Translational Chern--Simons Action and New Planar Particle Dynamics

Abstract

We consider a nonstandard $D=2+1$ gravity described by a translational Chern--Simons action, and couple it to the nonrelativistic point particles. We fix the asymptotic coordinate transformations in such a way that the space part of the metric becomes asymptotically Euclidean. The residual symmetries are (local in time) translations and rigid rotations. The phase space Hamiltonian $H$ describing two-body interactions satisfies a nonlinear equation $H={\cal H}(\vec{x},\vec{p};H)$ what implies, after quantization, a nonstandard form of the Schr\"{o}dinger equation with energy-dependent fractional angular momentum eigenvalues. Quantum solutions of the two-body problem are discussed. The bound states with discrete energy levels correspond to a confined classical motion (for the planar distance between two particles $r\leq r_0$) and the scattering states with continuous energy correspond to classical motion for $r>r_0$.