PonzanoRegge

The Ponzano–Regge model is a state-sum model of three-dimensional quantum gravity without a cosmological constant. Proposed by Giulio Ponzano and Tomas Regge in 1968, it expresses quantum amplitudes as sums over discrete geometries labeled by SU(2) spins assigned to the edges of a triangulation.

Construction: Let M be a closed oriented 3-manifold with a triangulation T. To every edge e assign a spin j_e (a half-integer). Each tetrahedron t with edge spins j1,…,j6 contributes a Wigner 6j-symbol {6j}(j1,…,j6). The partition function is Z(T) = sum_{j_e} ∏_tetrahedra {6j}(j(t)) ∏_edges (2j_e+1). In practice, normalization factors and gauge-fixing are used, and the sum is formally divergent for noncompact gauge symmetry, requiring a regulator or careful treatment.

Asymptotics and interpretation: In the large-spin limit, the 6j-symbol exhibits oscillatory behavior whose phase matches the Regge action for the triangulated 3-manifold: S_R = Σ_e l_e θ_e, with edge lengths l_e related to j_e and θ_e the dihedral angles around edge e. Thus the model provides a connection between quantum amplitudes and classical discrete gravity in the semiclassical regime.

Relation to cosmological constant and extensions: A q-deformed version based on quantum groups yields the Turaev–Viro model, which is finite and corresponds to 3D gravity with a positive cosmological constant. The Ponzano–Regge model is foundational in the study of spin foam formulations and has influenced approaches in loop quantum gravity for three-dimensional spacetimes.