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We apply the non-commutative Fourier transform for Lie groups to formulatethe non-commutative metric representation of the Ponzano-Regge spin foam modelfor 3d quantum gravity. The non-commutative representation allows to expressthe amplitudes of the model as a first order phase space path integral, whoseproperties we consider. In particular, we study the asymptotic behavior of thepath integral in the semi-classical limit. First, we compare the stationaryphase equations in the classical limit for three different non-commutativestructures corresponding to the symmetric, Duflo and Freidel-Livine-Majidquantization maps. We find that in order to unambiguously recover discretegeometric constraints for non-commutative metric boundary data through thestationary phase method, the deformation structure of the phase space must beaccounted for in the variational calculus. When this is understood, our resultsdemonstrate that the non-commutative metric representation facilitates aconvenient semi-classical analysis of the Ponzano-Regge model, which yields asthe dominant contribution to the amplitude the cosine of the Regge action inagreement with previous studies. We also consider the asymptotics of the ${\rmSU}(2)$ $6j$-symbol using the non-commutative phase space path integral for thePonzano-Regge model, and explain the connection of our results to the previousasymptotic results in terms of coherent states.

Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data