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Loop-Quantum-Gravity Vertex Amplitude
Open accessAbstract
Spin foam models are hoped to provide the dynamics of loop-quantum gravity. However, the most popular of these, the Barrett-Crane model, does not have the good boundary state space and there are indications that it fails to yield good low-energy n-point functions. We present an alternative dynamics that can be derived as a quantization of a Regge discretization of Euclidean general relativity, where second class constraints are imposed weakly. Its state space matches the SO(3) loop gravity one and it yields an SO(4)-covariant vertex amplitude for Euclidean loop gravity.
Checked 2026-10-06 · 22 references not checked yet
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Written from the paper’s full text.
TL;DR
The paper proposes a new SO(4)-covariant spinfoam vertex amplitude for loop quantum gravity, obtained by imposing the off-diagonal simplicity constraints weakly rather than strongly as in the Barrett-Crane model. Its boundary state space matches SO(3) loop quantum gravity and it can be derived as a quantization of a Regge discretization of Euclidean general relativity.
Key findings
- The model provides an SO(4)-covariant vertex amplitude for LQG whose boundary state space is a natural subspace of the SO(4) spin-network space and matches the SO(3) Hamiltonian spin-network space, so “no degrees of freedom are lost.”
- It modifies the Barrett-Crane model by replacing the single Barrett-Crane intertwiner \(i_{BC}\) with a space \(K_e\) of intertwiners, leaving intertwiner degrees of freedom free instead of constraining them entirely.
- The vertex is built from an SO(4) \(15j\) symbol, which is the product of two conventional Wigner SU(2) \(15j\) symbols, together with a linear map from SO(3) intertwiners to SO(4) intertwiners.
- The authors argue the model may yield the correct low-energy \(n\)-point functions because the degrees of freedom missing in Barrett-Crane are recovered, though this is presented as an expectation rather than a demonstrated result.
Methods
- The theory is defined on a fixed 4d triangulation \(\Delta\), with a spinfoam partition function summing over integer SO(3) spins \(j_f\) on faces and intertwiners \(i_e\) on edges; the amplitude uses the SO(4) \(15j\) symbol and the map \(f\) from SO(3) to SO(4) intertwiners.
- The derivation treats a Regge discretization of Euclidean GR: co-tetrad one-forms \(e^I(t)\) for tetrahedra and \(e^I(v)\) for simplices, SO(4) group elements \(V_{vt}\), face bivectors \(B_f(t)=\int_f \star(e(t)\wedge e(t))\), closure constraints, and simplicity constraints.
- The canonical boundary variables are identified with those of SO(4) lattice gauge theory; imposing the simplicity constraints weakly reduces the state space to the SO(3) LQG spin-network space, and the single 4-simplex amplitude is evaluated by integrals and transformed back to the spin-network basis.
Limitations
- The paper explicitly states: “We do not discuss here the issue of the recovery of triangulation independence,” and that it gives “only a rapid sketch of its derivation from Regge GR. Details will be given elsewhere.”
- Evident from the text: the model is presented on a fixed triangulation, and no numerical low-energy \(n\)-point functions or quantitative tests are reported; the claim that the vertex “may yield the correct low-energy \(n\)-point functions” is not demonstrated here.
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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
“Third, the vertex can be seen as a vertex over SO (3) spin networks or SO (4) spin networks, and is both SO (3) and SO (4) covariant.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA 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 present an alternative model, that can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity, and where the constraints are imposed weakly .”
- Also stated in: § The loop-quantum-gravity vertex-amplitude · p. 1 · plain
“by properly imposing some of the constraints weakly ( 〈 φC n ψ 〉 = 0)”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § The loop-quantum-gravity vertex-amplitude · 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
“In the model (1), instead, intertwiner degrees of freedom remain free.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe 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
“Its state space is a natural subspace of the SO (4) spin-network space and matches the SO (3) hamiltonian spin network space.”
- Also stated in: § The loop-quantum-gravity vertex-amplitude · p. 1 · boosted
“First, its boundary quantum state space matches exactly the one of SO (3) loop quantum gravity: no degrees of freedom are lost.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § The loop-quantum-gravity vertex-amplitude · p. 1 · boosted
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
“More precisely, the states (4) span a subspace K e of H e .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes? Based: unclearThe 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
“Its state space is a natural subspace of the SO (4) spin-network space and matches the SO (3) hamiltonian spin network space.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe 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
“Combining this 15 j SO (4) amplitude with the constraints discussed above, gives the model (1)-(2).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe 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
“Therefore the canonical boundary variables are precisely the same as those of SO (4) lattice gauge theory.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe 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
“Finally, it is easy to check that the off-diagonal simplicity constraints are all weakly zero in this space: this follows from the fact that they are antisymmetric in the i + , i -indices, while the states (4) are symmetric.”
Load-bearingStrength:boosted· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe 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 resulting space of states is not only mathematical isomorphic to the corresponding one of SO (3) loop quantum gravity, but it can also be physically identified with it, because we have an explicit identification of the quantum operators on the two spaces with the same classical analogues, such as the area of the faces.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
- Probably not a substantial claim. 1
- It may state two separate claims as one. 5
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
“First, the boundary states of the theory are spanned by trivalent graphs colored with SO (3) spins and intertwiners.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
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
“Third, the vertex can be seen as a vertex over SO (3) spin networks or SO (4) spin networks, and is both SO (3) and SO (4) covariant.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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 constraint (15) gives gaugeinvariance at each tetrahedron, and reduces the space of states to the space of the SO (4) spin networks on the graph dual to the boundary triangulation.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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 simplicity constraints (11), as seen above, reduce each SO (4) link representation to a simple one, and the intertwiners spaces to K e .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“In BC theory, the constraints C ff ′ = 0 are imposed strongly on H e , and the only solution of these constraint equations is i BC [17].”
What the sources say
- Not checked yetcites [17] MP Reisenberger, 'On relativistic spin network vertices', J Math Phys 40 (1999) 2046 [arXiv:gr-qc/9809067] source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
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 resulting space of states is not only mathematical isomorphic to the corresponding one of SO (3) loop quantum gravity, but it can also be physically identified with it, because we have an explicit identification of the quantum operators on the two spaces with the same classical analogues, such as the area of the faces.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper relies on this without support.
The Barrett-Crane model's boundary state space does not exactly match that of loop quantum gravity.
“Its boundary state space is similar, but does not exactly match, that of loop quantum gravity; in particular the volume operator is ill-defined.”
The paper relies on this without support.
The volume operator is ill-defined in the Barrett-Crane model.
“Its boundary state space is similar, but does not exactly match, that of loop quantum gravity; in particular the volume operator is ill-defined.”
The paper relies on this without support.
The spinfoam picture has never been fully implemented in 4d.
“Compelling and popular as it is, however, this picture has never been fully implemented in 4d.”
The paper relies on this without support.
The classical limit of the simplicity constraints yields precisely the constraints that change BF theory into general relativity.
“by means of constraints -called simplicity constraints- whose classical limit yields precisely the constraints that change BF theory into general relativity (GR).”
The paper relies on this without support.
The reduction of the intertwiner space to the sole i_BC vector is commonly motivated by the imposition of the off-diagonal simplicity constraints.
“The reduction of the intertwiner space to the sole i BC vector is commonly motivated by the imposition of the off-diagonal simplicity constraints.”
The paper relies on this without support.
A simple SO(4) representation (j,j) transforms under the SO(3) subgroup in the representation j⊗j = 0⊕...⊕2j.
“A simple SO (4) representation ( j, j ) transforms under the SO (3) subgroup in the representation j ⊗ j = 0 ⊕ ... ⊕ 2 j .”
The paper relies on this without support.
All SO(3) subgroups in SO(4) are conjugate to one another.
“because all SO (3) subgroups in SO (4) are conjugate to one another.”
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
“Imposing the constraints strongly, as in the Barrett-Crane model, implies the imposition of additional conditions that are not present in the classical theory.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [168]
In the citing paper
“There is a natural way to relax the constraints and this leads to a simple model that has a more clear relationship with the canonical picture provided by LQG.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [168]
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 [168]
In the citing paper
“The basic idea was to relax the imposition of the Plebanski constraints that reduce BF theory to general relativity in the quantum theory.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [168]
In the citing paper
“They also emphasize how the model [168] corresponds to the sector γ = ∞, which has been shown to be topological [246].”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [168]
In the citing paper
“And finally, its kinematics matches exactly the one of the canonical quantization of GR, as given by loop quantum gravity (LQG) [6].”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [6]
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Metadata
- Authors
- Engle, J., Pereira, R., & Rovelli, C.
- Affiliations
- Centre de Physique Théorique (FR)
- ORCID
- Engle, Jonathan
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2007-10-16
- Volume
- 99
- Issue
- 16
- Pages
- 161301-161301
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2006259415
- Semantic Scholar
- b60ab84f99b4c4926f04fb03e184589f877f456c
- Cited by
- 335
- References
- 53
- Citations by year
- 2026: 1 · 2025: 4 · 2024: 11 · 2023: 9 · 2022: 7 · 2021: 36 · 2020: 7 · 2019: 6 · 2018: 5 · 2017: 10 · 2016: 11 · 2015: 10 · 2014: 17 · 2013: 29 · 2012: 31
- FWCI
- 26.04
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HAL (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- Agence Nationale de la Recherche (ANR-06-BLAN-0050)
- 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, spin foam models, vertex amplitude, Barrett-Crane model, Regge calculus, second-class constraints, soft constraints
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