CoChat
Sign Up
Cited by 584Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryBlack Holes and Theoretical Physics

LQG vertex with finite Immirzi parameter

Open access
Engle, Jonathan, Livine, Etera, Pereira, Roberto, Rovelli, Carlo·2008·Nuclear Physics B
DOI 10.1016/j.nuclphysb.2008.02.018
Open this paper in CoChat — free accountOpen PDF

Verified 2026-10-06 · 21 references not checked yet

AI snapshot

Written from the paper’s full text.

TL;DR

The paper extends the flipped loop-quantum-gravity vertex to finite Immirzi parameter γ in both Euclidean and Lorentzian cases. It shows that the resulting boundary Hilbert space and area spectrum match loop quantum gravity, with the area spectrum discrete and γ-dependent, and the ad hoc symplectic flip no longer required.

Key findings

  • A spinfoam model is defined for finite γ in both Euclidean and Lorentzian signatures; the boundary space is spanned by SU(2) spin networks, matching LQG on a fixed graph.
  • The area operator has the same discrete spectrum as LQG, including the correct γ dependence, in both Euclidean and Lorentzian cases; imposing the simplicity constraints reduces the previously continuous covariant-LQG spectrum to the exact discrete LQG spectrum.
  • In the Euclidean case, the diagonal constraint fixes the ratio of Spin(4) Casimirs; solutions select the highest SU(2) irreducible for γ < 1 and the lowest for γ > 1, with γ = 1 as the turning point, and impose a quantization condition on γ because the labels j± are half-integers.
  • In the Lorentzian case, solutions are ρ = γn or ρ = −n/γ; the second constraint selects ρ = γn and k = n/2, the lowest SU(2) irreducible, so the continuous label ρ becomes effectively quantized because n is discrete. The ad hoc flip of the symplectic structure is not required for finite γ.

Methods

  • The theory is discretized on a fixed Regge-like triangulation and dual 2-complex, with boundary variables B_f(t), holonomies U_f(t,t′), and conjugate J_f(t); constraints include closure, diagonal simplicity, and off-diagonal simplicity, with the latter reformulated using an internal normal vector n^I to select the B = ⋆e ∧ e sector.
  • Quantization uses L²(G)^L with G = Spin(4) in the Euclidean case and G = SL(2,C) in the Lorentzian case; the diagonal constraint is imposed strongly, while the off-diagonal constraints are imposed via a master constraint, then projections to SU(2) spin networks and vertex amplitudes are derived.
  • The area operators A and A₃ are analyzed using the constraints to obtain the spectrum.

Limitations

  • The paper states that it leaves open whether the Lorentzian vertex is finite or needs regulation, whether spinfoam finiteness results extend to this model, the complete background-independent GFT construction, checking the graviton propagator, and understanding the results in terms of coherent states.
  • The paper also notes an ordering ambiguity in the Casimir operators that may favor an area spectrum with regular spacing such as j or j + 1/2 instead of the standard √(j(j+1)), and that it is unclear whether a non-timelike choice for n^I is viable in the Lorentzian case.
  • Evident from the supplied text: several displayed equations are omitted, so the algebraic derivations and exact formulas cannot be independently checked from this snapshot.

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

Integrity signals

Registry

Crossref · 10.1016/j.nuclphysb.2008.02.018

Cites

52 of 73 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 22 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § III. EUCLIDEAN THEORY · p. 5

    Group averaging over Spin(4) defines the physical intertwiner space for the node e.

    “Group averaging over Spin (4) defines then the physical intertwiner space for the node e .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
  • Findingreasoning
    § III. EUCLIDEAN THEORY · p. 4

    The off-diagonal simplicity constraints select the SU(2) irreducible with the spin determined by (27) in each of the two factors.

    “The off-diagonal simplicity constraints select the SU (2) irreducible with the spin determined by (27) in each of the two factors.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
  • Findingreasoning
    § III. EUCLIDEAN THEORY · p. 5

    The construction defines an embedding from the SU(2) spin networks to the Spin(4) spin networks on Γ.

    “This construction defines also an embedding from the SU (2) spin networks to the Spin (4) spin networks on Γ.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
  • Findingreasoning
    § III. EUCLIDEAN THEORY · p. 4

    The diagonal simplicity constraint restricts the direct sum to spins satisfying (25).

    “The diagonal simplicity constraint restricts the direct sum to spins satisfying (25).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
  • Findingreasoning
    § IV. LORENTZIAN THEORY · p. 5

    The continuous label ρ becomes quantized because n is discrete.

    “Notice also that the continuous label ρ becomes quantized, because n is discrete.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
  • May combine two claims. 13
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 10

  • May combine two claims.

    Findingreasoning
    § IV. LORENTZIAN THEORY · p. 5

    The second constraint (22) breaks the symmetry between the two sectors and selects the branch ρ=γn.

    “However, in our framework, the second constraint (22) breaks this symmetry and select the first branch ρ = γn .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

  • May combine two claims.

    Findingreasoning
    § A. Classical theory · p. 2

    For finite non-trivial Immirzi parameter, both sectors of solutions of constraints (5) and (6) yield general relativity, but with different Newton constant and Immirzi parameter.

    “For finite, non-trivial Immirzi parameter, both sectors in fact yield GR, but the value of the Newton constant and Immirzi parameter are different in each sector.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

  • May combine two claims.

    Findingnone given
    § A. Classical theory · p. 3

    The closure constraint for B_f is equivalent to the closure constraint for J_f, which is imposed automatically by the dynamics.

    “The closure for the B f is equivalent to the closure for the J f which, as noted above, will be imposed automatically by the dynamics.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Geometry is flat on each 4-simplex; curvature is concentrated on the bones and is coded in the holonomy around the link of each face.
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

  • May combine two claims.

    Findingreasoning
    § A. Classical theory · p. 3

    The reformulation of the simplicity constraint (6) by the existence of an internal vector n^I for each tetrahedron is stronger than (6) and selects only the B=⋆e∧e sector.

    “This condition is stronger than (6) since it selects only the desired B = ⋆ e ∧ e sector.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Geometry is flat on each 4-simplex; curvature is concentrated on the bones and is coded in the holonomy around the link of each face.
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

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

    Findingown data
    § LQG vertex with finite Immirzi parameter · p. 1

    The resulting dynamics is defined on a Hilbert space isomorphic to the one of loop quantum gravity; the boundary space matches LQG, spanned by SU(2) spin networks, for all values of γ on a fixed graph, in both euclidean and lorentzian cases.

    “We show that the resulting dynamics is defined on a Hilbert space isomorphic to the one of loop quantum gravity”
    • Also stated in: § I. INTRODUCTION · p. 1 · plain
      “Second, for all values of γ the state space precisely matches the one of LQG (on a fixed graph).”
    • Also stated in: § VI. CONCLUSION · p. 7 · plain
      “In both cases, the boundary space turns out to be the same as in LQG, spanned by SU (2) spin networks.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

  • May combine two claims.

    Findingnone given
    § B. Quantum kinematics · p. 4

    Constraints (14) do not close as a Poisson algebra and their imposition in the quantum theory is more subtle.

    “The constraints (14) on the other hand do not close as a Poisson algebra and their imposition in the quantum theory is more subtle.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Geometry is flat on each 4-simplex; curvature is concentrated on the bones and is coded in the holonomy around the link of each face.
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

  • May combine two claims.

    Findingnone given
    § B. Quantum kinematics · p. 3

    Constraint (12) commutes with the other constraints and can be imposed directly as a strong operator equation.

    “The constraint (12) commutes with the others and can be imposed directly as a strong operator equation.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Geometry is flat on each 4-simplex; curvature is concentrated on the bones and is coded in the holonomy around the link of each face.
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

  • May combine two claims.

    Findingreasoning
    § LQG vertex with finite Immirzi parameter · p. 1

    The area operator has the same discrete spectrum as in loop quantum gravity, including the correct dependence on γ, and this holds in the lorentzian case as well.

    “and that the area operator has the same discrete spectrum as in loop quantum gravity. This includes the correct dependence on γ , and, remarkably, holds in the lorentzian case as well.”
    • Also stated in: § I. INTRODUCTION · p. 1 · plain
      “Third, the spectrum of the area operator turns out to be discrete, and to be the same as in LQG, including the correct dependence on the Immirzi parameter γ .”
    • Also stated in: § VI. CONCLUSION · p. 7 · plain
      “The spectrum of the area operator too is the same as in LQG, both for the euclidean and the lorentzian sectors.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

  • Classified as probably not a substantial claim.

    Findingreasoning
    § IV. LORENTZIAN THEORY · p. 6

    The construction defines the projection from the SL(2,C) boundary Hilbert space to the SU(2) space.

    “This construction defines the projection from the SL (2 , C ) boundary Hilbert space to the SU (2) space.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

  • May combine two claims.

    Interpretationreasoning
    § I. INTRODUCTION · p. 1

    The area spectrum is continuous in spinfoams at the kinematical level, but becomes discrete after proper implementation of the second class constraints.

    “The solution is that the area spectrum is continuous in spinfoams at the kinematical level, but it turns out to become discrete after proper implementation of the (second class) constraints.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

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

    Interpretationnone given
    § V. AREA SPECTRA · p. 6

    The ordering of the Casimir operators required to have meaningful simplicity constraints does not use the usual ordering but seems to select an area spectrum with regular spacing j or j+1/2 instead of the standard sqrt(j(j+1)).

    “Finally, we would like to point out that the ordering of the Casimir operators for SU (2) and SL (2 , C ) required to have meaningful simplicity constraints do not use the usual ordering but seems to select an area spectrum with a regular spacing such as j (or j + 1 / 2) instead of the standard √ j ( j +1).”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.

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

    Premisecitation
    § I. INTRODUCTION · p. 2

    A group field theory yielding the new vertex amplitude is already considered in [12].

    “A GFT yielding the new vertex amplitude is indeed already considered in [12].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [12]

      Not checked in the free run.

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

    Premisecitation
    § I. INTRODUCTION · p. 1

    The vertex was defined in [2,3] only for the euclidean case and in the absence of an Immirzi parameter γ.

    “The vertex was defined in [2, 3] only for the euclidean case, and in the absence of an Immirzi parameter γ .”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [2] source

      Not checked in the free run.

    • cites [3] source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § I. INTRODUCTION · p. 1

    The spectrum of geometrical operators in the coherent-state formulation turns out to be quite different from the standard one of loop quantum gravity.

    “Furthermore, the spectrum of the geometrical operators in this formulation turns out to be quite different from the standard one of loop quantum gravity [14].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [14] source

      Not checked in the free run.

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

    Premisecitation
    § I. INTRODUCTION · p. 2

    Any spinfoam model admits a group field theory formulation, which is triangulation independent.

    “It was shown in [16] that any spinfoam model [17], as the one we define here, admits a group field theory (GFT) [18] formulation, which is triangulation independent.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [16] source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    The original model of [2,3] and the variant pointed out in [11] appear as limiting γ→0 and γ→∞ cases, respectively.

    “The original model of [2, 3] and the variant pointed out in [11] appear as limiting γ → 0 and γ → ∞ cases, respectively.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [2] source

      Not checked in the free run.

    • cites [3] source

      Not checked in the free run.

    • cites [11] source

      Not checked in the free run.

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

    Premisecitation
    § I. INTRODUCTION · p. 1

    A lorentzian version of the vertex amplitude was constructed in [7], still without γ.

    “A key step to extend the definition of this new vertex was taken in [7], where a lorentzian version of the vertex amplitude is constructed, still without γ .”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [7]

      Not checked in the free run.

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

    Premisecitation
    § I. INTRODUCTION · p. 1

    All these models are defined by the same vertex, the square of the SU(2) Wigner 15j symbol; they differ for the class of boundary states considered and their measure in the spinfoam sum.

    “All these models ([2, 3, 11, 12],) are defined by the same vertex, namely the square of the SU (2) Wigner 15 j symbol; they differ for the class of boundary states considered and their measure in the spinfoam sum.”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [2] source

      Not checked in the free run.

    • cites [3] source

      Not checked in the free run.

    • cites [11] source

      Not checked in the free run.

    • cites [12]

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § I. INTRODUCTION · p. 1

    The variant of the model has been extended to finite γ>1.

    “This variant has been extensively explored in [12], and extended to the case of finite γ > 1.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [12]

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § I. INTRODUCTION · p. 1

    The new vertex amplitude is essentially given by the square of an SU(2) Wigner 15j symbol.

    “A different vertex has been recently introduced in [2, 3]; its amplitude essentially given by the square of an SU (2) Wigner 15 j symbol.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [2] source

      Not checked in the free run.

    • cites [3] source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    The constraints can equivalently be solved using coherent states, yielding the same result as the master-constraint technique.

    “In [11], it was shown that these constraints can equivalently be solved using a different technique, based on coherent states, yielding the same result.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [11] source

      Not checked in the free run.

21 of 21

  • The authors state this as an assumption.

    Geometry is flat on each 4-simplex; curvature is concentrated on the bones and is coded in the holonomy around the link of each face.

    “Geometry is assumed to be flat on each 4-simplex; curvature is concentrated on the 'bones' f , and is coded in the holonomy around the 'link' of each f .”

  • The authors state this as an assumption.

    The Immirzi parameter γ is finite and not 0 or 1.

    “(see [20] and [13]) where we have assumed γ fi nite and = 0 , 1.”

  • The authors state this as an assumption.

    The internal vector n^I is fixed to δ^I_0, the general case is recovered by gauge invariance, and in the lorentzian case this restricts all tetrahedra to be spacelike.

    “In order to proceed, let us fix n I = δ 0 I . The general case will be recovered by gauge invariance. In the lorentzian case this choice restricts all tetrahedra to be spacelike”

  • The authors state this as an assumption.

    Constraint (3) is assumed to hold prior to varying the action.

    “(3) is assumed to hold prior to varying the action.”
4 of 4

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 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].”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [164]

  • The vertex amplitude for the EPRL/FK models is given by equation (30), with unbalanced spins j±_ab = γ± j_ab and γ± = (1 ± γ)/2.

    “where the unbalanced spins j + , j -are j ± ab = γ ± j ab , γ ± = 1 ± γ 2 . (31”

    From LQG propagator from the new spin foams (2009) · cites this paper as [19]

  • For Euclidean signature and Immirzi parameter 0 < γ < 1, the EPRL γ and FK γ spin foam models coincide.

    “We restrict attention to Euclidean signature and Immirzi parameter smaller than one: 0 < γ < 1.In this case the two models coincide.”

    From LQG propagator from the new spin foams (2009) · cites this paper as [19]

  • The definition of the n-valent Barrett-Crane intertwiner is equivalent to that of [10-12].

    “That definition is merely equivalent to that of [10-12], however we spell it out in our notation for the clarity and the sake of precision.”

    From Spin-foams for all loop quantum gravity (2010) · cites this paper as [10]

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.

  • Lorentzian γ is said to be restricted despite a continuous representation label

    Minor

    The Lorentzian section says γ is restricted as in the Euclidean case. But it states that ρ is real and selects ρ = γn; for any real γ and positive integer n, that choice supplies an allowed real ρ, so the stated conditions impose no analogous quantization of γ.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. Rovelli, 1995, Nucl. Phys. B, 456, 734
  2. DePietri, 2000, Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space, Nucl. Phys. B, 574, 785, 10.1016/s0550-3213(00)00005-5 · DOI · Paper page

Cited by

Sorted by citations

Related

Authors
Engle, J., Livine, E., Pereira, R., & Rovelli, C.
Affiliations
  • Université de Toulon (FR)
  • École Normale Supérieure de Lyon (FR)
Corresponding author
Livine, Etera
ORCID
Venue
Nuclear Physics B
Publisher
Elsevier BV
Published
2008-08
Volume
799
Issue
1-2
Pages
136-149
ISSN
0550-3213 (ISSN-L)
DOI
10.1016/j.nuclphysb.2008.02.018
OpenAlex
W2043256764
Cited by
584
References
73
Citations by year
2026: 11 · 2025: 18 · 2024: 23 · 2023: 25 · 2022: 17 · 2021: 27 · 2020: 18 · 2019: 23 · 2018: 24 · 2017: 24 · 2016: 39 · 2015: 20 · 2014: 41 · 2013: 66 · 2012: 55
FWCI
38.65
Type
journal article
Language
en
Open access
Licence
Locations
  • Nuclear Physics B (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • HAL (submittedVersion)
  • arXiv (Cornell University)
  • Nuclear Physics B - Proceedings Supplements
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, Immirzi parameter, symplectic structure, four-dimensional quantum gravity

About these checks

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.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review