Canonical quantization and braid invariance of (2+1)-dimensional gravity coupled to point particles

TL;DR

This paper explores the canonical quantization of 2+1 dimensional gravity coupled with point particles, revealing an extended gauge symmetry and braid invariance of observables due to phase factors in quantum transformations.

Contribution

It introduces auxiliary variables to achieve local Poincaré invariance and uncovers larger gauge constraints that induce braid invariance in the quantum theory.

Findings

Gauge group extends beyond Poincaré, including additional constraints.

Quantum phase factors lead to braid invariance of holonomies.

The approach mimics spacetime diffeomorphism invariance in a gauge framework.

Abstract

We investigate the canonical quantization of gravity coupled to pointlike matter in 2+1 dimensions. Starting from the usual point particle action in the first order formalism, we introduce auxiliary variables which make the action locally Poincar\'e invariant. A Hamiltonian analysis shows that the gauge group is actually larger than the Poincar\'e group -- certain additional gauge constraints are present which act on the matter degrees of freedom. These additional constraints are necessary to mimic the diffeomorphism invariance present if the theory is formulated with a spacetime metric. The additional gauge constraints are realized projectively in the quantum theory, with a phase in the composition law for finite gauge transformations. That phase is responsible for the braid invariance of physical observables (holonomies).