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.
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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.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
0 of 4 cited sources checked · 0 supported · 0 flagged
- Findingnone given§ The loop-quantum-gravity vertex-amplitude · p. 1
The vertex can be seen as a vertex over SO(3) spin networks or SO(4) spin networks.
“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.”
Quote in the paperSubstantial claim - Findingown data§ The loop-quantum-gravity vertex-amplitude · p. 1
In the presented model, the constraints are imposed weakly.
“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)”
Quote in the paperSubstantial claim - Also stated in: § The loop-quantum-gravity vertex-amplitude · p. 1 · plain
- Findingnone given§ The loop-quantum-gravity vertex-amplitude · p. 3
In the presented model, intertwiner degrees of freedom remain free.
“In the model (1), instead, intertwiner degrees of freedom remain free.”
Quote in the paperSubstantial claim - Findingreasoning§ The loop-quantum-gravity vertex-amplitude · p. 1
The state space of the presented model matches the SO(3) hamiltonian spin network space.
“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.”
Quote in the paperSubstantial claim - Also stated in: § The loop-quantum-gravity vertex-amplitude · p. 1 · boosted
- Findingnone given§ The loop-quantum-gravity vertex-amplitude · p. 3
The states (4) span a subspace K_e of H_e.
“More precisely, the states (4) span a subspace K e of H e .”
Quote in the paperNot a substantial claim
- Findingnone given§ The loop-quantum-gravity vertex-amplitude · p. 1
The state space of the presented model is a natural subspace of the SO(4) spin-network space.
“Its state space is a natural subspace of the SO (4) spin-network space and matches the SO (3) hamiltonian spin network space.”
Quote in the paperSubstantial claim - Findingreasoning§ The loop-quantum-gravity vertex-amplitude · p. 4
Combining the 15j SO(4) amplitude with the constraints discussed gives the model (1)-(2).
“Combining this 15 j SO (4) amplitude with the constraints discussed above, gives the model (1)-(2).”
Quote in the paperSubstantial claim - Findingreasoning§ The loop-quantum-gravity vertex-amplitude · p. 4
On the boundary of the triangulation, the canonical boundary variables are precisely the same as those of SO(4) lattice gauge theory.
“Therefore the canonical boundary variables are precisely the same as those of SO (4) lattice gauge theory.”
Quote in the paperSubstantial claim - Findingreasoning§ The loop-quantum-gravity vertex-amplitude · p. 3
The off-diagonal simplicity constraints are all weakly zero in K_e.
“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.”
Quote in the paperSubstantial claim - Findingreasoning§ The loop-quantum-gravity vertex-amplitude · p. 4
The resulting space of states is mathematically isomorphic to the corresponding one of SO(3) loop quantum gravity.
“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.”
Quote in the paperSubstantial claim - Findingreasoning§ The loop-quantum-gravity vertex-amplitude · p. 3
Imposing the constraints on each face selects from the tensor product of simple SO(4) representation spaces the space formed by the tensor product of the highest SO(3) irreducibles.
“Thus imposing the constraints on each face selects from ( H ( j f 1 ,j f 1 ) ⊗ ... ⊗ H ( j f 4 ,j f 4 ) ) the space formed by the tensor product of the highest SO (3) irreducibles.”
Quote in the paperSubstantial claim - Findingreasoning§ The loop-quantum-gravity vertex-amplitude · p. 3
What is obtained by this projection is precisely K_e.
“In fact, what we obtain is precisely K e .”
Quote in the paperSubstantial claim - Interpretationreasoning§ The loop-quantum-gravity vertex-amplitude · p. 1
The difficulties of the Barrett-Crane model can be traced to the SO(4) → SU(2) gauge fixing and the way certain second class constraints are imposed strongly.
“These difficulties can be traced to the SO (4) → SU (2) gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly .”
Quote in the paperSubstantial claim- Rests on an assumption: The Barrett-Crane model's boundary state space does not exactly match that of loop quantum gravity.
- Rests on an assumption: The volume operator is ill-defined in the Barrett-Crane model.
- Rests on an assumption: The spinfoam picture has never been fully implemented in 4d.
- Interpretationreasoning§ The loop-quantum-gravity vertex-amplitude · p. 1
The model provides a long sought SO(4)-covariant vertex amplitude for loop quantum gravity.
“The model provides a long sought SO (4)-covariant vertex amplitude for loop quantum gravity.”
Quote in the paperSubstantial claim - Premisecitation§ The loop-quantum-gravity vertex-amplitude · p. 3
Imposing second class constraints strongly is a well-known way of erroneously killing physical degrees of freedom.
“Imposing second class constraints strongly is a wellknown way of erroneously killing physical degrees of freedom in a theory.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] source
Not checked in the free run.
- Premisereasoning§ The loop-quantum-gravity vertex-amplitude · p. 3
The Barrett-Crane theory constrains entirely the intertwiner degrees of freedom.
“The Barrett-Crane theory therefore constrains entirely the intertwiner degrees of freedom.”
Quote in the paperSubstantial claim - Interpretationreasoning§ The loop-quantum-gravity vertex-amplitude · p. 3
The difference between the presented model and the Barrett-Crane model is in the intertwiner state space.
“The difference between the two theories is therefore in the intertwiner state space.”
Quote in the paperSubstantial claim - Interpretationcitation§ The loop-quantum-gravity vertex-amplitude · p. 1
The presented model can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity.
“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
“Finally, the theory can be obtained as a bona fide quantization of a discretization of euclidean GR on a Regge triangulation.”
- Also stated in: § The loop-quantum-gravity vertex-amplitude · p. 3 · plain
“We close by sketching the derivation of this model as a quantization of a discretization of GR (see [18]).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [18] A Perez: 'Spin foam quantization of SO(4) Plebanski's action', Adv Theor Math Phys 5 (2002) 947 [Erratum-ibid 6 (2003) 593]. MP Reisenberger, 'A lattice worldsheet sum for 4-d Euclidean general relativity', arXiv:gr-qc/9711052. L Freidel, K Krasnov, 'Spin Foam Models and the Classical Action Principle', Adv Theor Math Phys 2 (1999) 1183-1247.
Not checked in the free run.
- Also stated in: § The loop-quantum-gravity vertex-amplitude · p. 1 · plain
- Interpretationreasoning§ The loop-quantum-gravity vertex-amplitude · p. 3
The step from the single intertwiner i_BC to the space K_e is the essential modification made with respect to the Barrett-Crane model.
“The step from the single intertwiner i BC to the space K e is therefore the essential modification made with respect to the BC model.”
Quote in the paperSubstantial claim - Premisecitation§ The loop-quantum-gravity vertex-amplitude · p. 1
The euclidean Barrett-Crane model may fail to yield the correct low-energy n-point functions.
“Worse, recent calculations appear to indicate that some n -point functions fail to yield the correct low-energy limit [13].”
- Also stated in: § The loop-quantum-gravity vertex-amplitude · p. 1 · hedged
“there are recent indications that, consequently, it may fail to yield the correct low-energy n -point functions.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [13] source
Not checked in the free run.
- Also stated in: § The loop-quantum-gravity vertex-amplitude · p. 1 · hedged
- Premisereasoning§ The loop-quantum-gravity vertex-amplitude · p. 3
The off-diagonal simplicity constraints impose that the faces of the tetrahedron lie on a common 3d subspace of 4d spacetime.
“As noted, these constraints impose the faces of the tetrahedron to lie on a common 3d subspace of 4d spacetime.”
Quote in the paperSubstantial claim - Interpretationreasoning§ The loop-quantum-gravity vertex-amplitude · p. 3
Projecting to the SO(4) invariant-tensor space makes the dependence on the chosen SO(3) subgroup drop out because all SO(3) subgroups in SO(4) are conjugate.
“So far this depends on which SO (3) subgroup we have chosen; but if we project to the SO (4) invariant-tensor space, then the dependence drops out because all SO (3) subgroups in SO (4) are conjugate to one another.”
Quote in the paperSubstantial claim- Rests on an assumption: All SO(3) subgroups in SO(4) are conjugate to one another.
- Classified as probably not a substantial claim. 1
- May combine two claims. 5
Probably not a substantial claim: may not pass S4 (based).
Findingnone given§ The loop-quantum-gravity vertex-amplitude · p. 2The boundary states of the theory are spanned by trivalent graphs colored with SO(3) spins and intertwiners.
“First, the boundary states of the theory are spanned by trivalent graphs colored with SO (3) spins and intertwiners.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingnone given§ The loop-quantum-gravity vertex-amplitude · p. 1The vertex is both SO(3) and SO(4) covariant.
“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.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ The loop-quantum-gravity vertex-amplitude · p. 4The closure constraint gives gauge-invariance at each tetrahedron and reduces the space of states to the space of SO(4) spin networks on the graph dual to the boundary triangulation.
“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.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ The loop-quantum-gravity vertex-amplitude · p. 4The simplicity constraints reduce each SO(4) link representation to a simple one, and the intertwiner spaces to K_e.
“The simplicity constraints (11), as seen above, reduce each SO (4) link representation to a simple one, and the intertwiners spaces to K e .”
Quote in the paperSubstantial claimMay combine two claims.
Premisecitation§ The loop-quantum-gravity vertex-amplitude · p. 3In the Barrett-Crane theory, the off-diagonal simplicity constraints C_ff'=0 are imposed strongly on H_e, and the only solution is i_BC.
“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].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [17] MP Reisenberger, 'On relativistic spin network vertices', J Math Phys 40 (1999) 2046 [arXiv:gr-qc/9809067] source
Not checked in the free run.
May combine two claims.
Interpretationreasoning§ The loop-quantum-gravity vertex-amplitude · p. 4The resulting space of states can be physically identified with the corresponding one of SO(3) loop quantum gravity, because there is an explicit identification of quantum operators on the two spaces with the same classical analogues, such as the area of the faces.
“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.”
Quote in the paperSubstantial claim
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?
Imposing the constraints strongly, as in the Barrett-Crane model, implies additional conditions that are not present in the classical theory.
“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]
Relaxing the constraints naturally leads to a simple model that has a clearer relationship with the canonical picture provided by LQG.
“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]
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 [168]
The basic idea of the EPRL model was to relax the imposition of the Plebanski constraints that reduce BF theory to general relativity in the quantum theory.
“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]
The model of [168] corresponds to the sector γ = ∞, which has been shown to be topological.
“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]
The kinematics of the new vertex matches exactly the one of canonical loop quantum gravity.
“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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Spin Foam Models Of Riemannian Quantum GravityPreprint
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Spin foam quantization of $SO(4)$ Plebanski’s action
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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 · see citing works
Flagged first, then by citations
Research works printed but not linked: 3
- C. Rovelli, 1995, Nucl. Phys., B456, 734
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- G. Ponzano, 1968, Spectroscopy and Group Theoretical Methods in Physics
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Loop quantum cosmology: a status report
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The Spin-Foam Approach to Quantum Gravity
Perez, A. (2013) · Living Reviews in Relativity · doi:10.12942/lrr-2013-3 · Cited by 677 · see citing works
LQG vertex with finite Immirzi parameter
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Self-citationNew spinfoam vertex for quantum gravity
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A new spin foam model for 4D gravity
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Colored Tensor Models - a Review
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Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory
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Carlo Rovelli, C. (2006) · The frontiers collection · doi:10.1007/978-3-319-55420-4_10 · Cited by 347 · see citing works
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Colored group field theoryPreprint
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“Recently, a vertex amplitude that modifies the BC amplitude, and which addresses precisely the problems that we find here, has been proposed [ 16 , 17], see also [18].”
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“The attempt to correct this problem was part of the motivation for the lively search of new spin foam models with non-trivial intertwiner dependence [25, 26, 19, 20, 27].”
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Bonanno, A., Denz, T., Pawlowski, J. M., & Reichert, M. (2022) · SciPost Physics · doi:10.21468/scipostphys.12.1.001 · Cited by 111 · see citing works
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Spinor representation for loop quantum gravity
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Geloun, J. B., Gurău, R., & Rivasseau, V. (2010) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/92/60008 · Cited by 97 · see citing works
4-dimensional spin-foam model with quantum Lorentz group
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Quantum deformation of two four-dimensional spin foam models
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Quantum geometry from phase space reduction
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Carlo Rovelli, C. (2011) · Foundations of Physics · doi:10.1007/s10701-011-9561-4 · Cited by 95 · see citing works
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Carlo Rovelli, C. (2013) · doi:10.22323/1.140.0003 · Cited by 109 · see citing works
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Non-commutative flux representation for loop quantum gravity
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Simplicity and closure constraints in spin foam models of gravity
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Proposed proper Engle-Pereira-Rovelli-Livine vertex amplitude
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Self-citationDiscrete Gravity Models and Loop Quantum Gravity: a Short Review
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Holomorphic Lorentzian simplicity constraints
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- 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 › 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, spin foam models, vertex amplitude, Barrett-Crane model, Regge calculus, second-class constraints, soft constraints
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