  We apply the non-commutative Fourier transform for Lie groups to formulate
the non-commutative metric representation of the Ponzano-Regge spin foam model
for 3d quantum gravity. The non-commutative representation allows to express
the amplitudes of the model as a first order phase space path integral, whose
properties we consider. In particular, we study the asymptotic behavior of the
path integral in the semi-classical limit. First, we compare the stationary
phase equations in the classical limit for three different non-commutative
structures corresponding to the symmetric, Duflo and Freidel-Livine-Majid
quantization maps. We find that in order to unambiguously recover discrete
geometric constraints for non-commutative metric boundary data through the
stationary phase method, the deformation structure of the phase space must be
accounted for in the variational calculus. When this is understood, our results
demonstrate that the non-commutative metric representation facilitates a
convenient semi-classical analysis of the Ponzano-Regge model, which yields as
the dominant contribution to the amplitude the cosine of the Regge action in
agreement with previous studies. We also consider the asymptotics of the ${\rm
SU}(2)$ $6j$-symbol using the non-commutative phase space path integral for the
Ponzano-Regge model, and explain the connection of our results to the previous
asymptotic results in terms of coherent states.
