Planons and their Carroll-Galilei symmetries

TL;DR

This paper classifies the symmetries and dynamics of planons, particles restricted to a plane, revealing their relation to Carroll-Galilei symmetries and providing new physical interpretations of Galilei group orbits.

Contribution

It introduces a classification of planon symmetries via coadjoint orbits, linking them to Carroll-Galilei symmetries and interpreting massless Galilei orbits physically.

Findings

Massless coadjoint orbits are central to planon description.

Constructs classical and quantum dipoles as bound states of monopoles.

Crystals are identified as systems with experimentally realized Carroll symmetry.

Abstract

We study the dynamics of planons, particles whose mobility is restricted to a plane, through the classification of coadjoint orbits and unitary irreducible representations of the centrally extended planon group. Planons are closely related to Galilei/Bargmann symmetries and, remarkably, the often-ignored massless coadjoint orbits of the Galilei group play a central r\^ole in their description. We thereby provide a nontrivial physical interpretation of these orbits. We further construct classical and quantum dipoles as bound states of monopoles, where the restricted planar motion arises from a novel mixed Carroll-Galilei symmetry. We also argue that the simplest and already experimentally realised systems with Carroll symmetry are in crystals.