Coset symmetries and coadjoint orbits

TL;DR

This paper reviews two methods for deriving particle actions from symmetry group cosets—non-linear realizations and coadjoint orbits—highlighting their equivalence and introducing symplectic geometry and coadjoint orbit theory for the Poincaré group.

Contribution

It demonstrates the equivalence of non-linear realizations and coadjoint orbit methods in constructing particle actions and provides an accessible introduction to symplectic geometry and coadjoint orbits.

Findings

Non-linear realizations and coadjoint orbits produce identical particle actions.

Introduction to symplectic geometry relevant to coadjoint orbits.

Sketch of coadjoint orbit theory for the Poincaré group.

Abstract

In these lectures we review two approaches to constructing particle actions from coset spaces of symmetry groups: non-linear realisations and coadjoint orbits. At the level of particle actions, we observe that they coincide. We also provide an introduction to symplectic geometry and we sketch the theory of coadjoint orbits for the Poincar\'e group.