Planar Carrollean dynamics, and the Carroll quantum equation

TL;DR

This paper develops the symplectic geometry and quantum dynamics of planar Carroll particles, revealing new invariants and differences from higher dimensions, with implications for electromagnetic and gravitational couplings.

Contribution

It computes the extended Carroll group, describes planar Carroll particle dynamics, and derives the Carroll quantum equation using geometric quantization, highlighting novel invariants and dimensional differences.

Findings

Planar Carroll dynamics differ from 3+1 dimensions in electromagnetic fields due to new Casimir invariants.

Coupling to gravity results in trivial dynamics for Carroll particles.

The Carroll quantum equation is derived via geometric quantization, extending previous classical results.

Abstract

We expand on the known result that the Carroll algebra in $2+1$ dimensions admits two non-trivial central extensions by computing the associated Lie group, which we call extended Carroll group. The symplectic geometry associated to this group is then computed to describe the motion of planar Carroll elementary particles, in the free case, when coupled to an electromagnetic field, and to a gravitational field. We compare to the motions of Carroll particles in $3+1$ dimensions in the same conditions, and also give the dynamics of Carroll particles with spin. In an electromagnetic background, the planar Carroll dynamics differ from the known Carroll ones due to 2 new Casimir invariants, and turn out to be non-trivial. The coupling to a gravitational field leaves the dynamics trivial, however. Finally, we obtain the quantum equation obeyed by Carroll wave functions \textit{via} geometric quantization.