The DOI matches its Crossref record; 16 of the 25 references it cites were found in a registry or an index, and 9 are not checked yet; none of the 9 sources its claims cite is checked for support yet.
Relativistic spin networks and quantum gravity
Open accessAbstract
Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2)×SU(2). Relativistic quantum spins are related to the geometry of the two-dimensional faces of a 4-simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3-simplex. This leads us to suggest that there may be a four-dimensional state sum model for quantum gravity based on relativistic spin networks that parallels the construction of three-dimensional quantum gravity from ordinary spin networks.
Checked 2026-10-06 · 9 references not checked yet
Overview
AI snapshot
Written from the paper’s full text.
TL;DR
The paper defines relativistic spin networks using the spin covering SU(2) × SU(2) of SO(4), relating quantum spins to bivectors on the 2-faces of a 4-simplex. It proposes a 4-dimensional state sum model for quantum gravity as a constrained doubling of the Crane-Yetter topological state sum.
Key findings
- A geometric 4-simplex is characterized by 10 bivectors on its triangles satisfying six conditions; the theorem states that any set satisfying these conditions determines a geometric 4-simplex unique up to parallel translation and inversion through the origin. Condition (6) is supported by the calculation \(\mathrm{Tr}\, b_1[b_2,b_3] = \frac{9}{8} V^2\), where \(V\) is the volume of the tetrahedron.
- Relativistic spin networks label faces with irreducible representations of SU(2) × SU(2); the classical simplicity condition for bivectors is transcribed quantum-mechanically by requiring \(j=k\), called simple representations.
- The proposed state sum is a “constrained” doubling of the Crane-Yetter model; for each 4-simplex, weights are linear combinations of products of two 15j-q symbols with identical representations but opposite orientation.
- A canonical vertex is exhibited as the usual SU(2) vertex doubled, with the middle edge labelled by \((j,j)\), showing the vertex space is non-empty in many cases.
Methods
- Classical Euclidean 4-dimensional geometry: an abstract 4-simplex \((01234)\) is embedded in \(\mathbb{R}^4\), and oriented triangles determine bivectors \(b = \frac{1}{2} e_{01} \wedge e_{02} \in \mathbb{R}^4 \wedge \mathbb{R}^4\); the paper proves that six listed properties characterize the embedding.
- Quantum construction: bivectors are replaced by bivector operators in representations of SU(2) × SU(2), with simplicity imposed by \(j=k\); a state sum on a triangulated 4-manifold is defined by summing weights over labellings of 2-simplexes and tetrahedra.
- The analysis is mathematical and axiomatic; no empirical data or numerical simulations are used.
Limitations
- The paper states it has an axiomatic description of the space of vertices but not an explicit construction; if the only allowable vertex is the one explicitly constructed, the state sum does not seem to give a topological invariant of manifolds, so its relationship to topology is unclear.
- The authors state they have not analysed the quantum analogues of inequalities (5) and (6), do not yet have an analogue of the Ponzano-Regge asymptotic formula linking the model to the Einstein action, and lack a complete argument that it contains local degrees of freedom or propagating plane waves.
- The paper states that the key remaining issue is how probability amplitudes behave under refinement of the triangulation, analogous to ultraviolet problems; possible routes to a finite theory are suggested but not established.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
11 No basis given · 9 Own interpretation · 7 Not checked yet · 1 Own method
In the paper
“Condition (6) follows from the calculation Tr b 1 [ b 2 , b 3 ] = 9 8 V 2 , where V is the volume of the tetrahedron [23].”
What the sources say
- Not checked yetcites [23] A. Barbieri , “Quantum tetrahedra and simplicial spin networks,” gr-qc/9707010.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a 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
“The simple representations of SU(2) × SU(2) occur as the summands of the Hilbert space of functions on S 3 , which has the obvious action of SO(4).”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says 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
“The sum of the bivector operators for a pair is equal to the bivector operator for the middle edge in the vertex.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a 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
“The orthogonality relation of the 6j-symbol shows that the vertex is given by the same formula if the top two and bottom two pairs are chosen.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a 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
“However we can show that the space is non-empty in many cases, by exhibiting a canonical element.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a 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
“Two simple bivectors b 1 , b 2 determine two planes lying in a hyperplane (of dimension 3) if and only if b 1 + b 2 is simple.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
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
“Thus the model is a 'constrained' doubling of the Crane-Yetter model.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
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 new state sum has a very natural relationship with Regge calculus in a new formulation, in that the classical version of the theory has the metric as its variable in the same way as Regge calculus.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
In the paper
“However, it is already very close to a topological theory, in that the deviation from the topological theory [4] is only in the imposed constraints.”
What the sources say
- Not checked yetcites [4] 1997, State sum invariants of 4-manifolds, J. Knot Theory Ram. source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“In fact, the Plebanski Lagrangian [8,9] for general relativity has the form of BF theory [10] with gauge group SL(2 , C ), with an added term which is a constraint.”
What the sources say
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- It may state two separate claims as one. 11
- Probably not a substantial claim. 3
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
“Theorem. Each geometric 4-simplex determines a set of bivectors satisfying these conditions, and each set of bivectors satisfying these conditions determines a geometric 4-simplex unique up to parallel translation and inversion through the origin.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It 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
“Each term in the sum obviously satisfies (2) and (4), and satisfies (3) for the two preferred pairs.”
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It 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
“A geometric simplex has a metric geometry, which is determined by the lengths of its 10 edges. Thus two geometric 4-simplexes have the same edge lengths if and only if they are related by an isometry of R 4 .”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It 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 corresponding momentum in the classical case is l µν x µ p ν , for a point ( x, p ) ∈ T ∗ S 3 . This is the dual pairing of so(4) with the simple bivector x ∧ p in R 4 , which explains how simple bivectors are associated to triangles.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It 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
“Therefore the bivector data determines a set of edge lengths. Two sets of bivector data determine the same edge lengths if and only if the bivectors are all transformed simultaneously by an orthogonal rotation.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It 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 idea is that to pass from 3 dimensions to 4 dimensions one has to replace SO(3) with SO(4), or its spin covering, SU(2) × SU(2).”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
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
“In our picture, intrinsic spin occurs for non-simple representations, which would arise from sections of spinor bundles over S 3 .”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It 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
“Now in the relativistic case, the theory is for four-dimensional manifolds and the relativistic spin network vertices reside in a tetrahedron. There will be one bivector operator for each face of a tetrahedron, and so this will be a four-valent vertex.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“It is interesting that the sort of tensor categories which go into the state sum we are proposing are so similar to the ones invented in constructing string theories [29]. Terms in our state sum can be interpreted as diagrams in string perturbation theory, by connecting together the diagrams we are associating to the 4-simplexes, and interpreting elements in the representations as string states.”
What the sources say
- Not checked yetcites [29] 1989, Classical and quantum conformal field theory, Commun. Math. Phys. source
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It 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
“However, in Einstein gravity it is important that so(4) ∗ is identified with R 4 ∧ R 4 , as B is constrained to be of the form B = ∗ e ∧ e , where e is a vector valued one-form.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“In three dimensions, quantum gravity is constructed by the Ponzano-Regge state sum model in which representations of SU(2) label edges [12].”
What the sources say
- Not checked yetcites [12]
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“However giving the areas of the four faces of a tetrahedron does not specify the geometry of the tetrahedron uniquely, so that two 4-simplexes cannot be glued together by specifying the areas of the 16 triangles in their union [25].”
What the sources say
- Not checked yetcites [25] J. W. Barrett , M. Rocek, and R. M. Williams, “A note on area variables in Regge calculus,” gr-qc/9710056.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“By contrast, specifying the 10 areas does not specify the metric geometry uniquely [24], although it does for almost all configurations.”
What the sources say
- Not checked yetcites [24] 1994, First order Regge calculus, Class. Quantum Grav. source
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The authors state this as an assumption.
The frame field gives a linear embedding of the triangle in R^4.
“Assuming the frame field gives a linear embedding of the triangle in R 4 , the integral over this triangle,”
The paper relies on this without support.
In the classical limit, when the representation is large, vector operators behave as ordinary vectors in 3-dimensional space with probability near to 1.
“The intuition is that in the classical limit, when the representation is 'large', these behave as ordinary vectors in 3-dimensional space, with probability near to 1.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Indeed, although spinfoam models have been derived in a number of different manners [6, 7, 8, 21], most derivations involve peculiar procedures or intuitive and ad hoc steps.”
From Flipped spinfoam vertex and loop gravity (2008) · cites this paper as [21]
Experimental — accuracy not yet measuredIn the citing paper
“Case B. In this case, as in case A, faces are colored by simple representation N f of SO (4). Each edge, however, is colored by a single, fixed, intertwiner i BC , the ( normalized ) Barrett-Crane intertwiner [1].”
From Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space (2000) · cites this paper as [1]
In the citing paper
“The Barrett-Crane model has been obtained by 'quantizing a geometric 4-simplex' [1].”
From Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space (2000) · cites this paper as [1]
In the citing paper
“Simple representations emerge in this process: in the quantum theory, the condition on bivectors that they describe planes in R 4 translates into the the simplicity condition (A1) on the representation.”
From Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space (2000) · cites this paper as [1]
In the citing paper
“However, there is a simple model that can be obtained as a systematic quantization of the simplicial SO(4) Plebanski’s action.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [59]
In the citing paper
“The second possibility is to work in the sector where . We will show later that this corresponds to the famous Barret-Crane model [59] with the various limitations discussed in Section 10.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [59]
In the citing paper
“The Barrett-Crane models [60, 59] can easily be obtained in the present framework as a suitable limiting case γ → ∞ of the previous constructions.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [59]
In the citing 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”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [2]
In the citing 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.”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [2]
In the citing paper
“The original model of [2, 3] and the variant pointed out in [11] appear as limiting γ → 0 and γ → ∞ cases, respectively.”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [2]
In the citing paper
“The vertex was defined in [2, 3] only for the euclidean case, and in the absence of an Immirzi parameter γ .”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [2]
In the citing 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.”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [2]
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 [12]
Citations
Cites
2 + 1 dimensional gravity as an exactly soluble system
Witten, E. (1988) · Nuclear Physics B · doi:10.1016/0550-3213(88)90143-5 · Cited by 2,562
Topological quantum field theory
Witten, E. (1988) · Communications in Mathematical Physics · doi:10.1007/bf01223371 · Cited by 2,396
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 3×
“We are also making contact with the loop variables picture of Rovelli and Smolin [14], in which the fundamental operators live on surfaces.”
BackgroundInfluentialClassical and quantum conformal field theory
Moore, G., & Seiberg, N. (1989) · Communications in Mathematical Physics · doi:10.1007/bf01238857 · Cited by 1,327
On the separation of Einsteinian substructures
Plebański, J. F. (1977) · Journal of Mathematical Physics · doi:10.1063/1.523215 · Cited by 499
Exactly soluble diffeomorphism invariant theories
Horowitz, G. T. (1989) · Communications in Mathematical Physics · doi:10.1007/bf01218410 · Cited by 472
TOPOLOGICAL LATTICE MODELS IN FOUR DIMENSIONS
OOGURI, H. (1992) · Modern Physics Letters A · doi:10.1142/s0217732392004171 · Cited by 379
“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
Capovilla, R., Dell, J., Jacobson, T., & Mason, L. (1991) · Classical and Quantum Gravity · doi:10.1088/0264-9381/8/1/009 · Cited by 286
Linking topological quantum field theory and nonperturbative quantum gravity
Smolin, L. (1995) · Journal of Mathematical Physics · doi:10.1063/1.531251 · Cited by 234
Cited by
Sorted by citations
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
The Spin-Foam Approach to Quantum Gravity
Perez, A. (2013) · Living Reviews in Relativity · doi:10.12942/lrr-2013-3 · Cited by 677
LQG vertex with finite Immirzi parameter
Engle, J., Livine, E., Pereira, R., & Rovelli, C. (2008) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2008.02.018 · Cited by 584
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
New spinfoam vertex for quantum gravity
Livine, E. R., & Speziale, S. (2007) · Physical Review D · doi:10.1103/physrevd.76.084028 · Cited by 401
Group Field Theory: An Overview
Freidel, L. (2005) · International Journal of Theoretical Physics · doi:10.1007/s10773-005-8894-1 · Cited by 378
Loop-Quantum-Gravity Vertex Amplitude
Engle, J., Pereira, R., & Rovelli, C. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.161301 · Cited by 335
Carlo Rovelli, C. (2006) · The frontiers collection · doi:10.1007/978-3-319-55420-4_10 · Cited by 504
FUNDAMENTAL STRUCTURE OF LOOP QUANTUM GRAVITY
Han, M., Ma, Y., & HUANG, W. (2007) · International Journal of Modern Physics D · doi:10.1142/s0218271807010894 · Cited by 292
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
Self-citationPonzano–Regge model revisited: III. Feynman diagrams and effective field theory
Freidel, L., & Livine, E. R. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/23/6/012 · Cited by 266
A Lorentzian Signature Model for Quantum General RelativityPreprint
Barrett, J. W., & Crane, L. (2000) · Cited by 282
Self-citationGroup Field Theory: An overview
Freidel, L. (2005) · CERN Document Server (European Organization for Nuclear Research) · Cited by 249
Lorentzian spin foam amplitudes: graphical calculus and asymptotics
Barrett, J. W., Dowdall, R. J., Fairbairn, W. J., Hellmann, F., & Pereira, R. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/16/165009 · Cited by 247
Self-citationBarrett–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
Asymptotic analysis of the Engle–Pereira–Rovelli–Livine four-simplex amplitude
Barrett, J. W., Dowdall, R. J., Fairbairn, W. J., Gomes, H., & Hellmann, F. (2009) · Journal of Mathematical Physics · doi:10.1063/1.3244218 · Cited by 224
Self-citationThe microscopic dynamics of quantum space as a group field theory
Oriti, D. (2012) · Cambridge University Press eBooks · doi:10.1017/cbo9780511920998.012 · Cited by 303
Quantum symmetry, the cosmological constant and Planck-scale phenomenology
Amelino-Camelia, G., Smolin, L., & Starodubtsev, A. (2004) · Classical and Quantum Gravity · doi:10.1088/0264-9381/21/13/002 · Cited by 228
Semiclassical limit of 4-dimensional spin foam models
Conrady, F., & Freidel, L. (2008) · Physical Review D · doi:10.1103/physrevd.78.104023 · Cited by 213
Graviton Propagator from Background-Independent Quantum Gravity
Rovelli, C. (2006) · Physical Review Letters · doi:10.1103/physrevlett.97.151301 · Cited by 212
Loop quantum gravity: An outside viewPreprint
Nicolai, H., Peeters, K., & Zamaklar, M. (2005) · Cited by 230
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
so (4) Plebanski action and relativistic spin-foam model
Pietri, R. D., & Freidel, L. (1999) · Classical and Quantum Gravity · doi:10.1088/0264-9381/16/7/303 · Cited by 182
Discrete Structures in Gravity
Regge, T., & Williams, R. M. (2000) · CERN Document Server (European Organization for Nuclear Research) · Cited by 182
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
Ponzano–Regge model revisited: I. Gauge fixing, observables and interacting spinning particles
Freidel, L., & Louapre, D. (2004) · Classical and Quantum Gravity · doi:10.1088/0264-9381/21/24/002 · Cited by 238
Loop quantum gravity: an outside view
Nicolai, H., Peeters, K., & Zamaklar, M. (2005) · Classical and Quantum Gravity · doi:10.1088/0264-9381/22/19/r01 · Cited by 229
Graviton 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
Diffeomorphisms and spin foam models
Freidel, L., & Louapre, D. (2003) · Nuclear Physics B · doi:10.1016/s0550-3213(03)00306-7 · Cited by 155
Complete LQG propagator: Difficulties with the Barrett-Crane vertex
Alesci, E., & Rovelli, C. (2007) · Physical Review D · doi:10.1103/physrevd.76.104012 · Cited by 154
“by the Barrett-Crane (BC) vertex [11], where the dependence on the intertwiners is trivial.”
BackgroundDiscrete structures in gravity
Regge, T., & Williams, R. M. (2000) · Journal of Mathematical Physics · doi:10.1063/1.533333 · Cited by 257
Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles
Freidel, L., & Louapre, D. (2004) · CERN Document Server (European Organization for Nuclear Research) · Cited by 143
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
Loop Quantum Gravity: An Inside View
Thiemann, T. (2007) · Lecture notes in physics · doi:10.1007/978-3-540-71117-9_10 · Cited by 153
Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
Baratin, A., & Oriti, D. (2012) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.85.044003 · Cited by 147
Graviton propagator in loop quantum gravityPreprint
Eugenio Bianchi, A., Modesto, L., Carlo Rovelli, C., & Simone Speziale, D. (2008) · Cited by 156
Spin-foams for all loop quantum gravity
Kamiński, W., Kisielowski, M., & Lewandowski, J. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/9/095006 · Cited by 191
LQG propagator from the new spin foams
Bianchi, E., Magliaro, E., & Perini, C. (2009) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2009.07.016 · Cited by 126
“Previous attempts to derive the graviton propagator from LQG adopted the Barrett-Crane spin foam vertex [21] as model for the dynamics [7, 8, 9, 10, 11, 12, 13, 14, 15] (see also [22, 23, 24] for investigations in the three-dimensional case).”
MethodAsymptotics of relativistic spin networks
Barrett, J. W., & Steele, C. M. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/7/307 · Cited by 118
Self-citationAsymptotics of 6 j and 10 j symbols
Freidel, L., & Louapre, D. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/7/303 · Cited by 113
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
Rivasseau, V. (2013) · Fortschritte der Physik · doi:10.1002/prop.201300032 · Cited by 109
Edge modes of gravity. Part III. Corner simplicity constraints
Freidel, L., Geiller, M., & Pranzetti, D. (2021) · Journal of High Energy Physics · doi:10.1007/jhep01(2021)100 · Cited by 125
Coarse graining methods for spin net and spin foam models
Dittrich, B., Eckert, F. C., & Martín-Benito, M. (2012) · New Journal of Physics · doi:10.1088/1367-2630/14/3/035008 · Cited by 103
Casting loop quantum cosmology in the spin foam paradigm
Ashtekar, A., Campiglia, M., & Henderson, A. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/13/135020 · Cited by 103
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
Spin foam models for quantum gravity from lattice path integrals
Bonzom, V. (2009) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.80.064028 · Cited by 109
Ten questions on Group Field Theory (and their tentative answers)
Baratin, A., & Oriti, D. (2012) · Journal of Physics: Conference Series · doi:10.1088/1742-6596/360/1/012002 · Cited by 100
Reconstructing quantum geometry from quantum information: spin networks as harmonic oscillators
Girelli, F., & Livine, E. R. (2005) · Classical and Quantum Gravity · doi:10.1088/0264-9381/22/16/011 · Cited by 99
Path integral representation of spin foam models of 4d gravity
Conrady, F., & Freidel, L. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/24/245010 · Cited by 98
Feynman diagrams of generalized matrix models and the associated manifolds in dimension four
Pietri, R. D., & Petronio, C. (2000) · Journal of Mathematical Physics · doi:10.1063/1.1290053 · Cited by 98
4-dimensional spin-foam model with quantum Lorentz group
Han, M. (2011) · Journal of Mathematical Physics · doi:10.1063/1.3606592 · Cited by 108
Quantum deformation of two four-dimensional spin foam models
Fairbairn, W. J., & Meusburger, C. (2012) · Journal of Mathematical Physics · doi:10.1063/1.3675898 · Cited by 114
Quantum geometry from phase space reduction
Conrady, F., & Freidel, L. (2009) · Journal of Mathematical Physics · doi:10.1063/1.3257109 · Cited by 106
Projector on physical states in loop quantum gravity
Rovelli, C. (1999) · Physical Review D · doi:10.1103/physrevd.59.104015 · Cited by 95
Asymptotics of spinfoam amplitude on simplicial manifold: Lorentzian theory
Han, M., & Zhang, M. (2013) · Classical and Quantum Gravity · doi:10.1088/0264-9381/30/16/165012 · Cited by 107
Carlo Rovelli, C. (2013) · doi:10.22323/1.140.0003 · Cited by 114
Coarse graining in spin foam models
Markopoulou, F. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/5/301 · Cited by 100
Loop quantum gravity: the first 25 years
Carlo Rovelli, C. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/15/153002 · Cited by 140
Quantum geometry with intrinsic local causality
Markopoulou, F., & Smolin, L. (1998) · Physical Review D · doi:10.1103/physrevd.58.084032 · Cited by 95
Baratin, A., & Oriti, D. (2011) · New Journal of Physics · doi:10.1088/1367-2630/13/12/125011 · Cited by 96
Complete LQG propagator. II. Asymptotic behavior of the vertex
Alesci, E., & Rovelli, C. (2008) · Physical Review D · doi:10.1103/physrevd.77.044024 · Cited by 89
New variables for classical and quantum gravity in all dimensions: I. Hamiltonian analysis
Bodendorfer, N., Thiemann, T., & Thurn, A. (2013) · Classical and Quantum Gravity · doi:10.1088/0264-9381/30/4/045001 · Cited by 105
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
Haggard, H. M., Han, M., Kamiński, W., & Riello, A. (2015) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2015.08.023 · Cited by 88
Spin Foams and Canonical Quantization
Alexandrov, S. Y. (2012) · Symmetry Integrability and Geometry Methods and Applications · doi:10.3842/sigma.2012.055 · Cited by 101
A note on area variables in Regge calculus
Barrett, J. W., Rocek, M., & Williams, R. M. (1999) · Classical and Quantum Gravity · doi:10.1088/0264-9381/16/4/025 · Cited by 80
Self-citationSimplicity in simplicial phase space
Dittrich, B., & Ryan, J. P. (2010) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.82.064026 · Cited by 88
Asymptotics of the spin foam amplitude on simplicial manifold: Euclidean theory
Han, M., & Zhang, M. (2012) · Classical and Quantum Gravity · doi:10.1088/0264-9381/29/16/165004 · Cited by 92
Baez, J. C., Christensen, J. D., & Egan, G. (2002) · Classical and Quantum Gravity · doi:10.1088/0264-9381/19/24/315 · Cited by 80
Projected spin networks for Lorentz connection: linking spin foams and loop gravity
Livine, E. R. (2002) · Classical and Quantum Gravity · doi:10.1088/0264-9381/19/21/316 · Cited by 77
Group field theories for all loop quantum gravity
Oriti, D., Ryan, J. P., & Thürigen, J. (2015) · MPG.PuRe (Max Planck Society) · Cited by 77
Time evolution as refining, coarse graining and entangling
Dittrich, B., & Steinhaus, S. (2014) · New Journal of Physics · doi:10.1088/1367-2630/16/12/123041 · Cited by 82
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
Holonomy spin foam models: Definition and coarse graining
Bähr, B., Dittrich, B., Hellmann, F., & Kamiński, W. (2013) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.87.044048 · Cited by 76
Holomorphic simplicity constraints for 4D spinfoam models
Dupuis, M., & Livine, E. R. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/21/215022 · Cited by 75
Projected Spin Networks for Lorentz connection: Linking Spin Foams and Loop GravityPreprint
Livine, E. R. (2002) · Cited by 83
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
Cosmological deformation of Lorentzian spin foam models
Noui, K., & Roche, P. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/14/318 · Cited by 71
On relativistic spin network vertices
Reisenberger, M. P. (1999) · Journal of Mathematical Physics · doi:10.1063/1.532850 · Cited by 72
3D spinfoam quantum gravity: matter as a phase of the group field theory
Fairbairn, W. J., & Livine, E. R. (2007) · Classical and Quantum Gravity · doi:10.1088/0264-9381/24/20/021 · Cited by 81
The face amplitude of spinfoam quantum gravity
Eugenio Bianchi, A., Regoli, D., & Carlo Rovelli, C. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/18/185009 · Cited by 78
Group field theories for all loop quantum gravity
Oriti, D., Ryan, J. P., & Thürigen, J. (2015) · New Journal of Physics · doi:10.1088/1367-2630/17/2/023042 · Cited by 97
Effective Spin Foam Models for Four-Dimensional Quantum Gravity
Asante, S. K., Dittrich, B., & Haggard, H. M. (2020) · Physical Review Letters · doi:10.1103/physrevlett.125.231301 · Cited by 72
Towards a phase diagram for spin foams
Delcamp, C., & Dittrich, B. (2017) · Classical and Quantum Gravity · doi:10.1088/1361-6382/aa8f24 · Cited by 76
Numerical study of the Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude
Donà, P., Fanizza, M., Sarno, G., & Simone Speziale, D. (2019) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.100.106003 · Cited by 66
Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners
Nicolai, H., & Peeters, K. (2007) · Springer eBooks · doi:10.1007/978-3-540-71117-9_9 · Cited by 86
Holomorphic Lorentzian simplicity constraints
Dupuis, M., Freidel, L., Livine, E. R., & Speziale, S. (2012) · Journal of Mathematical Physics · doi:10.1063/1.3692327 · Cited by 64
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
Self-citationHypercuboidal renormalization in spin foam quantum gravity
Bähr, B., & Steinhaus, S. (2017) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.95.126006 · Cited by 69
Regularization and finiteness of the Lorentzian loop quantum gravity vertices
Engle, J., & Pereira, R. (2009) · Physical Review D · doi:10.1103/physrevd.79.084034 · Cited by 63
Bahr, B., Hellmann, F., Kamiński, W., Kisielowski, M., & Lewandowski, J. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/10/105003 · Cited by 63
QUANTUM GRAVITY ASYMPTOTICS FROM THE SU(2) 15j-SYMBOL
Barrett, J. W., Fairbairn, W. J., & Hellmann, F. (2010) · International Journal of Modern Physics A · doi:10.1142/s0217751x10049281 · Cited by 65
Self-citationInvestigation of the spinfoam path integral with quantum cuboid intertwiners
Bähr, B., & Steinhaus, S. (2016) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.93.104029 · Cited by 70
Ding, Y., Han, M., & Rovelli, C. (2011) · Physical Review D · doi:10.1103/physrevd.83.124020 · Cited by 60
Alesci, E., Thiemann, T., & Zipfel, A. (2012) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.86.024017 · Cited by 65
Holonomy spin foam models: asymptotic geometry of the partition function
Hellmann, F., & Kamiński, W. (2013) · Journal of High Energy Physics · doi:10.1007/jhep10(2013)165 · Cited by 67
Metadata
- Authors
- Barrett, J. W., & Crane, L.
- Affiliations
- University of Nottingham (GB)
- Kansas State University (US)
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 1998-06-01
- Volume
- 39
- Issue
- 6
- Pages
- 3296-3302
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W2069771760
- Semantic Scholar
- de1b77a7e917bd8c77d8f612728f6d4d4b3d470b
- Cited by
- 404
- References
- 25
- Citations by year
- 2026: 3 · 2025: 4 · 2024: 11 · 2023: 11 · 2022: 13 · 2021: 7 · 2020: 10 · 2019: 15 · 2018: 13 · 2017: 14 · 2016: 7 · 2015: 9 · 2014: 12 · 2013: 29 · 2012: 22
- FWCI
- 13.39
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- 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
- quantum gravity, 4-simplex, Ponzano-Regge model, state sum models, spin networks
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
