The Hamiltonian constraint in 3d Riemannian loop quantum gravity

TL;DR

This paper discretizes the Hamiltonian constraint in 3D Riemannian loop quantum gravity, linking it to geometric quantities and recursion relations of Wigner symbols, bridging canonical quantization and spin foam models.

Contribution

It introduces a discretized Hamiltonian that interprets extrinsic curvature in terms of dihedral angles, connecting canonical LQG with the Ponzano-Regge model.

Findings

The Hamiltonian computes extrinsic curvature from dihedral angles.

The Wheeler-DeWitt equation becomes a recursion relation for Wigner symbols.

The recursion relation matches the Biedenharn-Elliott identity on a tetrahedron boundary.

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.