Asymptotics of 10 j symbols
Open accessAbstract
The Riemannian 10 j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett–Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the ten faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4-simplices. As a consequence, one can compute the asymptotics of the Riemannian 10 j symbols by evaluating a 'degenerate spin network', where the rotation group SO (4) is replaced by the Euclidean group of isometries of 3 . We conjecture formulae for the asymptotics of a large class of Riemannian and Lorentzian spin networks in terms of these degenerate spin networks, and check these formulae in some special cases. Among other things, this conjecture implies that the Lorentzian 10 j symbols are asymptotic to 1/16 times the Riemannian ones.
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The interaction vertex (10j symbol, or Barrett-Crane vertex amplitude) has been shown to include, beside the 'good' term approximating the exponential of the Einstein-Hilbert action, two 'bad' terms: an exponential with opposite sign giving the cosine of the Regge action, and a dominant term depending on the existence of degenerate four-simplices.
“The physical correctness of these theories has been questioned because in the low-energy limit their interaction vertex (10 j symbol, or Barrett-Crane vertex amplitude) has been shown to include -beside the 'good' term approximating the exponential of the Einstein-Hilbert action [19]- also two 'bad' terms: an exponential with opposite sign, giving the cosine of Regge action [19] (analogous to the”
From Graviton Propagator from Background-Independent Quantum Gravity (2006) · cites this paper as [21]
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- Authors
- Baez, J. C., Christensen, J. D., & Egan, G.
- Affiliations
- University of California, Riverside (US)
- Western University (CA)
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2002-11-29
- Volume
- 19
- Issue
- 24
- Pages
- 6489-6513
- ISSN
- 0264-9381 (ISSN-L)
- OpenAlex
- W1514483304
- Cited by
- 80
- References
- 26
- Citations by year
- 2026: 1 · 2025: 1 · 2024: 1 · 2023: 2 · 2022: 1 · 2021: 2 · 2016: 1 · 2015: 2 · 2014: 2 · 2013: 8 · 2012: 3
- FWCI
- 2.88
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- spin networks, Barrett-Crane model, Regge action, asymptotic analysis, Euclidean group
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