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

In the paper

“On the other hand, Minassian has recently demonstrated [202] that quantum fluctuations do not “push singularities off to infinity” (as suggested in [158]), and that several classically singular (2 + 1)-dimensional quantum spacetimes also have singular “quantum b-boundaries”.”

What the sources say

• Supported

  Section 4 says that paths in the quantized BTZ spacetime reach a classically singular region within a finite QGAP, rather than being pushed to infinity. Sections 2.2 and 8 explain that finite quantum b-length indicates b-incompleteness and report persistent singularities in two distinct (2+1)-dimensional models. The conclusion is limited to the models and paths studied. (LLM, conf=0.90)

  cites [202] Minassian, E.A., “Spacetime Singularities in (2+1)-Dimensional Quantum Gravity”, Class. Quantum Grav., 19, 5877–5901, (2002). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0203026. source
• Disputed

  Section 5.2 (A Gaussian wavepacket; page not supplied) finds that the quantum generalized affine length diverges in its (2+1)-dimensional black-hole example, even for arbitrarily small mass fluctuations. Section 6 treats infinite length as smearing out the classical singularity—the opposite of the attributed finding that fluctuations do not push it away. The paper does not establish the claim about several singular quantum b-boundaries. (LLM, conf=0.89)

  Checked by a reasoning model — may be wrong.

  cites [158] Hosoya, A., “Quantum smearing of spacetime singularity”, Class. Quantum Grav., 12, 2967–2975, (1995). sourceExperimental — accuracy not yet measured

In the paper

“For example, there is evidence that the matter Hamiltonian is bounded above in (2 + 1)-dimensional gravity [27]; perhaps this could cut off radiative contributions to the cosmological constant at an interesting scale.”

What the sources say

• Disputed

  In section I (p. 3), the paper says the bounded quantity is the total Hamiltonian of the system, explicitly including gravity. Section I (p. 4) distinguishes that Hamiltonian from H₀, the scalar-field energy, so the cited result does not establish an upper bound on the matter Hamiltonian. (LLM, conf=0.84)

  Checked by a reasoning model — may be wrong.

  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. sourceExperimental — accuracy not yet measured

In the paper

“If a negative cosmological constant is present, black hole solutions can be found [41, 40].”

What the sources say

• Supported

  The opening passage (p. 1) states that the Einstein-Maxwell equations in 2+1 spacetime dimensions with a negative cosmological constant admit a black hole solution. This directly supports the claim. (LLM, conf=0.95)

  cites [41] Bañados, M., Teitelboim, C., and Zanelli, J., “The Black Hole in Three Dimensional Space Time”, Phys. Rev. Lett., 69, 1849–1851, (1992). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9204099. source
• Supported

  The Abstract describes black holes in 2+1-dimensional standard Einstein gravity with a negative cosmological constant. Section 2.5 identifies the solution’s black-hole horizon and relates its length scale to the negative cosmological constant. (LLM, conf=0.94)

  cites [40] Bañados, M., Henneaux, M., Teitelboim, C., and Zanelli, J., “Geometry of the 2+1 Black Hole”, Phys. Rev. D, 48, 1506–1525, (1993). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9302012. source

In the paper

“This approach has been used mainly to understand point particle dynamics, but recent progress has allowed a general description of topologically nontrivial compact spaces [169].”

What the sources say

• Partly supported

  The Introduction (pp. 1–2) says the paper develops explicit initial data for compact higher-genus spaces, supporting the claim of progress beyond particle dynamics. But it does not establish that the approach had been used mainly for point particles, and it conjectures—rather than proves—that its higher-genus solution space is complete. The Discussion (§5) also says including particles remains a next step for this work. (LLM, conf=0.84)

  cites [169] Kádár, Z., and Loll, R., “(2+1) gravity for higher genus in the polygon model”, Class. Quantum Grav., 21, 2465–2491, (2004). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0312043. sourceExperimental — accuracy not yet measured

In the paper

“It is also interesting that the symplectic structure obtained in this way is closely related to the symplectic structure on the abstract space of loops on Σ first discovered by Goldman [145, 146].”

What the sources say

• Partly supported

  The opening discussion describes Goldman's construction of a symplectic structure on spaces of surface-group representations. It mentions a related Lie algebra on spaces based on closed curves, but explicitly assigns that work to a forthcoming paper. This copy therefore supports a connection to loops, not the claim that this paper discovered a symplectic structure on the abstract space of loops. (LLM, conf=0.84)

  cites [145] Goldman, W.M., “The symplectic nature of fundamental groups of surfaces”, Adv. Math., 54, 200–225, (1984). sourceExperimental — accuracy not yet measured
• ?Not checked — no full text of the sourcecites [146] Goldman, W.M., “Invariant functions on Lie groups and Hamiltonian flows of surface group representations”, Invent. Math., 85, 263–302, (1986). source

In the paper

“For such solutions, dynamics at either the horizon or the boundary at infinity can lead to local degrees of freedom [78, 247, 59, 96, 51, 24, 91]”

What the sources say

• Partly supported

  The Introduction (p. 2) says that a boundary can make would-be gauge degrees of freedom dynamical, and applies this mechanism to the black-hole horizon. The Boundary Action (p. 5) describes dynamical fields at that horizon. Our copy does not establish the other location named in the claim—a boundary at infinity. (LLM, conf=0.86)

  cites [78] Carlip, S., “The Statistical Mechanics of the (2+1)-Dimensional Black Hole”, Phys. Rev. D, 51, 632–637, (1995). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9409052. sourceExperimental — accuracy not yet measured
• Partly supported

  Section 6, “Relation to previous derivations” (page unavailable), describes boundary degrees of freedom at the black-hole horizon and a conformal field theory at a boundary at infinity. But it explicitly says pure 2+1 gravity has no local degrees of freedom. The claim is supported for boundary degrees of freedom, not for unqualified local degrees of freedom. (LLM, conf=0.85)

  cites [247] Strominger, A., “Black Hole Entropy from Near-Horizon Microstates”, J. High Energy Phys., 02, 009, (1998). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9712251. sourceExperimental — accuracy not yet measured
• Partly supported

  The Introduction (p. 2) describes asymptotic Virasoro symmetries of three-dimensional anti-de Sitter gravity, and §3 (p. 5) uses their generators to count states. This supports a boundary-related part of the claim, but the paper does not say that horizon dynamics can produce local degrees of freedom, nor does it explicitly characterize the asymptotic states as local degrees of freedom. (LLM, conf=0.79)

  cites [59] Birmingham, D., Sachs, I., and Sen, S., “Entropy of Three-Dimensional Black Holes in String Theory”, Phys. Lett. B, 424, 275–280, (1998). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9801019. sourceExperimental — accuracy not yet measured
• Partly supported

  Section III (p. 6) establishes that dynamics at infinity is described by Liouville theory. Section II (p. 4) says three-dimensional black holes can have inner boundaries with their own surface dynamics, but the authors explicitly exclude horizons from their analysis; the paper does not establish local degrees of freedom at a black-hole horizon. (LLM, conf=0.84)

  cites [96] Coussaert, O., Henneaux, M., and van Driel, P., “The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant”, Class. Quantum Grav., 12, 2961–2966, (1995). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9506019. sourceExperimental — accuracy not yet measured
• Partly supported

  The paper identifies local boundary degrees of freedom for 2+1-dimensional AdS black holes: on p. 11, it relates the Fefferman–Graham field locally to a Liouville field and connects its components to BTZ black-hole mass and angular momentum. It does not establish the other attributed possibility, that dynamics at the horizon can lead to local degrees of freedom. (LLM, conf=0.87)

  cites [51] Bautier, K., Englert, F., Rooman, M., and Spindel, P., “The Fefferman-Graham Ambiguity and AdS Black Holes”, Phys. Lett. B, 479, 291–298, (2000). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0002156. sourceExperimental — accuracy not yet measured
• Partly supported

  The Introduction and discussion describes boundary dynamics in three-dimensional gravity, including a boundary conformal field theory arising from bulk gravitational dynamics under asymptotic boundary conditions. That supports the boundary-at-infinity part of the claim, but the passages do not establish the corresponding claim about black-hole horizons. (LLM, conf=0.82)

  cites [24] Arcioni, G., Blau, M., and O’Loughlin, M., “On the Boundary Dynamics of Chern-Simons Gravity”, J. High Energy Phys., 01, 067, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0210089. sourceExperimental — accuracy not yet measured
• Supported

  The Introduction (pp. 1–2) describes asymptotic dynamics as a boundary conformal field theory whose states may account for BTZ black-hole entropy. The Conclusion (pp. 29–30) also suggests that Liouville theory may arise near a black-hole horizon and illuminate its gravitational degrees of freedom. The claim’s tentative “can lead” is consistent with that qualification. (LLM, conf=0.84)

  cites [91] Chen, Y.-J., “Quantum Liouville theory and BTZ black hole entropy”, Class. Quantum Grav., 21, 1153–1180, (2004). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0310234. source

In the paper

“although these are certainly not yet completely understood [82].”

What the sources say

• Supported

  The Abstract says there is no consensus about which fields provide the relevant degrees of freedom in the (2+1)-dimensional BTZ black hole or where those excitations live. Section 6 likewise says the entropy calculation does not establish what the states are. These passages support the claim that the relevant degrees of freedom are not yet completely understood. (LLM, conf=0.88)

  cites [82] Carlip, S., “What We Don’t Know about BTZ Black Hole Entropy”, Class. Quantum Grav., 15, 3609–3625, (1998). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9806026. source

In the paper

“But despite this enormous effort, no one has yet succeeded in formulating a complete, self-consistent quantum theory of gravity [83].”

What the sources say

• Supported

  The Introduction states that, despite decades of research, no one had formulated a consistent and complete quantum theory of gravity. This directly supports the cited sentence as a statement about the field at the time of the source paper. (LLM, conf=0.95)

  cites [83] Carlip, S., “Quantum gravity: a Progress Report”, Rep. Prog. Phys., 64, 885–942, (2001). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0108040. source

In the paper

“several authors have further explored the quantum group structure of these brackets, which can be expressed in terms of the quantum double of the Lorentz group [37, 36, 61, 201].”

What the sources say

• Supported

  The Introduction (p. 2) describes the phase-space Poisson structure through a classical r-matrix and says the Lorentz double’s R-matrix has the corresponding classical limit. Section 5.1 (p. 23) identifies the Lorentz double as the quantum double of the Lorentz group’s universal cover. This supports the attributed quantum-group description, although the paper says it does not fully carry out the quantisation programme. (LLM, conf=0.87)

  cites [37] Bais, F.A., Muller, N.M., and Schroers, B.J., “Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity”, Nucl. Phys. B, 640, 3–45, (2002). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0205021. source
• Partly supported

  Section 9 (pp. 39, 42) identifies D(SO(2,1)), the quantum double of the 2+1-dimensional Lorentz group, as the symmetry governing the quantum theory of 2+1 gravity. The cited paper does not establish that the specific brackets in the citing sentence have that structure; its detailed treatment of the quantum double concerns SU(2). (LLM, conf=0.82)

  cites [36] Bais, F.A., and Muller, N.M., “Topological field theory and the quantum double of SU(2)”, Nucl. Phys. B, 530, 349–400, (1998). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9804130. sourceExperimental — accuracy not yet measured
• Partly supported

  In the ‘For any representation’ section (page unavailable), the paper connects the Poisson brackets to the quantum group Uq(sl(2,C)R) and expresses that group as a quantum double. But it specifies the double of Uq(su(2)), not the double of the Lorentz group. (LLM, conf=0.88)

  cites [61] Buffenoir, E., Noui, K., and Roche, P., “Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group”, Class. Quantum Grav., 19, 4953–5015, (2002). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0202121. sourceExperimental — accuracy not yet measured
• Supported

  The Conclusion and outlook identifies the quantum double of the Lorentz group as the quantum group relevant to the Poincaré group and the r-matrix used to derive this paper’s Poisson brackets (section 4.4). It reports that identification from earlier work, which supports the citing sentence’s paraphrase. (LLM, conf=0.91)

  cites [201] Meusburger, C., and Schroers, B.J., “Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity”, Class. Quantum Grav., 20, 2193–2234, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0301108. source

In the paper

“A similar ambiguity occurs for (2 + 1)-dimensional gravity with point particles, where, as Matschull has emphasized [194], it may imply a physical difference between the metric and Chern-Simons formulations of (2 + 1)-dimensional gravity.”

What the sources say

• Supported

  Section 2 explains that translations treated as gauge symmetries in Chern-Simons theory need not be gauge symmetries in Einstein gravity. The example on p. 2607 involves particles and shows two states that are gauge-equivalent in Chern-Simons theory but represent non-diffeomorphic spacetimes in Einstein gravity. This supports the claimed physical difference between the formulations. (LLM, conf=0.93)

  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

“Classical Regge calculus in 2 + 1 dimensions was investigated by Rocek and Williams [235], who showed that it gave exact results for point particle scattering.”

What the sources say

• Partly supported

  The discussion on p. 703 uses Regge calculus in (2+1) dimensions to derive results for gravitating point particles, including a collision in which two particles coalesce. It does not establish exact results for point-particle scattering generally; that wording is broader than the collision result shown. (LLM, conf=0.84)

  cites [235] Rocek, M., and Williams, R.M., “Three-dimensional Einstein gravity and Regge calculus”, Class. Quantum Grav., 2, 701–706, (1985). sourceExperimental — accuracy not yet measured

In the paper

“This mystery may have recently been solved by Freidel and Louapre [131], who show that a residual piece of the diffeomorphism symmetry has not been factored out of the Ponzano-Regge action. Because of this symmetry, the sum (67) overcounts physical configurations, and the regulator Λ(L) is simply the remaining gauge volume.”

What the sources say

• Supported

  Split into 2 atoms — 2 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  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. source

In the paper

“This new technology essentially allows one to consider holonomies (56) that lie in infinite-dimensional unitary representations of the Lorentz group, with a finite inner product defined by appropriate gauge-fixing.”

What the sources say

• Supported

  The Introduction says noncompact spin networks should use infinite-dimensional unitary representations, and §6.1 (p. 25) explicitly constructs unitary representations for SL(2,C). The Conclusion (p. 40) says careful gauge fixing yields a measure that defines a Hilbert-space inner product for each graph. This supports the citing sentence’s account of the construction. (LLM, conf=0.90)

  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. source
• Partly supported

  Section II (p. 5) constructs SO(2,1) spin networks using holonomies in infinite-dimensional unitary representations and describes an orthonormal basis of gauge-invariant states. It does not establish the attributed detail that an appropriate gauge-fixing defines a finite inner product; its displayed scalar-product construction concerns compact groups. (LLM, conf=0.82)

  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

“A candidate for a discrete version of the Wheeler-DeWitt equation in three dimensions has been found [47], for which the Ponzano-Regge wave functions are solutions.”

What the sources say

• Supported

  The Introduction says the authors present constraint equations for the Ponzano-Regge wavefunction and argue that they quantize a discrete Wheeler-DeWitt equation. Section 3, Quantization (p. 2119), gives an equation satisfied by that wavefunction and identifies the Hamiltonian analogue of the Wheeler-DeWitt equation with the Biedenharn-Elliott relation. (LLM, conf=0.94)

  cites [47] Barrett, J.W., and Crane, L., “An algebraic interpretation of the Wheeler-DeWitt equation”, Class. Quantum Grav., 14, 2113–2121, (1997). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9609030. source

In the paper

“In this sector, the transform is not very well-behaved: In fact, a dense set of Chern-Simons states transforms to zero [192].”

What the sources say

• Supported

  The Introduction (page unavailable) says that, in the boost sector corresponding to spacelike toroidal sections, the loop transform has a kernel dense in the connection representation and is far from continuous. Thus a dense set of connection-representation states transforms to zero, as claimed. (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

“an explicit quantization of the algebra holonomies exists for genus one (see below) and genus two [214], but more general results do not yet exist.”

What the sources say

• Supported

  The Introduction (p. 2) says the paper explicitly reduces the genus-two algebra to six independent operators, and the Representations section (p. 5) constructs its Hilbert-space representation. The citation is attached to the genus-two example; the sentence points elsewhere for genus one, so this source need not establish that case. (LLM, conf=0.92)

  cites [214] Nelson, J.E., and Regge, T., “Quantisation of 2+1 gravity for genus 2”, Phys. Rev. D, 50, 5125–5129, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9311029. source

In the paper

“such amplitudes are given explicitly in [118], where it is shown that they are peaked around the classical trajectory.”

What the sources say

• Supported

  Section 4 (pp. 10–11) explicitly calculates the transition amplitude and states that it is distributed around the classical orbit. The Summary and Discussion (p. 16) likewise says the computed amplitude has a peak on that orbit. (LLM, conf=0.94)

  cites [118] Ezawa, K., “Transition Amplitude in 2+1 dimensional Chern-Simons Gravity on a Torus”, Int. J. Mod. Phys. A, 9, 4727–4746, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9305170. source

In the paper

“A second possibility is to change the integration measure in the loop transform to make various integrals converge better [30].”

What the sources say

• Supported

  The Introduction (p. 5) explicitly proposes modifying the measure in the loop transform. Section 3.2 (p. 14) explains that a damped measure makes the transform well-defined, and the Conclusion (p. 22) says the integral with the natural measure is generally ill-defined. (LLM, conf=0.94)

  cites [30] Ashtekar, A., and Loll, R., “New loop representations for 2+1 gravity”, Class. Quantum Grav., 11, 2417–2434, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9405031. source

In the paper

“Spacetimes with Λ < 0 expand from a big bang and recollapse in a big crunch, while those with Λ ≥ 0 expand forever [200, 20].”

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.
• Partly supported

  Section 3 (p. 6) says that, for trivial solutions with spatial genus greater than 1, Λ < 0 gives a ‘big bang’ and ‘big crunch,’ while Λ ≥ 0 gives infinite expansion in one direction. Theorem 4.1 (section 4, p. 9) extends the area behavior to the specified genus-greater-than-1 vacuum developments, and section 3 (p. 8) reports completeness in the expanding direction for Λ ≥ 0. The citing sentence omits these restrictions and presents the pattern as applying to spacetimes generally. (LLM, conf=0.87)

  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

“Such states can be defined in 2 + 1 dimensions as well, and there has been some interesting recent work on their role as “kinematic” states [130].”

What the sources say

• Supported

  Section II, “2 + 1 Loop Quantum Gravity,” constructs SO(2,1) spin network functionals and says they form a complete basis of gauge-invariant cylindrical states. This supports both defining spin network states in 2+1 dimensions and studying their role as kinematic states. (LLM, conf=0.91)

  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. source

In the paper

“The model has been used to study conditional probabilities and the emergence of quasiclassical behavior in quantum gravity [223], but the cut-off dependence of these results makes their physical significance unclear.”

What the sources say

• Supported

  The citation follows the statement that the model was used to study conditional probabilities and the emergence of quasiclassical behavior. The Introduction (p. 3) says the paper examines conditional probabilities in a Ponzano-Regge minisuperspace model, and the Conclusions (p. 13) connect the study to the emergence of classical spacetime. The subsequent judgment about physical significance is outside the cited clause. (LLM, conf=0.91)

  cites [223] Petryk, R., and Schleich, K., “Conditional probabilities in Ponzano-Regge minisuperspace”, Phys. Rev. D, 67, 024019-1–13, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0109083. source

In the paper

“In the (3 + 1)-dimensional theory, loop states have been largely superseded by spin network states, states characterized by a graph Γ with edges labeled by representations and vertices labeled by intertwiners [239].”

What the sources say

• Supported

  In “Loop states and spin network states” (page not given), the paper says multiloop states were the main tool before the discovery of the spin network basis and calls the loop representation the “old” one. The same section defines a spin network by a graph Γ whose links carry SU(2) representations and whose nodes carry invariant, or intertwining, tensors. (LLM, conf=0.93)

  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. source

In the paper

“For noncompact groups such as those that appear in gravity, much less is known, though there has been some promising work [280, 155, 43, 150].”

What the sources say

• ?Not checked — no full text of the sourcecites [280] Witten, E., “Quantization of Chern-Simons gauge theory with complex gauge group”, Commun. Math. Phys., 137, 29–66, (1991). source
• ?Not checked — no full text of the sourcecites [155] Hayashi, N., “Quantum Hilbert Space of G(C) Cher-Simons-Witten Theory and Gravity”, Prog. Theor. Phys. Suppl., 114, 125–147, (1993). source
• ?Not checked — no full text of the sourcecites [43] Bar-Natan, D., and Witten, E., “Perturbative expansion of Chern-Simons theory with non-compact gauge group”, Commun. Math. Phys., 141, 423–440, (1991). source
• Supported

  The Introduction identifies this paper as a study of Chern–Simons theory with the complex gauge group SL(2,C) and its connections to three-dimensional quantum gravity. It therefore supports the citation as an example of work on the noncompact-group case; Section 4 also describes a complication absent for compact gauge groups. (LLM, conf=0.88)

  cites [150] Gukov, S., “Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial”, (2003). URL (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0306165. source

In the paper

“In an interesting recent paper, Colosi et al. have investigated the dynamics of a single tetrahedron [241], showing that a quantum description of the evolution can be given in terms of a boundary amplitude.”

What the sources say

• Supported

  The Abstract says the paper models 3D Euclidean quantum spacetime with a single equilateral tetrahedron and expresses its physical information as a boundary amplitude. Sections 4.3 and 6 describe its quantum dynamics through a propagator that assigns amplitudes to boundary measurements. (LLM, conf=0.94)

  cites [241] Rovelli, C., Colosi, D., Doplicher, L., Fairbairn, W., Modesto, L., and Noui, K., “Background independence in a nutshell”, (2004). URL (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0408079. source

In the paper

“Second, the discovery that the sum over topologies can lead to a divergent partition function has been extended to 3 + 1 dimensions, at least for Λ < 0, and it has been argued that this behavior might signal a phase transition that could prohibit a conventional cosmology with a negative cosmological constant [79, 80].”

What the sources say

• Partly supported

  Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  cites [79] Carlip, S., “Spacetime Foam and the Cosmological Constant”, Phys. Rev. Lett., 79, 4071–4074, (1997). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9708026. sourceExperimental — accuracy not yet measured
• Partly supported

  Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  cites [80] Carlip, S., “Dominant Topologies in Euclidean Quantum Gravity”, Class. Quantum Grav., 15, 2629–2638, (1998). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9710114. sourceExperimental — accuracy not yet measured

In the paper

“Similarly, (2 + 1)-dimensional gravity has recently been examined as an arena in which to test for a new partially discrete, constraint-free formulation of quantum gravity [138].”

What the sources say

• Supported

  The abstract (p. 1) says the authors leave spatial slices continuous, obtain a theory free of diffeomorphism and Hamiltonian constraints, and exhibit the technique in 2+1-dimensional gravity. The paper’s 2+1-dimensional illustration uses a BF theory equivalent to Euclidean general relativity in those dimensions. (LLM, conf=0.94)

  cites [138] Gambini, R., and Pullin, J., “Consistent discretization and loop quantum geometry”, (2004). URL (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0409057. source

In the paper

“Freidel et al. argue that spacelike intervals are continuous, while timelike intervals are discrete, with a spectrum of the form”

What the sources say

• Supported

  The Abstract states that spacelike spectra are continuous and timelike spectra are discrete in 2+1-dimensional Lorentzian loop quantum gravity. Section B, “Length spectrum” (page not specified), gives timelike eigenvalues T_{ε,n} = ε√[n(n−1)]; Section C, “Length operator” (page not specified), supplies the Planck-length factor lP. (LLM, conf=0.94)

  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. source

In the paper

“In this case, the Hartle-Hawking wave function has been examined as well, and it has been shown that the wave function acquires infinite peaks at certain specific spatial geometries: Again, topologically complicated manifolds whose individual contributions are small occur in large enough numbers to dominate the path integral, and “entropy” wins out over “action” [69].”

What the sources say

• Supported

  The Introduction (p. 3) says that, for a negative cosmological constant, the wave function is peaked at highly symmetric spatial geometries where infinitely many distinct topologies contribute. The Implications section (p. 7) states that the wave function diverges for at least one spatial metric, while the Abstract (p. 1) explains that the numerous higher-action contributions outweigh the leading contribution. (LLM, conf=0.92)

  cites [69] Carlip, S., “Entropy versus action in the (2+1)-dimensional Hartle-Hawking wave function”, Phys. Rev. D, 46, 4387–4395, (1992). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9205022. source

In the paper

“Dynamical triangulation has been proven to be quite useful in two-dimensional gravity, and some important steps have been taken in higher dimensions, especially with the recent progress in understanding Lorentzian triangulations.”

What the sources say

• ?Not checked — no full text of the sourcecites [8] Ambjørn, J., Carfora, M., and Marzuoli, A., The Geometry of Dynamical Triangulations, vol. m50 of Lecture Notes in Physics, (Springer, Berlin, Germany; New York, U.S.A., 1997). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9612069. source
• Partly supported

  The Dynamical triangulations Introduction calls the method a valuable tool in two-dimensional gravity, and the Summary describes results from four-dimensional simulations. The paper does not report recent progress in understanding Lorentzian triangulations; its Conclusions and Outlook instead call continuation of path-integral results to Lorentzian signature an unresolved problem. No page numbers are available in our copy. (LLM, conf=0.86)

  cites [179] Loll, R., “Discrete Approaches to Quantum Gravity in Four Dimensions”, Living Rev. Relativity, 1, lrr-1998-13, (1998). URL (cited on 5 January 2005): http://www.livingreviews.org/lrr-1998-13. sourceExperimental — accuracy not yet measured

In the paper

“Probably the most elegant derivation of a Lorentzian spin foam description starts with the first-order action (14), with Λ = 0, for a triangulated manifold [125, 128].”

What the sources say

• Supported

  In “3D Gravity” (p. 1), Freidel starts from a BF action for Lorentzian 3D gravity. “Euclidean calculation” (p. 3) describes discretizing the action on a triangulation, and “Evaluation of the Path integral” (pp. 5–6) carries that construction into the Lorentzian case. The description of the derivation is supported; “most elegant” is the citing authors’ assessment. (LLM, conf=0.86)

  cites [125] Freidel, L., “A Ponzano-Regge model of Lorentzian 3-dimensional gravity”, Nucl. Phys. B (Proc. Suppl.), 88, 237–240, (2000). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0102098. source
• Partly supported

  The Introduction and §2.1 describe deriving spin-foam models from a classical action on a triangulated manifold. But the Introduction explicitly restricts gravity to Euclidean signature, and §3.5 presents a Euclidean first-order gravity action, not a Lorentzian derivation. The cited passages do not establish the specified Λ = 0 condition. (LLM, conf=0.88)

  cites [128] Freidel, L., and Krasnov, K., “Spin Foam Models and the Classical Action Principle”, Adv. Theor. Math. Phys., 2, 1183–1247, (1999). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9807092. sourceExperimental — accuracy not yet measured

In the paper

“For example, the matrix model connection can be used to show that Newton’s constant and the cosmological constant are additively renormalized [14], and to analyze the apparent nonrenormalizability of ordinary field theoretical approach.”

What the sources say

• Supported

  The Abstract says the mapping to a two-matrix model shows additive renormalization of both the cosmological and gravitational coupling constants. Section 4 (Discussion) also explains how the non-perturbative approach addresses the gravitational coupling’s perturbative non-renormalizability. (LLM, conf=0.93)

  cites [14] Ambjørn, J., Jurkiewicz, J., and Loll, R., “Renormalization of 3d quantum gravity from matrix models”, Phys. Lett. B, 581, 255–262, (2004). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0307263. source

In the paper

“For example, there is some evidence that (2 + 1)-dimensional gravity is renormalizable in the 1/N expansion when coupled to scalar fields [174, 205].”

What the sources say

• ?Not matched

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

  cites [174] Kugo, T., unknown status. Kyoto preprint KUNS 1014 HE(TH)90/05 (1990).
• Supported

  The Introduction (pp. 1–2) reports that three-dimensional Einstein gravity coupled to scalar fields was shown to be formally power-counting renormalizable in the 1/N expansion. Section II (p. 7) adds that the proposed all-order proof does not actually work because a tachyonic pole makes higher-order calculations ill-defined. The claim’s cautious wording, “some evidence,” is consistent with that limitation. (LLM, conf=0.88)

  cites [205] Mizoguchi, S., and Yamamoto, H., “On the stability of renormalizable expansions in three-dimensional gravity”, Phys. Rev. D, 50, 7351–7362, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9307131. source

In the paper

“The path integral for such a system can be evaluated numerically, using Monte Carlo methods and a set of “moves” that systematically change an initial triangulation [12, 10]. One finds two phases.”

What the sources say

• Supported

  Section 3 (pp. 8–9) describes Monte Carlo simulation using local moves that change one Lorentzian triangulation into another. Section 5 (p. 26) describes two phases separated by a critical coupling: an extended-geometry phase and a phase in which spacetime decouples into uncorrelated slices. (LLM, conf=0.93)

  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. source
• Supported

  Section 2 describes a Monte Carlo simulation in which five local moves change one triangulation into another. Section 3 reports a first-order transition near k₀ ≈ 6.64 and describes the phases above and below the critical value. (LLM, conf=0.92)

  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. source

In the paper

“First, Witten has suggested a novel mechanism by which supersymmetry in 2 + 1 dimensions might cancel radiative corrections to Λ without requiring the equality of superpartner masses, essentially because even if the vacuum is supersymmetric, the asymptotics forbid the existence of unbroken supercharges for massive states [281, 52].”

What the sources say

• Supported

  Witten states that in 2+1 dimensions supersymmetry can make the cosmological constant vanish without equal boson and fermion masses (p. 1). He explains that massive states have asymptotically conical geometry, which prevents unbroken supercharges even when supersymmetry applies to the vacuum (p. 2). The paper does not use the phrase “radiative corrections,” but the cited sentence's tentative description of the proposed mechanism matches these passages. (LLM, conf=0.88)

  cites [281] Witten, E., “Is Supersymmetry Really Broken?”, Int. J. Mod. Phys. A, 10, 1247–1248, (1995). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9409111. source
• Supported

  The abstract (p. 1) explicitly attributes to Witten the argument that local supersymmetry in 2+1 dimensions can make the cosmological constant vanish without equal boson and fermion masses. On p. 2, the paper explains that supercharges are ill-defined for nonzero-energy states with conical geometries, allowing a supersymmetric vacuum without degenerate excited states. (LLM, conf=0.91)

  cites [52] Becker, K., Becker, M., and Strominger, A., “Three-Dimensional Supergravity and the Cosmological Constant”, Phys. Rev. D, 51, 6603–6607, (1995). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9502107. source

In the paper

“For ordinary “Euclidean” dynamical triangulations, few signs of such a continuum limit have been seen. The system appears to exhibit two phases — a “crumpled” phase, in which the Hausdorff dimension is extremely large, and a “branched polymer” phase — neither of which look much like a classical spacetime [179].”

What the sources say

• Supported

  Split into 2 atoms — 2 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  cites [179] Loll, R., “Discrete Approaches to Quantum Gravity in Four Dimensions”, Living Rev. Relativity, 1, lrr-1998-13, (1998). URL (cited on 5 January 2005): http://www.livingreviews.org/lrr-1998-13. source

In the paper

“This is indeed true, as shown in [77, 245] for spatially closed universes and [63] for geometries with point particles.”

What the sources say

• Supported

  The paper states that it demonstrates equivalence between the phase space path integral and canonical quantization of the reduced phase space (p. 1). It specifies a closed spatial surface as its setting (p. 2). The citing sentence attributes the point-particle case to a different reference, not to this paper. (LLM, conf=0.92)

  cites [77] Carlip, S., “A Phase Space Path Integral for (2+1)-Dimensional Gravity”, Class. Quantum Grav., 12, 2201–2208, (1995). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9504033. source
• Supported

  The Abstract and §4 (pages unavailable in our copy) state that, for spatially closed (2+1)-dimensional pure Einstein gravity, gauge fixing reduces the full phase-space partition function to one for the reduced system. §4 also describes equivalence between two reduced formulations at the quantum level under the specified time gauge. The separate point-particle claim is assigned to [63], not to this source. (LLM, conf=0.88)

  cites [245] Seriu, M., “Partition Function for (2+1)-Dimensional Einstein Gravity”, Phys. Rev. D, 55, 781–790, (1997). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9609052. source
• Partly supported

  For geometries with point particles, the paper derives a phase-space functional integral with the same expression as reduced particle dynamics (Additional information, eq. 71). But it says the resulting integral does not resolve operator ordering and that the relationship between the functional and operator approaches remains unclear (p. 846; §5). It therefore does not establish unqualified equivalence to canonical quantization. The spatially closed-universe part is attributed to other citations, not this paper. (LLM, conf=0.86)

  cites [63] Cantini, L., and Menotti, P., “Functional approach to 2+1 dimensional gravity coupled to particles”, Class. Quantum Grav., 20, 845–858, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0209061. sourceExperimental — accuracy not yet measured

In the paper

“This is apparently no longer the case when gravity is coupled to fermions and a U(1) Chern-Simons gauge theory [22], although Anselmi has argued that if coupling constants are tuned to exact values, renormalizability can be restored, and in fact the theory can be made finite [21].”

What the sources say

• Supported

  The citation to this paper is attached to the first clause, not the later claim about tuning couplings, which has a separate citation. Section 3 (pp. 6–7) describes the gravity–fermion–U(1) Chern-Simons model as power-counting non-renormalizable and reports a divergent four-fermion vertex requiring new couplings. The Conclusions (p. 15) state that the coupled model is not finite. (LLM, conf=0.90)

  cites [22] Anselmi, D., “Renormalization of quantum gravity coupled with matter in three dimensions”, Nucl. Phys. B, 687, 143–160, (2004). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0309249. source
• Supported

  The citation to this paper supports the clause about restoring renormalizability and finiteness. The Introduction says the finiteness equations determine the otherwise independent irrelevant couplings, and the Conclusions (p. 19) state that the resulting action is renormalizable without new parameters and is finite in that sense. The paper also says the gravity–fermion–Chern–Simons model is not finite as it stands (section 4, p. 13). (LLM, conf=0.92)

  cites [21] Anselmi, D., “Finiteness of quantum gravity coupled with matter in three spacetime dimensions”, Nucl. Phys. B, 687, 124–142, (2004). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0309250. source

In the paper

“Further complications come from the fact that the inner product on the space of solutions of the Wheeler-DeWitt equation must be gauge-fixed [282, 144]; again, little is understood about the resulting Hilbert space.”

What the sources say

• Partly supported

  The discussion of formula (13c) says that the inner product of invariant states must be gauge-fixed, directly supporting the first part of the claim. The paper discusses unresolved questions about observables and physical time, but does not say that little is understood about the resulting Hilbert space. (LLM, conf=0.85)

  cites [282] Woodard, R.P., “Enforcing the Wheeler-DeWitt constraint the easy way”, Class. Quantum Grav., 10, 483–496, (1993). sourceExperimental — accuracy not yet measured
• Partly supported

  The Introduction and §2.1 describe difficulties and ambiguities in defining an inner product and physical Hilbert space for solutions of quantum constraints. But §2.2 defines the inner product through a rigging map, not as something that must be gauge-fixed. The paper discusses constrained systems generally and does not establish the claim specifically for Wheeler–DeWitt solutions. (LLM, conf=0.85)

  cites [144] Giulini, D., and Marolf, D., “On the Generality of Refined Algebraic Quantization”, Class. Quantum Grav., 16, 2479–2488, (1999). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9812024. sourceExperimental — accuracy not yet measured

In the paper

“In particular, one may evaluate the expectation value 〈A(t)〉 of the spatial area at fixed time and the correlation 〈A(t)A(t + 1)〉 of successive areas; the results match the classical de Sitter behavior for a spacetime ℝ × S2 quite well.”

What the sources say

• Partly supported

  The Numerical Results section (p. 13) says the model measures spatial slice-volumes and correlations between them. In the extended phase (pp. 20–21), the measured correlator fits a classical Euclidean three-sphere modified with a thin stalk rather well. The paper does not report a match to Lorentzian de Sitter behavior on ℝ × S², nor establish the claimed match for both the fixed-time expectation value and successive-area correlation. (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
• Partly supported

  The results (pp. 3–4, Eq. 3 and Fig. 3) report a spatial-volume correlator that fits a classical positive-curvature S³ solution rather well. They do not report that the fixed-time expectation value of spatial area was separately matched to that solution. The paper also specifies periodic S¹ × S² topology, rather than ℝ × S², and notes that a Gaussian time profile fits comparably well. (LLM, conf=0.87)

  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

“There are also indications that string theory might have something to say about the sum over topologies [111].”

What the sources say

• Supported

  The Introduction and Summary says the paper rewrites a string-theory quantity to reflect a sum over geometries. The discussion on p. 41 considers different topological fillings and asks whether additional geometries might contribute to the Euclidean partition sum. This supports the citing sentence’s cautious claim that string theory may inform the sum over topologies. (LLM, conf=0.90)

  cites [111] Dijkgraaf, R., Maldacena, J.M., Moore, G.W., and Verlinde, E., “A Black Hole Farey Tail”, (2000). URL (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0005003. source

In the paper

“Carlip and Cosgrove [84], for example, explicitly compute amplitudes for a transition between a genus three surface and a pair of genus two surfaces.”

What the sources say

• Supported

  Section 5 identifies M2 as a manifold mediating a transition from a genus-three surface to two genus-two surfaces. Section 6 explicitly states that the authors calculate the amplitude ZM2 for that transition. (LLM, conf=0.94)

  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

“It has been known for several years how to generalize the Ponzano-Regge action for a single tetrahedron [48, 101, 191], and recently considerable progress has been made in constructing Lorentzian spin foam models [222, 125, 102].”

What the sources say

• Supported

  The cited paper is offered for the first clause, not the separate claim about recent Lorentzian spin foam models. In §4.4 it identifies the Regge action for a single tetrahedron; in the discussion of the state sum it extends the analysis to a complex of tetrahedra and an action of the same form. Page numbers are not available in our copy. (LLM, conf=0.88)

  cites [48] Barrett, J.W., and Foxon, T.J., “Semiclassical limits of simplicial quantum gravity”, Class. Quantum Grav., 11, 543–556, (1994). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9310016. source
• Supported

  The citation is attached to the sentence’s claim about progress toward Lorentzian spin foam models. The Introduction (p. 2) describes the paper’s SU(1,1) tetrahedron asymptotics as a step toward a three-dimensional Lorentzian state-sum model; it says a full model remains future work, which is consistent with a claim of progress rather than completion. (LLM, conf=0.86)

  cites [101] Davids, S., “Semiclassical Limits of Extended Racah Coefficients”, J. Math. Phys., 41, 924–943, (2000). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9807061. source
• Supported

  The Introduction (p. 446) describes earlier studies of Lorentzian tetrahedron asymptotics and a proposed state-sum model for (2 + 1)-dimensional Lorentzian quantum gravity. It also sets out this paper’s construction of Lorentzian spin-network amplitudes, supporting the cited clause about progress on Lorentzian spin foam models. (LLM, conf=0.87)

  cites [191] Manuel García-Islas, J., “(2+1)-dimensional quantum gravity, spin networks and asymptotics”, Class. Quantum Grav., 21, 445–464, (2004). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0307054. source

In the paper

“An alternative “Lorentzian” model, introduced by Ambjørn and Loll [16, 9, 12, 10, 180, 13], however, has much nicer properties, including a continuum limit that appears numerically to match a finite-sized, spherical “semiclassical” configuration.”

What the sources say

• Partly supported

  The Abstract (p. 1) introduces a two-dimensional Lorentzian quantum-gravity model and states that it has a continuum limit. The Conclusions (p. 25) describe that limit but propose numerical simulations as future work; they do not report a numerical match to a finite-sized, spherical semiclassical configuration. (LLM, conf=0.88)

  cites [16] Ambjørn, J., and Loll, R., “Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change”, Nucl. Phys. B, 536, 407–434, (1998). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9805108. sourceExperimental — accuracy not yet measured
• Partly supported

  The paper introduces a Lorentzian model and reports better-behaved geometry than in the Euclidean model (pp. 3, 10). But it describes a nontrivial continuum theory as a possibility, not a demonstrated numerical result (p. 10). Its two-sphere spatial topology is chosen for the model (p. 4); the paper does not report a continuum limit numerically matching a finite-sized spherical semiclassical configuration. (LLM, conf=0.88)

  cites [9] Ambjørn, J., Jurkiewicz, J., and Loll, R., “A non-perturbative Lorentzian path integral for gravity”, Phys. Rev. Lett., 85, 924–927, (2000). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0002050. sourceExperimental — accuracy not yet measured
• Supported

  In §4.2 (p. 21), the authors report that a spherical geometry fits their numerical data rather well and say this provides some evidence that the universes behave semiclassically at macroscopic scales. In §5 (p. 26), they describe a continuum limit taken while keeping the three-volume constant. The claim’s tentative wording is consistent with the paper’s qualifications. (LLM, conf=0.91)

  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. source

In the paper

“The appearance of both continuous and discrete labels has a nice physical interpretation [130]: Continuous representations describe spacelike edges, and seem to imply a continuous length spectrum, while discrete representations label timelike edges, and suggest discrete time.”

What the sources say

• Supported

  Split into 2 atoms — 2 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  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. source

In the paper

“It is only quite recently that similar results for sums over topologies have been found in four dimensions [79, 80, 229, 19].”

What the sources say

• Supported

  Section 3 (p. 5) derives factorial growth in the number of four-dimensional manifolds and says this guarantees that the sum over topologies diverges. The same section (p. 6) explicitly discusses that sum in four dimensions. (LLM, conf=0.92)

  cites [79] Carlip, S., “Spacetime Foam and the Cosmological Constant”, Phys. Rev. Lett., 79, 4071–4074, (1997). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9708026. source
• Supported

  Section 3 (page not provided) finds superexponential growth in the density of four-dimensional topologies and concludes that this entropy dominates the action in the sum over topologies. This supports the citing sentence’s broad claim that similar results had been found in four dimensions. (LLM, conf=0.92)

  cites [80] Carlip, S., “Dominant Topologies in Euclidean Quantum Gravity”, Class. Quantum Grav., 15, 2629–2638, (1998). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9710114. source
• Supported

  The Introduction says the paper extends a result about the Hartle–Hawking wave function from 2+1 to 3+1 dimensions. Section 2 (p. 2) describes that wave function as a sum over four-dimensional manifolds, and the paper finds that the sum diverges for a particular boundary geometry. This supports the cited statement about similar four-dimensional results for sums over topologies. (LLM, conf=0.93)

  cites [229] Ratcliffe, J.G., and Tschantz, S.T., “On the Growth of the Number of Hyperbolic Gravitational Instantons with Respect to Volume”, Class. Quantum Grav., 17, 2999–3007, (2000). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0009004. source
• Supported

  The Introduction (p. 2) says that, in four spacetime dimensions, sums over topologies produce sharp peaks in the Hartle–Hawking wave function. The Conclusions (p. 10) state that the paper demonstrates a clear effect in four spacetime dimensions, while noting limitations. This supports citing it as an example of similar results in four dimensions. (LLM, conf=0.93)

  cites [19] Anderson, M., Carlip, S., Ratcliffe, J.G., Surya, S., and Tschantz, S.T., “Peaks in the Hartle-Hawking Wave Function from Sums over Topologies”, Class. Quantum Grav., 21, 729–742, (2004). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0310002. source

In the paper

“The Lorentzian dynamical triangulation model can also be translated into a two-matrix model, the so-called ABAB model. The Feynman diagrams of the matrix model correspond to dual graphs of a triangulation, and matrix model amplitudes become particular sums of transfer matrix elements in the gravitational theory [11, 14, 15].”

What the sources say

• Partly supported

  Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  cites [11] Ambjørn, J., Jurkiewicz, J., and Loll, R., “Lorentzian 3d Gravity with Wormholes via Matrix Models”, J. High Energy Phys., 09, 022, (2001). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0106082. sourceExperimental — accuracy not yet measured
• Supported

  Split into 2 atoms — 2 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  cites [14] Ambjørn, J., Jurkiewicz, J., and Loll, R., “Renormalization of 3d quantum gravity from matrix models”, Phys. Lett. B, 581, 255–262, (2004). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0307263. source
• Partly supported

  Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  cites [15] Ambjørn, J., Jurkiewicz, J., Loll, R., and Vernizzi, G., “3D Lorentzian Quantum Gravity from the asymmetric ABAB matrix model”, Acta Phys. Pol. B, 34, 4667–4688, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0311072. sourceExperimental — accuracy not yet measured

In the paper

“This has led to a suggestion [100, 99] that the conformal contribution is canceled by a Faddeev-Popov determinant (see also [198]), and some preliminary supporting computations have been made in a proper time gauge [100].”

What the sources say

• Supported

  Section 6 (Conclusions) says that, under stated assumptions about renormalization and the DeWitt metric, a Faddeev–Popov term cancels the conformal divergence. Section 2 describes a perturbative calculation in proper-time gauge intended to illustrate that cancellation. The citing sentence’s tentative wording is consistent with these qualifications. (LLM, conf=0.93)

  cites [100] Dasgupta, A., and Loll, R., “A proper-time cure for the conformal sickness in quantum gravity”, Nucl. Phys. B, 606, 357–379, (2001). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0103186. source
• Supported

  Section 3.2 (“In higher dimensions,” eq. 3.43) argues that the divergent conformal determinant has the same functional form as the inverse of the scalar part of the Faddeev–Popov determinant and that the two can be canceled. The paper also describes its proper-time-gauge calculation of that determinant (section C). The copy provides section labels but no page numbers. (LLM, conf=0.92)

  cites [99] Dasgupta, A., “The Real Wick Rotations in Quantum Gravity”, J. High Energy Phys., 07, 062, (2002). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0202018. source
• ?Not checked — no full text of the sourcecites [198] Mazur, P.O., and Mottola, E., “The path integral measure, conformal factor problem and stability of the ground state of quantum gravity”, Nucl. Phys. B, 341, 187–212, (1990). source

In the paper

“With the decomposition of [206], the momentum constraints ℌi = 0 become trivial, while the Hamiltonian constraint becomes an elliptic differential equation that determines the scale factor γ in Equation (31) as a function of ḡij and pij”

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

In the paper

“The model has an extensive gauge freedom available in the choice of lattice.”

In the paper

“Provided the triad e is invertible, the latter are equivalent to diffeomorphisms on a shell; more precisely, the combination of transformations with parameters ρa = ξ · ea and τa = ξ · ωa is equivalent to the diffeomorphism generated by the vector field ξ.”

In the paper

“First, as noted above, the holonomies do not always determine a unique geometric structure. In particular, for Λ > 0 one may need an additional discrete variable to specify the geometry.”

In the paper

“This follows from the fact that the group ISO(2, 1) is the cotangent bundle of SO(2, 1).”

In the paper

“The excitations (5) are thus pure gauge, confirming the absence of propagating degrees of freedom.”

In the paper

“The holonomy group of this flat bundle can be shown to be isomorphic to the holonomy group of the geometric structure, and for (2 + 1)-dimensional gravity, the flat connection constructed from the geometric structure is that of the Chern-Simons theory.”

In the paper

“The Einstein field equations with Λ = 0 then imply that edges of polygons move at constant velocities and that edge lengths may change, subject to a set of consistency conditions.”

In the paper

“A flat spacetime, for instance, can always be described as a collection of patches, each isometric to Minkowski space, that are glued together by isometries of the flat metric; but the gluing is not unique, and may be dynamical.”

In the paper

“Concretely, in the first-order formalism of Section 2.3, the curvature equation (16) with Λ = 0 implies that ω is a flat SO(2, 1) connection; and if ω(s) is a curve in the space of such flat connections, the tangent vector e = dω(s)/ds satisfies the torsion equation (17).”

In the paper

“Complications occur when an edge shrinks to zero length or collides with a vertex, but these are completely understood.”

In the paper

“In 2 + 1 dimensions, however, the Weyl tensor vanishes identically, and the full curvature tensor is determined algebraically by the remaining pieces:”

In the paper

“This follows from the peculiar nature of the field equations in three dimensions: As discussed above, any vacuum solution can be patched together from finite pieces of constant curvature spacetime, and the dynamics occurs only in the patching.”

In the paper

“In particular, for a spacetime M ≃ ℝ × Σ, one can choose a lattice that is simply a 4g-sided polygon with edges identified; the resulting spacetime can be visualized as a polygonal tube cut out of Minkowski spacetime, with corners lying on straight worldlines and edges identified pairwise.”

In the paper

“The resulting Chern-Simons connections A(±) of Equation (23) are flat, and their holonomies reproduce the holonomies (38) of the geometric structure we began with.”

In the paper

“For Λ ≥ 0, the final big crunch disappears, and the torus universe expands forever from an initial big bang.”

In the paper

“These transformations are consistent with the relationships between the ADM and holonomy variables, and that they preserve all Poisson brackets.”

In the paper

“The area of a slice of constant t is essentially the Hamiltonian (44); it increases from 0 at t = 0 to a maximum at t = and then shrinks to zero at t = πℓ/2”

In the paper

“Most of the detailed, explicit results in (2 + 1)-dimensional quantum gravity hold only for the torus universe ℝ × T2.”

In the paper

“In particular, the relationship between the ADM and Chern-Simons quantizations in Section 3.4 relied on a particularly simple operator ordering; it is not obvious that such an ordering can be found for the higher genus case [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

“For particular quantization programs, (2 + 1)-dimensional models have also offered more specific guidance: Special properties of the loop operators (55), methods for treating noncompact groups in spin foam models, and properties of the sums over topologies described in Section 3.11 have all been generalized to 3 + 1 dimensions.”

In the paper

“We have discovered some rather unexpected features, such as the difficulties caused by spatial diffeomorphism invariance and the consequent nonlocality in Wheeler-DeWitt quantization, and the necessity of understanding the representations of the group of large diffeomorphisms in almost all approaches.”

In the paper

“Much of the recent work on lattice formulations of (2 + 1)-dimensional gravity have centered on spin foams and on random triangulations, both inherently quantum mechanical.”

In the paper

“in general relativity there is no preferred time-slicing on which to normalize probabilities.”

In the paper

“A related first-order Hamiltonian lattice model has been studied by Waelbroeck et al. [266, 267, 268, 270].”

What the sources say

• ?Not checked — no full text of the sourcecites [266] Waelbroeck, H., “2+1 lattice gravity”, Class. Quantum Grav., 7, 751, (1990). source
• ?Not checked — no full text of the sourcecites [267] Waelbroeck, H., “Time-dependent solutions of 2+1 gravity”, Phys. Rev. Lett., 64, 2222–2225, (1990). source
• ?Not checked — no full text of the sourcecites [268] Waelbroeck, H., “Solving the time-evolution problem in 2+1 gravity”, Nucl. Phys. B, 364, 475–494, (1991). source
• ?Not checked — no full text of the sourcecites [270] Waelbroeck, H., and Zapata, J.A., “Translation symmetry in 2+1 Regge calculus”, Class. Quantum Grav., 10, 1923–1932, (1993). source

In the paper

“The resulting brackets among the holonomies have been evaluated by Nelson, Regge, and Zertuche [210, 211] for Λ < 0, for which the two SL(2, ℝ) factors in the gauge group G may be taken to be independent. The brackets are nonzero only for holonomies of curves that intersect, and can be written in terms of holonomies of “rerouted” curves”

What the sources say

• ?Not checked — no full text of the sourcecites [210] Nelson, J.E., and Regge, T., “Homotopy groups and 2+1 dimensional quantum gravity”, Nucl. Phys. B, 328, 190–202, (1989). source
• ?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

“for spatially closed universes, the natural candidate for a Hamiltonian in quantum gravity is identically zero when acting on physical states.”

In the paper

“It is easy to see that the mapping class group of a spacetime M acts on π1(M), and therefore on the holonomies of Section 2.2.”

In the paper

“But if the spacetime manifold is topologically nontrivial, its group of diffeomorphisms may not be connected: M may admit “large” diffeomorphisms, which cannot be built up smoothly from infinitesimal deformations.”

In the paper

“The space of such quadratic differentials parametrizes the cotangent space of the moduli space [1], and the reduced phase space becomes, essentially, the cotangent bundle of the moduli space.”

What the sources say

• Not checked yetcites [1] Abikoff, W., The Real Analytic Theory of Teichmüller Space, vol. 820 of Lecture Notes in Mathematics, (Springer, Berlin, Germany; New York, U.S.A., 1980).

In the paper

“This reproduces the quotient space picture discussed by Mess in the context of geometric structures [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

“The holonomies of the geometric structure in (2 + 1)-dimensional gravity are examples of diffeomorphism-invariant observables, which, as we shall see below, are closely related to the Wilson loop observables in the Chern-Simons formulation.”

In the paper

“and may not be equivalent quantum mechanically.”

In the paper

“As a generally covariant theory of spacetime geometry, (2 + 1)-dimensional gravity has the same conceptual foundation as realistic (3+ 1)-dimensional general relativity, and many of the fundamental issues of quantum gravity carry over to the lower dimensional setting.”

In the paper

“Mess also demonstrates that the holonomy group H acts properly discontinuously on a region W ⊂ V2,1 of Minkowski space, and that M can be obtained as the quotient space W/H.”

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

“standard power-counting arguments — finally confirmed in 1986 by explicit computations [149] — indicate that the theory is nonrenormalizable.”

What the sources say

• ?Not checked — no full text of the sourcecites [149] Goroff, M.H., and Sagnotti, A., “The ultraviolet behavior of Einstein gravity”, Nucl. Phys. B, 266, 709–736, (1986). source

In the paper

“With a few exceptions, (2 + 1)-dimensional solutions are physically quite different from those in 3 + 1 dimensions, and the (2 + 1)-dimensional model is not very helpful for understanding the dynamics of realistic quantum gravity.”

In the paper

“the fundamental (diffeomorphism-invariant) physical observables of quantum gravity are necessarily nonlocal.”

In the paper

“The space of homomorphisms (11) is not connected [148], and, in general, only one connected component gives our desired geometry.”

What the sources say

• ?Not checked — no full text of the sourcecites [148] Goldman, W.M., “Topological components of spaces of representation”, Invent. Math., 93, 557–607, (1988). source

In the paper

“Fortunately, the description is closely related to the more familiar first-order Chern-Simons formalism [108, 277, 279, 2], which, in turn, can connect us back to the standard metric formalism.”

What the sources say

• ?Not checked — no full text of the sourcecites [108] Deser, S., and van Nieuwenhuizen, P., “Nonrenormalizability of the quantized Dirac-Einstein system”, Phys. Rev. D, 10, 411–420, (1974). 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
• ?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

In the paper

“In the “timelike sector,” in which the traces of the two holonomies are both less than two, the loop transform is a simple Fourier transformation, and Chern-Simons and loop quantization are equivalent.”

In the paper

“For a torus universe with zero or positive cosmological constant, similar constructions are possible.”

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

“It appears to be self-consistent, and like ordinary quantum mechanics, it is guaranteed to have the correct classical limit on the reduced phase space of Section 2.4.”

In the paper

“For Λ = 0, the relevant gauge group is ISO(2, 1), the (2 + 1)-dimensional Poincaré group, or its universal cover.”

In the paper

“Often, however, we can determine such an isomorphism perturbatively in the neighborhood of a known classical solution.”

In the paper

“For the torus universe, such an isomorphism is known.”

In the paper

“the true physical motion in the moduli space of the torus — the space of physical configurations with the large diffeomorphisms modded out — is much more complicated; there are arbitrarily long geodesics, and the flow is, in fact, ergodic [93].”

What the sources say

• ?Not checked — no full text of the sourcecites [93] Cornfeld, I.P., Fomin, S.V., and Sinai, Y.G., Ergodic theory, vol. 245 of Grundlehren der mathematischen Wissenschaften, (Springer, New York, U.S.A., 1982). source

In the paper

“The algebra (57) is not a Lie algebra, but it is related to the Lie algebra of the quantum group Uq(sl(2)) with q = exp 4iθ [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 behavior of the corresponding wave functions has been explored by Puzio [227], who argues that they are well-behaved and nonsingular at the boundaries of moduli space.”

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 Riemann moduli space of the torus is two dimensional, so from Section 2.4 we expect a four-dimensional space of solutions.”

In the paper

“This means that even apart from the “crunch” in volume, the geometry is singular: A real value of τ represents a torus that has collapsed to a line.”

In the paper

“For higher genus spaces, however — and certainly for realistic (3 + 1)-dimensional gravity — it is not [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

“The problem is in part that no simple form such as Equation (41) for the “standard” constant curvature metrics exists, and in part that the ADM Hamiltonian becomes a complicated, nonlocal function of the moduli.”

In the paper

“With this weighting, identities among 6j-symbols may be used to show that the amplitude is invariant under refinement — that is, subdivision of a tetrahedron into four smaller tetrahedra — suggesting that we are dealing with a “topological” theory that does not depend on the choice of triangulation.”

In the paper

“The operators (62) introduced here on the covariant canonical Hilbert space were obtained from a particular classical time-slicing, and answer questions about spatial geometry in that slicing.”

In the paper

“For the torus universe of the preceding Section 2.7, the dynamics can be read off from the metric (40).”

In the paper

“Since the space of solutions of the Einstein equations in 3 + 1 dimensions has a similar orbifold structure [163], we might expect a similar ambiguity in realistic (3 + 1)-dimensional quantum gravity.”

What the sources say

• ?Not checked — no full text of the sourcecites [163] Isenberg, J.A., and Marsden, J.E., “A slice theorem for the space of solutions of Einstein’s equations”, Phys. Rep., 89, 179–222, (1982). source

In the paper

“This is possible, but at some cost — the inner products between loop states become considerably more complex, as does the action of the mapping class group — and it is not obvious that there is a canonical choice of the new measure and algebra.”

In the paper

“Such questions would seem to require a different classical time-slicing, and thus a different — and perhaps inequivalent — quantum theory.”

In the paper

“As in reduced phase space quantization, matters simplify considerably for the torus universe ℝ × T2.”

In the paper

“A few numerical investigations of the Ponzano-Regge path integral have been undertaken [151], but the evidence of a continuum limit is thus far inconclusive.”

What the sources say

• ?Not checked — no full text of the sourcecites [151] Hamber, H.W., and Williams, R.M., “Simplicial quantum gravity in three dimensions: Analytical and numerical results”, Phys. Rev. D, 47, 510–532, (1993). source

In the paper

“This is a consistent procedure, but many of the resulting states in the Cauchy completion are no longer functions of loops in any clear sense; they correspond instead to “extended loops” [136], whose geometrical interpretation is not entirely clear.”

What the sources say

• ?Not checked — no full text of the sourcecites [136] Gambini, R., and Pullin, J., Loops, Knots, Gauge Theories and Quantum Gravity, Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1996). source

In the paper

“If Λ > 0 or point particles are present, the holonomies do not quite determine a unique geometric structure, and the Chern-Simons theory is not quite equivalent to general relativity.”

In the paper

“Such a description effectively reduces quantum gravity to quantum mechanics, allowing us to evade the complications of quantum field theory.”

In the paper

“When the spatial metric {g_{ij}} = {e_i}^a{e_{ja}} is nondegenerate, it is straightforward to show that these constraints are equivalent to the standard constraints of first order gravity, and the classical theories are identical.”

In the paper

“In 2 + 1 dimensions, however, the full constraints imply that such states have their support on flat connections, and only holonomies around noncontractible curves describe nontrivial physics.”

In the paper

“But it is far less clear how to ask for transition amplitudes between other spatial slices, on which tr K is not constant.”

In the paper

“The gauge-invariant observables — the traces of the holonomies — are automatically nonlocal and time-independent, and one obtains a “frozen time formalism,” or what Kuchař has called “quantum gravity without time” [173].”

What the sources say

• Not checked yetcites [173] Kuchař, K., in Kunstatter, G., Vincent, D.E., and Williams, J.G., eds., General Relativity and Relativistic Astrophysics, Proceedings of the 4th Canadian Conference, University of Winnipeg, 16–18 May, 1991, 211, (World Scientific, Singapore; River Edge, U.S.A., 1992).

In the paper

“The first-order Euclidean action with Λ > 0 can be written as a difference of SU(2) Chern-Simons actions, so 71”

In the paper

“The technique of Chern-Simons quantization relies on special features of general relativity in 2 + 1 dimensions, and does not readily generalize to higher dimensions.”

In the paper

“The torus moduli space is not a manifold, but rather has orbifold singularities, and quantization on an orbifold is generally not unique.”

In the paper

“the Wheeler-DeWitt equation, for example, apparently does not”

In the paper

“For Λ > 0, the relevant gauge group is SO(3, 1) or its cover SL(2, ℂ), a complex gauge group whose Chern-Simons theory is somewhat better understood [280, 155, 43].”

What the sources say

• ?Not checked — no full text of the sourcecites [280] Witten, E., “Quantization of Chern-Simons gauge theory with complex gauge group”, Commun. Math. Phys., 137, 29–66, (1991). source
• ?Not checked — no full text of the sourcecites [155] Hayashi, N., “Quantum Hilbert Space of G(C) Cher-Simons-Witten Theory and Gravity”, Prog. Theor. Phys. Suppl., 114, 125–147, (1993). source
• ?Not checked — no full text of the sourcecites [43] Bar-Natan, D., and Witten, E., “Perturbative expansion of Chern-Simons theory with non-compact gauge group”, Commun. Math. Phys., 141, 423–440, (1991). source

In the paper

“Undoubtedly, the biggest embarrassment in quantum gravity today is the apparent prediction, at least in effective field theory, that the cosmological constant should be some 120 orders of magnitude larger than the observed limit.”

In the paper

“The Ponzano-Regge and Turaev-Viro models are examples of “spin foam” models [35, 222], that is, a model based on simplicial complexes with faces, edges, and vertices labeled by group representations and intertwiners.”

What the sources say

• ?Not checked — no full text of the sourcecites [35] Baez, J.C., “An Introduction to Spin Foam Models of Quantum Gravity and BF Theory”, in Gausterer, H., Grosse, H., and Pittner, L., eds., Geometry and Quantum Physics, Proceedings of the 38. Internationale Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, January 9–16, 1999, vol. 543 of Lecture Notes in Physics, 25–64, (Springer, Berlin, Germany; New York, U.S.A., 2000). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/9905087. source
• ?Not checked — no full text of the sourcecites [222] Perez, A., “Spin foam models for quantum gravity”, Class. Quantum Grav., 20, R43–R104, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/gr-qc/0301113. source

In the paper

“Building on work by Boulatov [60], Freidel and Loupre have recently considered a variant of the Ponzano-Regge model, and have shown that although the sum over topologies diverges, it is Borel summable [132].”

What the sources say

• ?Not checked — no full text of the sourcecites [60] Boulatov, D., “A Model of Three-Dimensional Lattice Gravity”, Mod. Phys. Lett. A, 7, 1629–1646, (1992). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/9202074. source
• ?Not checked — no full text of the sourcecites [132] Freidel, L., and Louapre, D., “Non-perturbative summation over 3D discrete topologies”, Phys. Rev. D, 68, 104004, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0211026. source

In the paper

“As a consequence, the Hamiltonian constraint becomes a nonlocal functional differential equation, and very little is understood about its solutions, even for the simplest case of the torus universe.”

In the paper

“Here, most of the literature has focused on the Hawking’s Euclidean path integral [154] and the Hartle-Hawking “no boundary” proposal [152], which describes the universe in terms of a path integral over Riemannian metrics on manifolds with a single, connected boundary Σ.”

What the sources say

• ?Not checked — no full text of the sourcecites [154] Hawking, S.W., in Hawking, S.W., and Israel, W., eds., General Relativity: An Einstein Centenary Survey, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1979).
• ?Not checked — no full text of the sourcecites [152] Hartle, J.B., and Hawking, S.W., “Wave function of the Universe”, Phys. Rev. D, 28, 2960–2975, (1983). source

In the paper

“While spin foam models ordinarily assume a fixed spacetime topology, recent work has suggested a method for summing over all topologies as well, thus allowing quantum fluctuations of spacetime topology [132].”

What the sources say

• ?Not checked — no full text of the sourcecites [132] Freidel, L., and Louapre, D., “Non-perturbative summation over 3D discrete topologies”, Phys. Rev. D, 68, 104004, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0211026. source

In the paper

“These results should probably not yet be considered conclusive, since they require operators that do not commute with all of the constraints, but they are certainly suggestive.”

In the paper

“If one allows Riemannian signature, on the other hand, such a path integral makes sense for any spatial topology, and if one further requires that Σ be totally geodesic — that is, that the extrinsic curvature of Σ vanish — one can smoothly join on a Lorentzian metric at Σ [141].”

What the sources say

• ?Not checked — no full text of the sourcecites [141] Gibbons, G.W., and Hartle, J.B., “Real tunneling geometries and the large-scale topology of the universe”, Phys. Rev. D, 42, 2458–2468, (1990). source

In the paper

“To obtain a Lorentzian version, one must replace SU(2) by SO(2, 1). The corresponding Plancherel formula involves a sum over both the (continuous) principle series of representations of SO(2, 1) and the discrete series. Consequently, edges may now be labeled either by discrete or continuous spins.”

In the paper

“More than an “existence theorem”, though, the (2 + 1)-dimensional models also provide a “nonuniqueness theorem”: Many approaches to the quantum theory are possible, and they are not all equivalent.”

In the paper

“At least for Λ = 0, the first order constraints coming from Equations (15, 16) are at most linear in the momenta, and are thus uncomplicated to solve.”

In the paper

“The momentum constraint fixes in terms of the scale factor λ, but it does so nonlocally.”

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

In the paper

“In particular, the AdS/CFT correspondence may impose boundary conditions that limit the topologies allowed in the sum.”

In the paper

“Saddle points are then Einstein manifolds, with actions proportional to the volume.”

In the paper

“Indeed, there are general reasons to expect the expression to be ill-defined: A conformal excitation gμν → e2φgμν contributes to IE with the wrong sign, and the action is unbounded below [142].”

What the sources say

• ?Not checked — no full text of the sourcecites [142] Gibbons, G.W., Hawking, S.W., and Perry, M.J., “Path integrals and the indefiniteness of the gravitational action”, Nucl. Phys. B, 138, 141–150, (1978). source

In the paper

“Lorentzian path integrals allow us to compute interesting topology-changing amplitudes, in which the universe tunnels from one spatial topology to another.”

In the paper

“Section 9.2 of [81] describes a partial gauge-fixing, which takes advantage of the fact that every metric on a three-manifold is conformal to one of constant scalar curvature.”

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

“But while this leads to some simplification, we are still left with an infinite-dimensional integral about which very little can yet be said.”

In the paper

“The model considered in [132] is not exactly the Ponzano-Regge model, and it is not clear that it is really “ordinary” quantum gravity.”

What the sources say

• ?Not checked — no full text of the sourcecites [132] Freidel, L., and Louapre, D., “Non-perturbative summation over 3D discrete topologies”, Phys. Rev. D, 68, 104004, (2003). Related online version (cited on 5 January 2005): http://arXiv.org/abs/hep-th/0211026. source

In the paper

“Hartle and Hawking therefore propose a “ground state” wave function 75”

What the sources say

• ?Not checked — no full text of the sourcecites [152] Hartle, J.B., and Hawking, S.W., “Wave function of the Universe”, Phys. Rev. D, 28, 2960–2975, (1983). source

In the paper

“This approach does not require a topology ℝ × Σ, and could potentially describe topology-changing amplitudes.”