The Hamiltonian constraint in 3d Riemannian loop quantum gravity

Abstract

We discretize the Hamiltonian scalar constraint of three-dimensional Riemannian gravity on a graph of the loop quantum gravity phase space. This Hamiltonian has a clear interpretation in terms of discrete geometries: it computes the extrinsic curvature from dihedral angles. The Wheeler–DeWitt equation takes the form of difference equations, which are actually recursion relations satisfied by Wigner symbols. On the boundary of a tetrahedron, the Hamiltonian generates the exact recursion relation on the 6j-symbol which comes from the Biedenharn–Elliott (pentagon) identity. This fills the gap between the canonical quantization and the symmetries of the Ponzano–Regge state-sum model for 3d gravity.