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Cited by 677Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryBlack Holes and Theoretical Physics

The Spin-Foam Approach to Quantum Gravity

Open access
Perez, Alejandro·2013·Living Reviews in Relativity
DOI 10.12942/lrr-2013-3PMC PMC5255902
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Abstract

to the quantization of gravity. Special attention is payed to the pedagogical presentation of the recently-introduced new models for four-dimensional quantum gravity. The models are motivated by a suitable implementation of the path integral quantization of the Plebanski formulation of gravity on a simplicial regularization. The article also includes a self-contained treatment of 2+1 gravity. The simple nature of the latter provides the basis and a perspective for the analysis of both conceptual and technical issues that remain open in four dimensions.

AI snapshot

Written from the paper’s full text.

TL;DR

This review surveys the spin-foam program for quantum gravity, focusing on how path-integral BF theory is constrained to produce four-dimensional gravity models such as EPRL and Freidel–Krasnov. It presents 2+1 gravity as a solvable guide and reviews semiclassical evidence that the new models recover Regge-like geometry on fixed triangulations.

Key findings

  • The main four-dimensional candidates are the EPRL and Freidel–Krasnov models; their vertex amplitudes agree for Immirzi parameter γ < 1, while edge weights and measure choices differ. The Barrett–Crane model appears as the γ → ∞ limit.
  • In 2+1 Riemannian gravity, the spin-foam construction reproduces the Ponzano–Regge state sum with 6j-symbol vertex amplitudes; for a genus-g boundary surface, physical states are labeled by 6g − 6 quantum numbers.
  • Semiclassical large-spin analysis on a fixed triangulation indicates that EPRL/FK amplitudes are dominated by Regge-like configurations, with non-geometric configurations exponentially suppressed. Riemannian asymptotics involve two terms depending on the Immirzi parameter, while Lorentzian asymptotics involve a parity-related pair.
  • The Plebanski simplicity constraints do not close under commutation in the quantum theory, so they are imposed weakly or semiclassically; the EPRL construction is not a true projector, motivating the Warsaw modification that restores a projector in the Riemannian case.

Methods

  • Narrative review and synthesis of the spin-foam approach, not an empirical study. It develops canonical LQG, Plebanski/BF theory, coherent-state representations, and graphical calculus for spin-foam amplitudes.
  • The paper treats 2+1 Riemannian gravity as a fully solvable model and reviews 4D Riemannian and Lorentzian EPRL, Freidel–Krasnov, and Barrett–Crane models.
  • Main analysis tools are constraint quantization, BF state sums on 2-complexes, coherent states, and stationary-phase/asymptotic methods for large spins.

Limitations

  • The paper states many open problems: LQG dynamics and observables, the physical inner product, anomaly-free scalar constraint quantization, path-integral measure ambiguities, discretization dependence, and low-energy Lorentz invariance.
  • It notes that the EPRL amplitude is not a projector and that a Lorentzian generalization of the Warsaw projector modification is still lacking.
  • Evident from the text: the review is not original research, and much of the four-dimensional evidence relies on semiclassical/large-spin approximations on fixed triangulations rather than a demonstrated continuum limit.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.12942/lrr-2013-3

Cites

450 of 450 references verified0 citation errors · 0 not checkable

Statistics

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Claims the paper makes

0 of 203 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    The constraint algebra does not close.

    “For this reason it is said that the constraint algebra does not close.”
    Quote in the paperNot a substantial claim
  • Findingreasoning

    The previous action corresponds to the standard Hamiltonian formulation of general relativity in terms of SU(2) connection variables.

    “The previous action corresponds to the standard Hamiltonian formulation of general relativity in terms of SU(2) connection variables.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The densitized triad encodes the full background-independent Riemannian geometry of the spatial manifold Σ.

    “As it follows from (42), encodes the full background-independent Riemannian geometry of Σ.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    There are seven first-class constraints for the 18 phase-space variables.

    “We have seven (first class) constraints for the 18 phase-space variables ().”
    Quote in the paperNot a substantial claim
  • Findingnone given

    The constraint (49) coincides with the standard Gauss law of Yang-Mills theory.

    “The constraint (49) coincides with the standard Gauss law of Yang-Mills theory (e.g., in electromagnetism).”
    Quote in the paperSubstantial claim
  • May combine two claims. 98
  • Classified as probably not a substantial claim. 49
  • The quote only partly states this claim. 1

  • May combine two claims.

    Findingreasoning

    The reduced phase space is four dimensional, giving two physical degrees of freedom.

    “The reduced phase space is four dimensional and therefore the resulting number of physical degrees of freedom is two, as expected.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    For compact G, group integration yields the projector onto the invariant subspace, so SO(4) BF spin-foam amplitudes reduce to a sum over representations.

    “For compact G it is easy to prove using the invariance (and normalization) of the integration measure (84) that is the projector onto . In this way the spin-foam amplitudes of SO(4) BF theory reduce to 88”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The manifold is compact and orientable.

  • May combine two claims.

    Findingreasoning

    For γ = ±1, the linear simplicity constraint can be imposed strongly; for the plus sign it sets all left quantum numbers in the Spin(4) BF state sum to zero.

    “The first is to take γ = ±1, in which case the constraint (186) can be imposed strongly. More precisely, for the plus sign it reduces to the condition (or equivalently for the other sign ), which amounts to setting all the left quantum numbers in the BF state-sum for the group Spin(4) to zero.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The SU(2) subgroup of Spin(4) used to define the linear simplicity constraints is chosen arbitrarily at each tetrahedron and equivalently on each edge.

  • May combine two claims.

    Findingreasoning

    SU(2) coherent states have semiclassical properties: the expectation of the generators Ji is j n_i, the fluctuations are minimal with ΔJ^2 = ħ^2 j, and the fluctuations vanish in the limit ħ → 0 and j → ∞ with ħj held constant.

    “SU(2) coherent states have the usual semiclassical properties. Indeed, if one considers the operators Ji generators of su(2) one has 112 where ni is the corresponding three-dimensional unit vector for n ∈ S2. The fluctuations of are also minimal with ΔJ2 = ℏ2j, where we have restored ℏ for clarity.”
    • Also stated in: § The coherent-states representation · plain
      “The fluctuations go to zero in the limit ℏ → 0 and j → ∞ while ℏj is kept constant.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    In the Lorentzian model, unlike the Riemannian model, all spin values are allowed and no restrictions on the Immirzi parameter arise.

    “Unlike the Riemannian model, all values of spin j are allowed, and no restrictions on the Immirzi parameter arise.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    The commutation relation of the tetrahedron constraint implies that the constraint algebra is not closed and therefore cannot generally be imposed as operator equations on the states summed over in the BF partition function.

    “The previous commutation relation implies that the constraint algebra is not closed and therefore cannot be imposed as operator equations of the states summed over in the BF partition function in general.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The conditions p = γj and k = j that minimize the Master constraint solve its Lorentz-invariant version strongly, resolving the cylindrical-consistency issue.

    “Surprisingly, the conditions p = γj and k = j minimizing the Master constraint, solve its Lorentz invariant version strongly! This resolves the cylindrical consistency issue pointed out above.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The triangle Plebanski constraints are satisfied in the semiclassical sense; their path-integral expectation value is proportional to a quantity that vanishes in the limit ħ → 0, j → ∞ with ħj constant.

    “Direct evaluation shows that the previous diagram is proportional to , which vanishes in the semiclassical limit ℏ → 0, j → ∞ with ℏj = constant. We use the notation already adopted in (140) and call such a quantity . This concludes the proof that the triangle Plebanski constraints are satisfied in the semiclassical sense.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    In the Riemannian model with γ > 1, integer γ allows any half-integer spin, while rational γ values restrict spins as in the γ < 1 case.

    “Case γ > 1: In this case, γ ∈ ℕ would allow any j ∈ ℕ/2. Rational choices of γ have the same effect as in the previous case.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given

    In the Riemannian model with γ < 1, allowing arbitrary half-integer spins requires γ = 0 or γ = 1; rational γ values restrict the spins to a subclass.

    “Case γ < 1: If one wants the spin j ∈ ℕ/2 to be arbitrary, then the only possibilities are γ = 0 or γ = 1. Rational γ values would restrict the spins j to a subclass in ℕ/2.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    In the Riemannian Master-constraint criterion with γ > 1, the minimum eigenvalue is obtained for the stated representation and has mj = ħ2(γ2 − 1)j, which vanishes in the semiclassical limit ħ → 0, k → ∞ with ħk constant.

    “Case γ > 1: In this case, the minimum eigenvalue is obtained for 134 for which mj = ℏ2(γ2 − 1)j, and thus vanishes in the semiclassical limit ℏ → 0 and k → ∞ with ℏk = constant.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The Master constraint is invariant only under the SU(2) subgroup selected at each tetrahedron, and its lack of Lorentz invariance follows from its justification via canonical analysis in the time gauge.

    “Notice however, that the Master constraint is only invariant under the subgroup SU(2) ⊂ Spin(4) chosen at each tetrahedron. Its lack of Lorentz invariance is clear from the fact that its validity is justified from the canonical analysis in the time gauge.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    Setting jf = 0 in all faces yields the Barrett-Crane model and assigns all tetrahedra the unique Barrett-Crane intertwiner.

    “The solution is obtained by setting jf = 0 in all faces at the level of the amplitude (148). An immediate consequence of this is that and all tetrahedra are assigned a unique intertwiner: the Barrett-Crane intertwiner.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    In the Riemannian Master-constraint criterion with γ < 1, the minimum eigenvalue is obtained for the stated representation and has mj = ħ2(1 − γ2)j, which vanishes in the semiclassical limit ħ → 0, k → ∞ with ħk constant.

    “Case γ < 1: In this case the minimum eigenvalue is obtained for 133 i.e., one has where mj = ℏ2(1 − γ2)j, where we have restored the explicit dependence on ℏ so that it is apparent that the selected eigenvalue vanishes in the semiclassical limit ℏ → 0 and k → ∞ with ℏk = constant.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For γ > 1, the representations j+ = j− + j are of maximum weight, so a coherent-state resolution of the identity can be inserted to obtain the discrete action.

    “However, the representations j+ = j− + j are of maximum weight. Therefore, we can insert a coherent states resolution of the identity on the right representations and get: 177”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingcitation

    The Lorentzian EPRL vertex amplitude is divergent because of a redundant SL(2,C) integration, and it becomes finite when one arbitrary integration is removed.

    “The vertex amplitude is divergent due to the presence of a redundant integration over SL(2, ℂ), it becomes finite by dropping an arbitrary integration, i.e., removing any of the 5 boxes in the vertex expression [167].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [167] Engle, J. and Pereira, R., “Regularization and finiteness of the Lorentzian loop quantum gravity vertices”, Phys. Rev. D, 79, 084034, (2009). [DOI], [arXiv:0805.4696]. (Cited on pages 68 and 73.) source

      Not checked in the free run.

  • May combine two claims.

    Findingreasoning

    For γ < 1, the coherent states appearing in the EPRL amplitude satisfy the linear simplicity constraints in expectation values and therefore solve them in the usual semiclassical sense.

    “Moreover, by construction the coherent states participating in the previous amplitude satisfy the linear constraints (127) in expectation values, namely Thus the coherent states participating in the above representation of the EPRL amplitudes solve the linear simplicity constraints in the usual semiclassical sense.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The number of quantum numbers needed to label a basis element is 6g − 6, which equals the dimension of the moduli space of SU(2) flat connections on a genus-g Riemann surface and the number of classical degrees of freedom.

    “The number of quantum numbers necessary to label the basis element is 6g − 6, corresponding to the dimension of the moduli space of SU(2) flat connections on a Riemann surface of genus g. This is the number of degrees of freedom of the classical theory.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The limit ε → 0 exists and gives a closed expression for the physical inner product.

    “The limit ϵ → 0 exists and one can give a closed expression for the physical inner product.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    In two dimensions, the action of two neighboring plaquette sums on the vacuum is equivalent to a single larger plaquette action from fusion, implying trivial scaling under refinement and a well-defined ε → 0 limit.

    “In two dimensions the action of two neighboring plaquette-sums on the vacuum is equivalent to the action of a single larger plaquette action obtained from the fusion of the original ones. This implies the trivial scaling of the physical inner product under refinement of the regulator and the existence of a well-defined limit ϵ → 0.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For arbitrary spin networks, the vertex amplitude corresponds to 3nj-symbols and the physical inner product takes the same general form.

    “More generally, for arbitrary spin networks, the vertex amplitude corresponds to 3nj-symbols, and 〈s, s′〉p takes the same general form.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given

    Equation (242) is the discrete version of Eq. (240).

    “Notice that the previous equation corresponds to the discrete version of Eq. (240).”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The 3-dimensional manifold is compact and orientable.

  • May combine two claims.

    Findingreasoning

    The assignment of intertwiners to edges follows from integrating over the connection, and this is a general property of local spin foams.

    “Notice that the assignment of intertwiners to edges is a consequence of the integration over the connection. This is not a particularity of this example, but rather a general property of local spin foams.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The 3-dimensional manifold is compact and orientable.

  • May combine two claims.

    Findingcitation

    The physical inner product is finite and satisfies all properties of an inner product.

    “One can also show that it is finite,28 and satisfies all the properties of an inner product [107].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [107] Buffenoir, E. and Noui, K., “Unfashionable observations about three dimensional gravity”, arXiv, e-print, (2003). [arXiv:gr-qc/0305079]. (Cited on page 87.)

      Not checked in the free run.

  • May combine two claims.

    Findingreasoning

    The spin-foam amplitudes are purely combinatorial and can be computed explicitly from the action of the loop operator.

    “The spin-foam amplitudes are purely combinatorial and can be explicitly computed from the simple action of the loop operator in .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingcitation

    The Riemannian EPRL vertex asymptotics is the unbalanced square of the SU(2) asymptotics, and the resulting asymptotic formula is also valid for γ > 1.

    “implies the asymptotic formula for the vertex amplitude to be given by the unbalanced square of the above formula [62], namely One can write the previous expression as 217”
    • Also stated in: § Vertex amplitudes asymptotics · boosted
      “Remarkably, the above asymptotic formula is also valid for the case of γ > 1 [61].”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [62] Barrett, J.W., Dowdall, R.J., Fairbairn, W.J., Gomes, H. and Hellmann, F., “A Summary of the Asymptotic Analysis for the EPRL Amplitude”, in Kowalski-Glikman, J., Durka, R. and Szczachor, M., eds., The Planck Scale: Proceedings of the XXV Max Born Symposium, Wroclaw, Poland, 29 June–3 July 2009, AIP Conference Proceedings, 1196, pp. 36–43, (American Institute of Physics, Melville, NY, 2009). [DOI], [arXiv:0909.1882 [gr-qc]]. (Cited on pages 80 and 82.) source

      Not checked in the free run.

    • cites [61] Barrett, J.W., Dowdall, R.J., Fairbairn, W.J., Gomes, H. and Hellmann, F., “Asymptotic analysis of the EPRL four-simplex amplitude”, J. Math. Phys., 50, 112504, (2009). [DOI], [arXiv:0902.1170 [gr-qc]]. (Cited on pages 67, 80, 81, and 82.) source

      Not checked in the free run.

  • May combine two claims.

    Findingreasoning

    When Δ is a simplicial complex, the vertex amplitude is a 6j-symbol and the partition function takes the Ponzano-Regge form.

    “When Δ is a simplicial complex, all the edges in are 3-valent and vertices are 4-valent (one such vertex is emphasized in Figure 5, the intersection with the surrounding S2 is shown in dotted lines). Consequently, the vertex amplitude is given by the contraction of the corresponding four 3-valent intertwiners, i.e., a 6j-symbol. In that case, the partition function takes the familiar Ponzano-Regge”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [309] Ponzano, G. and Regge, T., “Semiclassical limit of Racah Coeficients”, in Bloch, F., Cohen, S.G., de Shalit, A., Sambursky, S. and Talmi, I., eds., Spectroscopy and Group Theoretical Methods in Physics: Racah Memorial Volume, pp. 1–58, (North-Holland, Amsterdam, 1968). (Cited on pages 43, 80, 92, and 95.)

      Not checked in the free run.

    • The authors state this as an assumption: The 3-dimensional manifold is compact and orientable.

  • May combine two claims.

    Findingreasoning

    The sum over spins in (247) is typically divergent, so discretization independence is only formal without a suitable regularization.

    “The sum over spins in (247) is typically divergent, as indicated by the previous equation. Divergences occur due to infinite volume factors corresponding to the topological gauge freedom (225) (see [186]).30 The factor τ in (248) represents such a volume factor. It can also be interpreted as a δ(0) coming from the existence of a redundant delta function in (242).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [186] Freidel, L. and Louapre, D., “Diffeomorphisms and spin foam models”, Nucl. Phys. B, 662, 279–298, (2003). [DOI], [arXiv:gr-qc/0212001]. (Cited on pages 43, 92, 95, 99, and 100.) source

      Not checked in the free run.

    • The authors state this as an assumption: The 3-dimensional manifold is compact and orientable.

  • May combine two claims.

    Findingreasoning

    One can partially gauge fix the topological gauge freedom at the level of the discretization, which eliminates bubbles from the 2-complex.

    “One can partially gauge fix this freedom at the level of the discretization. This has the effect of eliminating bubbles from the 2-complex.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The 3-dimensional manifold is compact and orientable.

  • May combine two claims.

    Findingcitation

    Gauge-fixing conditions generally exist in spin-foam models and are used to regularize the vertex amplitudes of the Lorentzian models.

    “These conditions generally exist in spin-foam models and are used to regularize the vertex amplitudes of the Lorentzian models (we have illustrated this at the end of Section 8.3; for a general treatment see [186]).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [186] Freidel, L. and Louapre, D., “Diffeomorphisms and spin foam models”, Nucl. Phys. B, 662, 279–298, (2003). [DOI], [arXiv:gr-qc/0212001]. (Cited on pages 43, 92, 95, 99, and 100.) source

      Not checked in the free run.

  • May combine two claims.

    Findingreasoning

    In the Lorentzian sector with internal gauge group SL(2, C), the internal gauge orbits have infinite volume and the lattice path integral would diverge without an appropriate gauge fixing condition.

    “In the Lorentzian sector (internal gauge group SL(2, ℂ)), the internal gauge orbits have infinite volume and the lattice path integral would diverge without an appropriate gauge fixing condition.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The transition amplitudes imply the skein relations that define the physical Hilbert space, and the spin-foam quantization is equivalent to the canonical one.

    “The transition amplitudes imply the skein relations that define the physical Hilbert space! The spin-foam quantization is equivalent to the canonical one.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    If the internal gauge group is compact, anomaly-free measures are defined using appropriate variables and invariant measures, and gauge fixing is not necessary for the amplitudes to be well defined.

    “If this gauge group is compact, then anomaly free measures are defined using appropriate variables and invariant measures. In this case, gauge fixing is not necessary for the amplitudes to be well defined.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    For 2+1 gravity with M = S2 × R (g = 0), the physical Hilbert space is one-dimensional, the classical theory has zero degrees of freedom, and there is no non-trivial Dirac observable.

    “However, a close look at the physical inner product defined by P permits one to conclude that the physical Hilbert space is one dimensional — the classical theory has zero degrees of freedom and so there is no non-trivial Dirac observable in the quantum theory.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingreasoning

    The physical Hilbert space can be constructed by considering equivalence classes under states with zero transition amplitude with all elements of the kinematical Hilbert space (null states).

    “We can construct the physical Hilbert space by considering equivalence classes under states with zero transition amplitude with all the elements of , i.e., null states.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    Transition amplitudes can be defined using a manifold with boundaries, and spin foams induce spin networks on the boundaries.

    “Transition amplitudes can be defined along similar lines using a manifold with boundaries. Given Δ, then defines graphs on the boundaries. Consequently, spin foams induce spin networks on the boundaries.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For simply connected Σ, the gauge fixing is complete, and one can eliminate bubbles and compute finite transition amplitudes.

    “In the case of simply connected Σ, the gauge fixing is complete. One can eliminate bubbles and compute finite transition amplitudes.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The 3-dimensional manifold is compact and orientable.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given

    It remains unclear whether these colored tensor models correspond to a physical theory.

    “However, it is not yet clear if there is a sense in which these models correspond to a physical theory.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given

    Little is known at present about four-dimensional gravity models in this approach.

    “Not much is known about gravity models in four dimensions at this stage.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given

    An anomaly-free measure can be multiplied by any gauge invariant function and yield a new anomaly-free measure, and this ambiguity is present in most spin-foam models defined so far.

    “In fact, according to our tentative definition, an anomaly-free measure can be multiplied by any gauge invariant function and yield a new anomaly free measure. This kind of ambiguity is not wanted; however, it is, in fact, present in most of the spin-foam models defined so far”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingreasoning

    For loop quantum gravity, a systematic first-principles analysis of the low-energy regime is still lacking.

    “In the context of LQG, a systematic first-principles analysis of such properties of the low-energy regime remains open.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For topological theories such as 3D gravity, the result is independent of the chosen cellular decomposition, and a single discretization suffices to capture the degrees of freedom.

    “Since the theory has no local excitations (no gravitons), the result is independent of the chosen cellular decomposition. A single discretization suffices to capture the degrees of freedom of the topological theory.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    The discretization restricts the allowed spin foams to those obtained by coloring the underlying 2-complex and restricts boundary spin-network states by fixing a finite boundary graph, acting as a truncation/regulator.

    “This cutoff is reflected by the fact that only a restrictive set of spin foams (spin network histories) is allowed in the path integral: those that can be obtained by all possible coloring of the underlying 2-complex. In addition, it restricts the number of possible 3-geometry states (spin network states) on the boundary by fixing a finite underlying boundary graph. This represents a truncation in”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Findingreasoning

    Transition amplitudes are discretization dependent, and a consistent definition of the path integral using spin foams should include a prescription to eliminate this dependence.

    “However, transition amplitudes are discretization dependent now. A consistent definition of the path integral using spin foams should include a prescription to eliminate this discretization dependence.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given

    Loop quantum gravity quantizes gravitational degrees of freedom covariantly, and current hints suggest its framework may remove both classical singularities and ultraviolet issues.

    “Loop quantum gravity (LQG) is a framework to address this task. The degrees of freedom of gravity are quantized in accordance to the principles of general covariance. At the present stage there are some indications that both the singularity problems of classical general relativity as well as the UV problem of standard QFTs might vanish in the framework.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given

    Conventional quantum field theories suffer ultraviolet divergences from high-energy processes; these are expected to vanish once gravitational quantum fluctuations are accounted for.

    “quantum field theory is plagued by singularities too (UV divergences) coming from the contribution of arbitrary high-energy quantum processes. This limitation of standard QFTs is expected to disappear once the quantum fluctuations of the gravitational field, involving the dynamic treatment of spacetime geometry, are appropriately taken into account.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning

    Because conventional quantum field theories need a fixed spacetime geometry and lack general covariance, they cannot handle quantum gravity.

    “standard QFTs are not generally covariant, as they can only be defined if a non-dynamical spacetime geometry is provided”
    • Also stated in: § Quantum Gravity: A Unifying Framework for Fundamental Physics · plain
      “the standard QFTs used to describe other fundamental forces are not appropriate to tackle the problem of quantum gravity.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisenone given

    In three dimensions the radial integration can be performed and the resulting amplitudes agree with the results of Section 14.5.

    “In 3 dimensions the radial integration can be done and the corresponding amplitudes coincide with the results of Section 14.5.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The dimensional reasoning that works for Yang-Mills does not apply to gravity, so the impact of these ambiguities on predictions requires dynamical study.

    “The simple dimensional argument given for Yang-Mills theory is no longer valid in the case of gravity, and a full dynamical investigation of their effect in physical predictions must be investigated.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    Proposed quantizations of the scalar constraint in loop quantum gravity yield well-defined operators without ultraviolet divergences, but whether they are anomaly-free remains an open question.

    “Concrete quantizations producing well-defined operators are available: no UV divergences are encountered by these proposals; the fundamental discreteness of quantum geometry in LQG plays the role of physical regulator at the Planck scale [354].”
    • Also stated in: § Loop quantum gravity · plain
      “make the issue of whether any of the proposed quantizations of the constraints are anomaly-free a difficult open question.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [354] Thiemann, T., “Quantum spin dynamics (QSD): V: Quantum gravity as the natural regulator of matter quantum field theories”, Class. Quantum Grav., 15, 1281–1314, (1998). [DOI], [arXiv:gr-qc/9705019]. (Cited on pages 14 and 32.) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The basic idea behind the Barrett-Crane model was the quantum tetrahedron introduced by Barbieri and generalized to four dimensions.

    “The basic idea behind the definition was that of the quantum tetrahedron introduced by Barbieri in [58] and generalized to 4D in [40].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [58] Barbieri, A., “Quantum tetrahedra and simplicial spin networks”, Nucl. Phys. B, 518, 714–728, (1998). [DOI], [arXiv:gr-qc/9707010]. (Cited on page 19.) source

      Not checked in the free run.

    • cites [40] Baez, J.C. and Barrett, J.W., “The Quantum Tetrahedron in 3 and 4 Dimensions”, Adv. Theor. Math. Phys., 3, 815–850, (1999). (Cited on page 19.) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationnone given

    In the Iwasaki model, the integration over the radial variables r involves products of Bessel functions and its behavior has not been analyzed in detail.

    “Unfortunately, the integration over the radial variables r involves products of Bessel functions and its behavior is not analyzed in detail.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisereasoning

    Perturbative gravity's non-renormalizability is commonly compared to Fermi's four-fermion description, motivating a deeper theory.

    “The non-renormalizability of perturbative gravity is often explained through an analogy with the (non-renormalizable) Fermi’s four-fermion effective description of the weak interaction.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisereasoning

    Loop quantum gravity is a background-independent, non-perturbative approach built from canonical quantization using connection variables.

    “The theory is explicitly formulated in a background-independent, and therefore, non-perturbative fashion. The theory is based on the Hamiltonian (or canonical) quantization of general relativity in terms of variables that are different from the standard metric variables.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    At present, loop quantum gravity's discreteness does not settle non-renormalizability; the central problem is instead ambiguity in the quantization procedure.

    “Is this a complete answer to the non-renormalizability problem of gravity mentioned above? Unfortunately, the answer to this question is in the negative at this stage. The reason is that the true answer to the question of renormalizabity or not renormalizability is not in the presence of divergences but in the degree of intrinsic ambiguity of the quantization recipe applied to a field theory.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    For generic quantum states, separating the metric into a fixed background and perturbations is not consistent, since such a background lacks intrinsic meaning and can yield conflicting causal structures.

    “the previous separation of degrees of freedom has no intrinsic meaning in general relativity”
    • Also stated in: § Why non-perturbative quantum gravity? · plain
      “the splitting in Eq. (1) is inconsistent when considering general states with their arbitrary quantum excitations at all scales.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    The kernel's properties have not been examined thoroughly.

    “The properties of the kernel of have not been studied in detail.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Premisenone given

    Defining the physical inner product in loop quantum gravity is still an unresolved issue.

    “The definition of the physical inner product is still an open issue.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    In four dimensions, the spin-foam approach remains an ongoing program with unresolved problems.

    “In four dimensions, the spin-foam approach is a program in progress, where various issues still remain open.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Gambini and Pullin introduced a very simple model obtained by modifying the BF theory skein relations.

    “Gambini and Pullin [193] introduced a very simple model obtained by modification of the BF theory skein relations.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [193] Gambini, R. and Pullin, J., “A finite spin-foam-based theory of three- and four-dimensional quantum gravity”, Phys. Rev. D, 66, 024020, (2002). [DOI], [gr-qc/0111089]. (Cited on page 20.) source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    Comparable ambiguities arise in loop quantum gravity's constraint regularization and also in the spin-foam framework.

    “Similar ambiguities are present in LQG; they appear in the regularization of the constraints in the canonical formulation [303]. These ambiguities are also present in the spin-foam formulation, which is the central subject of this review.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [303] Perez, A., “On the regularization ambiguities in loop quantum gravity”, Phys. Rev. D, 73, 044007, (2006). [DOI], [arXiv:gr-qc/0509118]. (Cited on page 12.) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The definition of the Gambini-Pullin spin-foam model is not explicit because the theory is directly defined by the physical skein relations.

    “The definition of this spin-foam model is not explicit since the theory is directly defined by the physical skein relations.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [193] Gambini, R. and Pullin, J., “A finite spin-foam-based theory of three- and four-dimensional quantum gravity”, Phys. Rev. D, 66, 024020, (2002). [DOI], [gr-qc/0111089]. (Cited on page 20.) source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    Loop quantum gravity predicts that geometric operators for area and volume have discrete spectra, with effects at the Planck scale.

    “The main prediction of LQG is the discreteness [344] of the spectrum of geometrical operators such as area and volume.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [344] Smolin, L., “Recent developments in nonperturbative quantum gravity”, in Pérez-Mercader, J., Sola, J. and Verdaguer, E., eds., Quantum Gravity and Cosmology, Proceedings of the XXII GIFT International Seminar on Theoretical Physics, Sant Feliu de Guíxols, Catalonia, Spain, 3–8 June 1991, (World Scientific, Singapore, 1992). [hep-th/9202022]. (Cited on pages 13 and 32.)

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Iwasaki defines a spin-foam model of self-dual Riemannian gravity by directly discretizing the continuous Ashtekar formulation of general relativity on a lattice.

    “Iwasaki defines a spin-foam model of self-dual Riemannian gravity9 by a direct lattice discretization of the continuous Ashtekar formulation of general relativity.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    For generally covariant systems, ordinary time evolution is gauge, and transition amplitudes between states at distinct times have no meaningful definition.

    “In generally-covariant systems, conventional time evolution is pure gauge”
    • Also stated in: § The path integral for generally covariant systems · plain
      “There is no meaningful notion of transition amplitude between states at different times t1 > t0 or equivalently a notion of “unitary time evolution” represented by an operator U(t1 − t0).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    The Gambini-Pullin modification forbids some skein relations, which frees local excitations of a field theory.

    “Gambini and Pullin define a model by modifying these amplitudes so that some of the skein relations are now forbidden. This simple modification frees local excitations of a field theory.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [193] Gambini, R. and Pullin, J., “A finite spin-foam-based theory of three- and four-dimensional quantum gravity”, Phys. Rev. D, 66, 024020, (2002). [DOI], [gr-qc/0111089]. (Cited on page 20.) source

      Not checked in the free run.

  • May combine two claims.

    Premisereasoning

    A non-renormalizable theory contains infinitely many parameters that observations must fix, so its predictive capacity is severely restricted.

    “In a non-renormalizable theory (e.g., perturbative quantum gravity) the situation is similar except for the fact that there are infinitely many parameters to be fixed by renormalization conditions. As the latter must be specified by observations, a non-renormalizable theory has very limited predictive power.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisereasoning

    A path-integral formulation for generally covariant systems is available and serves to impose quantum constraints.

    “Nevertheless, a path integral representation of generally covariant systems exists and arises as a tool for implementing the constraints in the quantum theory as we argue below.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: For the formal derivation of the projector, the paper assumes there is only one constraint.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisereasoning

    Iwasaki writes spin-foam amplitudes in which the integration over the connection and the e variables can be computed explicitly.

    “Iwasaki writes down an expression for the spin-foam amplitudes in which the integration over the connection and the e’s can be computed explicitly.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    Some indirect evidence of a possible non-trivial continuum limit has been obtained in some versions of the Markopoulou-Smolin model in 1+1 dimensions.

    “Some indirect evidence of a possible non-trivial continuum limit has been obtained in some versions of the model in 1+1 dimensions.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    The rules of the Markopoulou-Smolin model generate surface-like excitations similar to those in standard spin-foam models but endow the foam with a notion of causality.

    “These rules generate surface-like excitations of the same kind we encounter in the more standard spin-foam model, but endow the foam with a notion of causality.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [267] Markopoulou, F. and Smolin, L., “Causal evolution of spin networks”, Nucl. Phys. B, 508, 409–430, (1997). [DOI]. (Cited on page 19.) source

      Not checked in the free run.

    • cites [264] Markopoulou, F., “Dual formulation of spin network evolution”, arXiv, e-print, (1997). [arXiv:gr-qc/9704013]. (Cited on page 19.)

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    At the Planck scale, gravitational quantum fluctuations make observation impossible without disturbing the field, eliminating test observers and any fundamental spacetime picture.

    “At the Planck scale (ℓp) the quantum fluctuations of the gravitational field become so important that there is no way (not even in principle6) to make observations without affecting the gravitational field. The notion of test fields and test observer is lost and with it the usual spacetime representation.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    A difficulty in Reisenberger's model is that the operator algebra fails to close, making sharp constraint imposition overly restrictive.

    “However, there is a difficulty with this procedure: the algebra of operators do not close so that imposing the constraints sharply becomes a too strong condition on the BF configurations.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Loop quantum gravity offers strong hints that its mathematical structure regularizes divergences, yet resolving the related ambiguities is still pending.

    “In loop quantum gravity there is strong indications that the mathematical framework of the theory provides a regularization of divergences. It remains to settle the crucial issue of how to fix the associated ambiguities.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    Their construction covers BF theories with cosmological terms, two-dimensional Yang-Mills, and Riemannian self-dual gravity.

    “Their formulation can be used to describe various theories of interest such as BF theories with cosmological terms, Yang-Mills theories (in 2 dimensions) and Riemannian self-dual gravity.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    The real connection formulation of general relativity presented here is a peculiar property of four dimensions due to the special property (45).

    “To conclude this section, let us point out that the real connection formulation of general relativity presented here is a peculiar property of four dimensions due to the special property (45).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    The Freidel-Krasnov models are arguably the same as the EPRL model for a suitable range of the Immirzi parameter, with vertex amplitudes coinciding for γ < 1.

    “Indeed, these models are arguably the same as the EPRL model for a suitable range of the Immirzi parameter (the vertex amplitudes coincide for γ < 1 as we will discuss in the sequel).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [182] Freidel, L. and Krasnov, K., “A new spin foam model for 4D gravity”, Class. Quantum Grav., 25, 125018, (2008). [DOI], [arXiv:0708.1595 [gr-qc]]. (Cited on pages 21, 23, 36, 47, 49, 51, 65, and 74.) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The coherent intertwiner basis relevant for the definition of spin foams and the 4D quantum gravity amplitude was also derived by Livine and Speziale.

    “The coherent intertwiner basis relevant for the definition of spin foams and the 4D quantum gravity amplitude was also derived by Livine and Speziale in [254].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [254] Livine, E.R. and Speziale, S., “New spinfoam vertex for quantum gravity”, Phys. Rev. D, 76, 084028, (2007). [DOI], [arXiv:0705.0674 [gr-qc]]. (Cited on pages 21, 47, 65, and 74.) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    The fundamental difference between LQG and standard gauge theories is the absence of any non-dynamical background field.

    “However, the fundamental difference is the absence of any non-dynamical background field in the theory.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    LQG is defined using Dirac quantization.

    “LQG is defined using Dirac quantization.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The primary constraint is exactly the linear simplicity constraints that are rediscovered in the simplicial setting and imposed at all tetrahedra to satisfy the Plebanski constraints and produce a state sum model for gravity from BF theory.

    “This is exactly the linear simplicity constraints that we will rediscover in the simplicial setting of Sections 7 and 8, and which will be imposed in a suitable sense at all tetrahedra of the cellular decomposition in order to satisfy the Plebanski constraints and produce a state sum model for gravity out of that of BF theory.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    LQG is a proposal for implementing the quantization program established by Dirac, Wheeler, and DeWitt.

    “LQG is a proposal for the implementation of the quantization program established in the 1960s by Dirac, Wheeler, and DeWitt, among others (for recent reviews see [35, 356, 323]).”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [35] Ashtekar, A. and Lewandowski, J., “Background independent quantum gravity: A status report”, Class. Quantum Grav., 21, R53–R152, (2004). [DOI], [arXiv:gr-qc/0404018]. (Cited on pages 12, 14, 22, 28, and 29.) source

      Not checked in the free run.

    • cites [356] Thiemann, T., “Introduction to Modern Canonical Quantum General Relativity”, arXiv, e-print, (2001). [arXiv:gr-qc/0110034]. (Cited on pages 23 and 29.)

      Not checked in the free run.

    • cites [323] Rovelli, C., “Loop Quantum Gravity”, Living Rev. Relativity, 11, lrr-2008-5 (2008). URL (accessed 12 September 2011): http://www.livingreviews.org/lrr-2008-5. (Cited on page 29.)

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    The EPRL and Freidel-Krasnov models are the most developed candidates for spin-foam models in four dimensions.

    “They are the most developed candidates in four dimensions and we will review their properties in detail in the following part.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    The EPRL model is a modification of the Barrett-Crane model that was recently introduced and extended to arbitrary Immirzi parameter.

    “A modification of the Barrett-Crane model was recently introduced in [168, 169] and extended for arbitrary Immirzi parameter in [164].”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [168] Engle, J., Pereira, R. and Rovelli, C., “Loop-Quantum-Gravity Vertex Amplitude”, Phys. Rev. Lett., 99, 161301, (2007). [DOI], [arXiv:0705.2388 [gr-qc]]. (Cited on pages 20, 36, 49, and 74.) source

      Not checked in the free run.

    • cites [169] Engle, J., Pereira, R. and Rovelli, C., “Flipped spinfoam vertex and loop gravity”, Nucl. Phys. B, 798, 251–290, (2008). [DOI], [arXiv:0708.1236 [gr-qc]]. (Cited on pages 20 and 36.) source

      Not checked in the free run.

    • cites [164] Engle, J., Livine, E.R., Pereira, R. and Rovelli, C., “LQG vertex with finite Immirzi parameter”, Nucl. Phys. B, 799, 136–149, (2008). [DOI], [arXiv:0711.0146 [gr-qc]]. (Cited on pages 20, 36, 49, 58, and 68.) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    BF theory in any dimension can be coupled to topological defects.

    “As in the three-dimensional case, BF theory in any dimensions can be coupled to topological defects [41].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [41] Baez, J.C. and Perez, A., “Quantization of strings and branes coupled to BF theory”, Adv. Theor. Math. Phys., 11, 3, (2007). [arXiv:gr-qc/0605087]. (Cited on page 46.)

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    One-dimensional particles can be introduced in four-dimensional BF theory and gravity.

    “It is also possible to introduce one-dimensional particles in four-dimensional BF theory and gravity as shown in [180].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [180] Freidel, L., Kowalski-Glikman, J. and Starodubtsev, A., “Particles as Wilson lines of gravitational field”, Phys. Rev. D, 74, 084002, (2006). [DOI], [arXiv:gr-qc/0607014]. (Cited on page 46.) source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisereasoning

    The spin-foam approach is an attempt to define the path integral quantization of gravity using what has been learned from LQG.

    “The spin-foam approach is an attempt to define the path integral quantization of gravity using what we have learned from LQG.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Two-dimensional BF theory is related to usual two-dimensional gravity through constraints in a way similar to the four-dimensional case.

    “on the other hand, it is related to usual 2D gravity via constraints in a way similar to the one exploited in four dimensions (see Section 7).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [295] Oriti, D., Rovelli, C. and Speziale, S., “Spinfoam 2d quantum gravity and discrete bundles”, Class. Quantum Grav., 22, 85–108, (2005). [DOI], [arXiv:gr-qc/0406063]. (Cited on page 42.) source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    The quantum-deformed version of the three-dimensional amplitudes leads to the Turaev-Viro model, expected to correspond to three-dimensional Riemannian gravity with a positive cosmological constant.

    “The quantum deformed version of the above amplitudes lead to the Turaev-Viro model [364], which is expected to correspond to the quantization of three-dimensional Riemannian gravity in the presence of a non-vanishing positive cosmological constant.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [364] Turaev, V.G. and Viro, O.Y., “State sum invariants of 3-manifolds and quantum 6j-symbols”, Topology, 31, 865–902, (1992). [DOI]. (Cited on pages 43, 93, and 95.) source

      Not checked in the free run.

    • The authors state this as an assumption: The manifold is compact and orientable.

  • Classified as probably not a substantial claim.

    Premisenone given

    In LQG, the notion of a quantum state of 3-geometry is rigorously defined in terms of spin-network states.

    “The notion of a quantum state of 3-geometry is rigorously defined in terms of spin-network states.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    BF amplitudes are independent of the regulating triangulation, and with suitable regularization of bubble divergences the BF partition function is topologically invariant and discretization independent.

    “using this graphical notation one can easily prove the discretization independence of the BF amplitudes.”
    • Also stated in: § Special cases · plain
      “if an appropriate regularization of bubble divergences is provided, (86) is topologically invariant and the spin-foam path integral is discretization independent.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [286] Oeckl, R., Discrete gauge theory: From lattices to TQFT, (Imperial College Press, London, UK, 2005). [Google Books]. (Cited on page 40.) source

      Not checked in the free run.

    • cites [291] Oeckl, R. and Pfeiffer, H., “The dual of pure non-Abelian lattice gauge theory as a spin foam model”, Nucl. Phys. B, 598, 400–426, (2001). [DOI], [arXiv:hep-th/0008095]. (Cited on pages 40 and 103.) source

      Not checked in the free run.

    • cites [206] Girelli, F., Oeckl, R. and Perez, A., “Spin foam diagrammatics and topological invariance”, Class. Quantum Grav., 19, 1093–1108, (2002). [DOI], [arXiv:gr-qc/0111022]. (Cited on pages 40 and 93.) source

      Not checked in the free run.

    • The authors state this as an assumption: The manifold is compact and orientable.

  • May combine two claims.

    Premisereasoning

    In the generic case, the simplicity constraints do not close into a first-class algebra, so they cannot be imposed strongly and must be imposed in a weaker sense.

    “Due to the fact that the constraints do not form a closed (first class) algebra in the generic case, one needs to devise a weaker sense in which they are to be imposed.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    In four-dimensional BF theory, topological defects are string-like and can carry extra degrees of freedom such as topological Yang-Mills fields.

    “In the four-dimensional case, defects are string-like [175] and can carry extra degrees of freedom such as topological Yang-Mills fields [275].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [175] Fairbairn, W.J. and Perez, A., “Extended matter coupled to BF theory”, Phys. Rev. D, 78, 024013, (2008). [DOI], [arXiv:0709.4235 [gr-qc]]. (Cited on page 46.) source

      Not checked in the free run.

    • cites [275] Montesinos, M. and Perez, A., “Two-dimensional topological field theories coupled to four-dimensional BF theory”, Phys. Rev. D, 77, 104020, (2008). [DOI], [arXiv:0711.2875 [gr-qc]]. (Cited on page 46.) source

      Not checked in the free run.

  • May combine two claims.

    Premisereasoning

    Regularization choices for the scalar constraint have to be made and the result is not unique.

    “Regularization choices have to be made and the result is not unique.”
    Quote in the paperNot a substantial claim0 of 4 sources checked
    • cites [352] Thiemann, T., “Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity”, Phys. Lett. B, 380, 257–264, (1996). [DOI]. (Cited on page 32.) source

      Not checked in the free run.

    • cites [201] Gaul, M. and Rovelli, C., “A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements”, Class. Quantum Grav., 18, 1593–1624, (2001). [DOI]. (Cited on page 32.) source

      Not checked in the free run.

    • cites [135] Di Bartolo, C., Gambini, R., Griego, J. and Pullin, J., “Canonical quantum gravity in the Vassiliev invariants arena: II. Constraints, habitats and consistency of the constraint algebra”, Class. Quantum Grav., 17, 3239–3264, (2000). (Cited on pages 32 and 33.) source

      Not checked in the free run.

    • cites [136] Di Bartolo, C., Gambini, R., Griego, J. and Pullin, J., “Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants”, Phys. Rev. Lett., 84, 2314–2317, (2000). [DOI]. (Cited on pages 32 and 33.) source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    Unitary irreducible representations of SL(2, C) are infinite-dimensional and labelled by a positive real number p and a half-integer k.

    “Unitary irreducible representations of SL(2, ℂ) are infinite dimensional and are labelled by a positive real number p ∈ ℝ+ and a half-integer k ∈ ℕ/2.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [338] Ruhl, W., The Lorentz Group and Harmonic Analysis, (W.A. Benjamin, New York, 1970). (Cited on pages 69 and 72.)

      Not checked in the free run.

    • cites [203] Gel’fand, I.M., Graev, M.I. and Vilenkin, N.Y., Generalized Functions, Vol. 5: Integral geometry and representation theory, (Academic Press, New York; London, 1966). (Cited on pages 69 and 72.)

      Not checked in the free run.

    • cites [204] Gel’fand, I.M., Minlos, R.A. and Shapiro, Z.Y., Representations of the rotation and Lorentz groups and their applications, (Pergamon Press, London; MacMillan, New York, New York, 1963). Translated from the Russian edition (Moscow, 1958). (Cited on page 69.)

      Not checked in the free run.

  • May combine two claims. The quote only partly states this claim.

    Interpretationnone given

    The amplitude (167) fails the embedding-independence requirement because PEPRL is not a projector.

    “The amplitude (167) fails this requirement due to .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    Strong imposition of the Master constraint would require finding its kernel, but generically the associated positive operator has no zero eigenvalue because the constraint algebra is open.

    “Strong imposition of the constraints would amount to looking for the kernel of the Master constraint Mf. However, generically the positive operator associated with the Master constraint does not contain the zero eigenvalue in the spectrum due to the open nature of the constraint algebra (187).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    The model of [168] corresponds to the sector γ = ∞, which has been shown to be topological.

    “They also emphasize how the model [168] corresponds to the sector γ = ∞, which has been shown to be topological [246].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [168] Engle, J., Pereira, R. and Rovelli, C., “Loop-Quantum-Gravity Vertex Amplitude”, Phys. Rev. Lett., 99, 161301, (2007). [DOI], [arXiv:0705.2388 [gr-qc]]. (Cited on pages 20, 36, 49, and 74.) source

      Not checked in the free run.

    • cites [246] Liu, L., Montesinos, M. and Perez, A., “A topological limit of gravity admitting an SU(2) connection formulation”, Phys. Rev. D, 81, 064033, (2010). [DOI], [arXiv:0906.4524 [gr-qc]]. (Cited on page 74.) source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    From the Master-constraint viewpoint, the best choice is k = j, which gives p = γj.

    “As we will see in what follows the best choice from the point of view of the Master-constraint criterion is k = j, from which it follows that 189”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The Master-constraint criterion is advantageous for cylindrical consistency because the Gupta-Bleuler choice p = γ(j + 1) would give j = 0 spin-network links a non-trivial amplitude, conflicting with diffeomorphism invariance.

    “However, the advantage of the Master-constraint criterion is clear from the point of view of cylindrical consistency as the choice p = γ(j + 1) obtained in the previous subsection would imply that j = 0 spin-network links would be have a non-trivial amplitude in the path integral defined in Section 7.3. Such thing would be in sharp conflict with diffeomorphism invariance in the model.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning

    There are two exceptional possibilities: γ = ±i, where the constraint could in principle be imposed strongly but the self-dual sector is absent, and the sector corresponding to the Barrett-Crane model.

    “The first one is to take γ = ±i, in which case the constraint (186) could, in principle, be imposed strongly. However, the self-dual sector is not present in .The second possibility is to work in the sector where . Where, as in the Riemannian case, it corresponds to the Barret-Crane model [60] with the various limitations discussed in Section 10.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [60] Barrett, J.W. and Crane, L., “A Lorentzian Signature Model for Quantum General Relativity”, Class. Quantum Grav., 17, 3101–3118, (2000). [DOI]. (Cited on pages 70, 76, and 79.) source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    The restrictions on the Immirzi parameter in the Riemannian model are a special feature absent from the Lorentzian counterpart and appear to have little physical importance.

    “However, these restrictions seem so far of little physical importance as they are a special feature of the Riemannian model that is absent in its Lorentzian relative.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    The question of the path-integral measure in spin foams has not been completely settled.

    “As pointed out before, this question has not been completely settled yet in the spin-foam community.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given

    In the 3D toy example, the regulator can be removed without infinities and with the number of ambiguities under control.

    “A rigorous quantization is achieved if the regulator can be removed without the appearance of infinities, and if the number of ambiguities appearing in this process is under control. We shall see that all this can be done in the simple toy example in this section.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given

    The GFT formulation resolves the two fundamental conceptual problems of the spin-foam approach: diffeomorphism gauge symmetry and discretization dependence.

    “The GFT formulation resolves by definition the two fundamental conceptual problems of the spin-foam approach: diffeomorphism gauge symmetry and discretization dependence.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    In the large-spin limit, quantum geometry approximates classical geometry when tested with suitable states, while the geometry remains discrete because the limit is taken on a fixed regulating cellular structure.

    “In this limit the quantum geometry approximates the classical one when tested with suitable states (e.g., coherent states). However, the geometry remains discrete during this limiting process, as the limit is taken on a fixed regulating cellular structure.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The same relationship between boundary data and elements of the kinematical Hilbert space holds for the FK model as for γ < 1, but not for the γ > 1 model.

    “Due to their similarity to γ < 1, the same relationship between boundary data and elements of the kinematical Hilbert space hold for the FK model. However, such a simple relationship does not hold for the model in the case of γ > 1.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    In three-dimensional gravity, the gauge symmetries are so large that all solutions to the equations of motion are locally pure gauge, leaving only global or topological degrees of freedom.

    “The gauge symmetries are so large that all the solutions to the equations of motion are locally pure gauge. The theory has only global or topological degrees of freedom.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning

    Riemannian three-dimensional gravity has no local degrees of freedom and is a topological theory.

    “Riemannian gravity in 3 dimensions is a theory with no local degrees of freedom, i.e., a topological theory.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [112] Carlip, S., Quantum Gravity in 2+1 Dimensions, Cambridge Monographs on Mathematical Physics, (Cambridge University Press, Cambridge; New York, 1998). [Google Books]. (Cited on pages 43, 85, and 95.) source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationcitation

    Recursion relations for the 6j vertex amplitudes provide a tool for studying dynamics in spin foams of 3D gravity and might be useful in higher dimensions.

    “Recursion relations for the 6j vertex amplitudes have been investigated in [98, 157]. They provide a tool for studying dynamics in spin foams of 3D gravity and might be useful in higher dimensions [99].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [98] Bonzom, V. and Livine, E.R., “A New Recursion Relation for the 6j-Symbol”, Ann. Henri Poincare, 13, 1083–1099, (2012). [DOI], [arXiv:1103.3415]. (Cited on pages 44 and 95.) source

      Not checked in the free run.

    • cites [157] Dupuis, M. and Livine, E.R., “The 6j-symbol: Recursion, C. and Asymptotics”, Class. Quantum Grav., 27, 135003, (2010). [DOI], [arXiv:0910.2425]. (Cited on pages 44 and 95.) source

      Not checked in the free run.

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationreasoning

    Spin foams represent a gauge history of a kinematical state, and a sum over gauge histories defines P as a means for extracting true degrees of freedom from kinematical boundary states.

    “Spin foams represent a gauge history of a kinematical state. A sum over gauge histories is what defines P as a means for extracting the true degrees of freedom from those that are encoded in the kinematical boundary states.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given

    In addition to the invariance of the measure, appropriate gauge-fixing conditions are required for the amplitudes to be well defined.

    “It is well known that, in addition to the invariance of the measure, one must provide appropriate gauge-fixing conditions for the amplitudes to be well defined.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    The presence of the cosmological constant introduces a physical infrared cutoff.

    “Heuristically, the presence of the cosmological constant introduces a physical infrared cutoff.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The 3-dimensional manifold is compact and orientable.

  • May combine two claims.

    Premisecitation

    Equation (248) has been proven for simplicial decompositions, and its generalization to arbitrary cellular decomposition was obtained.

    “Eq. (248) has been proven in this case for the case of simplicial decompositions in [364]; see also [363, 238]. The generalization for arbitrary cellular decomposition was obtained in [206].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [364] Turaev, V.G. and Viro, O.Y., “State sum invariants of 3-manifolds and quantum 6j-symbols”, Topology, 31, 865–902, (1992). [DOI]. (Cited on pages 43, 93, and 95.) source

      Not checked in the free run.

    • cites [206] Girelli, F., Oeckl, R. and Perez, A., “Spin foam diagrammatics and topological invariance”, Class. Quantum Grav., 19, 1093–1108, (2002). [DOI], [arXiv:gr-qc/0111022]. (Cited on pages 40 and 93.) source

      Not checked in the free run.

    • The authors state this as an assumption: The 3-dimensional manifold is compact and orientable.

  • May combine two claims.

    Interpretationreasoning

    In general, the spacetime geometric interpretation of spin-foam configurations is subtle, and one cannot interpret the spin-foam sum of Eq. (238) as a sum over geometries in any obvious way.

    “This means that, in general, (even when local excitations are present, as in 4D) the spacetime geometric interpretation of the spin-foam configurations is subtle. This is an important point that is often overlooked in the literature: one cannot interpret the spin-foam sum of Eq. (238) as a sum over geometries in any obvious way.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The sum over spin foams in Eq. (238) is a sum over pure gauge degrees of freedom and has no physical interpretation.

    “This means that the sum over spin foams in Eq. (238) is nothing but a sum over pure gauge degrees of freedom and hence no physical interpretation can be associated to it.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Study design and conductreasoning

    The factorization of the remaining discrete symmetry is well defined because the number of equivalent spin foams can be characterized combinatorially and is finite for any spin foam on a finite discretization.

    “This factorization is well defined, since the number of equivalent spin foams can be characterized in a fully combinatorial manner, and is finite for any spin foam defined on a finite discretization.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The discretization can be interpreted as a gauge fixing of 3-diffeomorphisms on the boundary, but this gauge fixing is partial, leaving a discrete symmetry remnant that must be factored out when computing transition amplitudes.

    “The discretization can be interpreted, in this way, as a gauge fixing of 3-diffeomorphisms on the boundary. This gauge fixing is partial in the sense that, generically, there will remain a discrete symmetry remnant given by the discrete symmetries of the spin network. This remaining symmetry has to be factored out when computing transition amplitudes”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The underlying 2-complex on which spin foams are defined represents a partial gauge fixing for the configurations entering the path integral, and the remaining symmetry corresponds to the finite group of discrete symmetries of the spin foams.

    “The underlying 2-complex on which spin foams are defined represents a partial gauge fixing for the configurations (spin foams) entering in the path integral. The remaining symmetry, to be factored out in the computation of transition amplitudes, corresponds simply to the finite group of discrete symmetries of the corresponding spin foams.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

    Interpretationnone given

    No Euclidean formulation seems meaningful in a background-independent scenario.

    “Moreover, no Euclidean formulation seems meaningful in a background-independent scenario.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given

    Fundamental excitations are intrinsically discrete.

    “Fundamental excitations are intrinsically discrete.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationcitation

    Discretization independence is a problem for Regge gravity as well as for spin foams.

    “Finally, discretization independence is not only a problem of spin foams but also of Regge gravity.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    In the group field theory formulation, the difficulties move to the physical role of Δ and to convergence of the perturbative series.

    “The difficulties are shifted to the question of the physical role of Δ and the convergence of the corresponding perturbative series.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    Ignoring convergence, the sum of amplitudes of different orders that correspond to equivalent spin foams should define the physical amplitude of that spin foam.

    “From this perspective (and leaving aside the issue of convergence) the sum of different order amplitudes corresponding to equivalent spin foams should be interpreted as the definition of the physical amplitude of that particular spin foam.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Premisenone given

    Divergent series can often be assigned an asymptotic meaning and yield physical information.

    “This is not necessarily a definite obstruction as divergent series can often be given an asymptotic meaning and provide physical information.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    Because of this suppression, the effects are in principle negligible at low energies, so quantum-gravity effects would show up as corrections to well-tested standard QFT physics.

    “This makes them, in principle, negligible at low energies so that the potential quantum-gravity effects would be observed as corrections to the standard QFT well-tested physics.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    This lack is mainly caused by unsolved difficulties in quantitatively describing physical states.

    “This is mainly due to the presently-unsolved difficulties associated with a quantitative dynamical description of physical states.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    New means to eliminate the truncation introduced by the lattice need to be developed.

    “New means to eliminate the truncation introduced by the lattice need to be developed.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given

    In the background-independent context, the lattice contains only topological information and has no geometrical meaning associated to its components.

    “The lattice (triangulation) contains only topological information and there is no geometrical meaning associated to its components.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Interpretationnone given

    The theory is presumed to remain discrete at the fundamental level, and the smooth-manifold diffeomorphism-invariant description is expected to emerge in the low-energy continuum limit.

    “Even when the regulator (or the discretization dependence) eventually has to be removed (see next Section 17), the theory is presumed to remain discrete at the fundamental level. The smooth-manifold diffeomorphism-invariant description is expected to emerge in the (low energy) continuum limit.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The theory remains discrete at the fundamental level.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    It has been shown that large LIV effects can be avoided in Euclidean field theory by replacing the non-compact Poincaré group with the Euclidean group.

    “For example, such a possibility has been considered in [200], where it is shown that the large LIV effects can be avoided in the context of Euclidean field theory (the protecting mechanism being here the trading of the non-compact Poincaré group by the Euclidean group).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [200] Gambini, R., Rastgoo, S. and Pullin, J., “Small Lorentz violations in quantum gravity: do they lead to unacceptably large effects?”, Class. Quantum Grav., 28, 155005, (2011). [DOI], [arXiv:1106.1417 [gr-qc]]. (Cited on pages 101 and 105.) source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    In Zapata's treatment of the Turaev-Viro model, the amplitudes do not depend on the discretization because the theory is topological, so the refinement limit is trivial.

    “An example where this procedure is well defined is Zapata’s treatment of the Turaev-Viro model [372]. The key point in this case is that amplitudes are independent of the discretization (due to the topological character of the theory), so that the refinement limit is trivial.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [372] Zapata, J.A., “Spin foam model for 3D gravity in the continuum”, in García-Compeán, H., Mielnik, B., Montesinos, M. and Przanowski, M., eds., Topics in Mathematical Physics, General Relativity and Cosmology in Honor of Jerzy Plebański, Proceedings of 2002 International Conference, Cinvestav, Mexico City, 17–20 September 2002, pp. 495–502, (World Scientific, Singapore; Hackensack, NJ, 2006). [DOI]. (Cited on pages 93, 94, and 103.) source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    The regulator in spin foams is fundamentally different from the standard lattice gauge theory regulator because geometry is encoded in the coloring, which can take any spin values, so configurations involve fluctuations all the way to the Planck scale.

    “However, the nature of this regulator is fundamentally different from the standard concept in the background-independent framework: since geometry is encoded in the coloring (that can take any spin values) the configurations involve fluctuations all the way to Plank scale.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Freidel's proposal uses tree-level group field theory amplitudes to define the physical inner product of quantum gravity.

    “A prescription that avoids this last issue is the radical proposal of Freidel [179], where the tree-level GFT amplitudes are used to define the physical inner product of quantum gravity.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [179] Freidel, L., “Group field theory: An overview”, Int. J. Theor. Phys., 44, 1769–1783, (2005). [DOI], [arXiv:hep-th/0505016]. (Cited on page 103.) source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    The precise meaning of the gauge symmetries of such a theory would have to be formulated directly at the discrete level, and this can be achieved in the case of 3-dimensional gravity.

    “From this viewpoint, the precise meaning of the gauge symmetries of such a theory would have to be formulated directly at the discrete level. We have seen that this can be achieved in the case of 3-dimensional gravity (recall Section 14.5.1).”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The theory remains discrete at the fundamental level.

  • May combine two claims.

    Interpretationreasoning

    In the group field theory formulation, the discussion of Section 16 does not apply, so gauge fixing is unnecessary.

    “The discussion of Section 16 does not apply in the GFT formulation, i.e., there is no need for gauge fixing.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning

    The group field theory formulation could by definition solve the two main conceptual problems of spin foams: diffeomorphism gauge symmetry and discretization dependence.

    “The GFT formulation could resolve by definition the two fundamental conceptual problems of the spin-foam approach: diffeomorphism gauge symmetry and discretization dependence.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Early model calculations suggest that Lorentz-invariance-violation effects may arise from a quantum-gravity granularity of spacetime measured at rest by a preferred frame.

    “In the absence of detailed formulations, certain early pioneering model calculations [192] (see also general treatment [340, 341]) indicate the possibility for certain Lorentz-invariance-violation (LIV) effects associated to a quantum-gravity granularity of spacetime measured at rest by a preferred frame of observers.”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [192] Gambini, R. and Pullin, J., “Nonstandard optics from quantum space-time”, Phys. Rev. D, 59, 124021, (1999). [DOI], [arXiv:gr-qc/9809038 [gr-qc]]. (Cited on page 105.) source

      Not checked in the free run.

    • cites [340] Sahlmann, H. and Thiemann, T., “Towards the QFT on curved space-time limit of QGR: I. A general scheme”, Class. Quantum Grav., 23, 867–908, (2006). [DOI], [arXiv:gr-qc/0207030 [gr-qc]]. (Cited on page 105.) source

      Not checked in the free run.

    • cites [341] Sahlmann, H. and Thiemann, T., “Towards the QFT on curved space-time limit of QGR: II. A concrete implementation”, Class. Quantum Grav., 23, 909–954, (2006). [DOI], [arXiv:gr-qc/0207031 [gr-qc]]. (Cited on page 105.) source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    This implies that Planckian discreteness in LQG must enter physics through reference-frame-independent phenomena, as mass does in Klein-Gordon theory.

    “This would mean that the way the fundamental Planckian discreteness of LQG must enter physics is through reference-frame-independent phenomena (as mass enters a Klein-Gordon theory).”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    At the level of gauge-invariant observables, some discrete features might vanish.

    “on the other, at the level of gauge-invariant observables, some discrete aspects could disappear [154]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [154] Dittrich, B. and Thiemann, T., “Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?”, J. Math. Phys., 50, 012503, (2009). [DOI], [arXiv:0708.1721 [gr-qc]]. (Cited on page 106.) source

      Not checked in the free run.

140 of 140

  • The authors state this as an assumption.

    For the formal derivation of the projector, the paper assumes there is only one constraint.

    “Assuming for simplicity that there is only one constraint we have”

  • The authors state this as an assumption.

    There exists a global time function t whose level hypersurfaces foliate the spacetime manifold.

    “Now one performs the usual 3+1 decomposition by assuming the existence of a global time function t, whose level hypersurfaces Σt define a foliation of the spacetime manifold ℳ.”

  • The authors state this as an assumption.

    The manifold is compact and orientable.

    “For the moment, we assume to be compact and orientable.”

  • The authors state this as an assumption.

    The SU(2) subgroup of Spin(4) used to define the linear simplicity constraints is chosen arbitrarily at each tetrahedron and equivalently on each edge.

    “where (very importantly) the subgroup SU(2) ⊂ Spin(4) that is necessary for the definition of the above constraints is chosen arbitrarily at each tetrahedron, equivalent on each edge e ∈ Δ*.”

  • The paper relies on this without support.

    Arbitrary half-integer spins should be admissible in the state sum.

    “Now, in order for this to be so, one would like arbitrary spins j ∈ ℕ/2 to be admissible in the state sum.”

  • The paper relies on this without support.

    The cellular decomposition has no physical meaning and is only a subsidiary regulating structure to be removed when computing physical quantities.

    “The point is that the cellular decomposition Δ has no physical meaning and is to be interpreted as a subsidiary regulating structure to be removed when computing physical quantities.”

  • The paper relies on this without support.

    The tetrahedron state space should be compatible with the corresponding canonical LQG state space so that arbitrary kinematical LQG states have non-trivial amplitudes in the model.

    “In addition, one would add the requirement that the state-space of tetrahedra is compatible with the state-space of the analogous excitation in the canonical context of LQG so that arbitrary states in the kinematical state of LQG have non-trivial amplitudes in the model.”

  • The authors state this as an assumption.

    The 3-dimensional manifold is compact and orientable.

    “where, for the moment, we assume to be compact and orientable.”

  • The authors state this as an assumption.

    The theory remains discrete at the fundamental level.

    “the theory is presumed to remain discrete at the fundamental level.”
9 of 9

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • GFTs provide a complete definition of the covariant formulation of the dynamics of loop gravity spin networks, via field theory methods, in addition to being a 2nd quantized reformulation of its canonical, operator-based formulation.

    “The last fact makes clear how GFTs provide a complete definition of the covariant formulation of the dynamics of loop gravity spin networks [17], via field theory methods, in addition to being a 2nd quantized reformulation of its canonical, operator-based formulation.”

    From The universe as a quantum gravity condensate (2017) · cites this paper as [17]

    The sections “Group field theories associated to the new spin foams” and “Discretization Dependence” describe GFT as a field-theory formulation of spin-foam amplitudes and a combinatorial sum over discretizations. They do not establish a complete definition of loop-gravity dynamics: the paper flags convergence and other unresolved questions. Nor do these passages describe GFT as a second-quantized reformulation of canonical loop gravity. (LLM, conf=0.86)

    Experimental — accuracy not yet measured

  • The GFT formulation of the EPRL spin foam model for Lorentzian 4d gravity uses simplicial combinatorics in the interactions, GFT fields over SU(2)^4, and an embedding into SL(2,C) data via geometricity constraints implemented in the dynamics.

    “The starting point is the GFT formulation of the EPRL spin foam model [17, 29] for Lorentzian 4d gravity.”

    From The universe as a quantum gravity condensate (2017) · cites this paper as [17]

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

    Experimental — accuracy not yet measured

  • All tensorial GFT models analyzed so far are asymptotically free in the UV limit, rooted in the combinatorial nonlocality of the interaction.

    “Remarkably, all these models are shown to be asymptotically free in the UV limit which is deeply rooted in the combinatorial nonlocality of the interaction [20].”

    From Impact of nonlinear effective interactions on group field theory quantum gravity condensates (2016) · cites this paper as [20]

Cites

Flagged first, then by citations

2 of the 452 references couldn’t be matched to a known paper.

Research works printed but not linked: 5

  1. Bianchi, E., Han, M., Magliaro, E., Perini, C., Rovelli, C. and Wieland, W.M., “Spinfoam fermions”, arXiv, e-print, (2010). [arXiv:1012.4719]. (Cited on page 79.) · arXiv
  2. Livine, E.R., Perez, A. and Rovelli, C., “2D manifold-independent spinfoam theory”, Class. Quantum Grav., 20, 4425–4445, (2003). [DOI], [gr-qc/0102051]. (Cited on page 103.) · DOI
  3. Penrose, R., “Angular momentum: an approach to combinatorial space-time”, in Bastin, T., ed., Quantum Theory and Beyond, Essays and Discussions Arising from a Colloquium, pp. 151–180, (Cambridge University Press, Cambridge; New York, 1971). (Cited on page 30.)
  4. Penrose, R., “Applications of negative dimensional tensors”, in Welsh, D.J.A., ed., Combinatorial Mathematics and its Application, Proceedings of a conference held at the Mathematical Institute, Oxford, from 7–10 July, 1969, pp. 221–243, (Academic Press, London; New York, 1971). (Cited on page 30.)
  5. Velázquez, M., BF gravity, matter couplings, and related theories, Ph.D. thesis, (Cinvestav, Mexico City, 2011). (Cited on page 23.)

Cited by

Sorted by citations

Related

Authors
Perez, A.
Venue
Living Reviews in Relativity
Publisher
Springer Science and Business Media LLC
Published
2013-02-14
Volume
16
Issue
1
Pages
3-3
ISSN
  • 1433-8351 (ISSN-L)
  • 2367-3613
DOI
10.12942/lrr-2013-3
PMC
PMC5255902
OpenAlex
W1965001977
Semantic Scholar
c78d8dc3e7e6204c00d553671975928cf92cb765
Cited by
677
References
452
Citations by year
2026: 25 · 2025: 57 · 2024: 42 · 2023: 41 · 2022: 42 · 2021: 72 · 2020: 64 · 2019: 61 · 2018: 44 · 2017: 45 · 2016: 52 · 2015: 39 · 2014: 46 · 2013: 32 · 2012: 14
FWCI
3.69
Type
journal article
Language
en
Open access
Licence
cc-by
Locations
  • Living Reviews in Relativity (publishedVersion, cc-by)
  • Europe PMC (PubMed Central) (publishedVersion)
  • PubMed Central (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • HAL (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Keywords
spin foam models, quantum gravity, path integral quantization, three-dimensional gravity, four-dimensional quantum gravity, simplicial quantum gravity

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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.

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