Planar Carrollean dynamics, and the Carroll quantum equation

TL;DR

This work addresses planar Carroll dynamics by constructing the extended Carroll group in $2+1$ dimensions and deriving its symplectic structure to describe free, electromagnetic, and gravitational motion, including spin. It uses the orbit method and Cartan-geometric formalism to obtain explicit equations of motion and identifies two new central charges that nontrivially affect planar dynamics. The paper then derives Carroll quantum dynamics via three routes: Casimir invariants, Klein–Gordon limiting behavior, and geometric quantization, yielding a first-order Carroll equation and associated wavefunction representations. The findings reveal rich planar behavior, connections to BMS structures, and potential implications for horizon physics and holography, while leaving open the interpretation of the extra curvature terms and their physical origins.

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.

Figures (1)

• Figure 1: Visualization of a 1+2 dimensional Bargmann structure, and its link to Newton-Cartan and Carroll structures.