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

In the paper

“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].”

What the sources say

• ?Could 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)

  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. sourceExperimental — accuracy not yet measured
• Supported

  In §3, the paper identifies the S subsector by its boost-type holonomies, corresponding to the hyperbolic sector. In §7 (p. 17), it states that all eight sectors other than M(S,S) have timelike T² slices. This supports the claim for the SO(2,2)₀ gauge theory discussed there. (LLM, conf=0.91)

  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. source
• Disputed

  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.

  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. sourceExperimental — accuracy not yet measured
• ?Could 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• Disputed

  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.

  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. sourceExperimental — accuracy not yet measured
• Disputed

  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.

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• Disputed

  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.

  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. sourceExperimental — accuracy not yet measured
• Supported

  The Introduction says that the low-energy limit of quantum gravity in 2+1 dimensions is characterized by excitations transforming under a deformed Poincaré algebra. Section 2.2 states that this limit must have κ-Poincaré symmetry, while noting that the precise physical form of the algebra requires additional input. (LLM, conf=0.92)

  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. source
• Supported

  The paper states that the symmetry algebra of 2+1-dimensional quantum gravity with vanishing cosmological constant is κ-deformed Poincaré rather than ordinary Poincaré. It also identifies 2+1 gravity coupled to point particles as an example of doubly special relativity. These passages support the citing sentence’s paraphrase in the setting the paper analyzes. (LLM, conf=0.90)

  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. source

In the paper

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

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured
• Disputed

  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.

  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. sourceExperimental — accuracy not yet measured

In the paper

“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.”

What the sources say

• ?Not checked — no full text of the sourcecites [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). source
• Partly 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)

  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). sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [250] ’t Hooft, G., “Causality in (2+1)-dimensional gravity”, Class. Quantum Grav., 9, 1335–1348, (1992). source
• ?Not checked — no full text of the sourcecites [252] ’t Hooft, G., “Classical N-particle cosmology in 2+1 dimensions”, Class. Quantum Grav., 10, S79–S91, (1993). source
• ?Not checked — no full text of the sourcecites [253] ’t Hooft, G., “The evolution of gravitating point particles in 2+1 dimensions”, Class. Quantum Grav., 10, 1023–1038, (1993). source
• Partly 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)

  cites [124] Franzosi, R., and Guadagnini, E., “Topology and classical geometry in (2+1) gravity”, Class. Quantum Grav., 13, 433–460, (1996). sourceExperimental — accuracy not yet measured
• Partly 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)

  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. sourceExperimental — accuracy not yet measured
• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [107] Deser, S., Jackiw, R., and ’t Hooft, G., “Three dimensional Einstein gravity: Dynamics of at space”, Ann. Phys. (N.Y.), 152, 220, (1984). source
• Supported

  The Introduction states that (2+1)-dimensional general relativity does not reduce to two-dimensional Newtonian gravity in the weak-field limit. The Conclusions likewise report the nonexistence of a Newtonian weak-field limit, directly supporting the paraphrase. (LLM, conf=0.95)

  cites [49] Barrow, J.D., Burd, A.B., and Lancaster, D., “Three-dimensional classical spacetimes”, Class. Quantum Grav., 3, 551–567, (1986). source
• ?Not checked — no full text of the sourcecites [94] Cornish, N.J., and Frankel, N.E., “Gravitation in 2+1 dimensions”, Phys. Rev. D, 43, 2555–2565, (1991). source

In the paper

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

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• Supported

  The paper identifies its setting as 2+1 gravity on ℝ × T² (p. 1). It then gives a concrete example of the ambiguity for positive cosmological constant: infinitely many nondiffeomorphic spacetimes have the same SO(3,1) holonomy (p. 3). (LLM, conf=0.94)

  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. source

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [65] Carlip, S., “Exact quantum scattering in 2+1 dimensional gravity”, Nucl. Phys. B, 324, 106–122, (1989). source
• Partly 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)

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

In the paper

“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].”

What the sources say

• Supported

  The Outline (p. not available) says that the first-order Einstein–Hilbert action matches the Chern–Simons action, but Einstein gravity requires an invertible dreibein whereas Chern–Simons theory does not. Section 2 explains that dropping this restriction admits singular metrics, supporting the claimed loss of equivalence with the metric description in the paper’s 2+1-dimensional setting. (LLM, conf=0.88)

  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. source

In the paper

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

What the sources say

• Supported

  The Conclusions state that Loll presents a complete and independent set of Wilson loop variables for 2+1 gravity on compact spatial manifolds of arbitrary genus. The paper also says these invariants form complete, global parameters on Teichmüller space, supporting the description of them as configuration-space variables. (LLM, conf=0.94)

  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. source

In the paper

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

What the sources say

• Supported

  The Discussion (section 4) says that ’t Hooft’s polygon representation can be derived from the covariant lattice formulation by imposing gauge conditions. The concluding passage (p. 1762) says the formalisms represent the same classical solutions, while distinguishing their quantum theories. (LLM, conf=0.92)

  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. source

In the paper

“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]”

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

“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.”

What the sources say

• Supported

  Section 3.1 (p. 12) says the brackets induce the correct brackets for the moduli and momenta defined in equations (3.3)–(3.4). It also states that the Hamiltonian generates their classical time development through Hamilton’s equations. This supports the attributed claim; equation (42) is the citing paper’s numbering, not the source’s. (LLM, conf=0.91)

  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. source
• Partly 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)

  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). sourceExperimental — accuracy not yet measured

In the paper

“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].”

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

“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].”

What the sources say

• Supported

  Section 4.5 (p. 15) states that a flat spacetime is completely determined by its initial singularity and the linear part of its holonomy, represented by a surface F. It then identifies the initial singularity and F as the past and future asymptotic states of the cosmological-time level surfaces. The paper frames this result for maximal globally hyperbolic flat (2+1)-spacetimes with compact spatial surfaces of genus greater than one. (LLM, conf=0.95)

  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. source

In the paper

“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],”

What the sources say

• ?Not checked — no full text of the sourcecites [228] Rama, S.K., and Sen, S., “3-D manifolds, graph invariants, and Chern-Simons theory”, Mod. Phys. Lett. A, 7, 2065–2076, (1992). source
• ?Not matched

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

  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).
• Partly 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)

  cites [233] Roberts, J., “Skein theory and Turaev-Viro invariants”, Topology, 34, 771–788, (1995). sourceExperimental — accuracy not yet measured

In the paper

“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].”

What the sources say

• ?Not checked — no full text of the sourcecites [26] Ashtekar, A., Lectures on Non-Perturbative Canonical Gravity, vol. 6 of Advanced Series in Astrophysics and Cosmology, (World Scientific, Singapore, 1991). source
• Supported

  In “An example: the 2-torus” (p. L191), the paper explicitly describes loop-representation states as complex-valued functions of two integer homotopy labels, gives their symmetry condition and inner product, and specifies the basic operators. Its outline (p. L186) also says it obtains both representations explicitly for the torus. (LLM, conf=0.92)

  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). source

In the paper

“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.”

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

“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).”

What the sources say

• ?Not checked — no full text of the sourcecites [26] Ashtekar, A., Lectures on Non-Perturbative Canonical Gravity, vol. 6 of Advanced Series in Astrophysics and Cosmology, (World Scientific, Singapore, 1991). source
• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [26] Ashtekar, A., Lectures on Non-Perturbative Canonical Gravity, vol. 6 of Advanced Series in Astrophysics and Cosmology, (World Scientific, Singapore, 1991). source
• Supported

  In “An example: the 2-torus,” the paper says the torus loop transform can be explicitly evaluated and gives it as an integral (equation 12). Its introductory discussion (p. L186) also describes the torus transform as essentially a Fourier transform. (LLM, conf=0.94)

  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). source
• Supported

  Section 3, “The loop transform in the spacelike sector” (page unavailable), introduces the transform’s explicit form as an integral. Section 5 identifies it as a two-sided Laplace transform. The paper notes that the integral is not defined for every function, but that does not conflict with the claim. (LLM, conf=0.91)

  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. source

In the paper

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

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• ?Not matched

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

  cites [200] Mess, G., “Lorentz Spacetimes of Constant Curvature”, unknown status, (1990). Institut des Hautes Etudes Scientifiques preprint IHES/M/90/28.
• Supported

  For the higher-genus spacetimes studied in the paper, Proposition 4.4 (§4.1, p. 9) describes the initial singularity as a real tree rather than a spatial surface. Remark 4.28 (§4.5, p. 18) explains that as cosmological time approaches zero, parts of the spatial surfaces contribute negligibly to curve lengths; it says the result extends from simplicial examples to the general case. (LLM, conf=0.88)

  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. source

In the paper

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

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• Supported

  Section 4, “The evolution of torus,” states that the torus moduli trace a semicircle and that this semicircle is a geodesic for the standard Poincaré metric. The Figure 5 caption repeats that identification. (LLM, conf=0.94)

  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). source
• Supported

  In the Teichmüller-space discussion (§5; page unavailable), the paper says the torus moduli space has a Weil–Petersson metric coinciding with the Poincaré metric. It identifies the trajectory for the twisting-torus case, depicted in Fig. 4, as a geodesic. Although the text does not use the word “semicircular,” this supports the cited characterization of the curved trajectory. (LLM, conf=0.87)

  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). source

In the paper

“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.”

What the sources say

• Partly supported

  Section 3.2.4 (page unavailable) describes a projection onto physical states and the simplification of graphs by eliminating irreducible loops. But it says the resulting basis consists of spin network states, not loop states; the paper addresses three-dimensional gravity rather than Euclidean quantum gravity generally. (LLM, conf=0.88)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• ?Could 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)

  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. sourceExperimental — accuracy not yet measured
• ?Could 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• Partly 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)

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

In the paper

“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.”

What the sources say

• ?Not checked — no full text of the sourcecites [175] Lee, J., and Wald, R.M., “Local symmetries and constraints”, J. Math. Phys., 31, 725–473, (1990). source
• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [278] Witten, E., “Quantum field theory and the Jones polynomial”, Commun. Math. Phys., 121, 351–399, (1989). source
• ?Not checked — no full text of the sourcecites [243] Schwarz, A.S., “The partition function of degenerate quadratic functional and Ray-Singer invariants”, Lett. Math. Phys., 2, 247–252, (1978). source
• ?Not checked — no full text of the sourcecites [244] Schwarz, A.S., “The partition function of a degenerate functional”, Commun. Math. Phys., 67, 1–16, (1979). source
• ?Not checked — no full text of the sourcecites [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). source
• Supported

  The Introduction describes a rigorous construction of aspects of Hamiltonian Chern–Simons quantization. Section 2 formulates the gauge-theory problem for a compact Lie group, while the Canonical quantization section describes related constructions as relatively well understood and notes other rigorous approaches. Together, these support the citing sentence as a broad background statement. (LLM, conf=0.86)

  cites [34] Axelrod, S.E., DellaPietra, S., and Witten, E., “Geometric quantization of Chern-Simons gauge theory”, J. Differ. Geom., 33, 787–902, (1991). source

In the paper

“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].”

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured
• Supported

  The paper defines H(k) as the Turaev–Viro space of physical states (p. 7) and states that H(k = ∞) is isomorphic to the space of gauge-invariant functions over the moduli space of flat SU(2) connections (p. 7). It calls the limiting space the Ponzano–Regge space of physical states. (LLM, conf=0.91)

  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. source
• Partly 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)

  cites [242] Sasakura, N., “Exact three-dimensional lattice gravities”, Prog. Theor. Phys. Suppl., 110, 191–206, (1992). sourceExperimental — accuracy not yet measured

In the paper

“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.”

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• Supported

  Section 4 (page not supplied) says that for a negative cosmological constant, high-volume manifolds are exponentially suppressed, but the number of contributing manifolds can overcome that suppression. It also reports a divergent sum over topologies whose divergences may come from manifolds with fairly large volumes. (LLM, conf=0.91)

  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. source

In the paper

“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].”

What the sources say

• ?Could 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)

  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. sourceExperimental — accuracy not yet measured
• ?Could 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)

  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. sourceExperimental — accuracy not yet measured
• Supported

  Section I (p. 2) identifies the rotationally symmetric model as equivalent to an Einstein–Rosen-wave midi-superspace. Section I (pp. 3–4) says its total Hamiltonian is bounded above and describes it as a bounded, non-polynomial functional in the classical theory. (LLM, conf=0.93)

  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. source
• Supported

  Section 2.1 defines the model as axi-symmetric 2+1-dimensional gravity coupled to a scalar field. Section 4.2, “Hamiltonian and Time” (p. 15), explicitly finds that the physical Hamiltonian has spectrum [0, 1/4G] and remains bounded even as the corresponding free-field energy grows without bound. (LLM, conf=0.94)

  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. source
• ?Could 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)

  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. sourceExperimental — accuracy not yet measured
• ?Could 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)

  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. sourceExperimental — accuracy not yet measured
• Partly 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)

  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. sourceExperimental — accuracy not yet measured
• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

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

What the sources say

• Supported

  In “Canonical gravity in 2 + 1 dimensions,” Carlip distinguishes solving the constraints before quantization from imposing them on quantum states. The discussion of equation (2.6) says that imposing the constraints yields a complicated, non-local Wheeler–DeWitt equation; section 3 adds that explicit solutions are difficult even in this simple model. This supports the citing sentence’s characterization of the added complexity. (LLM, conf=0.88)

  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. source

In the paper

“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].”

What the sources say

• Supported

  The Discussions section (p. 19) states that in a Lorentzian cobordism the initial and final spaces have equal Euler characteristics. It defines those spaces as spatial boundaries of a manifold with a nondegenerate Lorentzian metric, and says the paper's selection rule reproduces this result for boundary components of genus at least two. (LLM, conf=0.90)

  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. source

In the paper

“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].”

What the sources say

• ?Not checked — no full text of the sourcecites [283] Wu, S., “Topological quantum field theories on manifolds with a boundary”, Commun. Math. Phys., 136, 157–168, (1991). source
• Supported

  Section 3, Eq. (3.1), defines a path integral Z_M[I*ω] that depends on the induced boundary connection. Section 2 (p. 3 and following) specifies the additional conditions: the normal component of ω is integrated over rather than fixed, while the normal component of the triad is fixed at the boundary. (LLM, conf=0.93)

  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. source

In the paper

“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.”

What the sources say

• Partly 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)

  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. sourceExperimental — accuracy not yet measured

In the paper

“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.”

What the sources say

• Supported

  Section 3.2 (p. 13) states that the selection rule requires the Euler characteristics of the boundary surfaces to sum to zero with assigned signs. For a process with a single spatial boundary Σ, this requires χ(Σ)=0; among the closed orientable surfaces considered in the paper, that means a torus. The Discussion (p. 19) says the rule also holds for Lorentzian cobordism. (LLM, conf=0.86)

  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. source

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

“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)”

In the paper

“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.”

In the paper

“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 the paper

“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 the paper

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

In the paper

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

In the paper

“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.”

In the paper

“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).”

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [278] Witten, E., “Quantum field theory and the Jones polynomial”, Commun. Math. Phys., 121, 351–399, (1989). source

In the paper

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

In the paper

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

In the paper

“then gives a representation of the loop algebra.”

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [211] Nelson, J.E., and Regge, T., “Homotopy groups and (2+1)-dimensional quantum de Sitter gravity”, Nucl. Phys. B, 339, 516–532, (1990). source

In the paper

“The initial spatial geometry is again degenerate.”

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

What the sources say

• ?Not checked — no full text of the sourcecites [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). source
• ?Not checked — no full text of the sourcecites [277] Witten, E., “2+1 dimensional gravity as an exactly soluble system”, Nucl. Phys. B, 311, 46–78, (1988). source
• ?Not checked — no full text of the sourcecites [279] Witten, E., “Topology-changing amplitudes in 2+1 dimensional gravity”, Nucl. Phys. B, 323, 113–140, (1989). source

In the paper

“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.”

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [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/.

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [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/.

In the paper

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

In the paper

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

What the sources say

• ?Not matched

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

  cites [200] Mess, G., “Lorentz Spacetimes of Constant Curvature”, unknown status, (1990). Institut des Hautes Etudes Scientifiques preprint IHES/M/90/28.

In the paper

“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.”

In the paper

“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]”

What the sources say

• ?Not checked — no full text of the sourcecites [213] Nelson, J.E., and Regge, T., “2+1 quantum gravity”, Phys. Lett. B, 272, 213–216, (1991). source
• ?Not checked — no full text of the sourcecites [212] Nelson, J.E., and Regge, T., “(2+1) gravity for genus ¿ 1”, Commun. Math. Phys., 141, 211–223, (1991). source

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [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 yetcites [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 — no full text of the sourcecites [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). source

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [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). source

In the paper

“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.”

What the sources say

• ?Not checked — no full text of the sourcecites [277] Witten, E., “2+1 dimensional gravity as an exactly soluble system”, Nucl. Phys. B, 311, 46–78, (1988). source
• ?Not checked — no full text of the sourcecites [279] Witten, E., “Topology-changing amplitudes in 2+1 dimensional gravity”, Nucl. Phys. B, 323, 113–140, (1989). source

In the paper

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

In the paper

“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.”

What the sources say

• ?Not matched

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

  cites [200] Mess, G., “Lorentz Spacetimes of Constant Curvature”, unknown status, (1990). Institut des Hautes Etudes Scientifiques preprint IHES/M/90/28.

In the paper

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

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

What the sources say

• ?Not checked — no full text of the sourcecites [262] Turaev, V.G., and Viro, O.Y., “State Sum Invariants of 3-Manifolds and Quantum 6j-Symbols”, Topology, 31, 865–902, (1992). source

In the paper

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

In the paper

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

In the paper

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

In the paper

“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”

What the sources say

• ?Not checked — no full text of the sourcecites [26] Ashtekar, A., Lectures on Non-Perturbative Canonical Gravity, vol. 6 of Advanced Series in Astrophysics and Cosmology, (World Scientific, Singapore, 1991). source

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

“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]).”

What the sources say

• ?Not checked — no full text of the sourcecites [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). source

In the paper

“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).”

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

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

In the paper

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

What the sources say

• Not checked yetcites [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).

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

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

What the sources say

• ?Not matched

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

  cites [200] Mess, G., “Lorentz Spacetimes of Constant Curvature”, unknown status, (1990). Institut des Hautes Etudes Scientifiques preprint IHES/M/90/28.

In the paper

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

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [207] Moncrief, V., “How solvable is (2+1)-dimensional Einstein gravity?”, J. Math. Phys., 31, 2978–2982, (1990). source

In the paper

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

In the paper

“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.”

In the paper

“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.”

What the sources say

• ?Not matched

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

  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).

In the paper

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

In the paper

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

In the paper

“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]”

What the sources say

• ?Not checked — no full text of the sourcecites [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). source

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [279] Witten, E., “Topology-changing amplitudes in 2+1 dimensional gravity”, Nucl. Phys. B, 323, 113–140, (1989). source

In the paper

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

In the paper

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

What the sources say

• ?Not checked — no full text of the sourcecites [227] Puzio, R.S., “On the square root of the Laplace-Beltrami operator as a Hamiltonian”, Class. Quantum Grav., 11, 609–620, (1994). source

In the paper

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

In the paper

“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].”

What the sources say

• ?Not checked — no full text of the sourcecites [277] Witten, E., “2+1 dimensional gravity as an exactly soluble system”, Nucl. Phys. B, 311, 46–78, (1988). source
• ?Not checked — no full text of the sourcecites [279] Witten, E., “Topology-changing amplitudes in 2+1 dimensional gravity”, Nucl. Phys. B, 323, 113–140, (1989). source
• ?Not checked — no full text of the sourcecites [230] Ray, D.B., and Singer, I.M., “R-torsion and the Laplacian on Riemannian manifolds”, Adv. Math., 7, 145–210, (1971). source

In the paper

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

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

“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.”

In the paper

“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.”

In the paper

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