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Spin foam models
Open accessAbstract
While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a `spin foam' going from one spin network to another. Just as a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators, a spin foam is a two-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. Spin foams arise naturally as higher-dimensional analogues of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory. When formulated as a `spin foam model', such a theory consists of a rule for computing amplitudes from spin foam vertices, faces and edges. The product of these amplitudes gives the amplitude for the spin foam, and the transition amplitude between spin networks is given as a sum over spin foams. After reviewing how spin networks describe `quantum 3-geometries', we describe how spin foams describe `quantum 4-geometries'. We conclude by presenting a spin foam model of four-dimensional Euclidean quantum gravity, closely related to the state sum model of Barrett and Crane, but not assuming the presence of an underlying spacetime manifold.
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“L 2 (A S /G S ) ∼ = j T ∈ 3 Inv(j ∂ 0 T ⊗ j ∂ 1 T ⊗ j ∂ 2 T ⊗ j ∂ 3 T )”
§ Spin foam models > 3. Spin networks as quantum 3-geometries
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“We cannot obtain L 2 (A S /G S ) simply by starting with a tensor product of copies of H, one for each triangle in , and then imposing a closure constraint for each tetrahedron.This gives too small a space of states.”
§ Spin foam models > J C Baez
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“Proposition 5.If F = (κ, ρ, ι) is a closed spin foam in M, there exists a unique spin network F | S = (γ, ρ, ι) in S such that:”
§ Spin foam models > (A S /G S ).
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“In fact, we obtain an orthonormal basis of states as ρ ranges over all labellings of the edges of * by irreducible representations of G and ι ranges over all labellings of the vertices by intertwiners chosen from some orthonormal basis.”
§ Spin foam models > (A S /G S ).
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“Any spin foam F : → in M determines an operator from L 2 (A S /G S ) to L 2 (A S /G S ), which we also denote by F , such that , F = , , for any states , .”
§ Spin foam models > (A S /G S ).
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“Note that this space is not a represention of SO(3), but only of its universal cover, SU (2).In this way quantum bivectors are naturally related to SU (2) gauge theory.”
§ Spin foam models > J C Baez
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“In this description, the constraint E 0 + E 1 + E 2 + E 3 = 0 is a consequence of the fact that the integral of the unit normal to a 2-sphere piecewise-linearly embedded in R 3 must vanish.”
§ Spin foam models > J C Baez
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“Thus if we pick a splitting of each tetrahedron in , a spin foam in M amounts to a labelling of each triangle and each tetrahedron in by a pair of spins.”
§ Spin foam models > 4. Spin foams as quantum 4-geometries
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“This gives the smaller subspace ρ T ∈ 3 Inv(ρ ∂ 0 T ⊗ ρ ∂ 1 T ⊗ ρ ∂ 2 T ⊗ ρ ∂ 3 T ) ∼ = L 2 (A S /G S )”
§ Spin foam models > J C Baez
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“As a consequence of all this, given a 3-manifold S with a triangulation and a splitting of each tetrahedron in , there is a basis of L 2 (A S /G S ) given by all labellings of triangles and tetrahedra in by spins satisfying conditions like those in equations ( 3) and ( 4) for each tetrahedron.”
§ Spin foam models > (A S /G S ).
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“Note that the quadratic constraints guarantee that all left-handed area operators equal the corresponding right-handed operators when restricted to X .”
§ Spin foam models > J C Baez
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“Thus when we quantize so(4) * with its Kirillov-Kostant Poisson structure, we get a tensor product of Hilbert spaces corresponding to the self-dual and anti-self-dual parts of the space of bivectors.”
§ Spin foam models > J C Baez
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“The key is to rewrite equation ( 9) as follows: Ê+ i (T ) • Ê+ i (T ) ψ = Ê- i (T ) • Ê- i (T ) ψ.”
§ Spin foam models > J C Baez
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“Namely, having already imposed (9) for i = i , the constraint for i = i is equivalent to Â+ ii (T )ψ = Âii (T )ψ.”
§ Spin foam models > J C Baez
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“In fact, one can show that vectors in X are invariant under the transformation P : L 2 (A S /G S ) → L 2 (A S /G S ) sending each vector ψ + ⊗ ψ -in L 2 A + S /G + S ⊗ L 2 A + S /G - S to P ψ + ⊗ ψ -= ψ -⊗ ψ + .”
§ Spin foam models > J C Baez
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“By the results of the previous section, this gives the subspace T ∈ 3 T + ⊗ T - where T + and T -are left-handed and right-handed copies of the Hilbert space for the quantum tetrahedron introduced in section 3.”
§ Spin foam models > J C Baez
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“States satisfying these constraints form Hilbert space of the quantum 4-simplex, X = ψ ∈ L 2 (A S /G S ): ( Êi (T ) ∧ Êi (T ))ψ = 0 for all i, i , T .”
§ Spin foam models > J C Baez
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“Note that a quantum bivector ψ ∈ H + ⊗ H -satisfies Ê+ • Ê+ ψ = ( Ê-• Ê-)ψ if and only if it lies in the subspace j j ⊗ j ⊆ H + ⊗ H -.”
§ Spin foam models > J C Baez
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“If E ab can be written in any of these four ways, it can be written so uniquely up to a parity transformation e a → -e a .”
§ Spin foam models > J C Baez
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“Since this formula does not depend on the ambient spacetime manifold, but only on the spin foam itself, we may consider it more generally as a formula for evaluating the amplitude of any Spin(4) spin foam with four faces meeting at each edge and with ten faces and five edges meeting at each vertex in a pattern dual to that of a 4-simplex.”
§ Spin foam models > Quantum gravity without a background spacetime
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“However, the operators corresponding to different splittings do not commute.”
§ Spin foam models > (A S /G S ).
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“By the same reasoning, the states in L 2 (A S /G S ) satisfying (9) for i = i are precisely those lying in ρ T ∈ 3 Inv(ρ ∂ 0 T ⊗ ρ ∂ 1 T ⊗ ρ ∂ 2 T ⊗ ρ ∂ 3 T ) where ρ ranges only over labellings of triangles in by representations of the form j ⊗ j .”
§ Spin foam models > J C Baez
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“In these terms the closure constraints become E 0 (T ) + E 1 (T ) + E 2 (T ) + E 3 (T ) = 0.”
§ Spin foam models > J C Baez
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“Thus the expectation value of the area of any triangle in M is well defined in the quantum 4-geometry described by any spin foam.The space S serves only as a disposable tool for studying the geometry of the spacetime M.”
§ Spin foam models > 4. Spin foams as quantum 4-geometries
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“To impose the closure constraint we apply symplectic reduction to each leaf [27], first taking the subvariety defined by the constraint E 0 + • • • + E 3 = 0, and then taking the quotient of this subvariety by the SU (2) action generated by the constraint.The latter step amounts to considering tetrahedra only modulo rotations.When nonempty, the resulting symplectic manifold is generically two”
§ Spin foam models > (A S /G S ).
What the sources say
- Not checked yetcites [27] Guillemin V, 1984
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“there is a basis of Inv(j 0 ⊗ • • • ⊗ j 3 ) labelled by spins j 4 satisfying”
§ Spin foam models > (A S /G S ).
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“(Unfortunately, Thiemann's formula for the Hamiltonian constraint lacks crossing symmetry.)”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [59] source
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“It is reassuring therefore that spin foams also arise naturally in lattice gauge theory, where these problems are absent.”
§ Spin foam models > Introduction
What the sources say
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“Under some assumptions on the form of the Hamiltonian constraint, Reisenberger and Rovelli are able to compute the transition amplitude , exp(-itH ) as a formal power series in t for any spin network states and .The coefficient of t n in this power series is a sum over certain equivalence classes of spin foams embedded in spacetime with as their initial slice, as their final slice, and a total of”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [51] Rourke C, 1972
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“Thanks in part to the introduction of spin network techniques [55], we now have a mathematically rigorous and intuitively compelling picture of the kinematical aspects of what used to be called the 'loop representation' of quantum gravity [54].”
§ Spin foam models > Introduction
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“First, perturbative string theory considers only 2-manifolds with their boundary mapped into spacetime, not more general piecewise-linear cell complexes.”
§ Spin foam models > Introduction
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“As we shall see, this suggests that the quantization of these theories should involve only two-dimensional complexes.”
§ Spin foam models > Definition 4.
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“and shown that we may naturally associate a Wess-Zumino-Witten model to the surfaces appearing in the spacetime formulation of the loop representation of quantum gravity [16].”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [16] Baez J Degenerate solutions of general relativity from topological field theory, Commun. Math. Phys.
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“Imposing the closure constraint for a given tetrahedron of has the same effect as imposing gauge invariance at the spin network vertex centred at that tetrahedron.”
§ Spin foam models > J C Baez
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“Barrett and Crane showed that these constraints take a particularly nice form if we describe them in terms of bivectors corresponding to all ten triangular faces of the 4-simplex.”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [19] Barrett J
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- Rests on an assumption: Not every collection of bivectors E_ab comes from a 4-simplex; extra constraints are required.
In the paper
“Quantizing these constraints, one obtains conditions that a Spin(4) spin foam must satisfy to describe a quantum 4-geometry built from flat quantum 4-simplices.”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [19] Barrett J
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- Rests on an assumption: Not every collection of bivectors E_ab comes from a 4-simplex; extra constraints are required.
- Rests on an assumption: If stationary points of the BF action subject to the extra constraints are found, the equations are equivalent to the vacuum Einstein equations when E is nondegenerate.
In the paper
“The work of Barbieri [18] clarifies the geometrical significance of this description of L 2 (A S /G S ): the space Inv(j 0 ⊗ j 1 ⊗ j 2 ⊗ j 3 ) describes the states of a 'quantum tetrahedron' whose ith face has area 1 2 √ j i (j i + 1).”
§ Spin foam models > 3. Spin networks as quantum 3-geometries
What the sources say
- Not checked yetcites [18] Barbieri A Quantum tetrahedra
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“Thanks in part to Barbieri's work on the quantum tetrahedron [18], the Hilbert space spanned by these states can be interpreted as the space of 'quantum 3-geometries' of the triangulated 3-manifold S.”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [18] Barbieri A Quantum tetrahedra
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In the paper
“Interestingly, these constraints also arise naturally from the relationship between general relativity and BF theory, a four-dimensional topological field theory [15,16].”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [15] Baez J, 1996
- Not checked yetcites [16] Baez J Degenerate solutions of general relativity from topological field theory, Commun. Math. Phys.
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- Rests on an assumption: Not every collection of bivectors E_ab comes from a 4-simplex; extra constraints are required.
- Rests on an assumption: In the cotetrad formulation, the equations of motion are equivalent to the vacuum Einstein equations when the cotetrad is nondegenerate.
- Rests on an assumption: If stationary points of the BF action subject to the extra constraints are found, the equations are equivalent to the vacuum Einstein equations when E is nondegenerate.
In the paper
“This establishes a one-to-one and onto map from linearly independent triples e i to linearly independent triples E i .”
§ Spin foam models > 3. Spin networks as quantum 3-geometries
What the sources say
- Not checked yetcites [18] Barbieri A Quantum tetrahedra
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- The authors state this as an assumption: Barbieri's orientation-fixing strategy is adopted, accepting that a linearly dependent triple E_i does not describe a unique tetrahedron.
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In the paper
“We review this work in what follows, but first, note an important consequence: the above equation describes L 2 (A S /G S ) as the Hilbert space of quantum 3-geometries of the simplicial complex .”
§ Spin foam models > 3. Spin networks as quantum 3-geometries
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In the paper
“Thus we may work in the 'dual picture' and think of a spin network state as a labelling of all the triangles and tetrahedra in S by spins [28,31,38,53].”
§ Spin foam models > Introduction
What the sources say
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“In a wide variety of lattice gauge theories, the time evolution operator can be computed as a sum of spin foam operators, weighted by amplitudes computed as a product of vertex, edge, and face amplitudes.”
§ Spin foam models > (A S /G S ).
What the sources say
- Not checked yetcites [48] source
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“The proof is trivial but the physical idea is important: a history on spacetime determines a state on any submanifold corresponding to space at a given time.”
§ Spin foam models > (A S /G S ).
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In the paper
“Restricting attention to the 'left-handed' copy of SU (2), such a spin foam determines an SU (2) spin network in any three-dimensional submanifold S ⊆ M compatible with the triangulation.”
§ Spin foam models > Introduction
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In the paper
“A spin foam in M gives each of the ten triangles in S an area and each of the five tetrahedra in S a volume.”
§ Spin foam models > Introduction
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In the paper
“Interestingly, these conditions also guarantee that the quantum geometries for three-dimensional submanifolds obtained using the right-handed copy of SU (2) agree with those coming from the left-handed copy.”
§ Spin foam models > Introduction
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In the paper
“The basic idea, due to Iwasaki [32] and Reisenberger [48], is that a path integral in lattice gauge theory can be represented as a sum over spin foams living in the lattice.”
§ Spin foam models > Introduction
What the sources say
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In the paper
“This suggests that some constraints must hold for S to be the boundary of a 'flat' 4-simplex.”
§ Spin foam models > Introduction
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- Rests on an assumption: The geometry of a 4-simplex affinely embedded in Euclidean R4 is determined by ten numbers, such as its edge lengths.
In the paper
“When the cosmological constant is nonzero, one obtains instead the Turaev-Viro model [62], where the quantum group SU q (2) takes the place of the gauge group SU (2), with the deformation parameter q”
§ Spin foam models > (A S /G S ).
What the sources say
- Not checked yetcites [62] source
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In the paper
“However, because there are usually infinitely many spin foams in M, this sum is likely to diverge.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
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In the paper
“The reason for these problems is that Spin(4) has infinitely many irreducible representations.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
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In the paper
“All this taken together suggests that the q-deformed version of Barrett and Crane's state sum is a discretization of Euclidean quantum gravity with cosmological constant.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
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In the paper
“The Ooguri model makes sense for any compact Lie group G and any compact oriented 4-manifold M equipped with a principal G-bundle P → M and a triangulation .”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
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In the paper
“The main point, however, is that if we think of an abstract spin foam as a kind of quantum 4-geometry, the above formula serves as a rule for computing the amplitude of any such quantum 4-geometry.”
§ Spin foam models > Quantum gravity without a background spacetime
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In the paper
“Since this projection operator appears at every vertex of the dual complex, this modification of the vertex amplitudes has the effect of picking out precisely those linear combinations of spin foams that correspond to reasonable quantum 4-geometries.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
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In the paper
“In the generic case, when the E ab (a < b) are linearly independent, it can be shown [20,22] that conditions (5) and (6) admit exactly three sorts of solutions in addition to those of the form”
§ Spin foam models > J C Baez
What the sources say
- Not checked yetcites [20] Bryant R
- Not checked yetcites [22] Capovilla R, 1991, Class. Quantum Grav. source
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In the paper
“They take as their vertex amplitude the quantity µ(P (F v )), where P : L 2 (A S /G S ) → X is the projection onto the Hilbert space of the quantum 4-simplex.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
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In the paper
“Following ideas of Barrett and Crane [19], we now describe the subspace”
§ Spin foam models > J C Baez
What the sources say
- Not checked yetcites [19] Barrett J
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Thus the Hilbert space X of a quantum 4-simplex is an infinite-dimensional subspace of L 2 (A S /G S ).”
§ Spin foam models > J C Baez
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In the paper
“To apply these analogies to the four-dimensional case, it helps to note that the Crane-Yetter model, based on triangulations, is equivalent to Turaev's earlier construction using spin foams having a 'skeleton' as their underlying complex [61].”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
What the sources say
- Not checked yetcites [61] Turaev V, 1994
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Barrett and Crane obtain their state sum model by taking the Ooguri model and imposing these constraints in their quantized form.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Since quantum groups have only finitely many irreducible representations when q is a suitable root of unity, one obtains well defined partition functions and time evolution operators this way.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
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In the paper
“The quadratic constraints E abc ∧ E abc = 0 have 1848 J C Baez a particularly nice geometric interpretation: they say the bivectors E abc can be written as wedge products of vectors in R 4 .”
§ Spin foam models > J C Baez
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In the paper
“Just as general relativity can be described as a constrained BF theory, so general relativity with nonzero cosmological constant can be described as a constrained BF theory with cosmological term.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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
“But again, in practice this sum is likely to diverge.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
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In the paper
“Thus the left-and right-handed versions of all nonchiral geometrical operators (e.g.volume operators) agree on states in X .”
§ Spin foam models > J C Baez
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Barrett and Crane make the all-important observation that conditions (5) and ( 6) can be rewritten in a simpler way if we use bivectors for all triangular faces of the 4-simplex, not just those having 0 as a vertex.”
§ Spin foam models > J C Baez
What the sources say
- Not checked yetcites [19] Barrett J
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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
“However, we show that in the end the physics is independent of this choice.”
§ Spin foam models > 4. Spin foams as quantum 4-geometries
boosted- Asserted: passes — The authors put it forward as true.
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- It may state two separate claims as one. 41
- The quote only partly states this claim. 8
- Probably not a substantial claim. 27
In the paper
“Then by a now familiar argument [13], defines a state in L 2 (A S /G S ), which we also call .Moreover, such spin network states span L 2 (A S /G S ).”
§ Spin foam models > (A S /G S ).
What the sources say
- Not checked yetcites [13] source
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“These operators commute and are simultaneously diagonalized on the subspaces Inv(j 0 ⊗ • • • ⊗ j 3 ); the eigenvalue of Âi on this subspace is 1 2 √ j i (j i + 1).”
§ Spin foam models > Quantizing these expressions, we define area operators on T by
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In the paper
“Not every triple of bivectors E i comes from a triple of vectors e i this way.It is necessary and sufficient that either the E i are linearly independent and satisfy”
§ Spin foam models > 3. Spin networks as quantum 3-geometries
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“In this case, the E i determine e i up to a parity transformation e i → -e i .The second case occurs when the vectors e i lie in a plane; in this case the E i determine e i up to an area-preserving (but possibly orientation-reversing) linear transformation of this plane.”
§ Spin foam models > 3. Spin networks as quantum 3-geometries
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“All these operators commute, and on the subspace of L 2 (A S /G S ) corresponding to a given labelling j , each area operator Âf has eigenvalue 1 2 j f (j f + 1).”
§ Spin foam models > Quantizing these expressions, we define area operators on T by
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“More precisely, this operator is first defined on H ⊗4 , but since it commutes with the action of so(3) on this space it preserves the subspace T .”
§ Spin foam models > Quantizing these expressions, we define area operators on T by
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In the paper
“Instead, we must first 'explode' the triangulation into a disjoint union of tetrahedra.The resulting complex has two triangles for every triangle in .”
§ Spin foam models > J C Baez
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Just as the space of states L 2 (A S /G S ) is spanned by spin network states, every operator from L 2 (A S /G S ) to L 2 (A S /G S ) will be a linear combination of spin foam operators if there is a spin foam F : → for every pair of spin networks and .”
§ Spin foam models > (A S /G S ).
hedgedconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“The Hilbert space of the quantum tetrahedron is then defined as the subspace T = ψ ∈ H ⊗4 : ( Ê0 + Ê1 + Ê2 + Ê3 )ψ = 0 on which the quantized version of the closure constraint holds.Since H ⊗4 is naturally a representation of so(3) and the components of the closure constraint generate the so(3) action, it follows that”
§ Spin foam models > J C Baez
plaincorrelational- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“The general solution of these equations is not yet known, but Barrett and Crane exhibit one solution for each labelling of triangles of by spins.”
§ Spin foam models > J C Baez
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“In fact, these correspond to the self-dual and anti-self-dual parts of the quantum bivector, either of which can be identified with a quantum bivector in three dimensions.”
§ Spin foam models > J C Baez
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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
“where both the 'left-handed' and 'right-handed' factors in this tensor product are spanned by SU (2) spin networks in the dual 1-skeleton of S. In particular, both factors are isomorphic.”
§ Spin foam models > 4. Spin foams as quantum 4-geometries
plaincorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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In the paper
“A 2 01 + A 2 02 + A 2 03 = A 2 0 + A 2 1 + A 2 2 + A 2 3 .”
§ Spin foam models > (A S /G S ).
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one. The quote only partly states this claim.
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
“It does not affect the computation of spin foam amplitudes; it only constrains the set of spin foams being considered, via the requirement that they live in the dual complex.”
§ Spin foam models > 5. Spin foam models
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Since the operators ÊI 0 + ÊI 1 generate an so(3) action on the space j 0 ⊗ • • • ⊗ j 3 and the operator Â01 commutes with the closure constraint, we can find a basis of J C Baez Inv(j 0 ⊗ • • • ⊗ j 3 ) consisting of eigenvectors of Â01 , with eigenvalues of the form 1 4 √ j 4 (j 4 + 1).”
§ Spin foam models > (A S /G S ).
plaincorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one. The quote only partly states this claim.
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
“Thus different splittings give different bases.The transformations between these bases are given by 6j symbols in a well known way.”
§ Spin foam models > (A S /G S ).
boosteddirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“One can check that with this definition F becomes a category.”
§ Spin foam models > J C Baez
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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
“E ab ∧ E cd = 0 if {a, b} ∩ {c, d} = ∅(5)”
- Also stated in: § J C Baez · plain
“E 12 ∧ E 34 = E 14 ∧ E 23 = E 13 ∧ E 42 .(6)”
§ Spin foam models > J C Baez
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- The authors state this as an assumption: Any antisymmetric collection of bivectors E_ab satisfying conditions (5) and (6) is taken as an adequate substitute for a cotetrad, ignoring the extra solutions.
- Also stated in: § J C Baez · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Since so(4) is isomorphic to so(3) ⊕ so(3), when we quantize this phase space we obtain the Hilbert space of a quantum bivector in four dimensions,”
§ Spin foam models > J C Baez
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one. The quote only partly states this claim.
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In the paper
“Again the right-hand side is actually independent of S, since it depends only on the data by which F labels the tetrahedron T and its faces.”
§ Spin foam models > J C Baez
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Since the group Spin(4) is a product of two copies of SU (2), the bundle P is a product of 'left-handed' and 'right-handed' SU (2) bundles P + and P -.This implies that the space A S of connections on P | S can be written as a product”
§ Spin foam models > 4. Spin foams as quantum 4-geometries
plaindirect- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: There is a principal bundle over M with structure group Spin(4).
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, smoothly and real-analytically embedded spin foams are also interesting, and one cannot define a smooth or real-analytic embedding of a surface defined only up to piecewise-linear homeomorphism!”
§ Spin foam models > J C Baez
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“A naive count of the degrees of freedom suggests that these conditions might be sufficient: it takes 36 numbers to specify the bivectors E ab , but there are 18 independent conditions of type (5) and 2 of type (6), leaving us 16 degrees of freedom, exactly the number of degrees of freedom in the vectors e a .Unfortunately, these conditions are not quite enough.”
§ Spin foam models > J C Baez
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In the paper
“This is to be expected in Euclidean quantum gravity, since it amounts to saying that applying a fourdimensional rotation to a spin foam vertex does not affect its amplitude.”
§ Spin foam models > Introduction
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In the paper
“Unfortunately, the dynamical aspects of quantum gravity remain much less well understood in this approach.”
§ Spin foam models > Introduction
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In the paper
“In models with 'crossing symmetry', the spin foam vertex amplitude can be computed using only the spin network obtained by intersecting the spin foam with a small sphere about the vertex, and result depends only on the isotopy type of this spin network.”
§ Spin foam models > Introduction
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In the paper
“Spin foams arise naturally as higher-dimensional analogues of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory.”
§ Spin foam models > Abstract
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In the paper
“Either way, a generic slice of a spin foam 'at fixed time' gives a spin network.”
- Also stated in: § Introduction · plain
“As we move the slice 'forwards in time', the spin network changes topology only when the slice passes a vertex of the spin foam.”
§ Spin foam models > Introduction
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- Also stated in: § Introduction · plain
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In the paper
“In fact, the constraint E 0 + • • • + E 3 = 0 is a kind of simplicial version of the Gauss constraint d A E = 0. Since the Gauss constraint generates SU (2) gauge transformations, it should not be surprising that the closure constraint generates rotations of the tetrahedron.”
§ Spin foam models > Quantizing these expressions, we define area operators on T by
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In the paper
“Labelling the triangles of by spins fixes the areas of the triangles and determines a space of states consistent with these areas for each tetrahedron in .”
§ Spin foam models > 3. Spin networks as quantum 3-geometries
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In the paper
“The area and volume operators first introduced by Rovelli and Smolin [56] differ from those later studied by Ashtekar, Lewandowski and collaborators [6][7][8][9]33].However, these differences do not arise in situations involving spin networks with generic 4-valent vertices (i.e.those for which no triple of incident edges have linearly dependent tangent vectors).In such situations, their formulae”
§ Spin foam models > (A S /G S ).
What the sources say
- Not checked yetcites [56] source
- Not checked yetcites [6][7][8][9]
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“We want this composition to be associative, and for each spin network we want a spin foam 1 : → serving as a left and right unit for composition.”
- Also stated in: § Definition 4. · plain
“To get a category, one must therefore take the morphisms to be certain equivalence classes of spin foams.”
§ Spin foam models > Definition 4.
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- Also stated in: § Definition 4. · plain
In the paper
“BF theory with gauge group SU (2) is three-dimensional Euclidean quantum gravity, and the lattice formulation of this theory is just the Ponzano-Regge model [46].”
§ Spin foam models > (A S /G S ).
What the sources say
- Not checked yetcites [46] Regge T, 1961, Nuovo Cimento source
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In the paper
“Indeed, since spin networks form a very convenient basis of kinematical states [13], they have largely replaced collections of loops as our basic picture of 'quantum 3-geometries'.”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [13] source
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In the paper
“We wish the 4-simplices in M to be flat because we want a picture similar to that of the Regge calculus, where spacetime is pieced together from flat 4-simplices, and curvature is concentrated along their boundaries [47].”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [47] Reisenberger M
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In the paper
“The author has already argued that the loop representation of quantum gravity can be viewed as a novel sort of background-free string theory [10]”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [10] Baez J, 1994
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In the paper
“Indeed, since spin networks form a very convenient basis of kinematical states [13], they have largely replaced collections of loops as our basic picture of 'quantum 3-geometries'.”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [13] source
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“and a closely related model in which no background spacetime manifold is assumed.”
§ Spin foam models > Introduction
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“In certain lattice gauge theories, spin networks describe states, while spin foams describe 'histories': the path integral can be computed as a sum over spin foams.”
§ Spin foam models > 2. Spin foams in lattice gauge theory
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“In spin foam models, interactions occur where spin foam faces meet at vertices, so there is no need for a fixed background metric.”
§ Spin foam models > Introduction
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In the paper
“For BF theory and the connection formulation of quantum gravity in three and four dimensions, the basic fields can all be thought of as vector-bundle-valued differential forms of degree 2.”
§ Spin foam models > Definition 4.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Since strings do not interact when they intersect in perturbative string theory, interactions arise primarily from the coupling of the string to the fixed metric on the ambient spacetime.”
§ Spin foam models > Introduction
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In the paper
“Second, transition amplitudes in perturbative string theory are computed with the aid of a fixed metric on the ambient spacetime, while spin foam models make no reference to such a metric.”
§ Spin foam models > Introduction
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Following Kirillov and Kostant, we can apply geometric quantization to this phase space [27].Only the 'integral' coadjoint orbits contribute, that is, those for which symplectic 2-form divided by 2π defines an integral cohomology class.These are the spheres of radius j = 0, 1 2 , 1, . . . .Over each such sphere there is a complex line bundle having the symplectic 2-form as its curvature, and if we”
§ Spin foam models > J C Baez
What the sources say
- Not checked yetcites [27] Guillemin V, 1984
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“However, for these quantum 3-geometries to fit together to form a sensible quantum 4-geometry, it appears that certain constraints must hold.”
§ Spin foam models > Introduction
hedgedconditional- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: The geometry of a 4-simplex affinely embedded in Euclidean R4 is determined by ten numbers, such as its edge lengths.
In the paper
“the recently proposed state sum model of Barrett and Crane (which may dually be formulated as a spin foam model)”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [19] Barrett J
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“These operators are closely related to the area and volume operators first introduced by Rovelli and Smolin [56], as well as the slightly different ones studied by Ashtekar, Lewandowski and collaborators [6][7][8][9]33].”
§ Spin foam models > Introduction
What the sources say
- Not checked yetcites [56] source
- Not checked yetcites [6][7][8][9]33]
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“It turns out that there is a category whose objects are spin networks and whose morphisms are spin foams.”
§ Spin foam models > Introduction
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“To obtain their model, Barrett and Crane take G = Spin(4) and use the same face and edge amplitudes as in the Ooguri model, while modifying the vertex amplitudes.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“It can be viewed as a rule to compute an amplitude for any spin foam F = (κ, ρ, ι) in M.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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
“We regard a linear combination of spin foams c i F i as describing a quantum 4-geometry for which X is flat when the state c i F i | S lies in this subspace.”
§ Spin foam models > J C Baez
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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 category we propose should be useful for studying smoothly or real-analytically embedded spin foams, as well as piecewise-linearly embedded ones.”
§ Spin foam models > J C Baez
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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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
“Moreover, as described in section 2, any spin foam F in M determines a state F | S ∈ L 2 (A S /G S ).Thus spin foams in M determine quantum 3-geometries for S.”
§ Spin foam models > 4. Spin foams as quantum 4-geometries
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In the paper
“This gives an 'abstract' spin foam model, where there is no picture of spacetime other than that provided by the spin foam itself.”
§ Spin foam models > Quantum gravity without a background spacetime
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In the paper
“The author has argued [12] that the Crane-Yetter model is a discrete formulation of BF theory with a cosmological term, i.e. with the action given by M tr E ∧ F + 1 12 E ∧ E , where the deformation parameter q is a function of the cosmological constant .”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
What the sources say
- Not checked yetcites [12] Baez J, 1996, Lett. Math. Phys. source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Previously, Crane and Yetter [26] constructed a q-deformed version of the Ooguri model and showed that with suitable normalizations the partition function is not only convergent but also independent of the choice of triangulation of M.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
What the sources say
- Not checked yetcites [26] Crane L, 1993
boosteddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The state sum model of Barrett and Crane is based on the observation that the Ooguri state sum model [42] is a discrete formulation of BF theory, together with the observation that general relativity can be viewed as BF theory with extra constraints forcing the E field to be of the form e ∧ e for some cotetrad field.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
What the sources say
- Not checked yetcites [42] Penrose R, 1971
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In the paper
“To make the connection, it is worth noting that for SU (2) spin networks on the 3-sphere, evaluation on the flat connection is the same as the Penrose evaluation, up to certain pesky but important signs.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“The first model is just another way of looking at the state sum model of Barrett and Crane [19].”
§ Spin foam models > 5. Spin foam models
What the sources say
- Not checked yetcites [19] Barrett J
plaincorrelational- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“In three dimensions one has the Ponzano-Regge [46] model based on SU (2) and the Turaev-Viro [62] model based on the quantum group SU q (2).”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
What the sources say
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. 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
“As we have seen, the quantized version of these constraints is exactly what guarantees that a spin foam represents a reasonable quantum 4-geometry, i.e. one for which every quantum 4-simplex is flat, and the left-and right-handed 3-geometries of any submanifold coincide.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
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In the paper
“As described above, Barbieri's work suggests another way out of this problem: changing the definition of E i in terms of the cotriad field e i , so that the above quantity is negative when the cotriad is left-handed.”
§ Spin foam models > (A S /G S ).
What the sources say
- Not checked yetcites [18] Barbieri A Quantum tetrahedra
hedgeddirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The former corresponds to Euclidean quantum gravity with = 0, but is divergent, while the latter is convergent and corresponds to Euclidean quantum gravity with = 0.”
§ Spin foam models > Quantum gravity on a spacetime with fixed triangulation
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Loll works in the piecewise-linear context, but she considers 6-valent spin networks dual to a cubical lattice [34][35][36][37].If one sets to zero two of the spins labelling the six edges incident to a vertex, her area and volume operators match those above up to constant factors, except for one difference: she does not include an absolute value in the definition of the volume operator.”
§ Spin foam models > (A S /G S ).
What the sources say
- Not checked yetcites [34][35][36][37]
plaincorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper relies on this without support.
The geometry of a 4-simplex affinely embedded in Euclidean R4 is determined by ten numbers, such as its edge lengths.
“However, the geometry of a 4-simplex affinely embedded in Euclidean R 4 is determined by only ten numbers, e.g. the lengths of its edges.”
The paper relies on this without support.
Not every collection of bivectors E_ab comes from a 4-simplex; extra constraints are required.
“The reason is that not every such E field can be written as e ∧ e for some cotetrad field e.For this to hold, it must satisfy some extra constraints.”
The paper relies on this without support.
In the cotetrad formulation, the equations of motion are equivalent to the vacuum Einstein equations when the cotetrad is nondegenerate.
“When e is nondegenerate these equations are equivalent to the vacuum Einstein equations.”
The paper relies on this without support.
If stationary points of the BF action subject to the extra constraints are found, the equations are equivalent to the vacuum Einstein equations when E is nondegenerate.
“If one finds stationary points of the above action subject to these constraints, one obtains equations equivalent to the vacuum Einstein equations, at least when E is nondegenerate.”
The authors state this as an assumption.
The principal bundle and triangulation over M restrict to the already given bundle and triangulation over S.
“We assume that P → M is a principal G-bundle over M that restricts to the already given bundle over S, and that is a triangulation of M that restricts to the already given triangulation of S.”
The authors state this as an assumption.
For a cobordism M:S→S', M has a boundary identified with the disjoint union of S and S', and there is a principal G-bundle and triangulation restricting to those on S and S'.
“Here we assume that M has a boundary which has been identified with the disjoint union of S and S .We assume there is a principal G-bundle P → M and a triangulation of M which restricts to triangulations and of S and S , respectively.”
The authors state this as an assumption.
A trivialization of P is fixed at every boundary vertex of the dual complex.
“We fix a trivialization of P at every vertex of * on the boundary of M.”
The authors state this as an assumption.
Barbieri's orientation-fixing strategy is adopted, accepting that a linearly dependent triple E_i does not describe a unique tetrahedron.
“While somewhat problematic, we tentatively adopt this strategy in what follows, since it allows us the convenience of working with arbitrary triples E i if we simply accept the fact that a linearly dependent triple E i does not describe a unique tetrahedron.”
The authors state this as an assumption.
Any antisymmetric collection of bivectors E_ab satisfying conditions (5) and (6) is taken as an adequate substitute for a cotetrad, ignoring the extra solutions.
“In what follows, we simply ignore these subtleties and take any collection of bivectors E ab antisymmetric in a, b and satisfying (5) and ( 6) as an adequate substitute for a cotetrad.”
The authors state this as an assumption.
There is a principal bundle over M with structure group Spin(4).
“We also assume there is a principal bundle P → M with structure group Spin(4).”
The paper relies on this without support.
The classical closure, equal-area, and quadratic constraints correctly characterize the geometry of a flat 4-simplex.
“Now we turn to quantizing the constraints that say when ten bivectors correspond to the faces of a flat 4-simplex.”
The paper relies on this without support.
The quantization procedure used (replacing classical bivectors with operators and imposing the constraints as operator equations) yields the correct quantum description of those constraints.
“To quantize the closure constraints this way, we start with a Hilbert space”
The paper relies on this without support.
There are infinitely many spin foams in a compact oriented 4-manifold M for the Barrett-Crane model.
“However, because there are usually infinitely many spin foams in M, this sum is likely to diverge.”
The paper relies on this without support.
Spin(4) has infinitely many irreducible representations.
“The reason for these problems is that Spin(4) has infinitely many irreducible representations.”
The paper relies on this without support.
Quantum groups have only finitely many irreducible representations when q is a suitable root of unity.
“Since quantum groups have only finitely many irreducible representations when q is a suitable root of unity, one obtains well defined partition functions and time evolution operators this way.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“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 [38]
In the citing paper
“Another difficulty associated with is the failure of the amplitudes of the EPRL model, as defined here, to be consistent with the abstract notion of spin foams as defined in [38].”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [38]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Spin foam models > J C Baez
“E abc = (e c -e b ) ∧ (e b -e a ). The bivector E abc corresponds to the triangular face abc; in particular, we have E 0bc = E bc .”
§ Spin foam models > Introduction
“E ab = e a ∧ e b”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Spin foam models > J C Baez
“each E i is normal to the ith face of the tetrahedron (the face missing the ith vertex), with magnitude given by the area of that face.”
§ Spin foam models > J C Baez
“Classically the area of the ith face of the tetrahedron is given by A i = 1 2 (E i • E i ) 1/2 .”
Citations
Cites
New Variables for Classical and Quantum Gravity
Ashtekar, A. (1986) · Physical Review Letters · doi:10.1103/physrevlett.57.2244 · Cited by 1,663
General relativity without coordinates
Regge, T. (1961) · Il Nuovo Cimento · doi:10.1007/bf02733251 · Cited by 1,477
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
State sum invariants of 3-manifolds and quantum 6j-symbols
Turaev, V. G., & Viro, O. Y. (1992) · Topology · doi:10.1016/0040-9383(92)90015-a · Cited by 1,145
Quantum geometry and black hole entropy
Ashtekar, A., Baez, J., Corichi, A., & Krasnov, K. (1998) · Physical Review Letters · doi:10.1103/physrevlett.80.904 · Cited by 1,105 · cited 1×
“A subcomplex of a complex κ is a subset of κ which is again a complex, and the k-skeleton of κ is the subcomplex consisting of all cells of dimension k or less.”
BackgroundSelf-citationIntroduction to Piecewise-Linear Topology
Rourke, C., & Sanderson, B. J. (1972) · Springer eBooks · doi:10.1007/978-3-642-81735-9 · Cited by 1,093
Lectures on Non-Perturbative Canonical Gravity
Ashtekar, A. (1991) · Advanced Series in Astrophysics and Cosmology · doi:10.1142/1321 · Cited by 872
Quantum theory of geometry: I. Area operators
Ashtekar, A., & Lewandowski, J. (1997) · Classical and Quantum Gravity · doi:10.1088/0264-9381/14/1a/006 · Cited by 852
Loop space representation of quantum general relativity
Rovelli, C., & Smolin, L. (1990) · Nuclear Physics B · doi:10.1016/0550-3213(90)90019-a · Cited by 806
Spin networks and quantum gravity
Rovelli, C., & Smolin, L. (1995) · Physical Review D · doi:10.1103/physrevd.52.5743 · Cited by 629
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
Quantum gravity: a progress report
Carlip, S. (2001) · Reports on Progress in Physics · doi:10.1088/0034-4885/64/8/301 · Cited by 449
Metadata
- Authors
- Baez, J. C.
- Affiliations
- University of California, Riverside (US)
- Corresponding author
- Baez, John C
- ORCID
- Baez, John C
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1998-07-01
- Volume
- 15
- Issue
- 7
- Pages
- 1827-1858
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2137409288
- Semantic Scholar
- 137704ee1801f82ad06fad03827a9242c499820e
- Cited by
- 381
- References
- 61
- Citations by year
- 2026: 4 · 2025: 5 · 2024: 9 · 2023: 7 · 2022: 5 · 2021: 10 · 2020: 3 · 2019: 7 · 2018: 9 · 2017: 6 · 2016: 10 · 2015: 6 · 2014: 15 · 2013: 20 · 2012: 18
- FWCI
- 22.35
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (9514240)
- 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
- spin foams, spin networks, quantum gravity, Barrett-Crane model, state sum models, Euclidean quantum gravity, Feynman diagrams, lattice gauge theory, transition amplitudes
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