The DOI matches its Crossref record; 52 of the 73 references it cites were found in a registry or an index, and 21 are not checked yet; none of the 22 sources its claims cite is checked for support yet.
LQG vertex with finite Immirzi parameter
Open accessChecked 2026-10-06 · 21 references not checked yet
Overview
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
Cites
Statistics
Verification
Claims the paper makes
0 of 22 cited sources checked · 0 supported · 0 flagged
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Group averaging over Spin (4) defines then the physical intertwiner space for the node e .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The off-diagonal simplicity constraints select the SU (2) irreducible with the spin determined by (27) in each of the two factors.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This construction defines also an embedding from the SU (2) spin networks to the Spin (4) spin networks on Γ.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The diagonal simplicity constraint restricts the direct sum to spins satisfying (25).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Notice also that the continuous label ρ becomes quantized, because n is discrete.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The ad hoc flip of the symplectic structure that was required to derive the flipped vertex is not anymore required for finite γ .”
- Also stated in: § I. INTRODUCTION · p. 2 · plain
“is not required in the finite γ case.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
- Also stated in: § I. INTRODUCTION · p. 2 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The constraint (17) fixes the ratio between these two Casimirs in term of the Immirzi parameter γ .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
None of the sources cited for this claim have been checked yet.
In the paper
“Following [2, 3, 7], the amplitude of a single vertex bounded by ten SU (2) spins j ab , a, b = 1 , ..., 5 and five SU (2) intertwiners i a is given by”
What the sources say
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Imposing C e = 0 strongly on the states in H 0 selects in each link the lowest (resp. highest) SU (2) irreducible.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It further imposes that k = n/ 2, where k again labels the subspaces diagonalizing L 2 .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
- It may state two separate claims as one. 13
- The quote only partly states this claim. 1
- Probably not a substantial claim. 10
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“However, in our framework, the second constraint (22) breaks this symmetry and select the first branch ρ = γn .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This condition is stronger than (6) since it selects only the desired B = ⋆ e ∧ e sector.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- 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.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
- Also stated in: § I. INTRODUCTION · p. 1 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The constraints (14) on the other hand do not close as a Poisson algebra and their imposition in the quantum theory is more subtle.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The constraint (12) commutes with the others and can be imposed directly as a strong operator equation.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
- Also stated in: § I. INTRODUCTION · p. 1 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This construction defines the projection from the SL (2 , C ) boundary Hilbert space to the SU (2) space.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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).”
Strength:hedged· not yet calibratedCausal:conditional· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
In the paper
“A GFT yielding the new vertex amplitude is indeed already considered in [12].”
What the sources say
- Not checked yetcites [12]
Strength:boosted· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“The vertex was defined in [2, 3] only for the euclidean case, and in the absence of an Immirzi parameter γ .”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may state two separate claims as one.
In the paper
“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].”
What the sources say
- Not checked yetcites [14] source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
In the paper
“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.”
What the sources say
- Not checked yetcites [16] source
Strength:boosted· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“The original model of [2, 3] and the variant pointed out in [11] appear as limiting γ → 0 and γ → ∞ cases, respectively.”
What the sources say
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“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 .”
What the sources say
- Not checked yetcites [7]
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
What the sources say
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may state two separate claims as one.
In the paper
“This variant has been extensively explored in [12], and extended to the case of finite γ > 1.”
What the sources say
- Not checked yetcites [12]
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
In the paper
“In [11], it was shown that these constraints can equivalently be solved using a different technique, based on coherent states, yielding the same result.”
What the sources say
- Not checked yetcites [11] source
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
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.”
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?
In the citing paper
“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]
In the citing paper
“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]
In the citing paper
“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]
In the citing paper
“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
MinorThe 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.
“Notice that there is restriction on the value of γ as there was in the Euclidean case.”
“The unitary representations in the principal series are labelled by ( n, ρ ), where n is a positive integer and ρ real [22, 23].”
Citations
Cites
Generalized Coherent States and Their Applications
Perelomov, A. M. (1986) · Springer eBooks · doi:10.1007/978-3-642-61629-7 · Cited by 3,566
Kiefer, C. (2004) · Cited by 2,981
Background independent quantum gravity: a status report
Ashtekar, A., & Lewandowski, J. (2004) · Classical and Quantum Gravity · doi:10.1088/0264-9381/21/15/r01 · Cited by 1,951
Discreteness of area and volume in quantum gravity
Rovelli, C., & Smolin, L. (1995) · Nuclear Physics B · doi:10.1016/0550-3213(95)00150-q · Cited by 1,365 · cited 2×
“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].”
BackgroundQuantum theory of geometry: I. Area operators
Ashtekar, A., & Lewandowski, J. (1997) · Classical and Quantum Gravity · doi:10.1088/0264-9381/14/1a/006 · Cited by 852
Loop space representation of quantum general relativity
Rovelli, C., & Smolin, L. (1990) · Nuclear Physics B · doi:10.1016/0550-3213(90)90019-a · Cited by 806
Introduction to Modern Canonical Quantum General RelativityPreprint
Thiemann, T. (2001) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0110034 · Cited by 685
Quantum theory of geometry II: Volume operators
Ashtekar, A., & Lewandowski, J. (1997) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.1997.v1.n2.a8 · Cited by 580
Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action
Holst, S. (1996) · Physical Review D · doi:10.1103/physrevd.53.5966 · Cited by 576
Spin foam models for quantum gravity
Perez, A. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/6/202 · Cited by 532
A new spin foam model for 4D gravity
Freidel, L., & Krasnov, K. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/12/125018 · Cited by 521
An Introduction to Spin Foam Models of BF Theory and Quantum Gravity
Baez, J. C. (2000) · Lecture Notes in Physics · doi:10.1007/3-540-46552-9_2 · Cited by 464
Knot Theory and Quantum Gravity
Rovelli, C., & Smolin, L. (1988) · Physical Review Letters · doi:10.1103/physrevlett.61.1155 · Cited by 463
Relativistic Spin Networks and Quantum GravityPreprint
Barrett, J. W., Crane, L., & Crane, J. W. B. L. (1998) · Cited by 423
Relativistic spin networks and quantum gravity
Barrett, J. W., & Crane, L. (1998) · Journal of Mathematical Physics · doi:10.1063/1.532254 · Cited by 404
New spinfoam vertex for quantum gravity
Livine, E. R., & Speziale, S. (2007) · Physical Review D · doi:10.1103/physrevd.76.084028 · Cited by 401
Self-citationWeaving a classical metric with quantum threads
Ashtekar, A., Rovelli, C., & Smolin, L. (1992) · Physical Review Letters · doi:10.1103/physrevlett.69.237 · Cited by 399
Baez, J. C. (1998) · Classical and Quantum Gravity · doi:10.1088/0264-9381/15/7/004 · Cited by 381
Group Field Theory: An Overview
Freidel, L. (2005) · International Journal of Theoretical Physics · doi:10.1007/s10773-005-8894-1 · Cited by 378
“Sum over surfaces” form of loop quantum gravity
Reisenberger, M. P., & Rovelli, C. (1997) · Physical Review D · doi:10.1103/physrevd.56.3490 · Cited by 372
Loop-Quantum-Gravity Vertex Amplitude
Engle, J., Pereira, R., & Rovelli, C. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.161301 · Cited by 335
Self-citationQuantum gravity and Regge calculus
Immirzi, G. (1997) · Nuclear Physics B - Proceedings Supplements · doi:10.1016/s0920-5632(97)00354-x · Cited by 287
A Lorentzian Signature Model for Quantum General RelativityPreprint
Barrett, J. W., & Crane, L. (2000) · Cited by 282
A Lorentzian signature model for quantum general relativity
Barrett, J. W., & Crane, L. (2000) · Classical and Quantum Gravity · doi:10.1088/0264-9381/17/16/302 · Cited by 275
The Phoenix Project: master constraint programme for loop quantum gravity
Thiemann, T. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/23/7/002 · Cited by 272
1 Quantum Gravity and Regge CalculusPreprint
Immirzi, G. (1997) · Cited by 268
Group Field Theory: An overview
Freidel, L. (2005) · CERN Document Server (European Organization for Nuclear Research) · Cited by 249
Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space
De Pietri, R., Freidel, L., Krasnov, K., & Rovelli, C. (2000) · Nuclear Physics B · doi:10.1016/s0550-3213(00)00005-5 · Cited by 244
Flipped spinfoam vertex and loop gravity
Engle, J., Pereira, R., & Rovelli, C. (2008) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2008.02.002 · Cited by 241
Self-citationGraviton Propagator from Background-Independent Quantum Gravity
Rovelli, C. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.151301 · Cited by 212
Spacetime as a Feynman diagram: the connection formulation
Reisenberger, M. P., & Rovelli, C. (2001) · Classical and Quantum Gravity · doi:10.1088/0264-9381/18/1/308 · Cited by 202
Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity
Oriti, D. (2001) · Reports on Progress in Physics · doi:10.1088/0034-4885/64/12/203 · Cited by 201
The group field theory approach to Quantum Gravity
Oriti, D. (2009) · Cambridge University Press eBooks · doi:10.1017/cbo9780511575549.020 · Cited by 193
The Lorentz group and harmonic analysis
Rühl, W. (1970) · CERN Document Server (European Organization for Nuclear Research) · Cited by 188
Three-dimensional loop quantum gravity: physical scalar product and spin-foam models
Noui, K., & Perez, A. (2005) · Classical and Quantum Gravity · doi:10.1088/0264-9381/22/9/017 · Cited by 181
Solving the simplicity constraints for spinfoam quantum gravity
Livine, E. R., & Simone Speziale, D. (2008) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/81/50004 · Cited by 177
Self-citationGraviton propagator in loop quantum gravity
Bianchi, E., Modesto, L., Rovelli, C., & Speziale, S. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/23/23/024 · Cited by 163
Graviton propagator in loop quantum gravityPreprint
Eugenio Bianchi, A., Modesto, L., Carlo Rovelli, C., & Simone Speziale, D. (2008) · Cited by 156
SU(2)loop quantum gravity seen from covariant theory
Alexandrov, S., & Livine, E. R. (2003) · Physical Review D · doi:10.1103/physrevd.67.044009 · Cited by 141
Self-citationAn Introduction to Loop Quantum Gravity Through CosmologyPreprint
Ashtekar, A. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0702030 · Cited by 137
Testing the master constraint programme for loop quantum gravity: I. General framework
Dittrich, B., & Thiemann, T. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/23/4/001 · Cited by 122
A spin foam model without bubble divergences
Perez, A., & Rovelli, C. (2001) · Nuclear Physics B · doi:10.1016/s0550-3213(01)00030-x · Cited by 113
A lattice worldsheet sum for 4-d Euclidean general relativityPreprint
Reisenberger, M. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9711052 · Cited by 113
Spin foam model for Lorentzian general relativity
Pérez, A., & Carlo Rovelli, C. (2001) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.63.041501 · Cited by 102
Self-citationQuantum spin dynamics: VIII. The master constraint
Thiemann, T. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/23/7/003 · Cited by 99
Hamiltonian analysis of Plebanski theory
Buffenoir, E., Marc Henneaux, A., Noui, K., & Roche, P. (2004) · Classical and Quantum Gravity · doi:10.1088/0264-9381/21/22/012 · Cited by 95
Particle Scattering in Loop Quantum Gravity
Modesto, L., & Rovelli, C. (2005) · Physical Review Letters · doi:10.1103/physrevlett.95.191301 · Cited by 93
Lorentzian loop quantum gravity vertex amplitude
Roberto Pereira, B. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/8/085013 · Cited by 85
Group integral techniques for the spinfoam graviton propagator
Livine, E. R., & Simone Speziale, D. (2006) · Journal of High Energy Physics · doi:10.1088/1126-6708/2006/11/092 · Cited by 83
Self-citationChoice of connection in loop quantum gravity
Alexandrov, S. (2001) · Physical Review D · doi:10.1103/physrevd.65.024011 · Cited by 80
SPIN FOAMS AS FEYNMAN DIAGRAMS
Reisenberger, M., & Carlo Rovelli, C. (2003) · Johns Hopkins Workshop on Current Problems in Particle Theory (JHW) · doi:10.1142/9789812791368_0023 · Cited by 80
Quantum Spin Dynamics VIII. The Master ConstraintPreprint
Thiemann, T. (2006) · Cited by 77
Testing the master constraint programme for loop quantum gravity: II. Finite-dimensional systems
Dittrich, B., & Thiemann, T. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/23/4/002 · Cited by 74
Finiteness of a spinfoam model for Euclidean quantum general relativity
Perez, A. (2001) · Nuclear Physics B · doi:10.1016/s0550-3213(01)00014-1 · Cited by 73
Linearized dynamics from the 4-simplex Regge action
Dittrich, B., Freidel, L., & Simone Speziale, D. (2007) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.76.104020 · Cited by 68
Perturbative Finiteness in Spin-Foam Quantum Gravity
Crane, L., Perez, A., & Rovelli, C. (2001) · Physical Review Letters · doi:10.1103/physrevlett.87.181301 · Cited by 64
Towards the graviton from spinfoams: Higher order corrections in the 3D toy model
Livine, E. R., Simone Speziale, D., & Willis, J. L. (2007) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.75.024038 · Cited by 60
Self-citationTowards the graviton from spinfoams: the 3d toy model
Speziale, S. (2006) · Journal of High Energy Physics · doi:10.1088/1126-6708/2006/05/039 · Cited by 59
A finiteness proof for the Lorentzian state sum spinfoam model for quantum general relativityPreprint
Crane, L., Pérez, A., & Carlo Rovelli, C. (2001) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0104057 · Cited by 58
Self-citationSpin foam model from canonical quantization
Alexandrov, S. Y. (2008) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.77.024009 · Cited by 56
Gluing 4-simplices: A derivation of the Barrett-Crane spin foam model for Euclidean quantum gravity
Oriti, D., & Williams, R. M. (2000) · Physical Review D · doi:10.1103/physrevd.63.024022 · Cited by 54
(3+1)-dimensional spin foam model of quantum gravity with spacelike and timelike components
Pérez, A., & Carlo Rovelli, C. (2001) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.64.064002 · Cited by 52
Towards the graviton from spinfoams: the 3d toy model
Speziale, S. (2005) · CERN Document Server (European Organization for Nuclear Research) · Cited by 49
Barrett-Crane spin foam model from generalizedBF-type action for gravity
Livine, R. E., & Oriti, D. (2002) · Physical Review D · doi:10.1103/physrevd.65.044025 · Cited by 46
Spin foam quantization of $SO(4)$ Plebanski’s action
Pérez, A. (2001) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.2001.v5.n5.a4 · Cited by 37
The perturbative Regge-calculus regime of loop quantum gravity
Bianchi, E., & Modesto, L. (2008) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2007.12.011 · Cited by 34
Coherent states, constraint classes and area operators in the new spin-foam models
Engle, J., & Roberto Pereira, B. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/10/105010 · Cited by 33
Self-citationLes Representations lineaires du Groupe de Lorentz
Naĭmark, M. A., & Corbeiller, P. L. (1964) · American Journal of Physics · doi:10.1119/1.1970103 · Cited by 31
Towards a Covariant Loop Quantum GravityPreprint
Livine, E. R. (2006) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0608135 · Cited by 26
Self-citationNumerical indications on the semiclassical limit of the flipped vertex
Magliaro, E., Perini, C., & Carlo Rovelli, C. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/9/095009 · Cited by 26
Corrections to “Spin foam quantization of $SO(4)$ Plebanski’s action”
Pérez, A. (2002) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.2002.v6.n3.e1 · Cited by 13
Livine, E. R., & Oriti, D. (2005) · Journal of High Energy Physics · doi:10.1088/1126-6708/2005/11/050 · Cited by 12
Self-citationCLEBSCH--GORDAN COEFFICIENTS FOR THE LORENTZ GROUP. I. PRINCIPAL SERIES.
Anderson, R. L., Rączka, R., Rashid, & Pavel Winternitz, A. (1967) · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · Cited by 5
Flagged first, then by citations
Research works printed but not linked: 2
- Rovelli, 1995, Nucl. Phys. B, 456, 734
- 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
Loop quantum cosmology: a status report
Ashtekar, A., & Singh, P. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/21/213001 · Cited by 1,205
Vagnozzi, S., Roy, R., Tsai, Y.-D., Visinelli, L., Afrin, M., Allahyari, A., Bambhaniya, P., Dey, D., Ghosh, S. G., Joshi, P. S., et al. (2023) · Classical and Quantum Gravity · doi:10.1088/1361-6382/acd97b · Cited by 924
The Spin-Foam Approach to Quantum Gravity
Perez, A. (2013) · Living Reviews in Relativity · doi:10.12942/lrr-2013-3 · Cited by 677
Quantum gravity phenomenology at the dawn of the multi-messenger era—A review
Addazi, A., Álvarez-Muñiz, J., Batista, R. A., Giovanni Amelino-camelia, A., Antonelli, V., Arzano, M., Asorey, M., Atteia, J. L., Bahamonde, S., Bajardi, F., et al. (2022) · Progress in Particle and Nuclear Physics · doi:10.1016/j.ppnp.2022.103948 · Cited by 580
Colored Tensor Models - a Review
Gurău, R. (2012) · Symmetry Integrability and Geometry Methods and Applications · doi:10.3842/sigma.2012.020 · Cited by 399
Metadata
- Authors
- Engle, J., Livine, E., Pereira, R., & Rovelli, C.
- Affiliations
- Université de Toulon (FR)
- École Normale Supérieure de Lyon (FR)
- Corresponding author
- Livine, Etera
- Venue
- Nuclear Physics B
- Publisher
- Elsevier BV
- Published
- 2008-08
- Volume
- 799
- Issue
- 1-2
- Pages
- 136-149
- ISSN
- 0550-3213 (ISSN-L)
- 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
- green
- arxiv.org
- 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 › Nuclear and High Energy 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.
A language model drew the claims from the paper on 2026-10-06, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-06 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
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
