Translational Chern–Simons Action and New Planar Particle Dynamics

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 = H(~x, ~ p;H) what implies, after quantization, a nonstandard form of the Schrödinger 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 ≤ r0) and the scattering states with continuous energy correspond to classical motion for r > r0.