The quantum tetrahedron in 3 and 4 dimensions
Open accessAbstract
Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the 'quantum tetrahedron'.Starting with a classical phase space whose points correspond to geometries of the tetrahedron in M 3 , we use geometric quantization to obtain a Hilbert space of states.This Hilbert space has a basis of states labeled by the areas of the faces of the tetrahedron together with one more quantum number, e-print archive: http.
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The basic idea behind the Barrett-Crane model was the quantum tetrahedron introduced by Barbieri and generalized to four dimensions.
“The basic idea behind the definition was that of the quantum tetrahedron introduced by Barbieri in [58] and generalized to 4D in [40].”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [40]
A general formalism for deriving any spin foam model from a generalized matrix model was developed in [90,91].
“A general formalism for deriving any kind of spin foam model from a generalized matrix model was developed in [90,91], proving the power and versatility of the formalism itself, and further study on this topic is in [92,93].”
From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [91]
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Cites
Elements of the Theory of Representations
Kirillov, A. A. (1976) · Grundlehren der mathematischen Wissenschaften · doi:10.1007/978-3-642-66243-0 · Cited by 1,493 · see citing works
State sum invariants of 3-manifolds and quantum 6j-symbols
Turaev, V. G., & Viro, O. Y. (1992) · Topology · doi:10.1016/0040-9383(92)90015-a · Cited by 1,145 · see citing works
(2010) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789814313735_0006 · Cited by 1,106 · see citing works
Quantization and unitary representations
Kostant, B. (1970) · Lecture notes in mathematics · doi:10.1007/bfb0079068 · Cited by 1,060 · see citing works
Guillemin, V., & Sternberg, S. (1977) · Mathematical surveys and monographs · doi:10.1090/surv/014 · Cited by 761 · see citing works
Bott, R. (1957) · The Annals of Mathematics · doi:10.2307/1969996 · Cited by 714 · see citing works
Woodhouse, N. M. J. (1992) · doi:10.1093/oso/9780198536734.001.0001 · Cited by 686 · see citing works
Geometric quantization and multiplicities of group representations
Guillemin, V., & Sternberg, S. (1982) · Inventiones mathematicae · doi:10.1007/bf01398934 · Cited by 637 · see citing works
Spin networks and quantum gravity
Rovelli, C., & Smolin, L. (1995) · Physical Review D · doi:10.1103/physrevd.52.5743 · Cited by 629 · see citing works
Angular Momentum in Quantum Physics: Theory and Application
Biedenharn, L. C., & Louck, J. D. (1984) · Cited by 491 · see citing works
Geometric Quantization and Quantum Mechanics
Śniatycki, J. (1980) · Applied mathematical sciences · doi:10.1007/978-1-4612-6066-0 · Cited by 472 · see citing works
Relativistic Spin Networks and Quantum GravityPreprint
Barrett, J. W., Crane, L., & Crane, J. W. B. L. (1998) · Cited by 423 · see citing works
Self-citationTOPOLOGICAL LATTICE MODELS IN FOUR DIMENSIONS
OOGURI, H. (1992) · Modern Physics Letters A · doi:10.1142/s0217732392004171 · Cited by 379 · see citing works
Reduction of Poisson manifolds
Marsden, J. E., & Ratiu, T. S. (1986) · Letters in Mathematical Physics · doi:10.1007/bf00398428 · Cited by 331 · see citing works
SEMICLASSICAL LIMIT OF RACAH COEFFICIENTS.
Ponzano, G., & Regge, T. (1969) · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · Cited by 315 · see citing works
The symplectic geometry of polygons in Euclidean space
Kapovich, M., & Millson, J. J. (1996) · Journal of Differential Geometry · doi:10.4310/jdg/1214459218 · Cited by 278 · see citing works
Baez, J. C. (1996) · Advances in Mathematics · doi:10.1006/aima.1996.0012 · Cited by 266 · see citing works
Self-citationQuantum theory of geometry III: Non-commutativity of Riemannian structuresPreprint
Ashtekar, A., Corichi, R., Zapata, J. A., & México, A. (1998) · Cited by 125 · see citing works
Spin networks are simplicial quantum gravity
Hasslacher, B., & Perry, M. J. (1981) · Physics Letters B · doi:10.1016/0370-2693(81)90185-4 · Cited by 105 · see citing works
Quantum tetrahedra and simplicial spin networks
Barbieri, A. (1997) · CERN Document Server (European Organization for Nuclear Research) · Cited by 99 · see citing works
Spin networks in nonperturbative quantum gravity
Baez, J. C. (1996) · Proceedings of symposia in applied mathematics · doi:10.1090/psapm/051/1372769 · Cited by 92 · see citing works
Self-citationPOLYGON SPACES AND GRASSMANNIANS
Hausmann, J.-C., & Knutson, A. E. (1997) · Archive ouverte UNIGE (University of Geneva) · doi:10.5169/seals-63276 · Cited by 77 · see citing works
Four-dimensional BF theory as a topological quantum field theory
Baez, J. C. (1996) · Letters in Mathematical Physics · doi:10.1007/bf00398315 · Cited by 75 · see citing works
Self-citationOn 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
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
Spectroscopic and group theoretical methods in physics
Dalitz, R. H. (1970) · Journal of the Franklin Institute · doi:10.1016/0016-0032(70)90275-9 · Cited by 35 · see citing works
Polygon spaces and GrassmanniansPreprint
Hausmann, J.-C., & Knutson, A. E. (1996) · arXiv (Cornell University) · doi:10.48550/arxiv.dg-ga/9602012 · Cited by 31 · see citing works
Quantum models of space-time based on recoupling theory
Moussouris, J. P. (1983) · OpenGrey (Institut de l'Information Scientifique et Technique) · doi:10.5287/ora-ovo4bykpp · Cited by 20 · see citing works
Imposing detE > 0 in discrete quantum gravity
Loll, R. (1997) · Physics Letters B · doi:10.1016/s0370-2693(97)00303-1 · Cited by 11 · see citing works
Quantum models of space-time based on recoupling theory
Penrose, R. (1984) · Cited by 8 · see citing works
Spin Networks in Quantum Gravity
Lorente, M. (2012) · doi:10.7546/jgsp-6-2006-85-100 · Cited by 8 · see citing works
Space of the vertices of relativistic spin networksPreprint
Barbieri, R. A. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9709076 · Cited by 7 · see citing works
Flagged first, then by citations
Research works printed but not linked: 4
- J. Baez, Spin foam models, Class. Quantum Grav., 15 (1998), 1827-1858.
- J.RM. Flude, Geometric Asymptotics of Spin, PhD thesis, Univer- sity of Nottingham, 1998.
- J.-C. Hausmann and A. Knutson, Polygon spaces and Grassman- nians, L'Enseignment Mathematique, 43 (1997), 173-198.
- R. Penrose, Angular momentum: an approach to combinatorial space-time, in 'Quantum Theory and Beyond', ed. T. Bastin, Cam- bridge U. Press, Cambridge, 1971.
Cited by
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The Spin-Foam Approach to Quantum Gravity
Perez, A. (2013) · Living Reviews in Relativity · doi:10.12942/lrr-2013-3 · Cited by 677 · see citing works
A new spin foam model for 4D gravity
Freidel, L., & Krasnov, K. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/12/125018 · Cited by 521 · 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
Asymptotic analysis of the Engle–Pereira–Rovelli–Livine four-simplex amplitude
Barrett, J. W., Dowdall, R. J., Fairbairn, W. J., Gomes, H., & Hellmann, F. (2009) · Journal of Mathematical Physics · doi:10.1063/1.3244218 · Cited by 224 · see citing works
Self-citationAlexandrov, S., & Livine, E. R. (2003) · Physical Review D · doi:10.1103/physrevd.67.044009 · Cited by 141 · see citing works
Asymptotics of 10 <i>j</i> symbols
Baez, J. C., Christensen, J. D., & Egan, G. (2002) · Classical and Quantum Gravity · doi:10.1088/0264-9381/19/24/315 · Cited by 80 · see citing works
Self-citationProjected spin networks for Lorentz connection: linking spin foams and loop gravity
Livine, E. R. (2002) · Classical and Quantum Gravity · doi:10.1088/0264-9381/19/21/316 · Cited by 77 · see citing works
Livine, R. E., & Oriti, D. (2002) · Physical Review D · doi:10.1103/physrevd.65.044025 · Cited by 46 · see citing works
Holomorphic Lorentzian simplicity constraints
Dupuis, M., Freidel, L., Livine, E. R., & Speziale, S. (2012) · Journal of Mathematical Physics · doi:10.1063/1.3692327 · Cited by 64 · see citing works
The perturbative Regge-calculus regime of loop quantum gravity
Bianchi, E., & Modesto, L. (2008) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2007.12.011 · Cited by 34 · see citing works
- Authors
- Baez, J. C., & Barrett, J. W.
- Affiliations
- University of California, Riverside (US)
- Corresponding author
- Baez, John C.
- Venue
- Advances in Theoretical and Mathematical Physics
- Publisher
- International Press of Boston
- Published
- 1999
- Volume
- 3
- Issue
- 4
- Pages
- 815-850
- ISSN
- 1095-0753 (ISSN-L)
- 1095-0761
- OpenAlex
- W2137570591
- Cited by
- 146
- References
- 33
- Citations by year
- 2026: 1 · 2025: 1 · 2024: 8 · 2023: 5 · 2022: 9 · 2021: 5 · 2020: 5 · 2019: 5 · 2018: 9 · 2017: 2 · 2016: 4 · 2015: 3 · 2014: 6 · 2013: 9 · 2012: 7
- FWCI
- 4.30
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Advances in Theoretical and Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Repository@Nottingham (University of Nottingham) (submittedVersion, cc-by)
- Repository@Nottingham (University of Nottingham) (submittedVersion, other-oa)
- 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
- state sum models, quantum gravity, geometric quantization, face area
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