Ab Initio Construction of Poincaré and AdS Particle

TL;DR

This work presents a covariant, ab initio construction of worldline actions for particles with Poincaré and AdS isometries by using coadjoint orbits. The method leverages the symplectic structure of orbits (via the Kirillov–Kostant–Souriau form) to derive actions from the symplectic potential S = ∫ ⟨φ, g^{-1}dg⟩, and enforces covariance through Hamiltonian constraints tied to the defining conditions of the isometries. By classifying stabiliser algebras for massive/massless and spinning cases in Minkowski and AdS spaces, the authors obtain universal action forms for SO(2,d−1) and ISO(1,d−1), along with explicit worldline actions for Poincaré and AdS spinning particles. The results illuminate a dual pair between stabiliser algebras and first-class constraints and pave the way for extensions to other symmetry groups, offering a systematic framework for covariant worldline theories with rich spin structure.

Abstract

We study the construction of a manifestly covariant worldline action from a coadjoint orbit. A coadjoint orbit is a submanifold in the dual vector space of a Lie algebra, generated by coadjoint actions. Since a coadjoint orbit is a symplectic space, we derive the worldline particle action from the symplectic two-form. One subtlety in formulating worldline particle actions from coadjoint orbits is choosing the coordinate system that sufficiently illustrates physical properties of the particles. We introduce Hamiltonian constraints by the defining conditions of the isometry. This allows us to write a manifestly covariant worldline action. We demonstrate our method for both massive and massless particles in Minkowski and AdS spacetime.