3d Quantum Gravity: Coarse-Graining and q-Deformation

TL;DR

This paper explores how q-deformation of the 6j-symbol in 3d quantum gravity models introduces a cosmological constant and examines the behavior of geometrical observables under coarse-graining through new identities.

Contribution

It derives new identities for length and volume observables in the Ponzano-Regge model, linking them to q-deformation and cosmological constant effects.

Findings

q-deformation introduces a cosmological constant in 3d quantum gravity

New identities describe geometrical observables under coarse-graining

Relations between classical and q-deformed 6j-symbols are established

Abstract

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 SU(2) representations). In this context, the invariance of the 6j-symbol under 4-1 Pachner moves, mathematically defined by the Biedenharn-Elliot 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-Elliot 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-Elliot 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.