Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 067, 32 pages      arXiv:1401.5819
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

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

Daniele Oriti a and Matti Raasakka b
a) Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam, Germany
b) LIPN, Institut Galilée, CNRS UMR 7030, Université Paris 13, Sorbonne Paris Cité, 99 av. Clement, 93430 Villetaneuse, France

Received February 04, 2014, in final form June 14, 2014; Published online June 26, 2014

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.

Key words: Ponzano-Regge model; non-commutative representation; asymptotic analysis.

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