Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity

TL;DR

This paper demonstrates that in (2+1)-dimensional quantum gravity, gravitational interactions deform the Poincare symmetry into a quantum group, and derives a formula for particle scattering cross sections using the Lorentz double quantum group.

Contribution

It introduces the quantum double of the Lorentz group as the symmetry in (2+1)D quantum gravity and derives a general scattering cross section formula.

Findings

Quantum group symmetry replaces Poincare symmetry in (2+1)D gravity.

Derived a universal R-matrix-based scattering formula.

Reproduces known semi-classical results in certain limits.

Abstract

Starting with the Chern-Simons formulation of (2+1)-dimensional gravity we show that the gravitational interactions deform the Poincare symmetry of flat space-time to a quantum group symmetry. The relevant quantum group is the quantum double of the universal cover of the (2+1)-dimensional Lorentz group, or Lorentz double for short. We construct the Hilbert space of two gravitating particles and use the universal R-matrix of the Lorentz double to derive a general expression for the scattering cross section of gravitating particles with spin. In appropriate limits our formula reproduces the semi-classical scattering formulae found by 't Hooft, Deser, Jackiw and de Sousa Gerbert.