Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles

TL;DR

This paper refines the Ponzano-Regge model for 3D quantum gravity by implementing gauge fixing, constructing transition amplitudes with spinning particles, and connecting spin foam and Hamiltonian quantizations.

Contribution

It provides explicit finite computations for transition amplitudes, introduces operators for gauge fixing and particle insertion, and links the model to Chern-Simons quantization.

Findings

Explicit gauge fixing of all symmetries in the Ponzano-Regge model.

Construction of transition amplitudes with interacting spinning particles.

Connection established between spin foam quantization and Chern-Simons theory.

Abstract

We show how to properly gauge fix all the symmetries of the Ponzano-Regge model for 3D quantum gravity. This amounts to do explicit finite computations for transition amplitudes. We give the construction of the transition amplitudes in the presence of interacting quantum spinning particles. We introduce a notion of operators whose expectation value gives rise to either gauge fixing, introduction of time, or insertion of particles, according to the choice. We give the link between the spin foam quantization and the hamiltonian quantization. We finally show the link between Ponzano-Regge model and the quantization of Chern-Simons theory based on the double quantum group of SU(2)