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Cited by 154Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryBlack Holes and Theoretical Physics

Complete LQG propagator: Difficulties with the Barrett-Crane vertex

Open access
Alesci, Emanuele, Rovelli, Carlo·2007·Physical Review D
DOI 10.1103/physrevd.76.104012
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Abstract

Some components of the graviton two-point function have been recently computed in the context of loop quantum gravity, using the spinfoam Barrett-Crane vertex. We complete the calculation of the remaining components. We find that, under our assumptions, the Barrett-Crane vertex does not yield the correct long-distance limit. We argue that the problem is general and can be traced to the intertwiner independence of the Barrett-Crane vertex, and therefore to the well-known mismatch between the Barrett-Crane formalism and the standard canonical spin networks. In another paper we illustrate the asymptotic behavior of a vertex amplitude that can correct this difficulty.

Verified 2026-10-06 · 17 references not checked yet

AI snapshot

Written from the paper’s full text.

TL;DR

The paper completes the calculation of the nondiagonal components of the graviton two-point function in loop quantum gravity using the Barrett-Crane spinfoam vertex. Under its assumptions, the Barrett-Crane vertex does not yield the correct long-distance limit, a failure traced to its intertwiner-independence and the mismatch between the Barrett-Crane formalism and standard canonical spin networks.

Key findings

  • The authors compute the remaining nondiagonal terms of the propagator \(G_{abcd}(x,y)\) and derive the full tensorial structure, finding that the Barrett-Crane vertex fails to give the correct tensorial structure in the large-distance limit.
  • In the large \(j_0 \to \infty\) limit, the problematic terms “go to zero fast” and therefore do not match the expected large-distance behavior of the propagator; including the normalization factor does not remove the pathology.
  • The failure is argued to be general and traced to the intertwiner-independence of the Barrett-Crane vertex, and to the well-known mismatch between the Barrett-Crane formalism and standard canonical spin networks. A phase factor \(i\frac{\pi}{2}\sum_p i_p\) present in the boundary state is not matched by a corresponding factor in the vertex amplitude, giving a rapidly oscillating term that suppresses the sum.
  • The companion paper is said to illustrate the asymptotic behavior of a vertex amplitude that can correct this difficulty.

Methods

  • Theoretical calculation in euclidean loop quantum gravity: the graviton two-point function is computed to first order in the coupling \(\lambda\), using a Barrett-Crane group field theory vertex and the pentagonal spin-network graph.
  • The boundary state is chosen to be peaked on a regular four-simplex geometry, with explicit intertwiner dependence; the calculation uses SU(2) recoupling theory, grasping operators \(E_{(ni)n}\cdot E_{(nj)n}\) on intertwiner spaces, \(6j\) symbols, and the asymptotic Regge action.
  • The main analysis evaluates sums over spins and intertwiners in the large \(j_0\) limit, including sums over pairings and orientations, and approximates them by Gaussian integrals.

Limitations

  • The paper states that the calculation is based on a number of assumptions on the form of the boundary state, and the authors acknowledge they do not have a real proof that a different boundary state could not give the correct low-energy behavior; they argue instead that the problem is more substantial and related to the Barrett-Crane vertex itself.
  • The authors state that they work entirely in the euclidean theory, and that the paper is not self-contained, referring to earlier work for full background.
  • Evident from the text: the negative result relies on the large-distance/large-\(j_0\) asymptotic expansion and on the specific boundary-state construction; no Lorentzian-signature calculation or empirical test is presented.

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

Integrity signals

Registry

Crossref · 10.1103/physrevd.76.104012

Cites

30 of 47 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 20 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § 3 The boundary state · p. 7

    In the large j0 limit, the state (24) transforms under change of pairing into a state with the same intertwiner mean value and the same variance σ, provided the spin-intertwiner gaussian terms are chosen as specified.

    “A detailed calculation (see below), shows indeed that in the large j 0 limit, the state (24) transforms under change of pairing into a state with the same intertwiner mean value and the same variance σ , provided we also choose”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Non-diagonal intertwiner-intertwiner terms are not included in the boundary state.
  • Findingreasoning
    § 3 The boundary state · p. 7

    For the regular 4-simplex, in the large distance limit A_ij = j0 and cos θ_ij = -1/3, which fixes i0.

    “For the regular 4-symplex, in the large distance limit we have A ij = j 0 , cos θ ij = -1 3 , which gives”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 3.2 Mean values and variances · p. 11

    The mean values of the state (39) reduce to the average of the mean values on each diagonal term in the large j0 limit because the interference terms vanish.

    “Because of the vanishing of the interference terms proven above, in large j 0 limit the mean values reduce to the average of the mean values on each diagonal term.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Abstract · p. 1

    Under the paper's assumptions, the Barrett-Crane vertex fails to give the correct tensorial structure of the propagator in the large-distance limit.

    “We find that, under our assumptions, the Barrett-Crane vertex does not yield the correct long distance limit.”
    • Also stated in: § 1 Introduction · p. 2 · plain
      “We find that under our assumptions the BC vertex fails to give the correct tensorial structure of the propagator in the large-distance limit.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The boundary functional W is given by a Barrett-Crane group field theory with coupling constant λ.
    • The authors state this as an assumption: Non-diagonal intertwiner-intertwiner terms are not included in the boundary state.
    • The authors state this as an assumption: The state (24) does not depend on the orientation.
  • Findingreasoning
    § 3.2 Mean values and variances · p. 11

    The state (39) satisfies the mean value conditions (58).

    “It is easy to show that the state (39) satysfies (58).”
    Quote in the paperSubstantial claim
  • May combine two claims. 11
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 9

  • May combine two claims.

    Findingown data
    § 3.1 Pairing independence · p. 7

    The state (31) does not respect the symmetries of the problem; its relative fluctuations of angle variables depend on the pair chosen.

    “A moment of reflection shows that the state (31) does not.”
    • Also stated in: § 3.1 Pairing independence · p. 7 · boosted
      “an explicit calculation confirms that the relative fluctuations of the angle variables determined by the state (31) depend on the pair chosen.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 3.1.1 Orthogonality of the terms in different bases in the large j 0 limit · p. 10

    The first term in the Taylor expansion of the Regge action is a rapidly oscillating phase factor in the j1n variable, which suppresses the Gaussian j1n integration, making the integral go to zero as j0→∞.

    “The key point now is that the first of these terms is a rapidly oscillating phase factor in the j 1 n variable. The Gaussian j 1 n integration in (48) is suppressed by this phase factor.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 2.2 Operators · p. 5

    The action of the E(i)·E(j) operator on the chosen intertwiner basis is diagonal when the grasped links are paired together and non-diagonal when they are not.

    “In these two cases the action of the operator is diagonal. If, instead, the grasped links are not paired together, the action of the operator is not diagonal in this basis.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § F Change of pairing on the boundary state · p. 29

    The auxiliary tetrahedron described by S_R is isosceles and σ_i_x1 = σ_i_y1 = j0/3.

    “Remarkably the auxiliary tetrahedron described by S R is isosceles and in this case σ i x 1 = σ i y 1 = j 0 / 3”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given
    § 4 Calculation of the propagator · p. 14

    The matrix M is invertible and independent from j0.

    “The matrix M is invertible and independent from j 0 .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 3.1.2 Change of basis · p. 11

    The value χ_n = π/2 makes the intertwiner phase equal, with opposite sign, to the term in the expansion of the Regge action linear in δi_y1, selecting one of the two exponentials in the asymptotic expansion.

    “In particular, the value χ n = π 2 makes the intertwiner phase equal, with opposite sign, to the term exp -i ( ∂S R ∂i y 1 ∣ ∣ ∣ j 0 ,i 0 δi y 1 -πδi y 1 ) , namely the term in the expansion of the Regge action S R linear in the variable δi y 1 .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § 4 Calculation of the propagator · p. 12

    In the large distance limit, ∏ n (2 i_n +1) = 2 i_0^5, hence W(j,i) = 2 i_0^5 W(j).

    “In the large distance limit, ∏ n (2 i n +1) = 2 i 5 0 , hence W ( j , i ) = 2 i 5 0 W ( j ) . (65)”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § 4 Calculation of the propagator · p. 14

    Terms of the kind (78) are still pathological in the limit.

    “Terms of the kind (78) are still pathological, since they give”
    Quote in the paperSubstantial claim

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

    Interpretationnone given
    § 1 Introduction · p. 2

    The results presented here are valid for all variants of the Barrett-Crane model.

    “The BC model exists in a number of variants [2, 14]; the results presented here are valid for all of them.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [2] [2] C. Rovelli, Quantum Gravity , (Cambridge University Press, Cambridge, 2004). -

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

    • cites [14] [14] R. De Pietri, L. Freidel, K. Krasnov and C. Rovelli, 'Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space,' Nucl. Phys. B 574 , 785 (2000) [arXiv:hep-th/9907154]. D Oriti, RM Williams, 'Gluing 4-simplices: a derivation of the Barrett-Crane spinfoam model for Euclidean quantum gravity', Phys Rev D 63 (2001) 024022. A. Perez and C. Rovelli, 'Finite SO(4)-state sum model of Euclidean GR,' Prepared for 9th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (MG 9), Rome, It source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § 3 The boundary state · p. 6

    The procedure of [6] is not well defined because of the mismatch between SO(4) linearity and SU(2) linearity.

    “However, it was also pointed out in [6] that this procedure is not well defined, because of the mismatch between SO (4) linearity and SU (2) linearity”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [6] [6] E. Bianchi, L. Modesto, C. Rovelli, S. Speziale, 'Graviton propagator in loop quantum gravity', Class. Quant. Grav. 23 (2006), 6989-7028; [arXiv: gr-qc/0604044]. -

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

  • Classified as probably not a substantial claim.

    Interpretationreasoning
    § 1 Introduction · p. 2

    The result opens the possibility of studying the conditions that an alternative vertex must satisfy in order to yield the correct long-distance behavior.

    “Finally, and more importantly, it opens the possibility of studying the conditions that an alternative vertex must satisfy, in order to yield the correct long-distance behavior.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § 1 Introduction · p. 2

    The failure of the Barrett-Crane vertex to give the correct long-distance limit is general and cannot be easily corrected by a different boundary state.

    “We argue that this result is general, and cannot be easily corrected, say by a different boundary state.”
    Quote in the paperSubstantial claim

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

    Premisenone given
    § 3.1 Pairing independence · p. 9

    The matrices of the change of basis are products of five 6j Wigner-symbols, as given by standard recoupling theory.

    “The matrices of the change of basis 〈 j , i | j , i m n 〉 are (products of five) 6j Wigner-symbols, as given by standard recoupling theory.”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § 3 The boundary state · p. 6

    The boundary state used in [6] was assumed to have a gaussian dependence on the spins and to be peaked on a particular intertwiner that projects trivially onto the Barrett-Crane intertwiner.

    “The boundary state utilized in [6] was assumed to have a gaussian dependence on the spins, and to be peaked on a particular intertwiner. This intertwiner was assumed to project trivially onto the BC intertwiner of the BC vertex.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [6] [6] E. Bianchi, L. Modesto, C. Rovelli, S. Speziale, 'Graviton propagator in loop quantum gravity', Class. Quant. Grav. 23 (2006), 6989-7028; [arXiv: gr-qc/0604044]. -

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

  • May combine two claims.

    Interpretationreasoning
    § 1 Introduction · p. 2

    The propagator calculations are nontrivial; they are not governed just by dimensional analysis, and they do test the dynamics of the theory.

    “First, it indicates that the propagator calculations are nontrivial; in particular they are not governed just by dimensional analysis, as one might have worried, and they do test the dynamics of the theory.”
    Quote in the paperSubstantial claim

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

    Interpretationnone given
    § F Change of pairing on the boundary state · p. 29

    The parameters (29) and (30) are determined by the requirement that the gaussian has the same shape in all bases.

    “Summarizing, the parameters (29) and (30) are determined by the requirement that the gaussian has the same shape in all bases.”
    Quote in the paperNot a substantial claim

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

    Interpretationcitation
    § 5 Conclusions · p. 15

    This structural difficulty is at the roots of the problem.

    “Second, as already mentioned, there is in fact a structural difficulty, already pointed out in [4, 5], with the definition (64) of the amplitude, and we think that this difficulty is at the roots of the problem.”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § B Analytic expressions for 6j symbols · p. 20

    The 6j symbol is invariant for interchange of any two columns, and also for interchange of the upper and lower arguments in each of any two columns.

    “The 6j symbol is invariant for interchange of any two columns, and also for interchange of the upper and lower arguments in each of any two columns:”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [25] [25] D. M. Brink, G. R. Satchler, Angular Momentum (Claredon Press, Oxford, 1968). -

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

  • Classified as probably not a substantial claim.

    Interpretationcitation
    § 5 Conclusions · p. 15

    In the companion paper [13], we show that a vertex with a suitable asymptotic behavior can overcome all these difficulties.

    “In the companion paper [13], we show that, perhaps surprisingly, a vertex with a suitable asymptotic behavior can overcame all these difficulties.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [13] [13] E. Alesci, C. Rovelli, 'The complete LQG propagator: II. Asymptotics of the vertex' to appear. -

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

19 of 19

  • The authors state this as an assumption.

    The boundary functional W is given by a Barrett-Crane group field theory with coupling constant λ.

    “it is assumed to be given here by a Barrett-Crane GFT [14, 24] with coupling constant λ .”

  • The authors state this as an assumption.

    Non-diagonal intertwiner-intertwiner terms are not included in the boundary state.

    “We do not include non-diagonal intertwiner-intertwiner terms here.”

  • The authors state this as an assumption.

    The state (24) does not depend on the orientation.

    “Since (by assumption) (24) does not depend on the orientation”

  • The paper relies on this without support.

    The variables associated to the angles between faces do not commute with one another.

    “Since the variables associated to the angles between faces do not commute with one another, the boundary state cannot be sharp on a classical configuration.”

  • The authors state this as an assumption.

    The boundary state has the specific form assumed in the calculation.

    “The calculation presented above is based on a number of assumptions on the form of the boundary state.”

  • The paper relies on this without support.

    The expected large distance behavior of the propagator is the correct low energy behavior.

    “These terms go to zero fast in the j 0 → ∞ limit, and therefore do not match the expected large distance behavior of the propagator.”
6 of 6

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?

  • The Barrett-Crane two-point function could not yield the expected result compatible with Regge gravity in the semiclassical limit.

    “It was then applied to the Barrett-Crane model in [5, 4, 77], where it was discovered that certain components of the two-point function could not yield the expected result compatible with Regge gravity in the semiclassical limit.”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [4]

  • The Barrett-Crane model's failure to give the correct propagator scaling can be traced back to a missing coherent cancellation of phases between the intertwiner wave function of the semiclassical boundary state and the intertwiner dependence of the model.

    “The problem can be traced back to a missing coherent cancellation of phases between the intertwiner wave function of the semiclassical boundary state and the intertwiner dependence of the model.”

    From LQG propagator from the new spin foams (2009) · cites this paper as [12]

  • The Barrett-Crane spin foam model fails to give the correct scaling behavior for off-diagonal components of the graviton propagator.

    “The analysis of [12,13] shows that the Barrett-Crane model fails to give the correct scaling behavior for off-diagonal components of the graviton propagator.”

    From LQG propagator from the new spin foams (2009) · cites this paper as [12]

  • In the recoupling basis for intertwiner space, the off-diagonal components of the metric operator generally have non-trivial matrix elements.

    “Using the recoupling basis for intertwiner space, we have that in general the off-diagonal components of the metric operator have non-trivial matrix elements”

    From LQG propagator from the new spin foams (2009) · cites this paper as [12]

  • The off-diagonal components are not suppressed as happened for the Barrett-Crane model [12,13] and have the same scaling as the diagonal ones.

    “The off-diagonal components are not suppressed as happened for the Barrett-Crane model [12,13] and have the same scaling as the diagonal ones.”

    From LQG propagator from the new spin foams (2009) · cites this paper as [12]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Only five tetrahedron edges are listed

    Minor

    The paper says the tetrahedron has six edges, but its stated list of edge lengths assigns values to only five distinct edges: 12, 13, 14, 34, and 23. The sixth edge, 24, is omitted.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 3

  1. C. Rovelli, 1995, Nucl. Phys., B456, 734
  2. A. Perez, 2002, Proceedings of the Ninth Marcel Grossmann Meeting on General Relativity
  3. D. M. Brink, 1968, Angular Momentum

Cited by

Sorted by citations

Related

Authors
Alesci, E., & Rovelli, C.
Affiliations
Roma Tre University (IT)
ORCID
Alesci, Emanuele
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2007-11-09
Volume
76
Issue
10
Pages
104012
ISSN
  • 0556-2821 (ISSN-L)
  • 1550-7998
  • 1550-2368
DOI
10.1103/physrevd.76.104012
OpenAlex
W2591970470
Semantic Scholar
06a80d1c823a7b9b20100948ee7c9932f12faf94
Cited by
154
References
46
Citations by year
2025: 1 · 2024: 3 · 2023: 3 · 2022: 2 · 2021: 2 · 2020: 2 · 2019: 3 · 2018: 3 · 2017: 4 · 2016: 6 · 2015: 4 · 2014: 9 · 2013: 14 · 2012: 13
FWCI
13.85
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • HAL (submittedVersion, other-oa)
  • SDL (submittedVersion)
  • arXiv (Cornell University)
Funders
Agence Nationale de la Recherche (ANR-06-BLAN-0050)
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
loop quantum gravity, propagators, spin networks, vertex amplitude, asymptotic behavior

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