The Kirillov picture for the Wigner particle

TL;DR

This paper explores the Kirillov method applied to massless Wigner particles, clarifying their classical phase space structure, symplectic geometry, and stabilizer subgroups, contributing to the understanding of their quantum representations.

Contribution

It provides a detailed classical phase space analysis of Wigner's massless infinite spin particles using Kirillov's orbit method, including Casimir functions and symplectic structures.

Findings

Identification of classical Casimir functions for Wigner particles

Explicit description of stabilizer subgroups and symplectic structures

Development of position coordinates on coadjoint orbits

Abstract

We discuss the Kirillov method for massless Wigner particles, usually (mis)named "continuous spin" or "infinite spin" particles. These appear in Wigner's classification of the unitary representations of the Poincar\'e group, labelled by elements of the enveloping algebra of the Poincar\'e Lie algebra. Now, the coadjoint orbit procedure introduced by Kirillov is a prelude to quantization. Here we exhibit for those particles the classical Casimir functions on phase space, in parallel to quantum representation theory. A good set of position coordinates are identified on the coadjoint orbits of the Wigner particles; the stabilizer subgroups and the symplectic structures of these orbits are also described.