Translational Chern--Simons Action and New Planar Particle Dynamics

TL;DR

This paper explores a novel 2+1 dimensional gravity model with a translational Chern--Simons action coupled to point particles, leading to unique quantum and classical two-body dynamics with energy-dependent fractional angular momentum.

Contribution

It introduces a new nonstandard gravity model in 2+1 dimensions and analyzes its impact on particle interactions, resulting in a nonlinear Hamiltonian and a modified Schrödinger equation.

Findings

Quantum solutions exhibit energy-dependent fractional angular momentum.

Bound states are confined with discrete energy levels.

Scattering states are classical with continuous energy.

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 $r>r_0$.