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Lessons from toy-models for the dynamics of loop quantum gravity
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

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We review some approaches to the Hamiltonian dynamics of (loop) quantum gravity, the main issues being the regularization of the Hamiltonian and the continuum limit. First, Thiemann's definition of the quantum Hamiltonian is presented, and then more recent approaches. They are based on toy models which provide new insights into the difficulties and ambiguities faced in Thiemann's construction. The models we use are parametrized field theories, the topological BF model of which a special case is three-dimensional gravity which describes quantum flat space, and Regge lattice gravity.
Keywords: Hamiltonian constraint, loop quantum gravity, parametrized field theories, topological BF theory, discrete gravity. 
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     author = {Valentin Bonzom and Alok Laddha},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a8/}
}
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Valentin Bonzom; Alok Laddha. Lessons from toy-models for the dynamics of loop quantum gravity. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a8/

[1] Alesci E., Noui K., Sardelli F., “Spin-foam models and the physical scalar product”, Phys. Rev. D, 78 (2008), 104009, 16 pp. ; arXiv: 0807.3561 | DOI

[2] Alesci E., Rovelli C., “Regularization of the Hamiltonian constraint compatible with the spinfoam dynamics”, Phys. Rev D, 82 (2010), 044007, 17 pp. ; arXiv: 1005.0817 | DOI

[3] Alesci E., Thiemann T., Zipfel A., Linking covariant and canonical LQG: new solutions to the Euclidean scalar constraint, arXiv: 1109.1290

[4] Ambjørn J., Durhuus B., Jónsson T., “Three-dimensional simplicial quantum gravity and generalized matrix models”, Modern Phys. Lett. A, 6 (1991), 1133–1146 | DOI | MR | Zbl

[5] Anderson R.W., Aquilanti V., Marzuoli A., “$3nj$ morphogenesis and semiclassical disentangling”, J. Phys. Chem. A, 113 (2009), 15106–15117 ; arXiv: 1001.4386 | DOI

[6] Aquilanti V., Bitencourt A.C.P., da S. Ferreira C., Marzuoli A., Ragni M., “Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity”, Phys. Scr., 78 (2008), 058103, 7 pp. ; arXiv: 0901.1074 | DOI | Zbl

[7] Aquilanti V., Haggard H.M., Hedeman A., Jeevanjee N., Littlejohn R., Yu L., Semiclassical mechanics of the Wigner $6j$-symbol, arXiv: 1009.2811

[8] Ashtekar A., Lewandowski J., “Background independent quantum gravity: a status report”, Classical Quantum Gravity, 21 (2004), R53–R152 ; arXiv: gr-qc/0404018 | DOI | MR | Zbl

[9] Ashtekar A., Pawlowski T., Singh P., “Quantum nature of the big bang: improved dynamics”, Phys. Rev. D, 74 (2006), 084003, 23 pp. ; arXiv: gr-qc/0607039 | DOI | MR | Zbl

[10] Baez J.C., “An introduction to spin foam models of BF theory and quantum gravity”, Geometry and Quantum Physics (Schladming 1999), Lecture Notes in Phys., 543, Springer, Berlin, 25–93 | DOI | MR | Zbl

[11] Baez J.C., Perez A., “Quantization of strings and branes coupled to BF theory”, Adv. Theor. Math. Phys., 11 (2007), 451–469 ; arXiv: gr-qc/0605087 | MR | Zbl

[12] Baez J.C., Wise D.K., Crans A.S., “Exotic statistics for strings in 4D BF theory”, Adv. Theor. Math. Phys., 11 (2007), 707–749 ; arXiv: gr-qc/0603085 | MR | Zbl

[13] Bahr B., Dittrich B., “(Broken) gauge symmetries and constraints in Regge calculus”, Classical Quantum Gravity, 26 (2009), 225011, 34 pp. ; arXiv: 0905.1670 | DOI | MR | Zbl

[14] Bahr B., Dittrich B., “Improved and perfect actions in discrete gravity”, Phys. Rev. D, 80 (2009), 124030, 15 pp. ; arXiv: 0907.4323 | DOI

[15] Bahr B., Dittrich B., He S., “Coarse graining free theories with gauge symmetries: the linearized case”, New J. Phys., 13 (2011), 045009, 34 pp. ; arXiv: 1011.3667 | DOI

[16] Bahr B., Dittrich B., Ryan J.P., Spin foam models with finite groups, arXiv: 1103.6264

[17] Bahr B., Dittrich B., Steinhaus S., “Perfect discretization of reparametrization invariant path integrals”, Phys. Rev. D, 83 (2011), 19 pp. ; arXiv: 1101.4775 | DOI | Zbl

[18] Baratin A., Girelli F., Oriti D., “Diffeomorphisms in group field theories”, Phys. Rev. D, 83 (2011), 104051, 22 pp. ; arXiv: 1101.0590 | DOI

[19] Barrett J.W., Crane L., “An algebraic interpretation of the Wheeler–DeWitt equation”, Classical Quantum Gravity, 14 (1997), 2113–2121 ; arXiv: gr-qc/9609030 | DOI | MR | Zbl

[20] Barrett J.W., Dowdall R.J., Fairbairn W.J., Gomes H., Hellmann F., Pereira R., “Asymptotics of 4d spin foam models”, Gen. Relativity Gravitation, 43 (2011), 2421–2436 ; arXiv: 1003.1886 | DOI | MR | Zbl

[21] Barrett J.W., Fairbairn W.J., Hellmann F., “Quantum gravity asymptotics from the ${\rm SU}(2)$ $15j$-symbol”, Internat. J. Modern Phys. A, 25 (2010), 2897–2916 ; arXiv: 0912.4907 | DOI | MR | Zbl

[22] Barrett J.W., Naish-Guzman I., “The Ponzano–Regge model”, Classical Quantum Gravity, 26 (2011), 155014, 48 pp. ; arXiv: 0803.3319 | DOI

[23] Bergeron M., Semenoff G.W., Szabo R.J., “Canonical BF-type topological field theory and fractional statistics of strings”, Nuclear Phys. B, 437 (1995), 695–721 ; arXiv: hep-th/9407020 | DOI | MR | Zbl

[24] Blau M., Thompson G., “A new class of topological field theories and the Ray–{S}inger torsion”, Phys. Lett. B, 228 (1989), 64–68 | DOI | MR

[25] Blau M., Thompson G., “Topological gauge theories of antisymmetric tensor fields”, Ann. Physics, 205 (1991), 130–172 | DOI | MR | Zbl

[26] Blohmann C., Fernandes M.C.B., Weinstein A., Groupoid symmetry and constraints in general relativity, arXiv: 1003.2857

[27] Bonzom V., “Spin foam models and the Wheeler–DeWitt equation for the quantum 4-simplex”, Phys. Rev. D, 84 (2011), 024009, 13 pp. ; arXiv: 1101.1615 | DOI

[28] Bonzom V., Fleury P., Asymptotics of Wigner $3nj$-symbols with small and large angular momenta: an elementary method, arXiv: 1108.1569

[29] Bonzom V., Freidel L., “The Hamiltonian constraint in 3d Riemannian loop quantum gravity”, Classical Quantum Gravity, 28 (2011), 195006, 24 pp. ; arXiv: 1101.3524 | DOI | MR | Zbl

[30] Bonzom V., Gurau R., Riello A., Rivasseau V., “Critical behavior of colored tensor models in the large $N$ limit”, Nuclear Phys. B, 853 (2011), 174–195 ; arXiv: 1105.3122 | DOI | MR | Zbl

[31] Bonzom V., Livine E.R., Yet another recursion relation for the $6j$-symbol, arXiv: 1103.3415

[32] Bonzom V., Livine E.R., Smerlak M., Speziale S., “Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model”, Nuclear Phys. B, 804 (2008), 507–526 ; arXiv: 0802.3983 | DOI | MR | Zbl

[33] Bonzom V., Livine E.R., Speziale S., “Recurrence relations for spin foam vertices”, Classical Quantum Gravity, 27 (2010), 125002, 32 pp. ; arXiv: 0911.2204 | DOI | MR | Zbl

[34] Bonzom V., Smerlak M., Bubble divergences: sorting out topology from cell structure, arXiv: 1103.3961

[35] Bonzom V., Smerlak M., “Bubble divergences from cellular cohomology”, Lett. Math. Phys., 93 (2010), 295–305 ; arXiv: 1004.5196 | DOI | MR | Zbl

[36] Bonzom V., Smerlak M., Bubble divergences from twisted cohomology, arXiv: 1008.1476

[37] Bonzom V., Smerlak M., Gauge symmetries in spinfoam gravity: the case for ‘cellular quantization’, arXiv: 1201.4996

[38] Carfora M., Marzuoli A., Rasetti M., “Quantum tetrahedra”, J. Phys. Chem. A, 113 (2009), 15376–15383 ; arXiv: 1001.4402 | DOI

[39] Cattaneo A.S., Cotta-Ramusino P., Fröhlich J., Martellini M., “Topological BF theories in 3 and 4 dimensions”, J. Math. Phys., 36 (1995), 6137–6160 ; arXiv: hep-th/9505027 | DOI | MR | Zbl

[40] Cattaneo A.S., Cotta-Ramusino P., Fucito F., Martellini M., Rinaldi M., Tanzini A., Zeni M., “Four-dimensional {Y}ang–{M}ills theory as a deformation of topological BF theory”, Comm. Math. Phys., 197 (1998), 571–621 ; arXiv: hep-th/9705123 | DOI | MR | Zbl

[41] Constantinidis C.P., Piguet O., Gieres F., Sarandy M.S., “On the symmetries of BF models and their relation with gravity”, J. High Energy Phys., 2002:1 (2002), 017, 25 pp. ; arXiv: hep-th/0111273 | DOI | MR

[42] David F., “A model of random surfaces with nontrivial critical behaviour”, Nuclear Phys. B, 257 (1985), 543–576 | DOI | MR

[43] De Pietri R., Freidel L., “${\rm so}(4)$ Plebański action and relativistic spin-foam model”, Classical Quantum Gravity, 16 (1999), 2187–2196 ; arXiv: gr-qc/9804071 | DOI | MR | Zbl

[44] Di Francesco P., Ginsparg P., Zinn-Justin J., “2D gravity and random matrices”, Phys. Rep., 254:1–2 (1995), 133 pp. ; arXiv: hep-th/9306153 | DOI | MR

[45] Dittrich B., Eckert F.C., Martin-Benito M., Coarse graining methods for spin net and spin foam models, arXiv: 1109.4927

[46] Dittrich B., Höhn P.A., “From covariant to canonical formulations of discrete gravity”, Classical Quantum Gravity, 27 (2010), 155001, 37 pp. ; arXiv: 0912.1817 | DOI | MR | Zbl

[47] Dittrich B., Ryan J.P., “Phase space descriptions for simplicial 4D geometries”, Classical Quantum Gravity, 28 (2011), 065006, 34 pp. ; arXiv: 0807.2806 | DOI | MR | Zbl

[48] Dittrich B., Ryan J.P., “Simplicity in simplicial phase space”, Phys. Rev. D, 82 (2010), 064026, 19 pp. ; arXiv: 1006.4295 | DOI

[49] Dittrich B., Thiemann T., “Testing the master constraint programme for loop quantum gravity. I. General framework”, Classical Quantum Gravity, 23 (2006), 1025–1065 ; arXiv: gr-qc/0411138 | DOI | MR | Zbl

[50] Dittrich B., Thiemann T., “Testing the master constraint programme for loop quantum gravity. II. Finite-dimensional systems”, Classical Quantum Gravity, 23 (2006), 1067–1088 ; arXiv: gr-qc/0411139 | DOI | MR | Zbl

[51] Dittrich B., Thiemann T., “Testing the master constraint programme for loop quantum gravity. III. ${\rm SL}(2,{\mathbb R})$ models”, Classical Quantum Gravity, 23 (2006), 1089–1120 ; arXiv: gr-qc/0411140 | DOI | MR | Zbl

[52] Dittrich B., Thiemann T., “Testing the master constraint programme for loop quantum gravity. IV. Free field theories”, Classical Quantum Gravity, 23 (2006), 1121–1142 ; arXiv: gr-qc/0411141 | DOI | MR | Zbl

[53] Dittrich B., Thiemann T., “Testing the master constraint programme for loop quantum gravity. V. Interacting field theories”, Classical Quantum Gravity, 23 (2006), 1143–1162 ; arXiv: gr-qc/0411142 | DOI | MR | Zbl

[54] Dupuis M., Livine E.R., “Pushing the asymptotics of the $6j$-symbol further”, Phys. Rev. D, 80 (2009), 024035, 14 pp. ; arXiv: 0905.4188 | DOI | MR

[55] Dupuis M., Livine E.R., “The $6j$-symbol: recursion, correlations and asymptotics”, Classical Quantum Gravity, 27 (2010), 135003, 15 pp. ; arXiv: 0910.2425 | DOI | MR | Zbl

[56] Fairbairn W.J., Perez A., “Extended matter coupled to BF theory”, Phys. Rev. D, 78 (2008), 024013, 21 pp. ; arXiv: 0709.4235 | DOI | MR

[57] Freidel L., “Group field theory: an overview”, Internat. J. Theoret. Phys., 44 (2005), 1769–1783 ; arXiv: hep-th/0505016 | DOI | MR | Zbl

[58] Freidel L., Krasnov K., Puzio R., “BF description of higher-dimensional gravity theories”, Adv. Theor. Math. Phys., 3 (1999), 1289–1324 ; arXiv: hep-th/9901069 | MR | Zbl

[59] Freidel L., Louapre D., “Ponzano–Regge model revisited. I. Gauge fixing, observables and interacting spinning particles”, Classical Quantum Gravity, 21 (2004), 5685–5726 ; arXiv: hep-th/0401076 | DOI | MR | Zbl

[60] Freidel L., Speziale S., On the relations between gravity and BF theories, arXiv: 1201.4247

[61] Freidel L., Speziale S., “Twisted geometries: a geometric parametrisation of SU(2) phase space”, Phys. Rev. D, 82 (2010), 084040, 16 pp. ; arXiv: 1001.2748 | DOI

[62] Freidel L., Starodubtsev A., Quantum gravity in terms of topological observables, arXiv: hep-th/0501191

[63] Frohman C., Kania-Bartoszynska J., Dubois' torsion, A-polynomial and quantum invariants, arXiv: 1101.2695

[64] Gambini R., Lewandowski J., Marolf D., Pullin J., “On the consistency of the constraint algebra in spin network quantum gravity”, Internat. J. Modern Phys. D, 7 (1998), 97–109 ; arXiv: gr-qc/9710018 | DOI | MR | Zbl

[65] Gieres F., Grimstrup J.M., Nieder H., Pisar T., Schweda M., “Topological field theories and their symmetries within the Batalin–Vilkovisky framework”, Phys. Rev. D, 66 (2002), 025027, 14 pp. ; arXiv: hep-th/0111258 | DOI | MR

[66] Giulini D., Marolf D., “On the generality of refined algebraic quantization”, Classical Quantum Gravity, 16 (1999), 2479–2488 ; arXiv: gr-qc/9812024 | DOI | MR | Zbl

[67] Gross M., “Tensor models and simplicial quantum gravity in $>2$-D”, Nuclear Phys. B Proc. Suppl., 25A (1992), 144–149 | DOI | MR | Zbl

[68] Gurau R., The complete $1/N$ expansion of colored tensor models in arbitrary dimension, arXiv: 1102.5759 | MR

[69] Gurau R., Ryan J.P., Colored tensor models – a review, arXiv: 1109.4812

[70] Hájícek P., Isham C.J., “The symplectic geometry of a parametrized scalar field on a curved background”, J. Math. Phys., 37 (1996), 3505–3521 ; arXiv: gr-qc/9510028 | DOI | MR

[71] Han M., Thiemann T., “On the relation between operator constraint, master constraint, reduced phase space and path integral quantization”, Classical Quantum Gravity, 27 (2010), 225019, 46 pp. ; arXiv: 0911.3428 | DOI | MR | Zbl

[72] Horowitz G.T., “Exactly soluble diffeomorphism invariant theories”, Comm. Math. Phys., 125 (1989), 417–437 | DOI | MR | Zbl

[73] Jeffrey L.C., “Chern–Simons–Witten invariants of lens spaces and torus bundles, and the semiclassical approximation”, Comm. Math. Phys., 147 (1992), 563–604 | DOI | MR | Zbl

[74] Kazakov V.A., “Bilocal regularization of models of random surfaces”, Phys. Lett. B, 150 (1985), 282–284 | DOI | MR

[75] Kitaev A.Y., “Fault-tolerant quantum computation by anyons”, Ann. Physics, 303 (2003), 2–30 ; arXiv: quant-ph/9707021 | DOI | MR | Zbl

[76] Kuchar K., “Parametrized scalar field on $R\times S^1$: dynamical pictures, spacetime diffeomorphisms, and conformal isometries”, Phys. Rev. D, 39 (1989), 1579–1593 | DOI | MR

[77] Laddha A., Varadarajan M., “Hamiltonian constraint in polymer parametrized field theory”, Phys. Rev D, 83 (2011), 025019, 27 pp. ; arXiv: 1011.2463 | DOI

[78] Laddha A., Varadarajan M., “Polymer quantization of the free scalar field and its classical limit”, Classical Quantum Gravity, 27 (2010), 175010, 45 pp. ; arXiv: 1001.3505 | DOI | MR | Zbl

[79] Laddha A., Varadarajan M., “The diffeomorphism constraint operator in loop quantum gravity”, Classical Quantum Gravity, 28 (2011), 195010, 29 pp. ; arXiv: 1105.0636 | DOI | MR | Zbl

[80] Levin M.A., Wen X.G., “String-net condensation: a physical mechanism for topological phases”, Phys. Rev. B, 71 (2005), 045110, 21 pp. ; arXiv: cond-mat/0404617 | DOI

[81] Lewandowski J., Marolf D., “Loop constraints: a habitat and their algebra”, Internat. J. Modern Phys. D, 7 (1998), 299–330 ; arXiv: gr-qc/9710016 | DOI | MR | Zbl

[82] Lewandowski J., Okolów A., Sahlmann H., Thiemann T., “Uniqueness of diffeomorphism invariant states on holonomy-flux algebras”, Comm. Math. Phys., 267 (2006), 703–733 ; arXiv: gr-qc/0504147 | DOI | MR | Zbl

[83] Livine E.R., Speziale S., “New spinfoam vertex for quantum gravity”, Phys. Rev. D, 76 (2007), 084028, 14 pp. ; arXiv: 0705.0674 | DOI | MR

[84] Livine E.R., Tambornino J., “Spinor representation for loop quantum gravity”, J. Math. Phys., 53 (2012), 012503, 33 pp. ; arXiv: 1105.3385 | DOI

[85] Lucchesi C., Piguet O., Sorella S.P., “Renormalization and finiteness of topological BF theories”, Nuclear Phys. B, 395 (1993), 325–353 ; arXiv: hep-th/9208047 | DOI | MR

[86] Maggiore N., Sorella S.P., “Perturbation theory for antisymmetric tensor fields in four dimensions”, Internat. J. Modern Phys. A, 8 (1993), 929–945 ; arXiv: hep-th/9204044 | DOI | MR | Zbl

[87] Nicolai H., Peeters K., Zamaklar M., “Loop quantum gravity: an outside view”, Classical Quantum Gravity, 22 (2005), R193–R247 ; arXiv: hep-th/0501114 | DOI | MR | Zbl

[88] Noui K., Perez A., “Three-dimensional loop quantum gravity: physical scalar product and spin-foam models”, Classical Quantum Gravity, 22 (2005), 1739–1761 ; arXiv: gr-qc/0402110 | DOI | MR | Zbl

[89] Noui K., Perez A., Pranzetti D., Canonical quantization of non-commutative holonomies in $2+1$ loop quantum gravity, arXiv: 1105.0439

[90] Ooguri H., “Partition functions and topology-changing amplitudes in the three-dimensional lattice gravity of Ponzano and Regge”, Nuclear Phys. B, 382 (1992), 276–304 ; arXiv: hep-th/9112072 | DOI | MR | Zbl

[91] Ooguri H., “Topological lattice models in four dimensions”, Modern Phys. Lett. A, 7 (1992), 2799–2810 ; arXiv: hep-th/9205090 | DOI | MR | Zbl

[92] Oriti D., The group field theory approach to quantum gravity: some recent results, arXiv: 0912.2441

[93] Perez A., Introduction to loop quantum gravity and spin foams, arXiv: gr-qc/0409061

[94] Perez A., “Regularization ambiguities in loop quantum gravity”, Phys. Rev. D, 73 (2006), 044007, 18 pp. ; arXiv: gr-qc/0509118 | DOI | MR

[95] Perez A., Pranzetti D., “On the regularization of the constraint algebra of quantum gravity in 2+1 dimensions with a nonvanishing cosmological constant”, Classical Quantum Gravity, 27 (2010), 145009, 20 pp. ; arXiv: 1001.3292 | DOI | MR | Zbl

[96] Ponzano G., Regge T., “Semi-classical limit of Racah coefficients”, Spectroscopic and Group Theoretical Methods in Physics, ed. F. Bloch, North-Holland, Amsterdam, 1968, 1–58

[97] Roberts J., “Classical $6j$-symbols and the tetrahedron”, Geom. Topol., 3 (1999), 21–66 ; arXiv: math-ph/9812013 | DOI | MR | Zbl

[98] Rovelli C., “A new look at loop quantum gravity”, Classical Quantum Gravity, 28 (2011), 114005, 24 pp. ; arXiv: 1004.1780 | DOI | MR | Zbl

[99] Rovelli C., Discretizing parametrized systems: the magic of Ditt-invariance, arXiv: 1107.2310

[100] Rovelli C., Speziale S., “On the geometry of loop quantum gravity on a graph”, Phys. Rev. D, 82 (2010), 044018, 6 pp. ; arXiv: 1005.2927 | DOI | MR

[101] Rozansky L., “A large $k$ asymptotics of Witten's invariant of Seifert manifolds”, Comm. Math. Phys., 171 (1995), 279–322 ; arXiv: hep-th/9303099 | DOI | MR | Zbl

[102] Sasakura N., “Tensor model for gravity and orientability of manifold”, Modern Phys. Lett. A, 6 (1991), 2613–2623 | DOI | MR | Zbl

[103] Schulten K., Gordon R.G., “Semiclassical approximations to $3j$- and $6j$-coefficients for quantum-mechanical coupling of angular momenta”, J. Math. Phys., 16 (1975), 1971–1988 | DOI | MR

[104] Smolin L., The classical limit and the form of the Hamiltonian constraint in nonperturbative quantum general relativity, arXiv: gr-qc/9609034

[105] Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007 ; arXiv: gr-qc/0110034 | DOI | MR | Zbl

[106] Thiemann T., “Quantum spin dynamics (QSD)”, Classical Quantum Gravity, 15 (1998), 839–873 ; arXiv: gr-qc/9606089 | DOI | MR | Zbl

[107] Thiemann T., “Quantum spin dynamics (QSD) II. The kernel of the Wheeler–DeWitt constraint operator”, Classical Quantum Gravity, 15 (1998), 875–905 ; arXiv: gr-qc/9606090 | DOI | MR | Zbl

[108] Thiemann T., “Quantum spin dynamics (QSD). III. Quantum constraint algebra and physical scalar product in quantum general relativity”, Classical Quantum Gravity, 15 (1998), 1207–1247 ; arXiv: gr-qc/9705017 | DOI | MR | Zbl

[109] Thiemann T., “Quantum spin dynamics (QSD). IV. $2+1$ Euclidean quantum gravity as a model to test 3+1 Lorentzian quantum gravity”, Classical Quantum Gravity, 15 (1998), 1249–1280 ; arXiv: gr-qc/9705018 | DOI | MR | Zbl

[110] Thiemann T., “Quantum spin dynamics. VIII. The master constraint”, Classical Quantum Gravity, 23 (2006), 2249–2265 ; arXiv: gr-qc/0510011 | DOI | MR | Zbl

[111] Thiemann T., “The Phoenix Project: master constraint programme for loop quantum gravity”, Classical Quantum Gravity, 23 (2006), 2211–2247 ; arXiv: gr-qc/0305080 | DOI | MR | Zbl

[112] Varshalovich D.A., Moskalev A.N., Khersonskii V.K., Quantum theory of angular momentum, World Scientific Publishing Co. Inc., Teaneck, NJ, 1988 | MR

[113] Witten E., “$2+1$ dimensional gravity as an exactly soluble system”, Nuclear Phys. B, 311 (1988), 46–78 | DOI | MR | Zbl

[114] Witten E., “On quantum gauge theories in two dimensions”, Comm. Math. Phys., 141 (1991), 153–209 | DOI | MR | Zbl

[115] Witten E., “Topology-changing amplitudes in $(2+1)$-dimensional gravity”, Nuclear Phys. B, 323 (1989), 113–140 | DOI | MR

[116] Yu L., Asymptotic limits of the Wigner $15j$-symbol with small quantum numbers, arXiv: 1104.3641

[117] Yu L., “Semiclassical analysis of the Wigner $12j$ symbol with one small angular momentum”, Phys. Rev. A, 84 (2011), 022101, 13 pp. ; arXiv: 1104.3275 | DOI

[118] Yu L., Littlejohn R.G., “Semiclassical analysis of the Wigner $9j$ symbol with small and large angular momenta”, Phys. Rev. A, 83 (2011), 052114, 16 pp. ; arXiv: 1104.1499 | DOI

[119] Yutsis A.P., Levinson I.B., Vanagas V.V., Mathematical apparatus of the theory of angular momentum, Israel Program for Scientific Translations, Jerusalem, 1962 | MR | Zbl