2014/01/31 by Daniele Oriti, Matti Raasakka
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Classical limit #Commutative property #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Path integral formulation #Phase space #Physics #Pure mathematics #Quantum #Quantum mechanics #gr-qc #hep-th
paper · pdf · doi:10.3842/sigma.2014.067
published as SIGMA 10 (2014), 067, 32 pages
arxiv created 2014/06/26 · openalex publication_date 2014/06/26 · arxiv updated 2014/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 noncommutative 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 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.