The Hamiltonian constraint in 3d Riemannian loop quantum gravity
Valentin Bonzom, Laurent Freidel
TL;DR
The work presents a discretized scalar constraint $H_{v,f}$ for 3d Riemannian gravity on a fixed LQG graph, interpreting it through discrete geometry and dihedral angles. Quantization yields a Wheeler-DeWitt equation that is a second-order difference equation on spins, whose solutions are fixed by the BE identity, i.e., by the 6j-symbol recursion that underpins the Ponzano-Regge model. This builds a concrete bridge between canonical LQG dynamics and spin-foam symmetries, showing how flatness constraints emerge as recursions on recoupling symbols and how the projector onto flat connections can be realized on a graph. The paper also outlines higher-spin generalizations and discusses implications for extending these ideas to 4d gravity and cosmological-constant-augmented models, suggesting a unified framework linking Hamiltonian dynamics and spin-foam recursions.
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.