Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data
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 integra...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
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| Zitieren: | Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data / D. Oriti, M. Raasakka // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 63 назв. — англ. |
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Oriti, D. Raasakka, M. 2019-02-10T10:16:30Z 2019-02-10T10:16:30Z 2014 Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data / D. Oriti, M. Raasakka // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 63 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 83C45; 81R60; 83C27; 83C80; 81S10; 53D55 DOI:10.3842/SIGMA.2014.067 https://nasplib.isofts.kiev.ua/handle/123456789/146624 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 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. This paper is a contribution to the Special Issue on Deformations of Space-Time and its Symmetries. The full collection is available at http://www.emis.de/journals/SIGMA/space-time.html. We are grateful for the anonymous referees for their constructive questions and comments, which led to several improvements to the original manuscript. We would like to thank A. Baratin for several useful discussions on the non-commutative Fourier transform and spin foam models. We also thank C. Guedes, F. Hellmann and W. Kaminski for several discussions. This work was supported by the A. von Humboldt Stiftung, through a Sofja Kovalevskaja Prize, which is gratefully acknowledged. The work of M. Raasakka was partially supported by Emil Aaltonen Foundation. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data |
| spellingShingle |
Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data Oriti, D. Raasakka, M. |
| title_short |
Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data |
| title_full |
Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data |
| title_fullStr |
Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data |
| title_full_unstemmed |
Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data |
| title_sort |
asymptotic analysis of the ponzano-regge model with non-commutative metric boundary data |
| author |
Oriti, D. Raasakka, M. |
| author_facet |
Oriti, D. Raasakka, M. |
| publishDate |
2014 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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 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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146624 |
| citation_txt |
Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data / D. Oriti, M. Raasakka // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 63 назв. — англ. |
| work_keys_str_mv |
AT oritid asymptoticanalysisoftheponzanoreggemodelwithnoncommutativemetricboundarydata AT raasakkam asymptoticanalysisoftheponzanoreggemodelwithnoncommutativemetricboundarydata |
| first_indexed |
2025-12-07T17:38:00Z |
| last_indexed |
2025-12-07T17:38:00Z |
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1850871990388260864 |