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

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.

18 pages

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• (3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces
• Modified cosmology from quantum deformed entropy
• Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries
• Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases
• Semiclassical Behavior of Spinfoam Amplitude with Small Spins and Entanglement Entropy
• Influence of exact Lorentz-violating mechanism on the critical exponents for massive $q$-deformed $\lambda {\varphi }^4$ scalar field theory
• Exact Lorentz-violating $q$-deformed O($N$) universality class
• Multistrand Eigenvalue conjecture and Racah symmetries
• Categorical Quantum Volume Operator