Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space
Open accessAbstract
Boulatov and Ooguri have generalized the matrix models of 2d quantum gravity to 3d and 4d, in the form of field theories over group manifolds. We show that the Barrett–Crane quantum gravity model arises naturally from a theory of this type, but restricted to the homogeneous space S3=SO(4)/SO(3), as a term in its Feynman expansion. From such a perspective, 4d quantum space-time emerges as a Feynman graph, in the manner of the 2d matrix models. This formalism provides a precise meaning to the “sum over triangulations”, which is presumably necessary for a physical interpretation of a spin-foam model as a theory of gravity. In addition, this formalism leads us to introduce a natural alternative model, which might have relevance for quantum gravity.
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TL;DR
The paper shows that the Barrett–Crane quantum gravity model (its first version, in the q→1 limit) arises as a term in the Feynman expansion of a Boulatov–Ooguri-type field theory over four copies of the homogeneous space S³ = SO(4)/SO(3). This gives a precise implementation of a “sum over triangulations” as a sum over colored 2-complexes/spin foams, and it introduces an alternative model (Case A) with possible relevance for quantum gravity.
Key findings
- The Barrett–Crane amplitude over a triangulation Δ is recovered as the Feynman amplitude of the dual 2-complex Γ(Δ): the paper states that the Barrett–Crane model over Δ is precisely the amplitude of Γ(Δ), and that the individual terms of the state sum match.
- In Case B, faces are colored by simple SO(4) representations N, each edge by the normalized Barrett–Crane intertwiner, the face amplitude is A_f(N) = dim N, with dim N = (N+1)² for simple representations, and the vertex amplitude is the 15j Wigner symbol built from ten simple representations and five Barrett–Crane intertwiners.
- The Feynman expansion generates a sum over colored 2-complexes s = (Γ,c), i.e. spin foams, rather than only a fixed triangulation; the 2-complex weight is w(Γ) = λ^{n(Γ)}/sym(Γ), and only 2-complexes with 4-valent edges and 5-valent vertices contribute.
- Case A gives a different model: faces are colored by simple SO(4) representations, edges by intertwiners in an orthonormal basis, face and edge amplitudes are 1, and the vertex amplitude is a 15j Wigner symbol; the paper argues this model provides another solution to the intersection constraints that reduce BF theory to general relativity.
Methods
- The authors define a real field φ̃(g₁,g₂,g₃,g₄) over four copies of G = SO(4), constant along SO(3) equivalence classes so it becomes a function on (S³)⁴, symmetric under permutations of its four arguments, and projected onto its SO(4)-invariant part in two inequivalent ways (Case A and Case B).
- The dynamics is given by an action whose fifth-order potential has the structure of a 4-simplex; the paper performs a Peter–Weyl mode expansion on S³ = SO(4)/SO(3), derives the propagator and vertex, and analyzes the Feynman expansion in powers of the coupling λ.
- The Feynman graphs are 4-strand, 5-valent graphs whose vertices, edges, and closed strands form colored 2-complexes; the authors compare these amplitudes to the Barrett–Crane state sum and identify the coloring of faces with spins and edges with intertwiners.
Limitations
- Stated: the sum over spin foams obtained is presumably divergent, with two sources—the sum over colorings and the sum over 2-complexes. The paper suggests regulating the coloring sum via quantum-group Clebsch–Gordan coefficients with q^k = 1, and notes that convergence in λ might occur below a critical value determined by the exponential growth of triangulations, but that the Feynman expansion generates triangulations of arbitrary manifolds and this might destroy convergence.
- Stated: not every 2-complex comes from a triangulation of a 4-manifold; the paper notes a characterization in 3d but says “We are not aware of related results in 4d.”
- Evident from the text: the analysis is formal and does not provide a rigorous convergence proof or a complete physical interpretation; the paper itself describes the sum as “presumably divergent” and the physical interpretation as a perspective rather than a derived result.
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- Findingreasoning§ I. INTRODUCTION · p. 1
The q→1 limit of the Barrett-Crane model can be derived from the Feynman expansion of a field theory over four copies of the homogeneous space S^3 = SO(4)/SO(3).
“Here, we show that the q → 1 limit of the Barrett-Crane model can be derived from the Feynman expansion of a field theory over four copies of the homogeneous space S 3 = SO(4) / SO(3).”
- Also stated in: § Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space · p. 1 · plain
“We show that the Barrett-Crane quantum gravity model arises naturally from a theory of this type, but restricted to the homogeneous space S 3 = SO (4) /SO (3), as a term in its Feynman expansion.”
Quote in the paperSubstantial claim - Also stated in: § Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space · p. 1 · plain
- Findingreasoning§ I. INTRODUCTION · p. 2
The discrete momenta c in the Feynman expansion correspond precisely to the colorings c of the triangulation.
“Next, we show that the discrete 'momenta' c in (4) correspond precisely to the colorings c of the triangulation in (1).”
Quote in the paperSubstantial claim - Findingreasoning§ IV. FEYNMAN GRAPHS · p. 9
For each graph, the abstract set formed by the vertices, edges, and cycles forms a 2-complex, in which the faces are the cycles.
“For each graph, the abstract set formed by the vertices, the edges, and the cycles forms a 2-complex, in which the faces are the cycles.”
Quote in the paperSubstantial claim - Findingreasoning§ IV. FEYNMAN GRAPHS · p. 9
In Case B, faces f are labeled by simple representations N_f and contribute a factor dim N_f.
“Thus, faces f are labeled by simple representations N f and contribute a factor dim N f .”
Quote in the paperSubstantial claim - Findingreasoning§ I. INTRODUCTION · p. 2
The Feynman expansion of the field theory generates many more terms than a Barrett-Crane model over a fixed triangulation.
“The Feynman expansion of field theory we consider generates many more terms than a Barrett-Crane model over a fixed triangulation.”
Quote in the paperSubstantial claim
- Findingreasoning§ IV. FEYNMAN GRAPHS · p. 9
Each strand of the 4-strand graph goes through several vertices and propagators and then closes to itself, forming a cycle.
“Each strand of the 4-strand graph, goes through several vertices and several propagators, and then closes to itself, forming a cycle.”
Quote in the paperSubstantial claim - Findingreasoning§ I. INTRODUCTION · p. 2
What emerges in the Feynman expansion is not a sum over actual triangulations, but rather a sum over colored 2-complexes s=(Γ,c).
“Thus, what emerges in the Feynman expansion is not a sum over actual triangulations , as one might have naively expected, but rather a sum over certain different objects: colored 2-complexes s = (Γ , c )”
Quote in the paperSubstantial claim- Rests on an assumption: The Barrett-Crane amplitude A_BC(Δ,c) depends only on the 2-complex Γ(Δ), its dual 2-skeleton, and not on the full combinatorial information of the triangulation Δ.
- Findingreasoning§ I. INTRODUCTION · p. 1
The single terms of the Barrett-Crane state sum match the corresponding terms in the Feynman expansion.
“More remarkably, the single terms of the sum (1) match.”
Quote in the paperSubstantial claim - Findingreasoning§ I. INTRODUCTION · p. 1
The Barrett-Crane model over a triangulation Δ is precisely the amplitude of the Feynman graph Γ(Δ) in the field theory.
“We show in this paper that the Barrett-Crane model over the triangulation ∆ is precisely the amplitude of the Feynman graph Γ(∆):”
- Also stated in: § II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5 · plain
“This is precisely the Barrett-Crane amplitude for a triangulation ∆ whose dual 2-skeleton is Γ.”
Quote in the paperSubstantial claim - Also stated in: § II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5 · plain
- Findingreasoning§ IV. FEYNMAN GRAPHS · p. 9
The labeling of the cycles by simple representations of SO(4) determines a coloring of the faces by spins, yielding a colored 2-complex, namely a spin foam.
“The labeling of the cycles by simple representations of SO (4) determines a coloring of the faces by spins. Thus, we obtain a colored 2-simplex, namely a spin foam.”
Quote in the paperSubstantial claim - Findingnone given§ II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5
The amplitude of a 2-complex in Case B is precisely the Barrett-Crane amplitude for a triangulation Δ whose dual 2-skeleton is Γ.
“This is precisely the Barrett-Crane amplitude for a triangulation ∆ whose dual 2-skeleton is Γ.”
Quote in the paperSubstantial claim - Findingreasoning§ IV. FEYNMAN GRAPHS · p. 9
In Case A, after fixing a basis of intertwiners i_e in K_N_e for every edge once and for all for each fixed 2-complex and using the vertex amplitude A(N_v,i_v), the matrix M_σ is absorbed into the vertex amplitude and the propagator is the identity.
“If we fix a basis of intertwiners i e in K ⃗ N e for every e , once and for all for each fixed 2-complex -that is, for each assigned permutation in (45)- and use vertex amplitude A ( ⃗ N v , ⃗ i v ) given in (31), the matrix M σ is absorbed into the vertex amplitude and the propagator is the identity.”
Quote in the paperSubstantial claim - Findingreasoning§ II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5
The Feynman expansion of the field theory yields a spin foam model.
“The Feynman expansion of our field theory yields a spin foam model.”
- Also stated in: § II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 4 · plain
“In the following sections, we show that the Feynman expansion (2) of this field theory is a sum over spin foams s as in (6).”
Quote in the paperSubstantial claim - Also stated in: § II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 4 · plain
- Findingreasoning§ I. INTRODUCTION · p. 2
The Barrett-Crane amplitude is precisely equal to the Feynman amplitude of the corresponding colored 2-complex (Γ,c).
“And, finally, we show that the Barrett-Crane amplitude is precisely equal to the Feynman amplitude of the corresponding 'colored' 2-complex (Γ , c )”
Quote in the paperSubstantial claim - Findingnone given§ VI. CONCLUSION · p. 11
The vertex of model A satisfies the intersection constraint in the sense that it satisfies the identity pictured in Fig. 4.
“The vertex of the model A satisfies the intersection constraint in this sense. Indeed, it satisfies the identity pictured in Fig. 4.”
Quote in the paperSubstantial claim - Findingreasoning§ IV. FEYNMAN GRAPHS · p. 9
In Case B, the resulting weight is given by equation (15).
“This yields the result (15).”
Quote in the paperSubstantial claim - Findingreasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 11
In these models, the propagator associated with a permutation gets multiplied by the character of this permutation in the given representation.
“The propagator associated with permutation gets multiplied by the character of this permutation in the given representation, and so on.”
Quote in the paperNot a substantial claim - Findingreasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10
In the modified field theory, the pairing of the strands at the vertex determines always an odd permutation.
“A moment of scrutiny of the vertex shows that the pairing of the strands discussed above (in defining the parity of the loops) determines always an odd permutation.”
Quote in the paperSubstantial claim - Findingreasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 11
Taking the field φ to be one of the φ_i leads to five different models having slightly different Feynman rules.
“Taking the field φ to be one of the φ i leads to five different models having slightly different Feynman rules.”
Quote in the paperNot a substantial claim- Rests on an assumption: The permutation group of four elements Σ4 has five irreducible representations: two one-dimensional (trivial and signature) and three of dimensions 2, 3 and 5.
- Findingreasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10
Therefore in closing the loop we obtain an even permutation.
“Therefore in closing the loop we obtain an even permutation.”
Quote in the paperSubstantial claim - Findingreasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10
Therefore the strands undergo an even number of odd permutations.
“Therefore the strands undergo an even number of odd permutations.”
Quote in the paperSubstantial claim - Findingreasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10
If the field theory is modified to require even permutation symmetry, the propagators (42) and (45) contain odd permutations only.
“If we do so, the propagators (42) and (45) contain odd permutations only.”
Quote in the paperSubstantial claim - Findingnone given§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10
In a loop of edges in the Feynman diagram, one crosses an equal number of vertices and edges.
“If we consider a loop of edges in the Feynman diagram, we cross an equal number of vertices and edges.”
Quote in the paperSubstantial claim - Findingreasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10
It is possible to modify the field theory so that it generates only orientable 2-complexes by replacing the requirement that φ is symmetric under any permutation of its arguments with a requirement that it is symmetric under even permutations and rewriting the action.
“It is not difficult to modify our field theory in such a way that it generates only orientable 2-complexes. To do that, it is sufficient to replace the requirement that φ is symmetric under any permutation of its arguments, eq. (8) with”
Quote in the paperNot a substantial claim - Findingreasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 11
Using the orthonormality of the characters, the original field is totally determined by the φ_i.
“Using the orthonormality of the characters, the original field is totally determined by the φ i”
Quote in the paperNot a substantial claim - Findingreasoning§ IV. FEYNMAN GRAPHS · p. 9
In Case B, vertices contribute a factor λ times B, where B is the Barrett-Crane weight (15), which depends on the ten simple representations of the faces adjacent to the vertex.
“Vertices contribute a factor λ times B , where B is the Barrett-Crane weight (15), which depends on the ten simple representations of the faces adjacent to the vertex.”
Quote in the paperSubstantial claim - Findingreasoning§ APPENDIX B: INTERTWINERS AND THEIR SPACES · p. 14
M_{N1 N2 N3 N4 Λ Λ'} is a 6-j symbol for SO(4).
“In fact, M N 1 N 2 N 3 N 4 Λ Λ ′ is a 6 -j symbol for SO (4).”
Quote in the paperNot a substantial claim - Findingnone given§ APPENDIX B: INTERTWINERS AND THEIR SPACES · p. 14
As Λ runs over the finite number of representations for which the (3-j)-symbols do not vanish, the C^{N1...N4 Λ}_{γ1...γ4} form an orthonormal basis of K_{N1...N4}.
“As Λ runs over the finite number of representations for which the (3 -j )-symbols do not vanish, the C N 1 ...N 4 Λ γ 1 ...γ 4 form an orthonormal basis of K N 1 ...N 4 .”
Quote in the paperNot a substantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 2
Some sort of sum over all triangulations is needed on physical grounds.
“Therefore some sort of 'sum over all triangulations' is needed on physical grounds.”
Quote in the paperSubstantial claim- Rests on an assumption: In spin foam models, the choice of a fixed triangulation breaks the full covariance of the theory and unphysically restricts the number of degrees of freedom.
- Interpretationreasoning§ I. INTRODUCTION · p. 3
Case A represents another solution to the problem of quantizing the constraints that transform BF theory into general relativity.
“In particular, it represents another solution to the problem of quantizing the constraints that transform BF theory into general relativity.”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 2
The Barrett-Crane state sum is actually a sum over all colorings of a fixed 2-complex.
“Thus, the Barrett-Crane state sum (1) is actually a sum over all colorings of a fixed 2-complex.”
Quote in the paperSubstantial claim- Rests on an assumption: The Barrett-Crane amplitude A_BC(Δ,c) depends only on the 2-complex Γ(Δ), its dual 2-skeleton, and not on the full combinatorial information of the triangulation Δ.
- Interpretationreasoning§ I. INTRODUCTION · p. 2
The field theory provides a precise implementation of the loose idea of summing over triangulations.
“Therefore, the field theory provides a precise implementation of the loose idea of 'summing over triangulations'.”
Quote in the paperSubstantial claim- Rests on an assumption: In spin foam models, the choice of a fixed triangulation breaks the full covariance of the theory and unphysically restricts the number of degrees of freedom.
- Rests on an assumption: A sum over triangulations is necessary for a physical interpretation of a spin foam model as a theory of gravity.
- Premisecitation§ I. INTRODUCTION · p. 2
Results in [11,22] support the idea that the Barrett-Crane model is related to quantum general relativity.
“Results in [11,22] support the idea that the Barrett-Crane model is indeed related to quantum general relativity.”
Quote in the paperNot a substantial claim0 of 2 sources checked - Premisecitation§ I. INTRODUCTION · p. 2
Barrett and Crane realized that the constraint reducing BF theory to general relativity can be implemented within an SO(4) Crane-Yetter state sum model by restricting the sum to simple representations only.
“Barrett and Crane realized that this constraint can be implemented within an SO(4) Crane-Yetter state sum model by (appropriately) restricting the sum to simple representations only.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [21] A. Barbieri, Nucl. Phys. B518 (1998) 714. L. Freidel and K. Krasnov, Class. Quantum Grav. 16 (1999) 351. J. Baez and J. Barrett, gr-qc/9903060.
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- Premisecitation§ I. INTRODUCTION · p. 2
Replacing SU(2) with the quantum group SU(2)_q, q a root of 1, yields a finite and well defined sum, the Ooguri-Crane-Yetter model.
“Replacing SU(2) with the quantum group SU(2) q ( q root of 1), yields a finite and well defined sum, the Ooguri-Crane-Yetter model [17]”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [17] L. Crane and D. Yetter, 'A Categorical construction of 4-D topological quantum field theories', in Quantum Topology , L. Kaufmann and R. Baadhio eds., World Scientific, Singapore 1993. hep-th/9301062.
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- Interpretationreasoning§ I. INTRODUCTION · p. 2
The additional summation in the Feynman expansion is precisely what is needed to address the main difficulty in developing a physical theory of quantum gravity starting from the Barrett-Crane model or from spin foam models in general.
“The additional summation is precisely what we need in order to address the main difficulty in developing a physical theory of quantum gravity starting from the Barrett-Crane model (or from spin foam models in general [4]).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [4] J. Baez, Class. Quant. Grav. 15 (1998) 1827.
Not checked in the free run.
- Rests on an assumption: In spin foam models, the choice of a fixed triangulation breaks the full covariance of the theory and unphysically restricts the number of degrees of freedom.
- Premisecitation§ I. INTRODUCTION · p. 2
The Ooguri-Crane-Yetter invariant can be shown to be the partition function of BF theory.
“The Ooguri-Crane-Yetter invariant can be shown to be the partition function of BF theory [19].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [19] G. Horowitz, Comm. Math. Phys. 125 (1989) 417. M. Blau and G. Thompson, Phys. Lett. B228 (1989) 64. J. Baez, Lett. Math. Phys. 38 (1996) 129.
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- Premisecitation§ I. INTRODUCTION · p. 2
Euclidean general relativity can be seen as an SO(4) BF theory with an added constraint.
“As euclidean general relativity can be seen as an SO(4) BF theory with an added constraint, see e.g. [20]”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [20] R. De Pietri and L. Freidel, Class. Quant. Grav. 16 (1999) 2187, gr-qc/9804071. M. Reisenberger, Class. Quant. Grav. 16 (1999) 1357, gr-qc/9804061.
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- Interpretationreasoning§ VI. CONCLUSION · p. 11
Thus, the model of case A provides another solution to the intersection constraints.
“Thus, the model of case A provides another solution to the intersection constraints.”
Quote in the paperSubstantial claim - Interpretationcitation§ VI. CONCLUSION · p. 11
In any dimension, Case B yields the higher-dimensional generalizations of the Barrett-Crane model discussed in [23].
“In any dimension, Case B yields the higher-dimensional generalizations of Barrett-Crane model discussed in [23].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [23] source
Not checked in the free run.
- Premisecitation§ VI. CONCLUSION · p. 11
The condition that these planes intersect along lines determines the Barrett-Crane intertwiner.
“In turn, the condition that these planes intersect along lines determines the Barrett-Crane intertwiner (see also [28]).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [28] M. Reisenberger, J. Math. Phys. 40 (1999) 2046. source
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- Interpretationnone given§ VI. CONCLUSION · p. 11
The divergence from the sum over colorings can be regulated by replacing the Clebsch-Gordan coefficients of the group with those of the corresponding quantum group, with q^k=1, as in the Barrett-Crane model.
“This divergence can be regulated by replacing the Clebsch-Gordan coefficients of the group with the ones of the corresponding quantum group, with q k = 1, as in the BarrettCrane model.”
Quote in the paperSubstantial claim - Interpretationnone given§ VI. CONCLUSION · p. 11
The sum over spin foam obtained in this paper is presumably divergent.
“The sum over spin foam we have obtained is presumably divergent.”
Quote in the paperSubstantial claim - Interpretationreasoning§ VI. CONCLUSION · p. 12
The field φ_{N1...N4 Λ} can be viewed as a quantum amplitude for a certain geometry of a tetrahedron or elementary chunk of space, with a certain volume, bounded by four elementary surfaces with a certain area.
“Therefore, we can view φ N 1 ...N 4 Λ as a quantum amplitude for a certain geometry of a tetrahedron or, more generally, of an elementary 'chunk' of space, with a certain volume, and bounded by four elementary surfaces with a certain area.”
Quote in the paperSubstantial claim - Premisereasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10
A 2-complex derived from a triangulation of an orientable manifold is orientable.
“A 2-complex derived from a triangulation of an orientable manifold, is orientable.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ VI. CONCLUSION · p. 11
The model described here as Case A can be seen as a different solution to the intersection constraint.
“The model described here as Case A can be seen as a different solution to this intersection constraint.”
Quote in the paperSubstantial claim - Interpretationreasoning§ VI. CONCLUSION · p. 12
The field φ(x1...x4) can be viewed as the Fourier transform of such an amplitude.
“Correspondingly, we can view the field φ ( x 1 . . . x 4 ) as the 'Fourier transform' of such amplitude.”
Quote in the paperSubstantial claim - Premisereasoning§ APPENDIX A: SOME FACTS ON REPRESENTATION THEORY · p. 13
The space of intertwiners of three representations of SO(4) is at most one-dimensional.
“The space of intertwiner of three representations of SO (4) is at most one dimensional.”
Quote in the paperNot a substantial claim - Interpretationnone given§ VI. CONCLUSION · p. 11
The second source of divergence is the sum over 2-complexes.
“The second source of divergences is the sum over 2-complexes.”
Quote in the paperSubstantial claim - Interpretationreasoning§ VI. CONCLUSION · p. 12
Four-dimensional spacetime can be viewed as a Feynman history of creations, destruction and interactions of these elementary quanta of space.
“Four dimensional spacetime can thus be viewed as a Feynman history of creations, destruction and interactions of these elementary quanta of space.”
Quote in the paperSubstantial claim - Premisecitation§ VI. CONCLUSION · p. 11
The number N of diff-inequivalent triangulations of a given four-dimensional manifold grows exponentially only with the number n of simplices; N(n) < exp{αn} for large n.
“The number N of diff-inequivalent triangulations of a given four dimensional manifold grows exponentially only with the number n of simplices [26]. That is N ( n ) < exp { αn } for large n .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [26] J. Ambjorn, M. Carfora, A. Marzuoli, The Geometry of Dynamical Triangulations , Lecture Notes in Physics, (SpringerVerlag, Berlin 1997). J. Ambjorn, M. Carfora, D. Gabrielli and A. Marzuoli, Nucl. Phys. B542 (1999) 349. source
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- Interpretationreasoning§ VI. CONCLUSION · p. 12
As a second quantized theory, it is a multiparticle theory in which particles are created and destroyed.
“As a second quantized theory, it is a 'multiparticle theory', in which particles are created and destroyed.”
Quote in the paperSubstantial claim - Interpretationnone given§ VI. CONCLUSION · p. 11
In dimension D, the field φ is defined over D copies of SO(D).
“The field φ is defined over D copies of the SO ( D ).”
Quote in the paperSubstantial claim - Interpretationnone given§ VI. CONCLUSION · p. 11
The first source of divergence is the sum over the colorings.
“The first is the sum over the colorings.”
Quote in the paperSubstantial claim - Premisecitation§ VI. CONCLUSION · p. 11
In the quantum theory, the condition on bivectors that they describe planes in R^4 translates into the simplicity condition (A1) on the representation.
“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.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [1] J. W. Barrett and L. Crane, J. Math. Phys. 39 (1998) 3296. source
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- Premisecitation§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10
In 3d, if and only if each cycle of the 2-complex is even, we can thicken faces, edges and vertices of the 2-complex, obtaining a 3 manifold with boundaries whose spine is the 2-complex.
“If, and only if, each cycle (a loop that bounds a face) of the 2-complex is even, we can 'thicken' faces, edges and vertices of the 2-complex, obtaining a 3 manifold with boundaries, whose spine is the 2-complex .”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [25] R. Benedetti and C. Petronio, Manuscripta Mathematica 88 (1995) 291-310. C. Petronio, Standard Spines and 3-Manifold , Collana Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, 1995. See also R. Benedetti and C. Petronio, Branched Standard Spine of 3-Manifolds , Lecture Notes in Mathematics N. 1653 (Springer Verlag, Berlin, 1997). source
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- Interpretationreasoning§ VI. CONCLUSION · p. 12
From this perspective, the quantum theory defined by action (11) is the second quantization of the quantum theory of the geometry of an elementary chunk of space.
“From this perspective, the quantum theory defined by the action (11) is the second quantization of the quantum theory of the geometry of an elementary chunk of space.”
Quote in the paperSubstantial claim
- May combine two claims. 21
- Classified as probably not a substantial claim. 25
May combine two claims.
Findingown data§ III. MODE EXPANSION · p. 6The invariant vectors w_β are real, and the matrix elements can also be chosen to be real.
“In the appendix, we show that the invariant vectors w β are real, and that the matrix elements can also be chosen to be real.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ III. MODE EXPANSION · p. 6The integral in the mode expansion projects the β index over a normalized SO(3) invariant vector w_β; because this exists only in simple representations and is unique, the sum over representations Λ reduces to a sum over simple representations N only.
“The integral in the last term projects the β index over a (normalized) SO (3) invariant vector, which we denote by w β .”
- Also stated in: § III. MODE EXPANSION · p. 6 · plain
“As this exists only in simple representations (and is unique, see the appendix), the sum over representations Λ reduces to a sum over the simple representations N only”
Quote in the paperSubstantial claim- Also stated in: § III. MODE EXPANSION · p. 6 · plain
May combine two claims.
Findingreasoning§ IV. FEYNMAN GRAPHS · p. 9The Feynman graphs generated are all possible 4-strand 5-valent graphs, where each edge is a collection of four strands and vertices are as shown in Fig. 3.
“The Feynman graphs we gets are all possible '4-strand' 5-valent graphs, where a '4-strand graph' is a graph whose edges are collections of four strands, and whose vertices are the ones shown in Fig. 3.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Findingreasoning§ II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 4The potential fifth-order term has the structure of a 4-simplex.
“The potential (fifth order) term has the structure of a 4-simplex.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Findingreasoning§ IV. FEYNMAN GRAPHS · p. 9In Case A, the weight of a 2-complex Γ is given by equation (14).
“The weight of 2-complex Γ is then given by (14).”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
Findingreasoning§ II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5In Case B, faces are colored by simple representations N_f of SO(4); each edge is colored by a single fixed normalized Barrett-Crane intertwiner i_BC; the face amplitude is A_f(N_f)=dim N_f=(N+1)^2; and the vertex amplitude is the 15j Wigner symbol constructed from ten simple representations and five Barrett-Crane intertwiners.
“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].”
- Also stated in: § II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5 · plain
“The amplitude of the vertex T v ( ⃗ N v , ⃗ ı v ) is the 15j Wigner symbol constructed from the ten simple representations ⃗ N v and the five BarrettCrane intertwiners i BC .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [1] J. W. Barrett and L. Crane, J. Math. Phys. 39 (1998) 3296. source
Not checked in the free run.
- Also stated in: § II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5 · plain
May combine two claims.
Findingreasoning§ III. MODE EXPANSION · p. 6The quantities H_N^α(g)=D^{(N)}_{αβ}(g)w^β are invariant under the right action of SO(3) and form an orthogonal basis for S^3 spherical harmonics.
“The quantities H N α ( g ) = D ( N ) αβ ( g ) w β are invariant under the (right) action of SO (3), and, thus, can be thought of as functions on the three-sphere: H N α ( g ) = H N α ( x ).”
- Also stated in: § III. MODE EXPANSION · p. 6 · plain
“In fact, they form an orthogonal basis for S 3 spherical harmonics.”
Quote in the paperSubstantial claim- Also stated in: § III. MODE EXPANSION · p. 6 · plain
May combine two claims.
Findingreasoning§ I. INTRODUCTION · p. 2The Feynman expansion extends the Barrett-Crane state sum to a sum over colored 2-complexes and fixes the relative weight of these.
“The expansion (2) extends the Barrett-Crane state sum (1) to a sum over colored 2-complexes, and fixes the relative weight of these.”
Quote in the paperSubstantial claim- Rests on an assumption: The Barrett-Crane amplitude A_BC(Δ,c) depends only on the 2-complex Γ(Δ), its dual 2-skeleton, and not on the full combinatorial information of the triangulation Δ.
May combine two claims.
Findingnone given§ II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5The 2-complex weight factor w(Γ) vanishes unless Γ has 4-valent edges and 5-valent vertices only; in this case w(Γ)=λ^{n(Γ)}/sym(Γ).
“The 2-complex weight factor w (Γ) vanishes unless Γ has 4-valent edges and 5-valent vertices only; in this case, it is given by w (Γ) = λ n (Γ) /sym (Γ).”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
Findingreasoning§ II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5In Case A, faces are colored by simple representations N_f of SO(4), edges by intertwiners i_e forming an orthonormal basis in K_{N_e}; the face amplitude is A_f=1, the edge amplitude is A_e=1, and the vertex amplitude is the 15j Wigner symbol constructed from ten simple representations and five intertwiners.
“Case A. We color faces of by simple representation N f of SO (4), and edges by intertwiners i e , forming an orthonormal basis in K ⃗ N e . The amplitude of the face is A f = 1. The amplitude of the edge is A e = 1.”
- Also stated in: § II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5 · plain
“The vertex amplitude A v ( ⃗ N v , ⃗ ı v ) is the 15j Wigner symbol A ( ⃗ N v , ⃗ ı v ) constructed from the ten simple representations ⃗ N v and five intertwiners i v .”
Quote in the paperSubstantial claim- Also stated in: § II. DEFINITION OF THE MODELS AND MAIN RESULT · p. 5 · plain
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Findingnone given§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10A 2-complex is orientable if all its loops (not just the cycles) are even.
“Clearly, a 2-complex is orientable if all its loops (not just the cycles) are even.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10Therefore all loops are even and the 2-complex is orientable.
“Therefore all loops are even and the 2-complex is orientable.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Findingreasoning§ APPENDIX B: INTERTWINERS AND THEIR SPACES · p. 14Using this technology, the integral of the product of four group elements is a resolution of the identity in K_{N1...N4} and can be written (B7).
“Using this technology, the integral of the product of four group elements is simply a resolution of the identity in K N 1 ...N 4 and can be written (for any choice of basis) as”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Premisecitation§ I. INTRODUCTION · p. 3Harmonic analysis on the homogeneous space S^3=SO(4)/SO(3) yields precisely the simple representations of SO(4).
“Recently, it was pointed out that harmonic analysis on the homogeneous space S 3 = SO(4) / SO(3) yields precisely the simple representations of SO(4) [23].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [23] source
Not checked in the free run.
May combine two claims.
Interpretationreasoning§ I. INTRODUCTION · p. 3The formalism leads to a natural alternative model, Case A, different from Barrett-Crane and somewhat closer to but distinct from the second version of the Barrett-Crane model.
“The first alternative (Case A, below), is far more natural from the group theoretical point of view. It yields a different model, somewhat closer to, but distinct from the second version of the Barrett-Crane model.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
Premisecitation§ I. INTRODUCTION · p. 2The Ooguri-Crane-Yetter model was shown by Crane, Kauffman and Yetter to be triangulation independent, and therefore an invariant of the 4-manifold.
“which was shown by Crane, Kauffman and Yetter [18] to be triangulation independent, and therefore to be an invariant of the 4-manifold.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [18] L. Crane, L. Kauffman and D. Yetter, J. Knot Theor. Ramifications 6 (1997) 177-234, hep-th/9409167.
Not checked in the free run.
May combine two claims.
Premisecitation§ I. INTRODUCTION · p. 2Ooguri extended the Boulatov construction to 4d; the Ooguri theory is a field theory over 4 copies of SU(2), and its Feynman expansion determines a state sum for a triangulated 4d spacetime summing over irreducible representations of SU(2).
“Ooguri has extended the Boulatov construction to 4d in [16]. The Ooguri theory is a field theory over 4 copies of SU(2). Its Feynman expansion determines a state sum for a triangulated 4d spacetime, in which the sum is over the irreducible representations of SU(2).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [16] H. Ooguri, Mod. Phys. Lett. A7 (1992) 2799.
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone given§ APPENDIX B: INTERTWINERS AND THEIR SPACES · p. 14The intertwiner from the tensor product of two representations N1, N2 to a representation Λ, if it exists, is unique.
“The intertwiner from the tensor product of two representations N 1 , N 2 to a representation Λ, if it exists is unique.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.
Premisenone given§ APPENDIX A: SOME FACTS ON REPRESENTATION THEORY · p. 13A spherical vector in a representation V_Λ of SO(D) exists if and only if the representation is simple; in that case it is unique up to normalization.
“Let V Λ be a representation of SO ( D ), we say that ω ∈ V Λ is a spherical vector if it is invariant under the action of SO ( D -1). Such a vector exists if and only if the representation is simple. In that case this vector is unique up to normalization.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Interpretationcitation§ VI. CONCLUSION · p. 11This fact suggests that the sum might converge for λ smaller than a critical value determined by α.
“This fact suggests that the sum might converge for λ smaller than a critical value, determined by α [27].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [27]
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ VI. CONCLUSION · p. 11The Barrett-Crane model has been obtained by 'quantizing a geometric 4-simplex'.
“The Barrett-Crane model has been obtained by 'quantizing a geometric 4-simplex' [1].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [1] J. W. Barrett and L. Crane, J. Math. Phys. 39 (1998) 3296. source
Not checked in the free run.
May combine two claims.
Interpretationnone given§ VI. CONCLUSION · p. 11In dimension D, the interaction term is of order D+1 and has the structure of a D-simplex.
“The interaction term is of order D + 1 and has the structure of a D -simplex.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone given§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10A 2-complex corresponds to a 4-strand graph.
“As we have seen, a 2-complex corresponds to a 4-strand graph.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation§ VI. CONCLUSION · p. 11From loop quantum gravity and other approaches, the N_i's are naturally related to the area of the faces, and their intertwiner to the volume.
“indeed, we know from loop quantum gravity [3,29] and from other approaches [21], that the N i 's are naturally related to the area of the faces, and their intertwiner to the volume.”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [3] source
Not checked in the free run.
- cites [29] C. Rovelli and L. Smolin, Nucl. Phys. B442 (1995) 593, Erratum-ibid. B456 (1995) 753. R. De Pietri and C. Rovelli, Phys. Rev. D54 (1996) 2664. source
Not checked in the free run.
- cites [21] A. Barbieri, Nucl. Phys. B518 (1998) 714. L. Freidel and K. Krasnov, Class. Quantum Grav. 16 (1999) 351. J. Baez and J. Barrett, gr-qc/9903060.
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Interpretationnone given§ VI. CONCLUSION · p. 11The results of this paper generalize easily to higher dimensions.
“The results of this paper generalize easily to higher dimensions.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation§ APPENDIX B: INTERTWINERS AND THEIR SPACES · p. 15The integral (B9) is computed for general SO(N) in [30], Chapter 9.4.11; in the case of SO(4) it is given by (B10).
“This integral is computed for the general case of SO( N ) in [30], Chapter 9.4.11. In the case of SO(4) it is given by:”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [30] N. J. Vilenkin and A. U. Klimyk, Representation of Lie Groups and Special Functions (Klewer Academic Publisher, Dordrecht, The Netherland, 1993), Vol. 2, volume 2: Class I Representations, Special Functions, and Integral Transforms.
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone given§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10The triangulation of an orientable manifold is orientable.
“Clearly, the triangulation of an orientable manifold is orientable.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.
Premisenone given§ APPENDIX A: SOME FACTS ON REPRESENTATION THEORY · p. 13The representations of SO(N) are real; it is always possible to choose a basis of V_Λ such that the representation matrices are real.
“The representation of SO ( N ) are real. This means that it is always possible to choose a basis of V Λ such that the representation matrices are real.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ VI. CONCLUSION · p. 11This regulation procedure cuts off the sum over representations and makes this sum finite.
“This procedure cuts off the sum over representations, and thus makes this sum finite.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
Interpretationnone given§ VI. CONCLUSION · p. 11In dimension D, one replaces the group SO(4) with SO(D) and the sphere S^3 with S^{D-1}=SO(D)/SO(D-1).
“In dimension D , we replace the group SO(4) with SO( D ), the sphere S 3 with S D -1 = SO( D ) / SO( D -1).”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S4 (based).
Interpretationnone given§ VI. CONCLUSION · p. 11There are two sources of divergences in the spin foam sum.
“There are two sources of divergences.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisenone given§ APPENDIX A: SOME FACTS ON REPRESENTATION THEORY · p. 13For Spin(N), a representation of half-integer spin is equivalent to its complex conjugate or dual, but the isomorphism is non-trivial.
“If we are interested by representation of half integer spin of Spin ( N ) it is still true that Λ is equivalent to its complex conjugate or dual. However the isomorphism is non trivial.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisenone given§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 9Given a triangulation ∆, the dual 2-skeleton Γ(∆) is a 2-complex.
“Given a triangulation ∆, the (abstract combinatorial structure of its) dual 2-skeleton, Γ(∆) is a 2-complex.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10If in addition each component of the boundary of this manifold is a sphere, we can fill the sphere with a ball, and obtain a triangulated manifold whose dual 2-skeleton is the 2-complex.
“If, in addition, each component of the boundary of this manifold is a sphere, we can then fill the sphere with a ball, and obtain a triangulated manifold, whose dual 2-skeleton is the 2-complex.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [25] R. Benedetti and C. Petronio, Manuscripta Mathematica 88 (1995) 291-310. C. Petronio, Standard Spines and 3-Manifold , Collana Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, 1995. See also R. Benedetti and C. Petronio, Branched Standard Spine of 3-Manifolds , Lecture Notes in Mathematics N. 1653 (Springer Verlag, Berlin, 1997). source
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing).
Premisereasoning§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 10The notion of orientability extends to 2-complexes with 4-valent edges and 5-valent vertices.
“Now, the notion of orientability extends to 2-complexes with 4-valent edges and 5-valent vertices.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ V. TRIANGULATIONS, ORIENTATION AND SYMMETRIES OF φ · p. 9In 3d, one can characterize the 2-complexes that are derived from triangulations.
“In 3d, one can nicely characterize the 2-complexes that are derived from triangulations [25].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [25] R. Benedetti and C. Petronio, Manuscripta Mathematica 88 (1995) 291-310. C. Petronio, Standard Spines and 3-Manifold , Collana Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, 1995. See also R. Benedetti and C. Petronio, Branched Standard Spine of 3-Manifolds , Lecture Notes in Mathematics N. 1653 (Springer Verlag, Berlin, 1997). source
Not checked in the free run.
- Rests on an assumption: Not every 2-complex comes from a triangulation of a 4-manifold in the way that the dual 2-skeleton of a triangulation does.
The paper relies on this without support.
In spin foam models, the choice of a fixed triangulation breaks the full covariance of the theory and unphysically restricts the number of degrees of freedom.
“In these models, indeed, the choice of a fixed triangulation breaks the full covariance of the theory and unphysically restricts the number of the degrees of freedom.”
The paper relies on this without support.
The Barrett-Crane amplitude A_BC(Δ,c) depends only on the 2-complex Γ(Δ), its dual 2-skeleton, and not on the full combinatorial information of the triangulation Δ.
“In fact, it depends only on the 2-complex Γ(∆), its dual 2-skeleton.”
The paper relies on this without support.
A sum over triangulations is necessary for a physical interpretation of a spin foam model as a theory of gravity.
“which is presumably necessary for a physical interpretation of a spin foam model as a theory of gravity.”
The paper relies on this without support.
Not every 2-complex comes from a triangulation of a 4-manifold in the way that the dual 2-skeleton of a triangulation does.
“In general, not every 2-complex comes from a triangulation triangulation of a 4-manifold in this way.”
The paper relies on this without support.
The permutation group of four elements Σ4 has five irreducible representations: two one-dimensional (trivial and signature) and three of dimensions 2, 3 and 5.
“In particular, there are 5 irreducible representation of the permutation group of 4 elements Σ 4 . Two of these are one-dimensional (the trivial and the signature), and the other three have dimensions 2,3 and 5.”
The paper relies on this without support.
The Feynman expansion generates triangulations of arbitrary manifolds.
“On the other hand, the Feynman expansion generates triangulations of arbitrary manifolds, and this might destroy the convergence.”
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?
Our work generalizes the results of Ooguri and De Pietri et al. who obtained field theory formulations for SU(2) BF theory and the Barrett-Crane model of Euclidean quantum gravity.
“Our work generalizes the results of Ooguri [17] and De Pietri et. al. [1] who obtained field theory formulations for two particular models (summed over spacetime 2-complexes), namely SU (2) BF theory [32] and the BarrettCrane model [27] of Euclidean quantum gravity.”
From Spacetime as a Feynman diagram: the connection formulation (2001) · cites this paper as [1]
Such a field theory was recently found for a particular gravity model.
“Such a field theory was recently found for a particular gravity model [1].”
From Spacetime as a Feynman diagram: the connection formulation (2001) · cites this paper as [1]
The work generalizes the result for a particular gravity model, Boulatov's and Ooguri's models of three and four dimensional topological field theories, and the old matrix models of two dimensional systems with dynamical topology.
“Our work generalizes this result as well as Boulatov's and Ooguri's models of three and four dimensional topological field theories, and ultimately the old matrix models of two dimensional systems with dynamical topology.”
From Spacetime as a Feynman diagram: the connection formulation (2001) · cites this paper as [1]
Spin-foam amplitudes equal Feynman diagram amplitudes of the group field theory.
“The duality holds in the sense that spin-foam amplitudes correspond to Feynman diagram amplitudes of the GFT.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [133]
Spin foams have a dual description as a field theory on a group manifold.
“The fundamental observation is that spin foams admit a dual formulation in terms of a field theory over a group manifold [133, 315, 314].”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [133]
The amplitude of a single vertex bounded by ten SU(2) spins and five SU(2) intertwiners is given by the square of the SU(2) Wigner 15j symbol.
“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 [3]
The original model of [2,3] and the variant pointed out in [11] appear as limiting γ→0 and γ→∞ cases, respectively.
“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 [3]
The new vertex amplitude is essentially given by the square of an SU(2) Wigner 15j symbol.
“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 [3]
All these models are defined by the same vertex, the square of the SU(2) Wigner 15j symbol; they differ for the class of boundary states considered and their measure in the spinfoam sum.
“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 [3]
The vertex was defined in [2,3] only for the euclidean case and in the absence of an Immirzi parameter γ.
“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 [3]
The results presented here are valid for all variants of the Barrett-Crane model.
“The BC model exists in a number of variants [2, 14]; the results presented here are valid for all of them.”
From Complete LQG propagator: Difficulties with the Barrett-Crane vertex (2007) · cites this paper as [14]
From experience with QCD, one can derive the persuasion that a nontrivial quantum field theory should be related to a natural lattice discretization of the corresponding classical field theory.
“From the experience with QCD, one can derive the persuasion that a nontrivial quantum field theory should be related to a natural lattice discretization of the corresponding classical field theory [22].”
From Flipped spinfoam vertex and loop gravity (2008) · cites this paper as [22]
Including matter or gauge fields is another step in developing the model, with recent results in [127-129].
“Another step in the development of the model is the inclusion of matter (or gauge fields) in it (recent results on this issue are in [127-129]).”
From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [129]
In GFT, the initial big-bang singularity is replaced by a regular bounce, followed by an era of accelerated expansion.
“In particular, the initial big-bang singularity is replaced by a regular bounce (see Ref. [1, 2]), followed by an era of accelerated expansion [2].”
From Cosmological implications of interacting group field theory models: Cyclic universe and accelerated expansion (2016) · cites this paper as [1]
The dynamics of an isotropic GFT condensate can be described by the effective action of Eq. (1).
“The dynamics of an isotropic GFT condensate can be described by means of the effective action [1]”
From Cosmological implications of interacting group field theory models: Cyclic universe and accelerated expansion (2016) · cites this paper as [1]
The equation of motion for the angular component of the GFT condensate field leads to a conserved charge Q, proportional to the scalar-field momentum πφ = ħ Q.
“Writing the complex field σ in polar form (as in Eq. (2), omitting indices) σ = ρ e iθ one finds (Ref. [1]) that the equation of motion for the angular component leads to the conservation law”
From Cosmological implications of interacting group field theory models: Cyclic universe and accelerated expansion (2016) · cites this paper as [1]
In the non-interacting GFT case, the Universe expands indefinitely and in the limit φ→∞ its dynamics follows the ordinary Friedmann equation for a flat Universe filled with a massless and minimally coupled scalar field.
“In that case one has that the Universe expands indefinitely and in the limit φ →∞ its dynamics follows the ordinary Friedmann equation for a flat Universe, filled with a massless and minimally coupled scalar field.”
From Cosmological implications of interacting group field theory models: Cyclic universe and accelerated expansion (2016) · cites this paper as [1]
The expectation value of the volume operator in GFT is given by the sum over spin representations of V_j ρ_j^2, as constructed from the first-quantized counterpart in LQG.
“Since the volume operator is diagonal in the basis of spin representations, one has for its expectation value”
From Cosmological implications of interacting group field theory models: Cyclic universe and accelerated expansion (2016) · cites this paper as [1]
Results already obtained in the free theory survive in the interacting case, in particular the occurrence of a bounce and an early epoch of accelerated expansion.
“Results that have already been obtained in the free theory (Refs. [1, 2]) survive in the interacting case, in particular for what concerns the occurrence of a bounce and an early epoch of accelerated expansion.”
From Cosmological implications of interacting group field theory models: Cyclic universe and accelerated expansion (2016) · cites this paper as [1]
The interaction terms considered can be understood as mimicking pseudosimplicial and pseudotensorial interactions of spin foam/tensorial GFT models.
“In this light, the particular type of interactions considered here can be understood as mimicking such types of interactions, which is the reason why we will refer to them as pseudosimplicial and pseudotensorial, respectively.”
From Cosmological implications of interacting group field theory models: Cyclic universe and accelerated expansion (2016) · cites this paper as [1]
Geometrogenesis is a possible way continuum spacetime and geometry could emerge from a quantum gravity substratum by means of at least one phase transition from a discrete pregeometric to a continuum geometric phase.
“It has been suggested, a possible way of how continuum spacetime and geometry could emerge from a quantum gravity substratum in such theories is by means of at least one phase transition from a discrete pregeometric to a continuum geometric phase. One refers to such a process as 'geometrogenesis' [1].”
From Impact of nonlinear effective interactions on group field theory quantum gravity condensates (2016) · cites this paper as [1]
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.
A purportedly different pairing repeats the original pairing
MinorThe appendix introduces an example of pairing the indices in a different manner, but gives exactly the same pairs as before.
“Clearly, there are other bases of this kind, obtained by pairing the indices in a different manner. For example, we can we pair the indices as ( N 1 , N 2 ) , ( N 3 , N 4 ) and define the basis”
“We pair the representations as ( N 1 , N 2 ) , ( N 3 , N 4 ). Then we define”
Dropping one point cannot leave n−2 points
MinorThe paper says an n-simplex has n+1 points. Dropping one therefore leaves n points, not the n−2 points stated for its (n−1)-simplices.
“Each of these is bounded by n-2 points, obtained by dropping one of the points of the n -simplex.”
“In n dimensions, an n -simplex S has n + 1 points (0-simplices) in its boundary.”
The citation lists haven’t been fetched yet
References 50 · Cited by 244
Research works printed but not linked: 11
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- Authors
- De Pietri, R., Freidel, L., Krasnov, K., & Rovelli, C.
- Affiliations
- École Normale Supérieure de Lyon (FR)
- Pennsylvania State University (US)
- Corresponding author
- Freidel, Laurent
- Krasnov, Kirill
- Venue
- Nuclear Physics B
- Publisher
- Elsevier BV
- Published
- 2000-05
- Volume
- 574
- Issue
- 3
- Pages
- 785-806
- ISSN
- 0550-3213 (ISSN-L)
- OpenAlex
- W2092692149
- Semantic Scholar
- 99a71c7da22239ebbd1de295c45bb796a6ef3a3f
- Cited by
- 244
- References
- 50
- Citations by year
- 2026: 1 · 2025: 1 · 2024: 7 · 2023: 4 · 2022: 5 · 2021: 9 · 2020: 3 · 2019: 4 · 2018: 5 · 2017: 5 · 2016: 10 · 2015: 7 · 2014: 10 · 2013: 21 · 2012: 10
- FWCI
- 11.97
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Nuclear Physics B (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- Barrett-Crane model, group manifolds, homogeneous spaces, 3-sphere, Feynman diagrams, quantum spacetime, spin foam models, three-dimensional quantum gravity, 4D quantum gravity
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.
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 ›
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