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 _fixing this Hilbert-Palatini action it leads to the Gauss constraints, the spatial diffeomorphism constraints, the Hamiltonian constraint and a new constraint C. The gauge symmetries generated by this new constraints are spacetime diffeomorphism 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 _finite spinfoam amplitudes. Then we establish a clear-cut link between the canonical quantization of the new theory to the spinfoam model (Ponzano-Regge model) defined in terms of the SU(2) spin networks.