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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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59 No basis given · 39 Not checked yet · 35 Own interpretation
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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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- The authors state this as an assumption: The principal bundle and triangulation over M restrict to the already given bundle and triangulation over S.
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In the paper
“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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- 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'.
- The authors state this as an assumption: A trivialization of P is fixed at every boundary vertex of the dual complex.
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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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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 ).
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- Not checked yetcites [13] source
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“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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“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
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“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
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“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
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“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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“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
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“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 ).
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“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
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“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
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“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
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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
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“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 ).
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“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
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“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 ).
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“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 ).
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“One can check that with this definition F becomes a category.”
§ Spin foam models > J C Baez
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“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
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- 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
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“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
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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
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“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
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“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
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“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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“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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“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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“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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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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“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]
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“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.
plaindirect- Asserted: passes — The authors put it forward as true.
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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
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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
plain- Asserted: passes — The authors put it forward as true.
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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
plaindirect- Asserted: passes — The authors put it forward as true.
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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
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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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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
“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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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In 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
plaindirect- Asserted: passes — The authors put it forward as true.
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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
plaindirect- Asserted: passes — The authors put it forward as true.
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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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Probably not a substantial claim. The paper may give no reason why it holds.
- 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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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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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Probably not a substantial claim.
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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Probably not a substantial claim. It may be too vague to check against evidence.
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
plaindirect- Asserted: passes — The authors put it forward as true.
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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
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
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
plaindirect- Asserted: passes — The authors put it forward as true.
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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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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
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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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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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It may state two separate claims as one.
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
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General relativity without coordinates
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“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
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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
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Items 101–109 of 109
Simple spin networks as Feynman graphs
Freidel, L., & Krasnov, K. (2000) · Journal of Mathematical Physics · doi:10.1063/1.533203 · Cited by 42
The physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory
Ding, Y., & Rovelli, C. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/20/205003 · Cited by 42
Spin foam quantization and anomalies
Bojowald, M., & Perez, A. (2010) · General Relativity and Gravitation · doi:10.1007/s10714-009-0892-9 · Cited by 35
The perturbative Regge-calculus regime of loop quantum gravity
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Eleven-dimensional supergravity as a constrained topological field theory
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Continuum spin foam model for 3d gravity
Zapata, J. A. (2002) · Journal of Mathematical Physics · doi:10.1063/1.1509850 · Cited by 24
The volume operator in covariant quantum gravity
Ding, Y., & Rovelli, C. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/16/165003 · Cited by 23
Asymptotics of loop quantum gravity fusion coefficients
Alesci, E., Bianchi, E., Magliaro, E., & Perini, C. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/9/095016 · Cited by 15
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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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