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LQG vertex with finite Immirzi parameter
Open accessChecked 2026-10-06 · 21 references not checked yet
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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.
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Claims the paper makes
30 No basis given · 13 Not checked yet · 5 Own interpretation · 1 Own result
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“Group averaging over Spin (4) defines then the physical intertwiner space for the node e .”
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- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
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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.”
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In the paper
“This construction defines also an embedding from the SU (2) spin networks to the Spin (4) spin networks on Γ.”
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- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
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In the paper
“The diagonal simplicity constraint restricts the direct sum to spins satisfying (25).”
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- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
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In the paper
“Notice also that the continuous label ρ becomes quantized, because n is discrete.”
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- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
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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.”
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- 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 .”
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- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
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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
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- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
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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.”
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- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
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In the paper
“It further imposes that k = n/ 2, where k again labels the subspaces diagonalizing L 2 .”
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- 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
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In the paper
“However, in our framework, the second constraint (22) breaks this symmetry and select the first branch ρ = γn .”
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It 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.
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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.”
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- The authors state this as an assumption: The Immirzi parameter γ is finite and not 0 or 1.
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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.”
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- 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.
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In the paper
“This condition is stronger than (6) since it selects only the desired B = ⋆ e ∧ e sector.”
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It 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.”
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It 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.”
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It 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.”
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It 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.”
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It 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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably 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.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It 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).”
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Probably 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]
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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In the paper
“The vertex was defined in [2, 3] only for the euclidean case, and in the absence of an Immirzi parameter γ .”
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Probably 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
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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Probably 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
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It 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]
plain- Asserted: passes — The authors put it forward as true.
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Probably 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
plain- Asserted: passes — The authors put it forward as true.
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Probably 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]
plain- Asserted: passes — The authors put it forward as true.
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Probably 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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably 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
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It 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.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ IV. LORENTZIAN THEORY · p. 5
“Notice that there is restriction on the value of γ as there was in the Euclidean case.”
§ IV. LORENTZIAN THEORY · p. 5
“The unitary representations in the principal series are labelled by ( n, ρ ), where n is a positive integer and ρ real [22, 23].”
Citations
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Perelomov, A. M. (1986) · Springer eBooks · doi:10.1007/978-3-642-61629-7 · Cited by 3,566
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Background independent quantum gravity: a status report
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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
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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
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
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Ashtekar, A., & Singh, P. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/21/213001 · Cited by 1,205
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Self-citation
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
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