State Sum Models and Observables

Abstract

From a historical viewpoint the Ponzano–Regge asymptotic formula for the 6j symbol of the group SU(2) (Ponzano and Regge, Semiclassical limit of Racah coefficients. In: Bloch et al (eds) Spectroscopic and group theoretical methods in physics. North–Holland, Amsterdam, pp 1–58, 1968), together with Penrose’s original idea of combinatorial spacetime out of coupling of angular momenta –or spin networks – Penrose (Angular momentum: an approach to combinatorial space–time. In: Bastin (ed) Quantum theory and beyond. Cambridge University Press, 151–180, 1971), is the precursor of the discretized approaches to 3–dimensional Euclidean quantum gravity collectively referred to as ‘state sum models’ after the 1992 paper by Turaev and Viro (State sum invariants and quantum 6j symbols. Topology 31:865–902, 1992). The prominent role here is played by the colored tetrahedron encoding the tetrahedral symmetry of the 6j symbol – reminiscent of the Platonic solid shown in the reproduction of Fig. 6.1 – and recognized in the semiclassical limit as a geometric 3–simplex whose edge lengths are irreps labels from the representation ring of either SU(2) or its universal enveloping algebra \({\mathcal {U}}_q(sl(2))\) with deformation parameter q = root of unity.