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A Lorentzian signature model for quantum general relativity
Open accessAbstract
We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra. We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028. As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the `10J' symbol needed in our model has a finite value.
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The Barrett-Crane models are obtained as the γ → ∞ limit of the previous constructions.
“The Barrett-Crane models [60, 59] can easily be obtained in the present framework as a suitable limiting case γ → ∞ of the previous constructions.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [60]
There are two exceptional possibilities: γ = ±i, where the constraint could in principle be imposed strongly but the self-dual sector is absent, and the sector corresponding to the Barrett-Crane model.
“The first one is to take γ = ±i, in which case the constraint (186) could, in principle, be imposed strongly. However, the self-dual sector is not present in .The second possibility is to work in the sector where . Where, as in the Riemannian case, it corresponds to the Barret-Crane model [60] with the various limitations discussed in Section 10.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [60]
The Turaev-Viro partition function was proved to give, in the asymptotic limit, 3D Euclidean gravity with a cosmological constant, and can be thought of as a quantization of BF theory, which is classically equivalent to 3D gravity and also equivalent to two copies of Chern-Simons theory.
“This partition function was then proved [51,52] to give, in the asymptotic limit, 3D Euclidean gravity with cosmological constant, and can be thought of as giving a quantization of BF theory [53,54], a topological field theory, classically equivalent to 3D gravity”
From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [55]
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References 21 · Cited by 275
Research works printed but not linked: 5
- Baez J, 2000, Adv. Theor. Math. Phys.
- Podles P, 1990, Commun. Math. Phys., 136, 399
- Gel'fand I M, 1947, Izv. Akad. Nauk SSSR, 11, 411
- Naimark M A, 1964, Am. Math. Soc. Transl. Ser. 2, 36, 101
- Gel'fand I M, 1959, Tr. Mosk. Mat. O.-va, 88, 321
Metadata
- Authors
- Barrett, J. W., & Crane, L.
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2000-08-02
- Volume
- 17
- Issue
- 16
- Pages
- 3101-3118
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2031282132
- Cited by
- 275
- References
- 21
- Citations by year
- 2026: 5 · 2025: 4 · 2024: 12 · 2023: 49 · 2022: 9 · 2021: 7 · 2020: 6 · 2019: 8 · 2018: 5 · 2017: 3 · 2016: 5 · 2015: 3 · 2014: 7 · 2013: 15 · 2012: 12
- FWCI
- 10.39
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Repository@Nottingham (University of Nottingham) (submittedVersion, cc-by)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- quantum general relativity, Lorentzian signature, discrete general relativity, quantum gravity, Euclidean signature, discrete spacetime
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