3d Quantum Gravity: Coarse-Graining and $$\varvec{q}$$q-Deformation

The Ponzano–Regge state-sum model provides a quantization of 3d gravity as a spin foam, providing a quantum amplitude to each 3d triangulation defined in terms of the 6j-symbol (from the spin-recoupling theory of $$\mathrm {SU}(2)$$SU(2) representations). In this context, the invariance of the 6j-symbol under 4-1 Pachner moves, mathematically defined by the Biedenharn–Elliott identity, can be understood as the invariance of the Ponzano–Regge model under coarse-graining or equivalently as the invariance of the amplitudes under the Hamiltonian constraints. Here, we look at length and volume insertions in the Biedenharn–Elliott identity for the 6j-symbol, derived in some sense as higher derivatives of the original formula. This gives the behavior of these geometrical observables under coarse-graining. These new identities turn out to be related to the Biedenharn–Elliott identity for the q-deformed 6j-symbol and highlight that the q-deformation produces a cosmological constant term in the Hamiltonian constraints of 3d quantum gravity.