Time Gauge Fixing For 3d Loop Quantum Gravity

Abstract

In this work, we review 3-dimensional gravity by canonically analyzing the Hilbert-

Palatini action. We apply a time gauge used in 4-dimensional loop quantum gravity

[14] to the simpler case of the 3-dimensional loop quantum gravity. By time

gauge xing this Hilbert-Palatini action it leads to the Gauss constraints, the spatial

dieomorphism constraints, the Hamiltonian constraint and a new constraint

C. The gauge symmetries generated by this new constraints are spacetime dieomorphism

and SO(2) gauge transformations. We solve the dynamics of the theory

by providing a regularization of the generalized projector operator in terms of the

Hamiltonian constraint. We provide the denition of the physical scalar product

which can be represented in terms of a sum over nite spinfoam amplitudes. Then

we establish a clear-cut link between the canonical quantization of the new theory

to the spinfoam model (Ponzano-Regge model) dened in terms of the SU(2) spin

networks.