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

TL;DR

This work identifies the Lorentz double $D(\tilde{L}_3^\uparrow)$ as the natural quantum-group symmetry underlying (2+1)-D gravity with zero cosmological constant, unifying spacetime and internal degrees of freedom for gravitating particles. It develops a momentum-space framework based on ribbon Hopf algebras to construct one- and two-particle Hilbert spaces and to derive scattering data from the universal $R$-matrix, producing a relativistic generalisation of Aharonov–Bohm-type effects. The authors obtain spin-quantisation conditions that depend on mass and derive a closed-form expression for the differential cross section, reproducing Deser–Jackiw–Deser results in appropriate limits while providing a broad, invariant generalisation for spinning gravitating particles. Through the Lorentz double, the paper offers a principled quantisation scheme for 2+1 gravity within combinatorial and quantum-group formalisms, with potential extensions to multi-particle systems and nontrivial topologies.

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.