Gluing 4-simplices: A derivation of the Barrett-Crane spin foam model for Euclidean quantum gravity
Open accessAbstract
We derive the Barrett-Crane spin foam model for Euclidean 4-dimensional quantum gravity from a discretized $\mathrm{BF}$ theory, imposing the constraints that reduce it to gravity at the quantum level. We obtain in this way a precise prescription of the form of the Barrett-Crane state sum, in the general case of an arbitrary manifold with boundary. In particular we derive the amplitude for the edges of the spin foam from a natural procedure of gluing different 4-simplices along a common tetrahedron. The generalization of our results to higher dimensions is also shown.
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Quantization of Regge calculus has been considered in the past and its relation to spinfoam theory has been pointed out.
“Quantization of Regge calculus has been considered in the past [24] and its relation to spinfoam theory has been pointed out (see [25] and references therein).”
From Flipped spinfoam vertex and loop gravity (2008) · cites this paper as [24]
A spin foam model of Lorentzian quantum gravity in 3D, generalizing the Ponzano-Regge-Turaev-Viro state sum, was constructed by Freidel [98].
“Again we concentrate on the Barrett-Crane model in 4D, but we point out that a spin foam model of Lorentzian quantum gravity in 3D, generalizing the Ponzano-Regge-Turaev-Viro state sum, was constructed by Freidel [98], using methods similar to those used in section 4.6.”
From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [98]
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Cites
Quantum theory of geometry: I. Area operators
Ashtekar, A., & Lewandowski, J. (1997) · Classical and Quantum Gravity · doi:10.1088/0264-9381/14/1a/006 · Cited by 852 · see citing works
Relativistic spin networks and quantum gravity
Barrett, J. W., & Crane, L. (1998) · Journal of Mathematical Physics · doi:10.1063/1.532254 · Cited by 404 · see citing works
Baez, J. C. (1998) · Classical and Quantum Gravity · doi:10.1088/0264-9381/15/7/004 · Cited by 381 · see citing works
<i>so</i> (4) Plebanski action and relativistic spin-foam model
Pietri, R. D., & Freidel, L. (1999) · Classical and Quantum Gravity · doi:10.1088/0264-9381/16/7/303 · Cited by 182 · see citing works
The Quantum Tetrahedron in 3 and 4 Dimensions
Baez, J. C., & Barrett, J. W. (1999) · arXiv · doi:10.48550/arxiv.gr-qc/9903060 · Cited by 146 · see citing works
Causal evolution of spin networks
Markopoulou, F., & Smolin, L. (1997) · Nuclear Physics B · doi:10.1016/s0550-3213(97)80019-3 · Cited by 134 · see citing works
Projector on physical states in loop quantum gravity
Rovelli, C. (1999) · Physical Review D · doi:10.1103/physrevd.59.104015 · Cited by 95 · see citing works
Quantum geometry with intrinsic local causality
Markopoulou, F., & Smolin, L. (1998) · Physical Review D · doi:10.1103/physrevd.58.084032 · Cited by 95 · see citing works
A lattice worldsheet sum for 4-d Euclidean general relativityPreprint
Reisenberger, M. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9711052 · Cited by 73 · see citing works
A left-handed simplicial action for Euclidean general relativity
Reisenberger, M. P. (1997) · Classical and Quantum Gravity · doi:10.1088/0264-9381/14/7/012 · Cited by 73 · see citing works
On relativistic spin network vertices
Reisenberger, M. P. (1999) · Journal of Mathematical Physics · doi:10.1063/1.532850 · Cited by 72 · see citing works
Dual formulation of spin network evolutionPreprint
Markopoulou, F. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9704013 · Cited by 66 · see citing works
Worldsheet formulations of gauge theories and gravityPreprint
Reisenberger, M. (1994) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9412035 · Cited by 60 · see citing works
A categorical construction of 4D TQFTsPreprint
Crane, L., & Yetter, D. N. (1993) · arXiv (Cornell University) · doi:10.48550/arxiv.hep-th/9301062 · Cited by 43 · see citing works
Spin foams as Feynman diagramsPreprint
Reisenberger, M., & Rovelli, C. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0002083 · Cited by 42 · see citing works
Simple spin networks as Feynman graphs
Freidel, L., & Krasnov, K. (2000) · Journal of Mathematical Physics · doi:10.1063/1.533203 · Cited by 42 · see citing works
R-torsion and linking numbers from simplicial abelian gauge theoriesPreprint
Adams, D. H. (1996) · arXiv (Cornell University) · doi:10.48550/arxiv.hep-th/9612009 · Cited by 40 · see citing works
State-Sum Invariants of 4-Manifolds IPreprint
Crane, L., Kauffman, L. H., & Yetter, D. N. (1994) · arXiv (Cornell University) · doi:10.48550/arxiv.hep-th/9409167 · Cited by 16 · see citing works
An Introduction to Spin Foam Models of Quantum Gravity and BF TheoryPreprint
Baez, J. C. (1999) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9905087 · Cited by 14 · see citing works
BF Description of Higher-Dimensional Gravity TheoriesPreprint
Freidel, L., Krasnov, K. V., & Puzio, R. (1999) · arXiv (Cornell University) · doi:10.48550/arxiv.hep-th/9901069 · Cited by 14 · see citing works
A surface theoretic model of quantum gravityPreprint
Iwasaki, J. (1999) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9903112 · Cited by 11 · see citing works
A lattice quantum gravity model with surface-like excitations in 4-dimensional spacetimePreprint
Iwasaki, J. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0006088 · Cited by 5 · see citing works
The asymptotics of an amplitude for the 4-simplexPreprint
Barrett, J. W., & Williams, R. M. (1998) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9809032 · Cited by 4 · see citing works
Self-citationBoundary actions in Ponzano-Regge discretization, Quantum groups and AdS(3)Preprint
O’Loughlin, M. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0002092 · Cited by 2 · see citing works
The classical evaluation of relativistic spin networksPreprint
Barrett, J. W. (1998) · arXiv (Cornell University) · doi:10.48550/arxiv.math/9803063
Spin Networks in Nonperturbative Quantum GravityPreprint
Baez, J. C. (1995) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9504036
Invariants of spin networks with boundary in Quantum Gravity and TQFT'sPreprint
Carbone, G., Mauro Carfora, B., & Marzuoli, A. (1999) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9905076
Flagged first, then by citations
Research works printed but not linked: 33
- [1][1] J.Baez, Class. Quant. Grav. 15, 1827 (1998), gr-qc/9709052 - · arXiv
- [3][3] M. Reisenberger, C. Rovelli, Phys. Rev. D56, 3490 (1997), gr-qc/9612035 - · arXiv
- [5][5] C. Rovelli, Phys. Rev. D59, 1040 (1999), gr-qc/9806121 - · arXiv
- [6][6] C. Rovelli, L. Smolin, Phys. Rev. D53, 5743 (1995), gr-qc/9505006 - · arXiv
- [8][8] A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao, T. Thiemann, J. Math. Phys 36, 6456 (1995), gr-qc/9504018 - · arXiv
- [9][9] A. Ashtekar, J. Lewandowski, Class. Quant. Grav.14, A55-A82, (1997), gr-qc/9602046 - · arXiv
- [10][10] A. Ashtekar, J. Lewandowski, gr-qc/9602035 - · arXiv
- [12][12] V. Turaev, O. Viro, Topology 31, 865 (1992) -
- [19][19] J. W. Barrett, L. Crane, J. Math. Phys. 39, 3296 (1998), gr-qc/9709028 - · arXiv
- [20][20] R. De Pietri, L. Freidel, Class. Quant. Grav. 16, 2187 (1999), gr-qc/9804071 - · arXiv
- [21][21] J. F. Plebanski, J. Math. Phys. 12, 2511 (1977) -
- [22][22] R. Capovilla, J. Dell, T. Jacobson, L. Mason, Class. Quant. Grav. 16, 41 (1991) -
- [23][23] D. Birmingham, M. Blau, M. Rakowski, G. Thompson, Phys. Rep. 209, 129 (1991) -
- [25][25] J. W. Barrett, M. Rocek, R. M. Williams, Class. Quant. Grav.16,1373 (1999), gr-qc/9710056 - · arXiv
- [26][26] L. Freidel, K. Krasnov, Adv. Theor. Math. Phys. 2, 1183 (1999), hepth/9807092 -
- [27][27] N. Kawamoto, N. Sato, Y. Uchida, Nucl. Phys. B 574, 809 (2000), hepth/9911228 -
- [29][29] N.J. Vilenkin, A.U. Klimyk, Representations of Lie Groups and special functions , Kluwer Academic publisher (1993), vol.2 -
- [31][31] R. De Pietri, L. Freidel, K. Krasnov, C. Rovelli, Nucl. Phys. B574, 785 (2000), hep-th/9907154 - · arXiv
- [32][32] A. Perez, C. Rovelli, gr-qc/0006107 - · arXiv
- [33][33] M. Reisenberger, J. Math. Phys. 40, 2046 (1999), gr-qc/9809067 - · arXiv
- [36][36] G. Carbone, M. Carfora, A. Marzuoli, gr-qc/0008011 - · arXiv
- [39][39] K. Ezawa, Prog. Theor. Phys. 95, 863 (1996), hep-th/9511047 - · arXiv
- [40][40] Y. Ling, L. Smolin, hep-th/0003285 - · arXiv
- [41][41] Y. Ling, L. Smolin, hep-th/0009018 - · arXiv
- [42][42] L. Freidel, K. Krasnov, J. Math. Phys. 41, 1681 (2000), hep-th/9903192 - · arXiv
- [43][43] V. Suneeta, R.K. Kaul, T.R. Govindarajan, Mod. Phys. Lett. A14, 349 (1999), gr-qc/9811071 - · arXiv
- [45][45] M. Reisenberger, C. Rovelli, gr-qc/0002095 - · arXiv
- [46][46] J. W. Barrett, L. Crane, Class. Quant. Grav. 17, 3101 (2000), grqc/9904025 -
- [47][47] A. Perez, C. Rovelli, gr-qc/0009021 - · arXiv
- [48][48] F. Markopoulou, L. Smolin, Nucl. Phys. B 508, 409 (1997) -
- [50][50] F. Markopoulou, L. Smolin, Phys. Rev. D58, 084032 (1998), gr-qc/9712067 - · arXiv
- [52][52] A. Perez, gr-qc/0011058 - · arXiv
- [53][53] A. Perez, C. Rovelli, gr-qc/0011037 · arXiv
Cited by
Sorted by citations
LQG vertex with finite Immirzi parameter
Engle, J., Livine, E., Pereira, R., & Rovelli, C. (2008) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2008.02.018 · Cited by 584 · see citing works
Loop-Quantum-Gravity Vertex Amplitude
Engle, J., Pereira, R., & Rovelli, C. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.161301 · Cited by 335 · see citing works
Group Field Theory: An Overview
Freidel, L. (2005) · International Journal of Theoretical Physics · doi:10.1007/s10773-005-8894-1 · Cited by 378 · see citing works
Complete LQG propagator: Difficulties with the Barrett-Crane vertex
Alesci, E., & Rovelli, C. (2007) · Physical Review D · doi:10.1103/physrevd.76.104012 · Cited by 154 · see citing works
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Modesto, L., & Rovelli, C. (2005) · Physical Review Letters · doi:10.1103/physrevlett.95.191301 · Cited by 93 · see citing works
Finiteness of a spinfoam model for Euclidean quantum general relativity
Perez, A. (2001) · Nuclear Physics B · doi:10.1016/s0550-3213(01)00014-1 · Cited by 73 · see citing works
Perturbative Finiteness in Spin-Foam Quantum Gravity
Crane, L., Perez, A., & Rovelli, C. (2001) · Physical Review Letters · doi:10.1103/physrevlett.87.181301 · Cited by 64 · see citing works
Livine, R. E., & Oriti, D. (2002) · Physical Review D · doi:10.1103/physrevd.65.044025 · Cited by 46 · see citing works
Self-citationThe physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory
Ding, Y., & Rovelli, C. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/20/205003 · Cited by 42 · see citing works
The volume operator in covariant quantum gravity
Ding, Y., & Rovelli, C. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/16/165003 · Cited by 23 · see citing works
- Authors
- Oriti, D., & Williams, R. M.
- Affiliations
- University of Cambridge (GB)
- Corresponding author
- Oriti, Daniele
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2000-12-29
- Volume
- 63
- Issue
- 2
- ISSN
- 0556-2821 (ISSN-L)
- 1089-4918
- OpenAlex
- W2113433756
- Cited by
- 54
- References
- 27
- Citations by year
- 2024: 1 · 2022: 1 · 2016: 3 · 2015: 1 · 2013: 1
- FWCI
- 4.07
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (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 › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- Barrett-Crane model, spin foam models, Euclidean quantum gravity, BF theory, tetrahedron, state sums, boundary conditions, higher-dimensional gravity, discrete gravity
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