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Cited by 212Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryBlack Holes and Theoretical Physics

Graviton Propagator from Background-Independent Quantum Gravity

Open access
Rovelli, Carlo·2006·Physical Review Letters
DOI 10.1103/physrevlett.97.151301
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Abstract

We study the graviton propagator in Euclidean loop quantum gravity. We use spin foam, boundary-amplitude, and group-field-theory techniques. We compute a component of the propagator to first order, under some approximations, obtaining the correct large-distance behavior. This indicates a way for deriving conventional spacetime quantities from a background-independent theory.

AI snapshot

Written from the paper’s full text.

TL;DR

Rovelli computes one component of the graviton propagator in Euclidean loop quantum gravity using the spinfoam/group-field-theory formalism, to first order and under approximations. The result has the correct large-distance spacetime dependence, and the “bad” terms of the Barrett-Crane vertex are suppressed by the phase of a vacuum state peaked on extrinsic geometry.

Key findings

  • The computed propagator component agrees with the conventional linearized graviton propagator, with the correct \(1/|x-y|^2\) spacetime dependence; the numerical coefficients are determined by \(\alpha = 4\pi^2/\sqrt{3}\, W^{-1} - i\sqrt{3}/4\, K\).
  • Only the “good” term of the Barrett-Crane vertex amplitude—the exponential of the Regge action—contributes; the opposite-sign/cosine term and the degenerate four-simplex term are suppressed by the rapidly oscillating phase of the vacuum state peaked on the boundary’s extrinsic geometry.
  • For a regular four-simplex of side \(L\), the triangle area is \(A_L=\sqrt{3}L^2/4\), \(j_L=8\pi\hbar G A_L\) in the large-\(L\) limit, and the dihedral angles satisfy \(\cos(\Phi)=-1/4\); points at centers of adjacent tetrahedra are at distance \(L/4\).
  • The paper concludes that it appears possible to compute \(n\)-point functions from background-independent quantum field theories, and that the propagator is likely cut off at the Planck scale, making it finite.

Methods

  • Uses the boundary-amplitude expression for the connected two-point function, with \(W[s]\) from the GFT/B spinfoam model, a coherent boundary state \(\Psi_q[s]\) peaked on intrinsic and extrinsic geometry, and a densitized metric fluctuation operator.
  • Restricts to first order in \(\lambda\): one vertex and a five-valent boundary spinnetwork dual to the boundary of a four-simplex; evaluates the large-\(j\) limit, expands the \(10j\)/Barrett-Crane vertex around \(j_L\), and approximates the sum by a Gaussian integral.
  • Compares the result with conventional linearized quantum general relativity.

Limitations

  • The paper states the computation is first order, for one component of the propagator, and under a number of approximations; other components and second-order terms are to be presented elsewhere.
  • Validity is for large \(|x-y|\); when \(|x-y|\) approaches the Planck length, quantum gravitational corrections appear, though the sum can then be computed numerically.
  • The text states it assumes for simplicity that \(\Psi_q[s]\) is nonvanishing only for Barrett-Crane intertwiners, with another symmetric choice to be explored elsewhere; also, several formulas are omitted/not decoded in the provided text, so intermediate expressions cannot be verified from this copy.

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

Integrity signals

Registry

Crossref · 10.1103/physrevlett.97.151301

Cites

7 of 7 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 12 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § Graviton propagator from background-independent quantum gravity · p. 5

    If the four indices i, j, k, l are all distinct, n and ̃n are orthogonal.

    “If the four indices i, j, k, l are all distinct, it is easy to see that n and ˜ n are orthogonal; then the propagator is easily computed to be [10]”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [10] source

      Not checked in the free run.

  • Findingreasoning
    § Graviton propagator from background-independent quantum gravity · p. 1

    In the large distance limit, only the 'good' term of the vertex amplitude, the exponential of the Regge action, contributes to the propagator.

    “In the large distance limit, the only term of the vertex amplitude that contributes is the exponential of the Regge action: the other terms, that have raised doubts on the physical viability of the model, are suppressed by the phase of the vacuum state, which is determined by the extrinsic geometry of the boundary.”
    • Also stated in: § Graviton propagator from background-independent quantum gravity · p. 1 · plain
      “We show here that only the 'good' term contributes to the propagator.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Graviton propagator from background-independent quantum gravity · p. 6

    The first term is suppressed by the rapidly oscillating factor exp 2i Φ ∑ nm j_nm.

    “The first term is suppressed (by the rapidly oscillating factor exp 2 i Φ ∑ nm j nm )”
    Quote in the paperSubstantial claim
  • Findingnone given
    § Graviton propagator from background-independent quantum gravity · p. 6

    The rapidly oscillating term exp i Φ ∑ nm j_nm tends to suppress the sum.

    “The rapidly oscillating term exp i Φ ∑ nm j nm tends to suppress the sum.”
    Quote in the paperNot a substantial claim
  • Findingown data
    § Graviton propagator from background-independent quantum gravity · p. 1

    To first order, the computation produces a 4d version of the 'nutshell' 3d model studied in [6].

    “To this order, the computation produces a 4d version of the 'nutshell' 3d model studied in [6].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [6] source

      Not checked in the free run.

  • May combine two claims. 8
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 2

  • May combine two claims.

    Findingnone given
    § Graviton propagator from background-independent quantum gravity · p. 5

    The components W_linearized_(ij)(ij) and W_linearized_(ij)(ik) are vacuum expectation values at fixed 'time': the first is the fluctuation of the area square of a triangle, and the second is the vacuum correlation between the fluctuations of the area squares of two adjacent triangles in the same tetrahedron.

    “On the other hand, the components W linearized ( ij )( ij ) and W linearized ( ij )( ik ) are vacuum expectation values at fixed 'time': the first is the flucuation of the area square of a triangle, and the second is the vacuum correlation between the fluctuations of the area squares of two adjacent triangles in the same tetrahedron.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Graviton propagator from background-independent quantum gravity · p. 7

    The propagator computed approximates the perturbative one, and this is correct for large |x-y| where the approximations used in evaluating the sum (18) hold.

    “The propagator we have computed approximates the perturbative one. This is correct for large | x -y | , where the approximations used in evaluating the sum (18) hold.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The gaussian suppresses higher-order terms in the expansion of the summand in fluctuations δj_ij.
    • The authors state this as an assumption: The dim j terms vary slowly over the range where the gaussian is peaked and can be considered constant.
    • The authors state this as an assumption: The sum can be approximated by a gaussian integral.

  • May combine two claims.

    Findingreasoning
    § Graviton propagator from background-independent quantum gravity · p. 5

    W_(12)(34) = W_3 = i, while W_1 and W_2 are real numbers of the order of unity.

    “In particular, W (12)(34) = W 3 = i , while W 1 and W 2 are real numbers of the order of unity, easily obtained from the linear theory.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingnone given
    § Graviton propagator from background-independent quantum gravity · p. 3

    In the large j limit, the dominant contribution to first order in λ is obtained by choosing the spinfoam σ and the spinnetwork s = ∂σ defined by the dual of a four-simplex and its boundary.

    “The dominant contribution for large j is obtained by choosing the spinfoam σ and the spinnetwork s = ∂σ defined by the dual of a four-simplex and its boundary.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The triangulation is fine enough for the gravitational field to be approximately constant on a tetrahedron.

  • May combine two claims.

    Findingreasoning
    § Graviton propagator from background-independent quantum gravity · p. 6

    The rapidly oscillating exp{i Φ ∑ nm j_nm} term in the second term of the large-j expansion cancels with the rapidly oscillating term in (19), so the second term contributes in a non-negligible way to the sum (19).

    “The key observation is now the fact that the rapidly oscillating exp { i Φ ∑ nm j nm } term in the second term of this expression cancels with the rapidly oscillating term in (19). Therefore the second term of the last expression contributes in a non-negligible way to the sum (19).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The gaussian suppresses higher-order terms in the expansion of the summand in fluctuations δj_ij.
    • The authors state this as an assumption: The dim j terms vary slowly over the range where the gaussian is peaked and can be considered constant.
    • The authors state this as an assumption: The sum can be approximated by a gaussian integral.

  • May combine two claims.

    Findingreasoning
    § Graviton propagator from background-independent quantum gravity · p. 1

    The 'bad' terms of the Barrett-Crane vertex amplitude are suppressed or cancelled when the correct vacuum state, appropriately peaked on the extrinsic geometry, is used.

    “The others are suppressed by the rapidly oscillating phase in the vacuum state that peaks the state on its correct extrinsic geometry.”
    • Also stated in: § Graviton propagator from background-independent quantum gravity · p. 7 · plain
      “Our main results are: (i) the 'bad' terms of the Barrett-Crane vertex amplitude are cancelled when the correct vacuum state, appropriately peaked on the extrinsic geometry, is used;”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Graviton propagator from background-independent quantum gravity · p. 7

    At least to this order, the component computed of the propagator is the standard free-theory one at large distances, with corrections at the Planck scale, which appear to make it finite.

    “Therefore we obtain the result that, at least to this order, the (component we have computed of the) propagator is the standard free-theory one at large distances, with corrections at the Planck scale, wilch appear to make it finite.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The gaussian suppresses higher-order terms in the expansion of the summand in fluctuations δj_ij.
    • The authors state this as an assumption: The dim j terms vary slowly over the range where the gaussian is peaked and can be considered constant.
    • The authors state this as an assumption: The sum can be approximated by a gaussian integral.

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

    Findingnone given
    § Graviton propagator from background-independent quantum gravity · p. 5

    In a flat background metric, two points in the center of adjacent tetrahedra, in a surface with the boundary geometry chosen, are at a distance L/4.

    “In a flat background metric, two points in the center of adjacent tetrahedra, in a surface with the boundary geometry chosen, are at a distance | x u 1 -x d 1 | q = L/ 4.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data
    § Graviton propagator from background-independent quantum gravity · p. 1

    The computed component of the graviton propagator agrees with the conventional or linearized graviton propagator to this order, with the correct spacetime dependence.

    “At low energy, this agrees with the conventional graviton propagator.”
    • Also stated in: § Graviton propagator from background-independent quantum gravity · p. 7 · boosted
      “This is precisely the value (15-16) of the propagator computed from the linearized theory, with the correct 1 / | x -y | 2 spacetime dependence.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The gaussian suppresses higher-order terms in the expansion of the summand in fluctuations δj_ij.
    • The authors state this as an assumption: The dim j terms vary slowly over the range where the gaussian is peaked and can be considered constant.
    • The authors state this as an assumption: The sum can be approximated by a gaussian integral.

  • Classified as probably not a substantial claim.

    Findingreasoning
    § Graviton propagator from background-independent quantum gravity · p. 4

    The state (10) is a coherent state peaked both on the background values of the spins (intrinsic geometry) and on the background values of the angles (extrinsic geometry).

    “This amounts to select a vacuum state which is a coherent state peaked both on the background values of the spins (the intrinsic geometry of the boundary surface), and on the background values of the angles (the intrinsic geometry of the boundary surface).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The boundary state Ψ_q[s] is non-vanishing only for spinnetworks whose intertwiners are i_BC intertwiners.

  • May combine two claims.

    Premisecitation
    § Graviton propagator from background-independent quantum gravity · p. 1

    The interaction vertex (10j symbol, or Barrett-Crane vertex amplitude) has been shown to include, beside the 'good' term approximating the exponential of the Einstein-Hilbert action, two 'bad' terms: an exponential with opposite sign giving the cosine of the Regge action, and a dominant term depending on the existence of degenerate four-simplices.

    “The physical correctness of these theories has been questioned because in the low-energy limit their interaction vertex (10 j symbol, or Barrett-Crane vertex amplitude) has been shown to include -beside the 'good' term approximating the exponential of the Einstein-Hilbert action [19]- also two 'bad' terms: an exponential with opposite sign, giving the cosine of Regge action [19] (analogous to the”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [19] source

      Not checked in the free run.

    • cites [20] source

      Not checked in the free run.

    • cites [21] source

      Not checked in the free run.

11 of 11

  • The authors state this as an assumption.

    The triangulation is fine enough for the gravitational field to be approximately constant on a tetrahedron.

    “Assuming that the triangulation is fine enough for this field to be approximately constant on a tetrahedron, with constant value e I µ , associate the 4d vector e I s = e I µ ∆ x µ s to the segment s of the triangulation, where ∆ x µ s is the coordinate difference between the initial and final extremes of s.”

  • The authors state this as an assumption.

    The boundary state Ψ_q[s] is non-vanishing only for spinnetworks whose intertwiners are i_BC intertwiners.

    “We assume here for simplicity that Ψ q [ s ] is non vanishing only for s whose intertwiners are i BC intertwiners.”

  • The authors state this as an assumption.

    The gaussian suppresses higher-order terms in the expansion of the summand in fluctuations δj_ij.

    “We expand the summand in the fluctuations δj ij = ( j ij -j L ), and keep only the lowest term, assuming that the gaussian suppress the higher terms.”

  • The authors state this as an assumption.

    The dim j terms vary slowly over the range where the gaussian is peaked and can be considered constant.

    “We assume that the dim j terms vary slowly over the range where the gaussian is peaked, and can be considered constant.”

  • The authors state this as an assumption.

    The sum can be approximated by a gaussian integral.

    “Approximating the sum by a gaussian integral gives”
5 of 5

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 phase of the boundary state selects a classical contribution to the asymptotics of a spin foam model, a fact first noticed by Rovelli for the Barrett-Crane model in [7].

    “This is the feature of the phase of the boundary state: it selects a classical contribution to the asymptotics of a spin foam model, a fact first noticed by Rovelli for the Barrett-Crane model in [7].”

    From LQG propagator from the new spin foams (2009) · cites this paper as [7]

  • The introduction of a topological term is required in order to have a theory of connections on the boundary: without it, the connection variable does not survive the Legendre transform.

    “The introduction of a topological term is required in order to have a theory of connections on the boundary: without it, as shown by Ashtekar, the connection variable does not survive the Legendre transform [34].”

    From Flipped spinfoam vertex and loop gravity (2008) · cites this paper as [34]

  • The introduction of the topological term is required in order to have a theory of connections on the boundary; without it, the connection variable does not survive the Legendre transform.

    “Recall that the introduction of this topological term is required in order to have a theory of connections on the boundary: without it, as shown by Ashtekar, the connection variable does not survive the Legendre transform [34].”

    From Flipped spinfoam vertex and loop gravity (2008) · cites this paper as [34]

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.

  • The retained vertex-amplitude term changes from second to first

    Minor

    The text says the second term contributes while the first is suppressed, then says the calculation keeps only the first term.

  • Angles are called both extrinsic and intrinsic geometry

    Minor

    The paper identifies dihedral angles with the boundary's extrinsic geometry, but later calls the background values of those angles intrinsic geometry.

  • Large-distance spin formula contradicts the area relation

    Minor

    The stated area relation implies that, for large j L, j L is approximately A L divided by 8 π ̄ hG, not multiplied by it.

3 of 3

Cites

Flagged first, then by citations

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  6. 10.1016/0550-3213(90)90019-A · DOI · Paper page
  7. 10.1103/PhysRevLett.61.1155 · DOI · Paper page
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  10. A. Ashtekar, 1997, Adv. Theor. Math. Phys., 1, 388, 10.4310/ATMP.1997.v1.n2.a8 · DOI · Paper page
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  15. J. W. Barrett, 1999, Adv. Theor. Math. Phys., 3, 209, 10.4310/ATMP.1999.v3.n2.a1 · DOI · Paper page
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  34. 10.1088/1126-6708/2006/05/039 · DOI · Paper page

Cited by

Sorted by citations

Authors
Rovelli, C.
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
2006-10-10
Volume
97
Issue
15
Pages
151301-151301
ISSN
  • 0031-9007 (ISSN-L)
  • 1079-7114
DOI
10.1103/physrevlett.97.151301
OpenAlex
W2076583809
Semantic Scholar
da0ca5a88b6e3eb587e9a9d8103a6682a6bee28e
Cited by
212
References
7
Citations by year
2026: 1 · 2024: 5 · 2023: 4 · 2022: 7 · 2021: 4 · 2020: 2 · 2019: 2 · 2018: 3 · 2017: 7 · 2016: 6 · 2015: 6 · 2014: 7 · 2013: 12 · 2012: 14
FWCI
18.57
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review Letters (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • HAL (submittedVersion, other-oa)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • 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
graviton propagator, loop quantum gravity, spin foams, group field theory, background independence, large-distance behavior, quantum gravity

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