abstract

We apply the nonlinear realizations method for constructing new Galilean conformal mechanics models. Our starting point is the Galilean conformal algebra which is a non-relativistic contraction of its relativistic counterpart. We calculate Maurer-Cartan one-forms, examine various choices of the relevant coset spaces and consider the geometric inverse Higgs-type constraints which reduce the number of the independent coset parameters and, in some cases, provide dynamical equations. New Galilean conformally invariant actions are derived in arbitrary space-time dimension D=d+1 (no central charges), as well as in the special dimension D=2+1 with one "exotic" central charge. We obtain new classical mechanics models which extend the standard (D=0+1) conformal mechanics in the presence of d non-vanishing space dimensions.