Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe

Only the hyperbolic-hyperbolic sector corresponds to a spacetime in which the T2 slices are spacelike.

“It may be shown that only the hyperbolic-hyperbolic sector corresponds to a spacetime in which the T2 slices are spacelike [117, 119, 182, 209].”

• cites [117] Ezawa, K., “Addendum to “Classical and Quantum Evolutions of the de Sitter and the anti-de Sitter Universes in 2+1 dimensions””, Phys. Rev. D, 50, 2935–2938, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9403160. sourceCould not verify

  The discussion on p. 2 identifies standard and flat sectors for torus spatial hypersurfaces. On p. 4, the paper compares its standard sector with a spacelike sector in the Λ = 0 case, but it does not say that only a hyperbolic-hyperbolic sector permits spacelike T² slices. Searches for the hyperbolic-hyperbolic sector and the causal character of torus slices found no passage establishing the claim. (LLM, conf=0.74)

  Experimental — accuracy not yet measured
• cites [119] Ezawa, K., “Chern-Simons quantization of (2+1)-anti-de Sitter gravity on a torus”, Class. Quantum Grav., 12, 373–391, (1995). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9409074. sourceSupported
• cites [182] Louko, J., and Marolf, D., “Solution space of 2+1 gravity on R × T2 in Witten’s connection formulation”, Class. Quantum Grav., 11, 311–330, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9308018. sourceDisputed

  Section IV (pp. 13–15) says that spacetimes with positive-definite induced metrics on T² fall into two classes: one with purely translational holonomies in M₀ and another in Mₛ. It explicitly says the tori in the M₀ class are spacelike in case (4.1), contrary to the claim that only the hyperbolic-hyperbolic sector permits spacelike T² slices. (LLM, conf=0.88)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured
• cites [209] Nelson, J.E., and Picken, R.F., “Parametrization of the moduli space of flat SL(2,R) connections on the torus”, Lett. Math. Phys., 59, 215–226, (2002). Related online version (cited on 5 January 2005): http://arXiv.org/abs/math-ph/0105015. sourceCould not verify

  Section 2 (p. 3) classifies the connection matrices, and the Discussion (pp. 8–9) describes the resulting sectors. Neither those passages nor searches for spacelike torus slices and spacetime address which sector permits spacelike T² slices. (LLM, conf=0.75)

  Experimental — accuracy not yet measured

At strong coupling, the Lorentzian dynamical triangulation system splits into uncorrelated two-dimensional spaces, each well-described by two-dimensional gravity.

“At strong coupling, the system splits into uncorrelated two-dimensional spaces, each well-described by two-dimensional gravity.”

• cites [12] Ambjørn, J., Jurkiewicz, J., and Loll, R., “Nonperturbative 3D Lorentzian quantum gravity”, Phys. Rev. D, 64, 044011-1–17, (2001). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0011276. sourceDisputed

  Section 4.1 (pp. 15, 18) describes uncorrelated two-dimensional spatial geometries for k₀ above its critical value and suggests they behave as uncoupled 2D gravity models. But the Introduction (p. 5) defines k₀ as proportional to the inverse gravitational coupling, so this is the weak-gravitational-coupling regime, not the strong-coupling regime claimed. (LLM, conf=0.86)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured
• cites [10] Ambjørn, J., Jurkiewicz, J., and Loll, R., “Computer simulations of 3-d Lorentzian quantum gravity”, Nucl. Phys. B (Proc. Suppl.), 94, 689–692, (2001). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-lat/0011055. sourceDisputed

  The Introduction identifies k₀ as proportional to the inverse gravitational coupling. The Results (p. 3) place the uncorrelated 2D-gravity phase above the critical k₀, where gravitational coupling is weaker; below it, at stronger gravitational coupling, the paper reports extended 3D configurations. (LLM, conf=0.86)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured

(2+1)-dimensional gravity automatically displays a deformation of the Poincaré algebra associated with doubly special relativity.

“It has recently been pointed out that (2 + 1)-dimensional gravity automatically displays such a deformation [197, 18, 126].”

• cites [197] Matschull, H.-J., and Welling, M., “Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity”, Class. Quantum Grav., 15, 2981–3030, (1998). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9708054. sourceDisputed

  The Symmetries section (page unavailable) says explicitly that it is not the Poincaré group itself that is deformed, but its action on phase-space variables. Its generator brackets (eq. 4.30) retain the ordinary Poincaré algebra; the deformation described is in how translations act. (LLM, conf=0.90)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured
• cites [18] Amelino-Camelia, G., Smolin, L., and Starodubtsev, A., “Quantum symmetry, the cosmological constant and Planck scale phenomenology”, Class. Quantum Grav., 21, 3095–3110, (2004). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0306134. sourceSupported
• cites [126] Freidel, L., Kowalski-Glikman, J., and Smolin, L., “2+1 gravity and Doubly Special Relativity”, Phys. Rev. D, 69, 044001, (2004). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0307085. sourceSupported

At weak coupling, a semiclassical regime appears that resembles the picture obtained from other approaches to (2+1)-dimensional gravity.

“At weak coupling, however, a “semiclassical” regime appears that resembles the picture obtained from other approaches to (2 + 1)-dimensional gravity.”

• cites [12] Ambjørn, J., Jurkiewicz, J., and Loll, R., “Nonperturbative 3D Lorentzian quantum gravity”, Phys. Rev. D, 64, 044011-1–17, (2001). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0011276. sourcePartly supported

  Section 4.2 (pp. 19–21) reports semiclassical macroscopic behavior, but places it in the phase of intermediate gravitational coupling, not the weak-coupling phase. Section 4.1 (p. 18) says the weak-coupling phase consists of uncorrelated two-dimensional geometries rather than a three-dimensional gravity theory. The discussion of canonical approaches in Section 5 (p. 27) says the authors lack observables for a comparison with their spherical-slice model, so resemblance to a picture from other approaches is not established. (LLM, conf=0.86)

  Experimental — accuracy not yet measured
• cites [10] Ambjørn, J., Jurkiewicz, J., and Loll, R., “Computer simulations of 3-d Lorentzian quantum gravity”, Nucl. Phys. B (Proc. Suppl.), 94, 689–692, (2001). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-lat/0011055. sourceDisputed

  The Introduction (p. 2) defines k0 as proportional to the inverse bare gravitational coupling. The Results (pp. 3–4) place the extended, semiclassical three-dimensional regime below the critical k0, while above it the model reduces to uncorrelated two-dimensional gravity models. Thus the paper places that regime at stronger, not weaker, gravitational coupling. (LLM, conf=0.87)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured

The metric formalism can be used to give a full description of the solutions of the vacuum field equations, at least for spacetimes with topology R×Σ.

“As Moncrief [206] and Hosoya and Nakao [159] have shown, this metric formalism can also be used to give a full description of the solutions of the vacuum field equations, at least for spacetimes with the topology ℝ × Σ.”

• cites [206] Moncrief, V., “Reduction of the Einstein equations in 2+1 dimensions to a Hamiltonian system over Teichmuller space”, J. Math. Phys., 30, 2907–2914, (1989). sourceNot checked — no full text of the source
• cites [159] Hosoya, A., and Nakao, K., “(2+1)-dimensional pure gravity for an arbitrary closed initial surface”, Class. Quantum Grav., 7, 163–176, (1990). sourcePartly supported

  The Introduction describes an ADM metric approach for vacuum spacetimes with topology Σ × ℝ, and the Summary says the authors have completely solved the classical equations. But the Summary also says they could not explicitly demonstrate the motion of the moduli in the general case, so the claim of a full description for all such spacetimes overstates what the paper establishes. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

The first discrete formulation designed explicitly for 2+1 dimensions was developed by 't Hooft et al.

“The first discrete formulation designed explicitly for 2 + 1 dimensions was developed by’ t Hooft et al. [250, 252, 253, 124, 274, 156].”

• cites [250] ’t Hooft, G., “Causality in (2+1)-dimensional gravity”, Class. Quantum Grav., 9, 1335–1348, (1992). sourceNot checked — no full text of the source
• cites [252] ’t Hooft, G., “Classical N-particle cosmology in 2+1 dimensions”, Class. Quantum Grav., 10, S79–S91, (1993). sourceNot checked — no full text of the source
• cites [253] ’t Hooft, G., “The evolution of gravitating point particles in 2+1 dimensions”, Class. Quantum Grav., 10, 1023–1038, (1993). sourceNot checked — no full text of the source
• cites [124] Franzosi, R., and Guadagnini, E., “Topology and classical geometry in (2+1) gravity”, Class. Quantum Grav., 13, 433–460, (1996). sourcePartly supported

  The Introduction identifies ’t Hooft’s formalism as a formulation of (2+1)-dimensional gravity, and §2 describes its polygon tessellation, supporting the attribution of a discrete formulation to him. The paper does not establish that it was the first such formulation. (LLM, conf=0.85)

  Experimental — accuracy not yet measured
• cites [274] Welling, M., “The Torus Universe in the Polygon Approach to 2+1-Dimensional Gravity”, Class. Quantum Grav., 14, 929–943, (1997). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9606011. sourcePartly supported

  Section 2 (p. 5) says ’t Hooft invented the polygon approach as a Hamiltonian description of 2+1-dimensional gravity, using piecewise-flat patches. It does not establish that this was the first discrete formulation designed for those dimensions. (LLM, conf=0.85)

  Experimental — accuracy not yet measured
• cites [156] Hollmann, H.R., and Williams, R.M., “Hyperbolic geometry in ’t Hooft’s approach to (2+1)-dimensional gravity”, Class. Quantum Grav., 16, 1503–1518, (1999). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9810021. sourcePartly supported

  The Introduction says ’t Hooft introduced a polygon approach to (2+1)-dimensional gravity, and Section 2 describes its tessellation into polygonal tiles. This supports the attribution of a discrete formulation to ’t Hooft, but the paper does not establish that it was the first such formulation. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

(2+1)-dimensional gravity does not have a good Newtonian limit.

“In particular, the theory does not have a good Newtonian limit [107, 49, 94].”

• cites [107] Deser, S., Jackiw, R., and ’t Hooft, G., “Three dimensional Einstein gravity: Dynamics of at space”, Ann. Phys. (N.Y.), 152, 220, (1984). sourceNot checked — no full text of the source
• cites [49] Barrow, J.D., Burd, A.B., and Lancaster, D., “Three-dimensional classical spacetimes”, Class. Quantum Grav., 3, 551–567, (1986). sourceSupported
• cites [94] Cornish, N.J., and Frankel, N.E., “Gravitation in 2+1 dimensions”, Phys. Rev. D, 43, 2555–2565, (1991). sourceNot checked — no full text of the source

York time-slicing, in which mean extrinsic curvature is used as a time coordinate, is a good global coordinate choice for classical solutions of the vacuum field equations.

“Andersson et al. have shown that this is a good global coordinate choice for classical solutions of the vacuum field equations [20].”

• cites [20] Andersson, L., Moncrief, V., and Tromba, A.J., “On the global evolution problem in 2+1 gravity”, J. Geom. Phys., 23, 191, (1997). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9610013. sourcePartly supported

  Theorem 4.1 and Corollary 4.2 (Global existence, p. 9) establish global evolution in constant-mean-curvature time and a global CMC foliation. The result is limited to specified 2+1-dimensional vacuum spacetimes with compact spatial surfaces of genus greater than one and CMC initial data; it does not establish the unqualified claim for classical vacuum solutions generally. (LLM, conf=0.88)

  Experimental — accuracy not yet measured

An explicit example of the ambiguity for de Sitter structures has been given by Ezawa for the topology R×T2.

“An explicit example of the resulting ambiguity has been given by Ezawa [117] for the case of a topology ℝ × T2”

• cites [117] Ezawa, K., “Addendum to “Classical and Quantum Evolutions of the de Sitter and the anti-de Sitter Universes in 2+1 dimensions””, Phys. Rev. D, 50, 2935–2938, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9403160. sourceSupported

In Chern-Simons scattering computations, imposing invariance under the mapping class group is necessary to recover the correct classical limit.

“In each case, it is only when one imposes invariance under the mapping class group that one recovers the correct classical limit.”

• cites [65] Carlip, S., “Exact quantum scattering in 2+1 dimensional gravity”, Nucl. Phys. B, 324, 106–122, (1989). sourceNot checked — no full text of the source
• cites [259] Troost, J., and Tsuchiya, A., “Towards black hole scattering”, Phys. Lett. B, 574, 301–308, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0307158. sourcePartly supported

  In “Scattering off a black hole” (page unavailable), the paper says that the braiding projection makes a probe wave-function periodic and reconstructs its semiclassical form on a BTZ black-hole background. This supports the role of projection in that scattering calculation, but our copy does not establish the broader claim that mapping-class-group invariance is necessary in each case to recover the correct classical limit. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites [67] Carlip, S., “Measuring the metric in (2+1)-dimensional quantum gravity”, Class. Quantum Grav., 8, 5–17, (1991). sourceNot checked — no full text of the source

If the invertibility condition for e is dropped, the first-order formalism is no longer quite equivalent to the metric formalism.

“The invertibility condition for e is important; if it is dropped, the first-order formalism is no longer quite equivalent to the metric formalism [194].”

• cites [194] Matschull, H.-J., “On the relation between 2+1 Einstein gravity and Chern Simons theory”, Class. Quantum Grav., 16, 2599–2609, (1999). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9903040. sourceSupported

Loll has found a complete set of configuration space variables.

“and Loll has found a complete set of “configuration space” variables [178].”

• cites [178] Loll, R., “Independent Loop Invariants for 2+1 Gravity”, Class. Quantum Grav., 12, 1655–1662, (1995). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9408007. sourceSupported

With a different gauge choice, Waelbroeck's model is classically equivalent to 't Hooft's.

“With a different gauge choice, Waelbroeck’s model is classically equivalent to’ t Hooft’s [271]”

• cites [271] Waelbroeck, H., and Zapata, J.A., “(2+1) covariant lattice theory and’ t Hooft’s formulation”, Class. Quantum Grav., 13, 1761–1768, (1996). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9601011. sourceSupported

An alternative quantization works directly with the holonomy matrices, imposing a quantum-group-like quantization condition.

“An alternative quantization, also differing by terms of order ℏ3, works directly with the holonomy matrices (38), imposing a quantum-group-like quantization condition [208]”

• cites [208] Nelson, J.E., and Picken, R.F., “Quantum Holonomies in (2+1)-Dimensional Gravity”, Phys. Lett. B, 471, 367–372, (2000). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9911005. sourcePartly supported

  The introductory discussion following equation (7) describes a new quantization approach that works directly with holonomy matrices satisfying a q-commutation relation, and calls the resulting structure quantum-group-like. The Discussion (p. 7) likewise contrasts matrix variables with earlier trace-based approaches. Our copy does not establish the citing sentence’s additional claim that this quantization differs by terms of order ℏ³. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

Hamilton's equations from the Poisson brackets reproduce the time dependence (42) of the moduli.

“It may be shown that the version of Hamilton’s equations of motion coming from these brackets reproduces the time dependence (42) of the moduli; see [87, 135] for details.”

• cites [87] Carlip, S., and Nelson, J.E., “Comparative quantizations of (2+1)-dimensional gravity”, Phys. Rev. D, 51, 5643–5653, (1995). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9411031. sourceSupported
• cites [135] Fujiwara, Y., and Soda, J., “Teichmüller Motion of (2+1)-Dimensional Gravity with the Cosmological Constant”, Prog. Theor. Phys., 83, 733–748, (1990). sourcePartly supported

  The paper gives Hamiltonian time-evolution equations and derives the time dependence of torus moduli, including the Teichmüller parameter in Eq. (5·5). But our copy does not show a derivation from the cited Poisson brackets or compare its result with the citing paper’s Eq. (42). The relevant passages are the Hamilton-equations passage and the Teichmüller-motion discussion; page numbers are unavailable in the indexed copy. (LLM, conf=0.83)

  Experimental — accuracy not yet measured

For (2+1)-dimensional gravity, there is evidence that one can always find operator orderings of the Hamiltonians that ensure consistent evolution.

“For (2 + 1)-dimensional gravity, on the other hand, there is evidence that one can always find operator orderings of the Hamiltonians that ensure consistent evolution [95].”

• cites [95] Cosgrove, R., “Consistent evolution with different time slicings in quantum gravity”, Class. Quantum Grav., 13, 891–919, (1996). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9511059. sourcePartly supported

  The Part 3 introduction and §14 offer a non-rigorous argument for consistent evolution across time slicings in (2+1)-dimensional gravity. The Conclusion says consistent evolution could be achieved through operator orderings if quantization after evolution is equivalent to Heisenberg evolution, but establishes this only for at least an infinite-parameter family of slicings—not always or in complete generality. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

When Λ = 0, a set of parameters describing the initial singularity and a second set describing the asymptotic future geometry together completely determine the spacetime.

“Benedetti and Guadagnini have shown that when Λ = 0, a set of parameters describing the initial singularity and a second set describing the geometry in the asymptotic future together completely determine the spacetime [55].”

• cites [55] Benedetti, R., and Guadagnini, E., “Cosmological Time in (2+1) — Gravity”, Nucl. Phys. B, 613, 330–352, (2001). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0003055. sourceSupported

For a closed three-manifold M, the Turaev-Viro amplitude, interpreted as a partition function, is equal to the absolute square of the partition function of an SU(2) Chern-Simons theory with coupling constant k.

“For a closed three-manifold M, the Turaev-Viro amplitude — now interpreted as a partition function — is equal to the absolute square of the partition function of an SU(2) Chern-Simons theory with coupling constant k [228, 260, 233],”

• cites [228] Rama, S.K., and Sen, S., “3-D manifolds, graph invariants, and Chern-Simons theory”, Mod. Phys. Lett. A, 7, 2065–2076, (1992). sourceNot checked — no full text of the source
• cites [260] Turaev, V.G., “Quantum invariants of 3-manifolds and a glimpse of shadow topology”, C. R. Acad. Sci. Ser. I, 313, 395–398, (1991).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [233] Roberts, J., “Skein theory and Turaev-Viro invariants”, Topology, 34, 771–788, (1995). sourcePartly supported

  The Introduction states that the Turaev–Viro invariant equals the squared modulus of the Reshetikhin–Turaev invariant, and §3.4 gives a proof in the paper’s skein-theoretic formulation. The copy identifies the SU(2) case but does not explicitly identify that invariant as a Chern–Simons partition function with coupling constant k; that part of the attributed claim is not established by the copy. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

For the torus universe, a complete, explicit description of loop representation states is possible.

“The loop representation is complicated by the existence of Mandelstam identities [190] among holonomies of loops, but for the case of the torus universe, a complete, explicit description of the states is again possible [26, 29].”

• cites [26] Ashtekar, A., Lectures on Non-Perturbative Canonical Gravity, vol. 6 of Advanced Series in Astrophysics and Cosmology, (World Scientific, Singapore, 1991). sourceNot checked — no full text of the source
• cites [29] Ashtekar, A., Husain, V., Rovelli, C., Samuel, J., and Smolin, L., “2+1 quantum gravity as a toy model for the 3+1 theory”, Class. Quantum Grav., 6, L185–L193, (1989). sourceSupported

Freidel and Louapre further show that the symmetry can be gauge-fixed, leading to a sum over a restricted and considerably simplified class of "collapsed" triangulations.

“Freidel and Louapre further show that the symmetry can instead be gauge-fixed, leading to a sum over a restricted and considerably simplified class of “collapsed” triangulations.”

• cites [131] Freidel, L., and Louapre, D., “Diffeomorphisms and spin foam models”, Nucl. Phys. B, 662, 279–298, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0212001. sourcePartly supported

  The Discussion (p. 13) says gauge fixing the translational symmetry amounts to computing the partition function on a collapsed triangulation. The Gauge fixing procedure (pp. 11–12) explains how collapsing edges along a maximal tree produces such a triangulation. Our copy does not describe a sum over a class of collapsed triangulations; it describes a partition function on a collapsed triangulation. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

In 2+1 dimensions with Λ=0, loop quantum gravity is closely related to the first order formalism of Section 2.3 but takes loop variables T0[γ] and T1[γ] as fundamental observables.

“In 2 + 1 dimensions with Λ = 0, this approach is closely related to the first order formalism of Section 2.3, but takes as its fundamental observables the loop variables T0[γ] and T1[γ] of Equation (55).”

• cites [26] Ashtekar, A., Lectures on Non-Perturbative Canonical Gravity, vol. 6 of Advanced Series in Astrophysics and Cosmology, (World Scientific, Singapore, 1991). sourceNot checked — no full text of the source
• cites [239] Rovelli, C., “Loop Quantum Gravity”, Living Rev. Relativity, 1, lrr-1998-1, (1998). URL (cited on 5 January 2005): http://www.livingreviews.org/lrr-1998-1. sourcePartly supported

  The Formalism > Loop quantum gravity identifies the loop algebra as the theory’s elementary observables, and Physical Results > Technical says the loop representation has been applied to 2+1 gravity. But these passages do not establish the claimed relationship to the first-order formalism at Λ=0 or identify T0[γ] and T1[γ] as the observables in that setting. Page numbers are unavailable in our copy. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

For field theories, even in flat spacetime, different time-slicings do not always evolve an initial wave function to the same final wave function.

“For field theories, even in flat spacetime, this will not always happen [258].”

• cites [258] Torre, C.G., and Varadarajan, M., “Functional evolution of free quantum fields”, Class. Quantum Grav., 16, 2651–2668, (1999). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9811222. sourcePartly supported

  The Discussion (pp. 14–15) says that evolution between certain Cauchy surfaces in flat spacetime cannot be implemented unitarily in the usual Fock-space Schrödinger picture. It does not show that two time-slicings produce different final wave functions from the same initial wave function; rather, it says that the failure to define the evolution makes that question of foliation independence academic. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

The quantum 6j-symbols are related to spherical tetrahedra rather than flat tetrahedra.

“Correspondingly, the quantum 6j-symbols are related to spherical tetrahedra rather than flat tetrahedra [255].”

• cites [255] Taylor, Y.U., and Woodward, C.T., “6j symbols for U q (sl2) and non-Euclidean tetrahedra”, (2003). URL (cited on 5 January 2005): http://arXiv.org/abs/math.QA/0305113. sourcePartly supported

  The Introduction (p. 1) relates the semiclassical asymptotics of quantum 6j-symbols to spherical tetrahedra when q is a root of unity. The claim omits that condition: the paper also relates the symbols to hyperbolic tetrahedra for positive real q and discusses a Euclidean formula for q = 1 (p. 3). (LLM, conf=0.86)

  Experimental — accuracy not yet measured

For the torus universe, the loop transform can be obtained explicitly and written as a simple integral transform.

“For the torus universe, this “loop transform” can be obtained explicitly [26, 29, 192], and written as a simple integral transform.”

• cites [26] Ashtekar, A., Lectures on Non-Perturbative Canonical Gravity, vol. 6 of Advanced Series in Astrophysics and Cosmology, (World Scientific, Singapore, 1991). sourceNot checked — no full text of the source
• cites [29] Ashtekar, A., Husain, V., Rovelli, C., Samuel, J., and Smolin, L., “2+1 quantum gravity as a toy model for the 3+1 theory”, Class. Quantum Grav., 6, L185–L193, (1989). sourceSupported
• cites [192] Marolf, D., “Loop representations for 2+1 gravity on a torus”, Class. Quantum Grav., 10, 2625–2647, (1993). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9303019. sourceSupported

A different choice of time-slicing might lead to a different quantum theory, as is known in quantum field theory.

“a different choice might lead to a different quantum theory, as it is known to do in quantum field theory [258].”

• cites [258] Torre, C.G., and Varadarajan, M., “Functional evolution of free quantum fields”, Class. Quantum Grav., 16, 2651–2668, (1999). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9811222. sourcePartly supported

  In “B. Quantum Field Theory” (p. 8), the paper says that evolution between Cauchy surfaces can yield states whose Hilbert-space representations are not unitarily equivalent. That supports the possibility of a difference arising from time-slicing, but the paper does not say that the resulting representations are different quantum theories; it describes evolution between arbitrary surfaces within one algebraic formulation (p. 6). (LLM, conf=0.84)

  Experimental — accuracy not yet measured

The degeneration of the spatial geometry at the initial singularity carries over to the higher genus case.

“Moreover, the degeneration of the spatial geometry at the initial singularity carries over to the higher genus case [200, 55].”

• cites [200] Mess, G., “Lorentz Spacetimes of Constant Curvature”, unknown status, (1990). Institut des Hautes Etudes Scientifiques preprint IHES/M/90/28.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [55] Benedetti, R., and Guadagnini, E., “Cosmological Time in (2+1) — Gravity”, Nucl. Phys. B, 613, 330–352, (2001). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0003055. sourceSupported

In the limit of large j, the 6j-symbol times a phase is asymptotic to a sum of two exponentials involving the Regge action and volume, as in Equation (66).

“Ponzano and Regge noticed, and Roberts later proved rigorously [234], that in the limit of large j,”

• cites [234] Roberts, J.D., “Classical 6j-symbols and the tetrahedron”, Geom. Topol., 3, 21–66, (1999). Related online version (cited on 5 January 2005): http://arXiv.org/abs/math-ph/9812013. sourcePartly supported

  Theorem 1 (Introduction) rigorously gives a large-label asymptotic formula with a volume-dependent amplitude and a cosine of the dihedral-angle phase; the cosine can be written as a sum of two exponentials. But the theorem gives that oscillatory formula only for an admissibly labelled Euclidean tetrahedron. It instead states exponential decay for the Minkowskian case, a condition the claim omits. (LLM, conf=0.87)

  Experimental — accuracy not yet measured

The semicircular curve is a geodesic in the natural Weil-Petersson (or Poincaré) metric on the torus moduli space.

“Such a curve is a geodesic in the natural Weil-Petersson (or Poincaré) metric on the torus moduli space [159, 135].”

• cites [159] Hosoya, A., and Nakao, K., “(2+1)-dimensional pure gravity for an arbitrary closed initial surface”, Class. Quantum Grav., 7, 163–176, (1990). sourceSupported
• cites [135] Fujiwara, Y., and Soda, J., “Teichmüller Motion of (2+1)-Dimensional Gravity with the Cosmological Constant”, Prog. Theor. Phys., 83, 733–748, (1990). sourceSupported

The quantities T0 and T1 can be represented as Hermitian operators on the space of holonomies or on a larger space of spin networks.

“The quantities T0 and T1 can be represented as Hermitian operators on this space of holonomies (or on a larger space of spin networks).”

• cites [129] Freidel, L., and Livine, E.R., “Spin Networks for Non-Compact Groups”, J. Math. Phys., 44, 1322–1356, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0205268. sourceCould not verify

  Section 5.2 (p. 23) says that the Laplacian operators Δe are Hermitian on a Hilbert space whose eigenvectors are spin networks. Searches for T0, T1, and Hermitian holonomy operators found no passage identifying T0 and T1 or saying they have the claimed representations; the Laplacian result does not establish this claim. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [130] Freidel, L., Livine, E.R., and Rovelli, C., “Spectra of length and area in (2+1) Lorentzian loop quantum gravity”, Class. Quantum Grav., 20, 1463–1478, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0212077. sourceCould not verify

  Section II discusses holonomy states, spin networks, and Hermitian operators representing the frame field. But our copy does not identify quantities T0 and T1 or say that those quantities have Hermitian representations on either space. Searches for T0/T1 and the claimed operator formulation found no decisive passage; page numbers were unavailable. (LLM, conf=0.74)

  Experimental — accuracy not yet measured

For large but finite k, the Turaev-Viro quantum group regularization introduces a cosmological constant to the Regge action, Λ(4π/k)^2.

“For large but finite k, the Turaev-Viro quantum group regularization introduces a cosmological constant to the Regge action [204, 203],”

• cites [204] Mizoguchi, S., and Tada, T., “3-dimensional Gravity from the Turaev-Viro Invariant”, Phys. Rev. Lett., 68, 1795–1798, (1992). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9110057. sourcePartly supported

  The abstract says the Turaev–Viro path integral effectively includes a cosmological term, and the discussion following Eq. (35) describes it as a regularization counterterm in the large-k approximation. But the abstract gives the cosmological constant as 4π²/k² + O(k⁻⁴), not (4π/k)²; Eq. (35) is consistent with the abstract’s value. (LLM, conf=0.90)

  Experimental — accuracy not yet measured
• cites [203] Mizoguchi, S., and Tada, T., “3-dimensional Gravity and the Turaev-Viro Invariant”, Prog. Theor. Phys. Suppl., 110, 207, (1992). sourceNot checked — no full text of the source

The space of solutions has a natural symplectic structure equivalent to the standard symplectic structure on phase space.

“the space of solutions has natural symplectic structure [175, 272], which can be shown to be equivalent to the standard symplectic structure on phase space.”

• cites [175] Lee, J., and Wald, R.M., “Local symmetries and constraints”, J. Math. Phys., 31, 725–473, (1990). sourceNot checked — no full text of the source
• cites [272] Wald, R.M., “Black hole entropy is Noether charge”, Phys. Rev. D, 48, R3427–R3431, (1993). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9307038. sourcePartly supported

  The passage labeled Abstract (p. 6) defines a symplectic current from variations of the dynamical fields, supporting the claim’s symplectic-structure component. The passage labeled Abstract (p. 9) relates the construction to Hamiltonian evolution on phase space, but our copy does not establish equivalence to the standard phase-space symplectic structure. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

For compact gauge groups, the quantization of Chern-Simons theory is well understood.

“For compact gauge groups, the quantization of Chern-Simons theory is well understood [278, 243, 244, 116, 34].”

• cites [278] Witten, E., “Quantum field theory and the Jones polynomial”, Commun. Math. Phys., 121, 351–399, (1989). sourceNot checked — no full text of the source
• cites [243] Schwarz, A.S., “The partition function of degenerate quadratic functional and Ray-Singer invariants”, Lett. Math. Phys., 2, 247–252, (1978). sourceNot checked — no full text of the source
• cites [244] Schwarz, A.S., “The partition function of a degenerate functional”, Commun. Math. Phys., 67, 1–16, (1979). sourceNot checked — no full text of the source
• cites [116] Elitzur, S., Moore, G.W., Schwimmer, A., and Seiberg, N., “Remarks on the canonical quantization of the Chern-Simons-Witten theory”, Nucl. Phys. B, 326, 108–134, (1989). sourceNot checked — no full text of the source
• cites [34] Axelrod, S.E., DellaPietra, S., and Witten, E., “Geometric quantization of Chern-Simons gauge theory”, J. Differ. Geom., 33, 787–902, (1991). sourceSupported

In the large k limit, the Turaev-Viro Hilbert space is isomorphic to the space of gauge-invariant functions of flat SU(2) connections.

“In the large k limit, the Turaev-Viro Hilbert space is isomorphic to the space of gauge-invariant functions of flat SU(2) connections [219, 220, 242].”

• cites [219] Ooguri, H., “Partition Functions and Topology-Changing Amplitudes in the 3D Lattice Gravity of Ponzano and Regge”, Nucl. Phys. B, 382, 276–304, (1992). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9112072. sourcePartly supported

  The Introduction (p. 5) describes the Ponzano–Regge model as the k → ∞ limit relevant to the Turaev–Viro conjecture, and Wave-Functions (p. 11) establishes an isomorphism of physical Hilbert spaces. But Wave-Functions (p. 8) identifies the corresponding states as half-densities on the moduli space of flat SO(3) connections, not simply gauge-invariant functions of flat SU(2) connections. (LLM, conf=0.88)

  Experimental — accuracy not yet measured
• cites [220] Ooguri, H., and Sasakura, N., “Discrete and Continuum Approaches to Three-Dimensional Quantum Gravity”, Mod. Phys. Lett. A, 6, 3591–3600, (1991). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9108006. sourceSupported
• cites [242] Sasakura, N., “Exact three-dimensional lattice gravities”, Prog. Theor. Phys. Suppl., 110, 191–206, (1992). sourcePartly supported

  Section 4 identifies the k→∞ Turaev–Viro model with the Ponzano–Regge model and describes the relevant continuum space as gauge-invariant functions of flat SU(2) connections. It concludes that this space is isomorphic to the large-k physical space, but explicitly assumes that the stated Wilson-line-network relations exhaust all linear dependencies. The citing sentence omits that assumption; page numbers were not available in our copy. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

In (2+1)-dimensional Lorentzian dynamical triangulation models, the wrong-sign contributions are unimportant because they are overwhelmed by the larger number of well-behaved geometries.

“In the (2 + 1)-dimensional Lorentzian dynamical triangulation models of Section 3.7, however, it is known that these wrong sign contributions are unimportant[12]; they are overwhelmed by the much larger number of well-behaved geometries in the path integral.”

• cites [12] Ambjørn, J., Jurkiewicz, J., and Loll, R., “Nonperturbative 3D Lorentzian quantum gravity”, Phys. Rev. D, 64, 044011-1–17, (2001). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0011276. sourcePartly supported

  In §5 (p. 26), the paper reports an extended phase with a well-defined ground state and says entropy contributions apparently outbalance potential conformal divergences from the Wick-rotated action. This supports the proposed mechanism, but the citing sentence presents it as an established result for the models generally, whereas the paper qualifies it as an inference about a particular phase. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

For Λ < 0, manifolds with larger volumes are numerous enough to lead to a divergence in the partition function.

“Here, too, however, manifolds with larger volumes — although individually exponentially suppressed — are numerous enough to lead to a divergence in the partition function [71].”

• cites [71] Carlip, S., “The Sum over Topologies in Three-Dimensional Euclidean Quantum Gravity”, Class. Quantum Grav., 10, 207–218, (1993). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9206103. sourceSupported

Work on circularly symmetric midi-superspace models has led to unexpected bounds on the Hamiltonian.

“Certain matter couplings in supergravity have been studied [104, 196], and work on circularly symmetric “midi-superspace models” has led to some surprising results, including unexpected bounds on the Hamiltonian [27, 32, 137, 53, 265, 224].”

• cites [104] de Wit, B., Matschull, H.-J., and Nicolai, H., “Physical States in d=3,N=2 Supergravity”, Phys. Lett. B, 318, 115–121, (1993). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9309006. sourceCould not verify

  The paper’s discussion of torus moduli and supermoduli describes a non-Hausdorff space, not circularly symmetric midi-superspace models or bounds on the Hamiltonian. Searches of our copy for circular symmetry and Hamiltonian bounds found no passage addressing the attributed result. (LLM, conf=0.73)

  Experimental — accuracy not yet measured
• cites [196] Matschull, H.-J., and Nicolai, H., “Canonical quantum supergravity in three dimensions”, Nucl. Phys. B, 411, 609–649, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9306018. sourceCould not verify

  Section 3 discusses when a sigma-model Hamiltonian is positive definite, but does not connect that result to circularly symmetric midi-superspace models or unexpected Hamiltonian bounds. Searches for circular symmetry, midi-superspace, and Hamiltonian bounds found no passage addressing the attributed claim. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [27] Ashtekar, A., “Large quantum gravity effects: Unexpected limitations of the classical theory”, Phys. Rev. Lett., 77, 4864–4867, (1996). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9610008. sourceSupported
• cites [32] Ashtekar, A., and Pierri, M., “Probing Quantum Gravity Through Exactly Soluble Midi-Superspaces I”, J. Math. Phys., 37, 6250–6270, (1996). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9606085. sourceSupported
• cites [137] Gambini, R., and Pullin, J., “Large quantum gravity effects: backreaction on matter”, Mod. Phys. Lett. A, 12, 2407–2414, (1997). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9703088. sourceCould not verify

  The paper discusses a 2+1-dimensional gravity model coupled to a scalar field (p. 1), but its reported surprising result concerns large metric fluctuations, not bounds on the Hamiltonian (pp. 1, 4). Searches for Hamiltonian bounds and circular symmetry found no passage in our copy that establishes the attributed result. (LLM, conf=0.74)

  Experimental — accuracy not yet measured
• cites [53] Beetle, C., “Midi-Superspace Quantization of Non-Compact Toroidally Symmetric Gravity”, Adv. Theor. Math. Phys., 2, 471–495, (1998). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9801107. sourceCould not verify

  Section 2.1 describes a toroidally symmetric midi-superspace model, not a circularly symmetric one. Section 2.2 says its Hamiltonian vanishes on the constraint surface, while Section 2.3 identifies the reduced Hamiltonian with that of a scalar field; neither passage reports unexpected bounds on the Hamiltonian. Searches for bounded or upper-bounded Hamiltonians found no supporting passage in our copy. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [265] Varadarajan, M., “On the metric operator for quantum cylindrical waves”, Class. Quantum Grav., 17, 189–199, (2000). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9910043. sourcePartly supported

  The discussion (p. 12) identifies the cylindrical-wave model as a midisuperspace and says its regulated metric operator can be viewed as a spatially smeared Hamiltonian. But it treats whether that operator has a lower bound as an open question; this paper does not establish the claimed Hamiltonian bound. (LLM, conf=0.85)

  Experimental — accuracy not yet measured
• cites [224] Pierri, M., “Probing Quantum General Relativity Through Exactly Soluble Midi-Superspaces II: Polarized Gowdy Models”, Int. J. Mod. Phys. D, 11, 135, (2002). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0101013. sourcePartly supported

  Section 2.1 (p. 3) identifies the polarized Gowdy model as an axisymmetric midi-superspace with U(1) symmetry. But the Hamiltonian discussion in Section 3.2 (p. 12) describes its operator and physical states without reporting an unexpected bound; Section 4 instead emphasizes that this model’s Hamiltonian arises through deparametrization. The claimed bound is therefore not supported by this paper. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

In the second order formalism, imposing the Wheeler-DeWitt constraints is considerably more complicated.

“In the second order formalism, matters become considerably more complicated [73].”

• cites [73] Carlip, S., “Notes on the (2+1)-Dimensional Wheeler-DeWitt Equation”, Class. Quantum Grav., 11, 31, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9309002. sourceSupported

If the initial and final surfaces are nondegenerate and spacelike, their topologies are restricted: they must have equal Euler numbers.

“If we demand that the initial and final surfaces be nondegenerate and spacelike, their topologies are severely restricted: Amano and Higuchi have shown that Σi and Σf must have equal Euler numbers [6].”

• cites [6] Amano, A., and Higuchi, S., “Topology change in ISO(2,1) Chern-Simons gravity”, Nucl. Phys. B, 377, 218–235, (1992). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9201075. sourceSupported

For manifolds with boundary, the path integral gives a transition function depending on specified boundary data, most simply the induced spin connection ω, with restrictions on the normal component of ω and the triad E.

“The path integral then gives a transition function that depends on specified boundary data — most simply, the induced spin connection ω, with some additional restrictions on the normal component of ω and the triad E [283, 84].”

• cites [283] Wu, S., “Topological quantum field theories on manifolds with a boundary”, Commun. Math. Phys., 136, 157–168, (1991). sourceNot checked — no full text of the source
• cites [84] Carlip, S., and Cosgrove, R., “Topology Change in (2+1)-Dimensional Gravity”, J. Math. Phys., 35, 5477–5493, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9406006. sourceSupported

For Λ > 0, the number of topologically distinct lens spaces with volumes less than vol(S3) grows fast enough that these spaces dominate, leading to a divergent partition function for closed three-manifolds.

“As shown in [71], though, the number of topologically distinct lens spaces with volumes less than volḡ(S3) grows fast enough that these spaces dominate, leading to a divergent partition function for closed three-manifolds.”

• cites [71] Carlip, S., “The Sum over Topologies in Three-Dimensional Euclidean Quantum Gravity”, Class. Quantum Grav., 10, 207–218, (1993). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9206103. sourcePartly supported

  In §3 (pp. 211–213), Carlip identifies distinct lens spaces with volumes proportional to 1/p and finds that the sum is dominated by manifolds with increasingly large fundamental groups and small volumes. But the paper calls lens-space dominance a suggestion, and its discussion of approximations says not all saddle points or topologies have been included. The claim therefore overstates the result as an established divergence of the partition function for all closed three-manifolds. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

In 2+1 dimensions, the selection rules imply that such a process can be described in Lorentzian signature only if χ(Σ)=0, that is, only for Σ a torus.

“In 2 + 1 dimensions, the selection rules of [6] imply that such a process can be described in Lorentzian signature only if χ(Σ) = 0, that is, only for Σ a torus.”

• cites [6] Amano, A., and Higuchi, S., “Topology change in ISO(2,1) Chern-Simons gravity”, Nucl. Phys. B, 377, 218–235, (1992). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9201075. sourceSupported

For g>1, the space of homomorphisms from π1(Σ) to G modulo conjugation has dimension 12g−12.

“For g > 1, this space of homomorphisms has dimension 12g − 12”

In 2+1 dimensions, a discrete description can be exact.

“In 2 + 1 dimensions, though, one has the added feature that a discrete description can be exact.”

If one starts with a flat three-manifold M and cuts out a ball, one obtains a new flat manifold without affecting the holonomy.

“If we start with a flat three-manifold M and simply cut out a ball, we can obtain a new flat manifold without affecting the holonomy.”

If Λ≠0, a vacuum spacetime can be built by patching together pieces of de Sitter or anti-de Sitter space by appropriate isometries: SO(3,1) for Λ>0 and SO(2,2) or SL(2,R)×SL(2,R)/Z2 for Λ<0.

“Similarly, if Λ ≠ 0, a vacuum spacetime can be built by patching together pieces of de Sitter or anti-de Sitter space by appropriate isometries: SO(3, 1) for Λ > 0 and SO(2, 2) or SL(2, ℝ) × SL(2, ℝ)/ℤ2 for Λ < 0.”

The use of the term holonomy in the Chern-Simons formulation is equivalent to that in the geometric structure description.

“This use of the term “holonomy” is somewhat different from that of Section 2.2, but the two are equivalent.”

In n dimensions, the phase space of general relativity has n(n−3) physical degrees of freedom per spacetime point.

“We are thus left with n(n − 1) − 2n = n(n − 3) physical degrees of freedom per spacetime point.”

The action (29) simplifies to a reduced phase space action involving only the physical degrees of freedom.

“The action (29) then simplifies to a “reduced phase space” action, involving only the physical degrees of freedom”

Not all homomorphisms from π1(Σ) to G give geometric structures that correspond to smooth manifolds.

“Second, not all homomorphisms from π1(Σ) to G give geometric structures that correspond to smooth manifolds.”

The first-order action allows computation of the symplectic structure on the space of solutions.

“The first-order action allows us an additional step that was unavailable in the geometric structure formalism — we can compute the symplectic structure on the space of solutions.”

Waelbroeck's model is a discretized version of the first-order formalism, with triads assigned to faces of a two-dimensional lattice and Lorentz transformations assigned to edges.

“This model is a discretized version of the first-order formalism of Section 2.3, with triads assigned to faces of a two-dimensional lattice and Lorentz transformations assigned to edges.”

Classical solutions to the vacuum field equations in 2+1 dimensions are all locally diffeomorphic to spacetimes of constant curvature: Minkowski, de Sitter, or anti-de Sitter.

“Classical solutions to the vacuum field equations are all locally diffeomorphic to spacetimes of constant curvature, that is, Minkowski, de Sitter, or anti-de Sitter space.”

Three-dimensional gravity reduces to a finite-dimensional system, albeit one with a complicated time-dependent Hamiltonian.

“Three-dimensional gravity again reduces to a finite-dimensional system, albeit one with a complicated time-dependent Hamiltonian.”

For a vanishing cosmological constant, the relevant gauge group is the Poincaré group ISO(2,1).

“Consider first the case of a vanishing cosmological constant. The relevant gauge group — the group G of the geometric structure — is then the Poincaré group ISO(2, 1)”

If Λ=0, for G=ISO(2,1) and M≃R×Σ, the space (11) of geometric structures is itself a cotangent bundle whose base space is the space of hyperbolic structures on Σ.

“If Λ = 0, this correspondence can be made more explicit: For G = ISO(2, 1) and M ≃ ℝ × Σ, the space (11) of geometric structures is itself a cotangent bundle, whose base space is the space of hyperbolic structures on Σ.”

If the spacetime manifold M is topologically trivial, the vacuum field equations imply that (M,g) is simply a subset of ordinary Minkowski space.

“If the spacetime manifold M is topologically trivial, then by Equation (1) the vacuum field equations imply that (M, g) is simply a subset of ordinary Minkowski space (V2,1, η).”

In three spacetime dimensions, general relativity simplifies to a topological theory with no propagating local degrees of freedom.

“In three spacetime dimensions, general relativity drastically simplifies, becoming a “topological” theory with no propagating local degrees of freedom.”

• Also stated in: § Introduction · plain

  “As I will explain below, general relativity in 2 + 1 dimensions has no local dynamical degrees of freedom.”

• Also stated in: § Why (2 + 1)-dimensional gravity is simple · plain

  “Physically, a (2 + 1)-dimensional spacetime has no local degrees of freedom: There are no gravitational waves in the classical theory, and no propagating gravitons in the quantum theory.”

In four dimensions, the phase space counting gives four phase space degrees of freedom, two gravitational wave polarizations and their conjugate momenta.

“In four dimensions, this gives the usual four phase space degrees of freedom, two gravitational wave polarizations and their conjugate momenta.”

The standard metrics ḡ_ij are labeled by moduli m_α that parametrize the Riemann moduli space of Σ.

“These standard metrics are labeled by a set of moduli mα that parametrize the Riemann moduli space of Σ.”

A constant negative curvature surface of genus g>1 is described by 6g−6 parameters.

“now, since PSL(2, ℝ) is three-dimensional, we find that a constant negative curvature surface of genus g > 1 is described by 6g − 6 parameters.”

Constant curvature metrics can be described in terms of a geometric structure-for genus g>1 an (H2, PSL(2,R)) structure-with moduli parametrizing the homomorphisms.

“As in Section 2.2, such constant curvature metrics can be described in terms of a geometric structure — for genus g > 1 an (ℍ2, PSL(2, ℝ)) structure — with moduli parametrizing the homomorphisms (10).”

Up to a diffeomorphism, the trace-free part of πij can be written as a holomorphic quadratic differential pij, a transverse traceless tensor with respect to the covariant derivative compatible with ḡij.

“Up to a diffeomorphism, the trace-free part of πij can be written as a holomorphic quadratic differential pij, that is, a transverse traceless tensor with respect to the covariant derivative compatible with ḡij.”

For a spacetime R×Σ with Σ compact genus g, the space of holonomies is the space of homomorphisms from π1(Σ) to G modulo overall conjugation.

“By Equation (11), the space of holonomies is the space of homomorphisms from π1(Σ) to G (where G is ISO(2, 1) for Λ = 0, SO(3, 1) for Λ > 0, or SO(2, 2) for Λ < 0) modulo overall conjugation.”

Waelbroeck's and 't Hooft's models are related by a nonlocal change of variables.

“but the two models are related by a nonlocal change of variables”

For a surface of genus g>1, this gives 12g−12 degrees of freedom, matching the results of Section 2.2.

“For a surface of genus g > 1, this gives us 12g − 12 degrees of freedom, matching the results of Section 2.2.”

The action (14) can be written as the standard Chern-Simons action for the group ISO(2,1) with k=1/(4G).

“Equation (22) may be recognized as the standard Chern-Simons action [278] for the group ISO(2, 1).”

• cites [278] Witten, E., “Quantum field theory and the Jones polynomial”, Commun. Math. Phys., 121, 351–399, (1989). sourceNot checked — no full text of the source

One obtains a dynamical description parametrized by a set of lengths and rapidities, which turn out to be canonically conjugate.

“One obtains a dynamical description parametrized by a set of lengths and rapidities, which turn out to be canonically conjugate.”

The theory of elliptic equations ensures that Equation (32) determines a unique scale factor λ.

“The theory of elliptic equations ensures that Equation (32) determines a unique scale factor λ.”

The action (64) gives a representation of the loop algebra.

“then gives a representation of the loop algebra.”

The modular group does not act properly discontinuously on the configuration space of square integrable functions of R2±.

“The modular group does not act properly discontinuously on this configuration space.”

As time increases, the modulus (42) moves along a semicircle in the upper half of the complex plane, with a center on the real axis.

“It is not hard to check that as time increases, the modulus (42) moves along a semicircle in the upper half of the complex plane, with a center on the real axis.”

For small Λ, the commutators differ from the naive quantization of the classical brackets by terms of order ħ3.

“for Λ small, these commutators differ from the naive quantization of the classical brackets (45) by terms of order ℏ3.”

At the big bang and big crunch the modulus (42) is purely real, τ2 = 0.

“At the “big bang” and “big crunch” the modulus (42) is purely real, τ2 = 0.”

The Poisson brackets among the traces of the holonomies reproduce the Poisson algebra of Nelson, Regge, and Zertuche.

“reproducing the Poisson algebra of Nelson, Regge, and Zertuche [211].”

• cites [211] Nelson, J.E., and Regge, T., “Homotopy groups and (2+1)-dimensional quantum de Sitter gravity”, Nucl. Phys. B, 339, 516–532, (1990). sourceNot checked — no full text of the source

The initial spatial geometry is again degenerate.

“The initial spatial geometry is again degenerate.”

The induced metric (40) is a space of constant negative curvature.

“An easy calculation confirms that this is a space of constant negative curvature.”

Canonical quantization gives a perfectly consistent description of a universe with a fixed spatial topology in 2+1 dimensions.

“The answer in 2 +1 dimensions is unequivocally no: Canonical quantization gives a perfectly consistent description of a universe with a fixed spatial topology.”

The sum over topologies is generally dominated by an infinite number of complicated topologies, each individually exponentially suppressed.

“The results of (2 + 1)-dimensional quantum gravity indicate that such claims should be treated with skepticism; as discussed in Section 3.11, the sum over topologies is generally dominated by an infinite number of complicated topologies, each individually exponentially suppressed.”

For the Euclidean Ponzano-Regge action, using the Plancherel formula to express each δ(gα) as a sum over characters of finite-dimensional representations of SU(2) permits an exact evaluation of the remaining integrals over gα, reproducing the 6j symbols.

“For the Euclidean Ponzano-Regge action, g ∈ SU(2), and the key trick is now to use the Plancherel formula to express each δ(gα) as a sum over the characters of finite-dimensional representations of SU(2). Fairly straightforward arguments then permit an exact evaluation of the remaining integrals over the gα, reproducing the 6j symbols in the Ponzano-Regge action.”

(2+1)-dimensional quantum gravity points toward a solution to the problem of observables, allowing construction of families of local and time-dependent observables that commute with all constraints.

“As we saw in Section 3.4, (2 + 1)-dimensional quantum gravity points toward a solution, allowing the construction of families of “local” and “time-dependent” observables that nevertheless commute with all constraints.”

The first-order action (14) is equivalent to that of a Chern-Simons theory.

“As first noted by Achúcarro and Townsend [2] and further developed by Witten [277, 279], the first-order action (14) is equivalent to that of a Chern-Simons theory.”

• cites [2] Achúcarro, A., and Townsend, P.K., “A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories”, Phys. Lett. B, 180, 89–92, (1986). sourceNot checked — no full text of the source
• cites [277] Witten, E., “2+1 dimensional gravity as an exactly soluble system”, Nucl. Phys. B, 311, 46–78, (1988). sourceNot checked — no full text of the source
• cites [279] Witten, E., “Topology-changing amplitudes in 2+1 dimensional gravity”, Nucl. Phys. B, 323, 113–140, (1989). sourceNot checked — no full text of the source

Even after accounting for the caveats, there is a (12g−12)-dimensional space of solutions that can in principle be described completely.

“Even once these caveats are taken into account, though, we still have a (12g − 12)-dimensional space of solutions that can, in principle, be described completely.”

H is unique up to conjugation by an element h ∈ G.

“it may be shown that H is unique up to such conjugation [256].”

• cites [256] Thurston, W.P., The Geometry and Topology of Three-Manifolds, Princeton Lecture Notes, (Princeton University Press, Princeton, U.S.A., 1979). Related online version (cited on 5 January 2005): http://www.msri.org/publications/books/gt3m/.Not checked — no full text of the source

The holonomy of a curve γ depends only on its homotopy class.

“It may be shown that the holonomy of a curve γ depends only on its homotopy class [256].”

• cites [256] Thurston, W.P., The Geometry and Topology of Three-Manifolds, Princeton Lecture Notes, (Princeton University Press, Princeton, U.S.A., 1979). Related online version (cited on 5 January 2005): http://www.msri.org/publications/books/gt3m/.Not checked — no full text of the source

The composition of loops implicit in the brackets makes it difficult to find small closed subalgebras needed for quantization.

“The composition of loops implicit in the brackets (27) makes it difficult to find small closed subalgebras of the sort needed for quantization.”

For de Sitter structures, the holonomies do not uniquely determine the geometry.

“For de Sitter structures, on the other hand, the holonomies do not uniquely determine the geometry [200].”

• cites [200] Mess, G., “Lorentz Spacetimes of Constant Curvature”, unknown status, (1990). Institut des Hautes Etudes Scientifiques preprint IHES/M/90/28.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In 2+1 dimensions there is a strong argument in favor of treating the mapping class group as a genuine invariance.

“In 2 + 1 dimensions, though, there seems to be a strong argument in favor of treating the mapping class group as a genuine invariance, as follows.”

Nelson and Regge have constructed a small but complete (actually overcomplete) set of holonomies on a surface of arbitrary genus that form a closed algebra.

“However, Nelson and Regge have succeeded in constructing a small but complete (actually overcomplete) set of holonomies on a surface of arbitrary genus that form a closed algebra [213, 212]”

• cites [213] Nelson, J.E., and Regge, T., “2+1 quantum gravity”, Phys. Lett. B, 272, 213–216, (1991). sourceNot checked — no full text of the source
• cites [212] Nelson, J.E., and Regge, T., “(2+1) gravity for genus ¿ 1”, Commun. Math. Phys., 141, 211–223, (1991). sourceNot checked — no full text of the source

The global geometry of vacuum spacetimes in 2+1 dimensions is described mathematically by the theory of geometric structures.

“The global geometry of vacuum spacetimes in 2 + 1 dimensions is described mathematically by the theory of geometric structures [256, 62, 147]”

• cites [256] Thurston, W.P., The Geometry and Topology of Three-Manifolds, Princeton Lecture Notes, (Princeton University Press, Princeton, U.S.A., 1979). Related online version (cited on 5 January 2005): http://www.msri.org/publications/books/gt3m/.Not checked — no full text of the source
• cites [62] Canary, R.D., Epstein, D.B.A., and Green, P., in Epstein, D.B.A., ed., Analytical and Geometric Aspects of Hyperbolic Space: Warwick and Durham 1984, Papers presented at two symposia held at the Universities of Warwick and Durham, vol. 111 of London Mathematical Society Lecture Notes Series, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1987).Not checked yet

  Not checked in the free run.

• cites [147] Goldman, W.M., in Goldman, W.M., and Magid, A.R., eds., Geometry of Group Representations, Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held July 5–11, 1987, vol. 74 of Contemporary Mathematics, (American Mathematical Society, Providence, U.S.A., 1988). sourceNot checked — no full text of the source

Any (G,X) structure on a manifold M determines a corresponding flat G bundle.

“Indeed, any (G, X) structure on a manifold M determines a corresponding flat G bundle [147]”

• cites [147] Goldman, W.M., in Goldman, W.M., and Magid, A.R., eds., Geometry of Group Representations, Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held July 5–11, 1987, vol. 74 of Contemporary Mathematics, (American Mathematical Society, Providence, U.S.A., 1988). sourceNot checked — no full text of the source

Nontrivial spatial or spacetime topologies in 2+1 gravity have moduli that can become dynamical.

“One can consider nontrivial spatial or spacetime topologies [277, 279]. Such “cosmological” solutions have moduli — a finite number of parameters that distinguish among geometrically inequivalent constant curvature manifolds — and these can become dynamical.”

• cites [277] Witten, E., “2+1 dimensional gravity as an exactly soluble system”, Nucl. Phys. B, 311, 46–78, (1988). sourceNot checked — no full text of the source
• cites [279] Witten, E., “Topology-changing amplitudes in 2+1 dimensional gravity”, Nucl. Phys. B, 323, 113–140, (1989). sourceNot checked — no full text of the source

Vacuum general relativity in 2+1 dimensions is equivalent-up to considerations of the invertibility of e-to a gauge theory.

“Vacuum general relativity in 2 + 1 dimensions is thus equivalent — again up to considerations of the invertibility of e — to a gauge theory.”

For vanishing cosmological constant, Mess has shown that the holonomy group determines a unique maximal spacetime M, specifically a domain of dependence of a spacelike surface Σ.

“For the case of a vanishing cosmological constant, Mess [200] has investigated this question for spacetimes with topologies ℝ × Σ. He shows that the holonomy group determines a unique “maximal” spacetime M — specifically, a domain of dependence of a spacelike surface Σ.”

• cites [200] Mess, G., “Lorentz Spacetimes of Constant Curvature”, unknown status, (1990). Institut des Hautes Etudes Scientifiques preprint IHES/M/90/28.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

For a manifold made of a collection of tetrahedra, the full Regge action occurs in a product of 6j-symbols.

“For a manifold made of a collection of such tetrahedra, the full Regge action will occur in a product of such 6j-symbols.”

Three-dimensional Euclidean gravity can be written as an ISU(2) Chern-Simons theory with a configuration space of flat SU(2) connections.

“Just as (2 + 1)-dimensional Lorentzian gravity can be written as an ISO(2, 1) Chern-Simons theory with a configuration space of flat SO(2, 1) connections, three-dimensional Euclidean gravity can be written as an ISU(2) Chern-Simons theory with a configuration space of flat SU(2) connections.”

The construction of dynamical observables and time-dependent states in covariant canonical quantum theory requires an explicit isomorphism between the phase space and the space of classical solutions.

“the construction of dynamical observables and time-dependent states in covariant canonical quantum theory requires an explicit isomorphism between the phase space and the space of classical solutions.”

Turaev and Viro discovered an improved regularization based on quantum groups.

“The “topological” feature of the Ponzano-Regge model was made more precise by Turaev and Viro [262], who discovered an improved regularization, based on the technology of quantum groups.”

• cites [262] Turaev, V.G., and Viro, O.Y., “State Sum Invariants of 3-Manifolds and Quantum 6j-Symbols”, Topology, 31, 865–902, (1992). sourceNot checked — no full text of the source

Understanding of quantum Chern-Simons gravity depends strongly on the sign of the cosmological constant.

“our understanding of the quantum Chern-Simons gravity depends strongly on the sign of the cosmological constant.”

The phase space of a well-behaved classical theory is isomorphic to the space of classical solutions.

“the phase space of a well-behaved classical theory is isomorphic to the space of classical solutions.”

The loop variables T0[γ] depend only on the configuration space variable ω.

“Observe now that the loop variables T0[γ] depend only on the “configuration space” variable ω.”

Loop quantum gravity starts with a Hamiltonian formulation of the first order formalism, with constraints (63) written in analogy to the (3+1)-dimensional case.

“More precisely, loop quantum gravity starts with a Hamiltonian formulation of the first order formalism, with constraints, written in analogy to the (3 + 1)-dimensional case [26], that take the form 63”

• cites [26] Ashtekar, A., Lectures on Non-Perturbative Canonical Gravity, vol. 6 of Advanced Series in Astrophysics and Cosmology, (World Scientific, Singapore, 1991). sourceNot checked — no full text of the source

The principal drawback of reduced phase space quantization is that it relies on a classical choice of time coordinate, which occurs as part of gauge-fixing.

“The principal drawback is that the method relies on a classical choice of time coordinate, which occurs as part of the gauge-fixing needed to solve the constraints.”

For (2+1)-dimensional gravity, the space of classical solutions is the space of geometric structures.

“For (2 + 1)-dimensional gravity, the space of classical solutions is the space of geometric strictures of Section 2.2.”

It is not clear that reduced phase space quantization preserves general covariance.

“In other words, it is not clear that this approach to quantum gravity preserves general covariance.”

The loop representation appears to be rather drastically different from the Chern-Simons formulation.

“The loop representation thus appears to be rather drastically different from the Chern-Simons formulation.”

Puzio's results suggest that the classical singularity may be better-behaved in the quantum theory.

“Puzio’s results suggest that the classical singularity may be better-behaved in the quantum theory.”

The choice of operator ordering may have dramatic effects on the physics, since the spectra of the various Maass Laplacians are quite different.

“Nevertheless, the choice of ordering may have dramatic effects on the physics, since the spectra of the various Maass Laplacians are quite different.”

In this setting, Chern-Simons quantization is covariant canonical quantization.

“Thus in this setting, Chern-Simons quantization is covariant canonical quantization.”

For the torus universe, quantizing the algebra of loop variables T0 and T1 simply reproduces the theory obtained in the Chern-Simons formulation.

“For the torus universe, it is not hard to show that such a quantization simply reproduces the theory we already obtained in the Chern-Simons formulation (see, for example, Section 7.2 of [81]).”

• cites [81] Carlip, S., Quantum Gravity in 2+1 Dimensions, Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1998). sourceNot checked — no full text of the source

For Λ = −1/l2 < 0, the group G, equivalently the Chern-Simons gauge group, is SO(2,2).

“The group G of Section 2.2, or, equivalently, the gauge group in the Chern-Simons formalism of Section 2.3, is then SO(2, 2).”

For a physically interesting geometry, one must go to the spacelike sector, in which the traces of the holonomies are both greater than two.

“For a physically interesting geometry, one must go to the “spacelike sector,” in which the traces of the holonomies are both greater than two.”

The main reason 2+1 dimensions offer an attractive setting for quantum gravity is that the classical solutions can be completely described by a finite set of parameters.

“the main reason that 2 + 1 dimensions offer such an attractive setting for quantum gravity is that the classical solutions can be completely described by a finite set of parameters.”

The mapping class group cannot be ignored.

“it is at least clear that the mapping class group cannot be ignored.”

A constant t slice is also a slice of constant York time.

“A constant t slice is thus also a slice of constant York time.”

AdS3 is isometric to the group manifold of SL(2,R).

“This gives an isometry between AdS3 and the group manifold of SL(2, ℝ).”

With appropriate substitutions, the Ponzano-Regge amplitude (67) becomes well-defined without any regularization.

“With appropriate substitutions (e.g., “quantum” 6j-symbols [90]), the Ponzano-Regge amplitude (67) becomes well-defined without any regularization.”

• cites [90] Carter, J.S., Flath, D.E., and Saito, M., The classical and quantum 6j-symbols, vol. 43 of Mathematical Notes, (Princeton University Press, Princeton, U.S.A., 1995).Not checked yet

  Not checked in the free run.

A recent step toward projecting out physical states in Euclidean quantum gravity has the ultimate effect of reducing spin network states to loop states.

“An interesting step toward projecting out the physical states has recently been taken in [216], in the context of Euclidean quantum gravity; the ultimate effect is to reduce spin network states to loop states of the sort we have considered above.”

• cites [216] Noui, K., and Perez, A., “Three dimensional loop quantum gravity: physical scalar product and spin foam models”, (2004). URL (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0402110.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

For spacetimes R × Σ with Σ a surface of genus g > 1, no explicit metrics analogous to Equation (40) are known, except for solutions with constant moduli.

“For spacetimes ℝ × Σ with Σ being a surface of genus g > 1, no explicit metrics analogous to Equation (40) are known, except for the special case of solutions with constant moduli.”

For Λ = 0, the phase space has a natural cotangent bundle structure, allowing holonomies of the spin connection to be identified as generalized positions and their derivatives as generalized momenta.

“In this case, one has the nice feature that the phase space has a natural cotangent bundle structure, allowing us to immediately identify the holonomies of the spin connection ω as generalized positions, and their derivatives as generalized momenta.”

The problems in the physical sector can be traced back to the fact that SL(2, R) is a noncompact group.

“The problems in the physical sector can be traced back to the fact that SL(2, ℝ) is a noncompact group.”

The holonomy group is generated by two commuting SO(2,2) matrices, unique up to overall conjugation.

“The holonomy group (11) is therefore generated by two commuting SO(2, 2) matrices, unique up to overall conjugation.”

The fundamental group π1(R × T2) has two generators satisfying a single commutation relation.

“The fundamental group π1(ℝ × T2) has two generators, [γ1] and [γ2], satisfying a single relation similar to Equation (12)”

For genus one, a full quantization is possible.

“For genus one, on the other hand, a full quantization is possible.”

For spacetimes R × Σ with Σ closed and Λ ≤ 0, the holonomies of a geometric structure determine a unique maximal domain of dependence.

“the holonomies of a geometric structure determine a unique maximal domain of dependence [200]”

• cites [200] Mess, G., “Lorentz Spacetimes of Constant Curvature”, unknown status, (1990). Institut des Hautes Etudes Scientifiques preprint IHES/M/90/28.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

For Λ < 0, the relevant gauge group is SO(2,2) or its cover SL(2,R) × SL(2,R).

“For Λ < 0, the relevant gauge group is SO(2, 2) or its cover SL(2, ℝ) × SL(2, ℝ).”

For spatial surfaces of genus g > 1, the complexity of the constraint makes reduced phase space quantization impractical.

“For spatial surfaces of genus g > 1, the complexity of the constraint (32) seems to make this approach to quantization impractical [207].”

• cites [207] Moncrief, V., “How solvable is (2+1)-dimensional Einstein gravity?”, J. Math. Phys., 31, 2978–2982, (1990). sourceNot checked — no full text of the source

For Λ < 0, a complete, in fact overcomplete, set of observables is given by the traces of the holonomies.

“A complete — in fact, overcomplete — set of observables is given by the traces (47) of the holonomies”

The model allows computation of transition amplitudes between the spatial geometry of a time slice of constant mean curvature T1 and a later slice of constant mean curvature T2.

“The model naturally allows us to compute the transition amplitude between the spatial geometry of a time slice of constant mean curvature − tr K = T1 and the geometry of a later slice of constant mean curvature − tr K = T2.”

Ponzano and Regge observed that the Regge action in 2+1 dimensions can be re-expressed in terms of Wigner-Racah 6j-symbols.

“The first hint that (2 + 1)-dimensional gravity might be exceptional came from the observation by Ponzano and Regge [225] that the Regge action in 2 + 1 dimensions can be re-expressed in terms of Wigner-Racah 6j-symbols.”

• cites [225] Ponzano, G., and Regge, T., in Bloch, F. et al., ed., Spectroscopic and group theoretical methods in physics: Racah memorial volume, (North-Holland, Amsterdam, Netherlands, 1968).Not checked yet

  Not checked in the free run.

The timelike sector is not physically relevant because it does not correspond to a geometric structure with spacelike T2 slices.

“Unfortunately, though, this is not the physically relevant sector: It does not correspond to a geometric structure with spacelike T2 slices.”

The simplest nontrivial vacuum cosmology occurs for a spacetime with topology R × T2.

“The simplest nontrivial vacuum cosmology occurs for a spacetime with the topology ℝ × T2, where T2 is the two-dimensional torus.”

In the first order formalism, Dirac quantization is equivalent to the Chern-Simons quantum theory.

“In the first order formalism, it is straightforward to show that Dirac quantization is equivalent to the Chern-Simons quantum theory we have already seen. Details can be found in Chapter 8 of [81]”

• cites [81] Carlip, S., Quantum Gravity in 2+1 Dimensions, Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1998). sourceNot checked — no full text of the source

Lorentzian path integrals do not directly address the problem of describing the birth of a universe from nothing.

“They do not, however, directly address a principle issue of quantum cosmology, the problem of describing the birth of a universe from “nothing”.”

When M admits more than one flat SO(2,1) connection, Equation (74) must be integrated over the moduli space of such connections.

“When M admits more than one such flat connection, Equation (74) must be integrated over the moduli space of such connections.”

In the discrete path integral, integrating over the Lie algebra elements eα produces a delta function δ(gα) for each edge, which translates to the geometric statement that the constraints require the connection to be flat and have trivial holonomy around a contractible curve surrounding an edge.

“As in the continuum path integral of Section 3.10, the integral over the eα produces a delta function δ(gα) for each edge. This translates back to the geometric statement that the constraints require the connection w to be flat, and thus to have trivial holonomy around a contractible curve surrounding an edge.”

The integral over the moduli space of flat connections sometimes diverges.

“This integral sometimes diverges [279]; the significance of that divergence is not understood.”

• cites [279] Witten, E., “Topology-changing amplitudes in 2+1 dimensional gravity”, Nucl. Phys. B, 323, 113–140, (1989). sourceNot checked — no full text of the source

The leading contribution to Equation (75) is a sum of terms of the form of Equation (76).

“An easy computation shows that the leading contribution to Equation (75) is a sum of terms of the form 76”

• AssumptionThe authors state this as an assumption: The conformal factor problem is solved, so that a saddle point evaluation of the path integral is a good approximation.

Puzio showed that a wave packet initially concentrated away from the singular points in moduli space will remain non-singular.

“Puzio, for example, has shown that a wave packet initially concentrated away from the singular points in moduli space will remain non-singular [227].”

• cites [227] Puzio, R.S., “On the square root of the Laplace-Beltrami operator as a Hamiltonian”, Class. Quantum Grav., 11, 609–620, (1994). sourceNot checked — no full text of the source

The path integral (75) cannot in general be evaluated exactly, even in 2+1 dimensions.

“The path integral (75) cannot, in general, be evaluated exactly, even in 2 + 1 dimensions.”

The path integral for the covariant first-order action (14) reduces to a ratio of determinants that has a topological interpretation as the analytic or Ray-Singer torsion.

“The path integral for this action was first fully analyzed in two seminal papers by Witten [277, 279], who showed that it reduced to a ratio of determinants that has an elegant topological interpretation as the analytic or Ray-Singer torsion [230].”

• cites [277] Witten, E., “2+1 dimensional gravity as an exactly soluble system”, Nucl. Phys. B, 311, 46–78, (1988). sourceNot checked — no full text of the source
• cites [279] Witten, E., “Topology-changing amplitudes in 2+1 dimensional gravity”, Nucl. Phys. B, 323, 113–140, (1989). sourceNot checked — no full text of the source
• cites [230] Ray, D.B., and Singer, I.M., “R-torsion and the Laplacian on Riemannian manifolds”, Adv. Math., 7, 145–210, (1971). sourceNot checked — no full text of the source

The path integral can give transition amplitudes between states on surfaces with different topologies.

“But the path integral can also give transition amplitudes between states on surfaces Σi and Σf with different topologies.”

General relativity can be quantized in 2+1 dimensions.

“Perhaps the most important lesson of (2 + 1)-dimensional quantum gravity is that general relativity can, in fact, be quantized.”

The partition function for a closed three-manifold with Λ = 0 is given by Equation (74).

“The partition function for a closed three-manifold with Λ = 0 takes the form 74”

In Dirac quantization, one is not allowed to choose a nice time-slicing such as York time because that would be a form of gauge-fixing.

“The first difficulty is that we are no longer allowed to choose a nice time-slicing such as York time; that would be a form of gauge-fixing, and is not permitted in Dirac quantization.”

Equation (74) can be extended to manifolds with boundary in a straightforward manner.

“Although it was originally derived for closed manifolds, Equation (74) can be extended to manifolds with boundary in a straightforward manner.”

For a manifold with topology R × Σ, the path integral results agree with covariant canonical quantization: the transition amplitude between two surfaces with prescribed spin connections is nonzero only if the holonomies agree.

“For a manifold with the topology ℝ × Σ, the results agree with those of covariant canonical quantization: The transition amplitude between two surfaces with prescribed spin connections is nonzero only if the holonomies agree.”

Frozen time quantum gravity is a Heisenberg picture corresponding to a fixed-time-slicing Schrödinger picture.

“The idea that “frozen time” quantum gravity is a Heisenberg picture corresponding to a fixed-time-slicing Schrödinger picture is a central insight of (2 + 1)-dimensional gravity.”

The phase space path integral for the first order formulation similarly reproduces the corresponding canonically quantized theory.

“The phase space path integral for the first order formulation similarly reproduces the corresponding canonically quantized theory.”