Spin foam models

May combine two claims.

Spin network states whose graph is the 1-skeleton of the dual complex define states in L2(A_S/G_S), and these states span that space.

“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 ).”

• cites [13] sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The area operators on the quantum tetrahedron commute, are simultaneously diagonalized on Inv subspaces, and have eigenvalue 1/2 sqrt(j_i(j_i+1)).

“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).”

May combine two claims.

Not every triple of bivectors E_i arises as wedge products of a triple of vectors e_i; necessary and sufficient conditions are either linear independence with triple product >0 or all being multiples of a fixed bivector.

“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”

May combine two claims.

In the generic case of linearly independent e_i, the bivectors E_i determine e_i up to a parity transformation; in the planar case, up to an area-preserving linear transformation of the plane.

“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.”

May combine two claims. The quote only partly states this claim.

On L2(A_S/G_S), there are area operators for each triangle and volume operators for each tetrahedron; all commute, and on a fixed spin labelling each area operator has eigenvalue 1/2 sqrt(j_f(j_f+1)).

“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).”

May combine two claims.

The volume operator on the quantum tetrahedron commutes with the so(3) action and preserves the closure-constraint subspace T.

“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 .”

May combine two claims. The quote only partly states this claim.

To obtain L2(A_S/G_S), one must first explode the triangulation into disjoint tetrahedra, then impose equal-area constraints for paired triangles and closure constraints for each tetrahedron.

“Instead, we must first 'explode' the triangulation into a disjoint union of tetrahedra.The resulting complex has two triangles for every triangle in .”

The quote only partly states this claim.

Every operator from L2(A_S/G_S) to L2(A_S'/G_S') is a linear combination of spin foam operators if there is a spin foam for every pair of spin networks.

“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 .”

May combine two claims.

The Hilbert space of the quantum tetrahedron is the subspace of H⊗4 satisfying the quantized closure constraint, and it is isomorphic to the direct sum over j0..j3 of Inv(j0⊗...⊗j3).

“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”

May combine two claims.

The general solution of the quadratic constraints is not yet known, but Barrett and Crane exhibit one solution for each labelling of triangles by spins.

“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.”

May combine two claims.

The left- and right-handed factors H+ and H- correspond to the self-dual and anti-self-dual parts of the quantum bivector, each identifiable with a quantum bivector in three dimensions.

“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.”

May combine two claims.

Both the left-handed and right-handed factors in the tensor product are spanned by SU(2) spin networks in the dual 1-skeleton of S, and they are isomorphic.

“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.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims. The quote only partly states this claim.

The relation A01^2 + A02^2 + A03^2 = A0^2 + A1^2 + A2^2 + A3^2 holds classically and continues to hold at the quantum level.

“A 2 01 + A 2 02 + A 2 03 = A 2 0 + A 2 1 + A 2 2 + A 2 3 .”

May combine two claims.

In the first model, the spacetime manifold does not affect the computation of spin foam amplitudes; it only constrains the set of spin foams via the requirement that they live in the dual complex.

“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.”

May combine two claims. The quote only partly states this claim.

The parallelogram area operator A_ij has eigenvectors forming a basis of Inv(j0⊗...⊗j3), with eigenvalues 1/4 sqrt(j4(j4+1)) for the corresponding splitting.

“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).”

May combine two claims.

Different splittings of a tetrahedron give different bases of Inv, and the transformations between these bases are given by 6j symbols.

“Thus different splittings give different bases.The transformations between these bases are given by 6j symbols in a well known way.”

Probably not a substantial claim: may not pass S3 (load-bearing).

With the definition of composition, F becomes a category.

“One can check that with this definition F becomes a category.”

May combine two claims.

For a classical 4-simplex, the bivectors E_ab satisfy E_ab∧E_cd=0 when the pairs are disjoint and E12∧E34=E14∧E23=E13∧E42.

“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)”

• AssumptionThe 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.

May combine two claims. The quote only partly states this claim.

Quantizing a single bivector in four dimensions yields the Hilbert space H⊗H, isomorphic to the direct sum over j+,j- of j+⊗j-, which is a representation of Spin(4) rather than SO(4).

“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,”

The quote only partly states this claim.

The expectation value of the 3-volume of any tetrahedron in M is well defined in the quantum 4-geometry described by any spin foam, independent of S.

“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.”

May combine two claims. The quote only partly states this claim.

Since Spin(4) is a product of two SU(2) groups, the Spin(4) bundle over S splits into left- and right-handed SU(2) bundles, and L2(A_S/G_S) is isomorphic to the tensor product of the corresponding left and right L2 spaces.

“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”

• AssumptionThe authors state this as an assumption: There is a principal bundle over M with structure group Spin(4).

Probably not a substantial claim: may not pass S4 (based).

One cannot define a smooth or real-analytic embedding of a surface defined only up to piecewise-linear homeomorphism.

“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!”

May combine two claims.

A naive degree-of-freedom count suggests conditions (5) and (6) might be sufficient to characterize bivectors from a cotetrad, but they are not quite enough.

“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.”

Classified as probably not a substantial claim.

Crossing symmetry is expected in Euclidean quantum gravity because it means that applying a four-dimensional rotation to a spin foam vertex does not change its amplitude.

“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.”

Classified as probably not a substantial claim.

The dynamical aspects of quantum gravity remain much less well understood in the spin network approach.

“Unfortunately, the dynamical aspects of quantum gravity remain much less well understood in this approach.”

May combine two claims.

In models with crossing symmetry, the vertex amplitude can be computed from the spin network obtained by intersecting the spin foam with a small sphere around the vertex, and it depends only on the isotopy type of that spin network.

“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.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Spin foams arise naturally as higher-dimensional analogues of Feynman diagrams in quantum gravity and other continuum gauge theories, and in lattice gauge theory.

“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.”

May combine two claims.

A generic time slice of a spin foam gives a spin network; its edges come from foam faces and its vertices from foam edges; topology changes only when the slice crosses a foam vertex.

“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.”

May combine two claims.

The closure constraint is a simplicial version of the Gauss constraint d_A E=0, and it generates rotations of the tetrahedron.

“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.”

May combine two claims.

Labelling triangles by spins fixes their areas and determines a space of states consistent with those areas for each tetrahedron.

“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 .”

May combine two claims.

The area and volume operators of Rovelli and Smolin differ from those of Ashtekar, Lewandowski and collaborators, but for spin networks with generic 4-valent vertices their formulae agree up to constant factors.

“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”

• cites [56] sourceNot checked yet

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• cites [6][7][8][9]Not checked yet

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Classified as probably not a substantial claim. May combine two claims.

Spin foams can be composed, and composition is associative with units up to equivalence; to obtain a category one takes morphisms to be equivalence classes of spin foams.

“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.”

May combine two claims.

BF theory with gauge group SU(2) is three-dimensional Euclidean quantum gravity, and its lattice formulation is the Ponzano-Regge model.

“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].”

• cites [46] Regge T, 1961, Nuovo Cimento sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

Spin networks form a convenient basis of kinematical states.

“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'.”

• cites [13] sourceNot checked yet

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Classified as probably not a substantial claim.

The paper wants the 4-simplices in M to be flat to resemble Regge calculus, where spacetime is built from flat 4-simplices and curvature is concentrated on their boundaries.

“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].”

• cites [47] Reisenberger MNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

The loop representation of quantum gravity can be viewed as a novel background-free string theory.

“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]”

• cites [10] Baez J, 1994Not checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

Spin networks have largely replaced collections of loops as the basic picture of quantum 3-geometries.

“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'.”

• cites [13] sourceNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing).

There is a closely related spin foam model for four-dimensional Euclidean quantum gravity in which no background spacetime manifold is assumed.

“and a closely related model in which no background spacetime manifold is assumed.”

Classified as probably not a substantial claim. May combine two claims.

In certain lattice gauge theories, spin networks describe states and spin foams describe histories, with the path integral computed as a sum over spin foams.

“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.”

May combine two claims.

In spin foam models, interactions occur where spin foam faces meet at vertices, so no fixed background metric is needed.

“In spin foam models, interactions occur where spin foam faces meet at vertices, so there is no need for a fixed background metric.”

Probably not a substantial claim: may not pass S4 (based).

For BF theory and the connection formulation of quantum gravity in three and four dimensions, the basic fields can be regarded as vector-bundle-valued differential forms of degree 2.

“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.”

May combine two claims.

In perturbative string theory, strings do not interact when they intersect, so interactions arise mainly from coupling to the fixed ambient metric.

“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.”

May combine two claims.

Transition amplitudes in perturbative string theory use a fixed metric on the ambient spacetime, whereas spin foam models make no reference to such a metric.

“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.”

May combine two claims.

Geometric quantization of integral coadjoint orbits of so(3)* yields the spin-j representation of so(3), and their direct sum is the Hilbert space of a quantum bivector in three dimensions.

“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”

• cites [27] Guillemin V, 1984Not checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S4 (based).

For the quantum 3-geometries induced on submanifolds to fit together into a sensible quantum 4-geometry, certain constraints must hold.

“However, for these quantum 3-geometries to fit together to form a sensible quantum 4-geometry, it appears that certain constraints must hold.”

• AssumptionRests on an assumption: The geometry of a 4-simplex affinely embedded in Euclidean R4 is determined by ten numbers, such as its edge lengths.

Probably not a substantial claim: may not pass S3 (load-bearing).

The Barrett-Crane state sum model may dually be formulated as a spin foam model.

“the recently proposed state sum model of Barrett and Crane (which may dually be formulated as a spin foam model)”

• cites [19] Barrett JNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

Area and volume operators for triangles and tetrahedra in this picture are closely related to those introduced by Rovelli and Smolin and by Ashtekar, Lewandowski and collaborators.

“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].”

• cites [56] sourceNot checked yet

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• cites [6][7][8][9]33]Not checked yet

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Probably not a substantial claim: may not pass S4 (based).

There is a category whose objects are spin networks and whose morphisms are spin foams.

“It turns out that there is a category whose objects are spin networks and whose morphisms are spin foams.”

May combine two claims.

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.

“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.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The Ooguri model can be viewed as a rule to compute an amplitude for any spin foam F = (κ, ρ, ι) in M.

“It can be viewed as a rule to compute an amplitude for any spin foam F = (κ, ρ, ι) in M.”

Classified as probably not a substantial claim.

A linear combination of spin foams describes a quantum 4-geometry for which a 4-simplex X is flat when the induced state on its boundary lies in X.

“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.”

Probably not a substantial claim: may not pass S2 (checkable).

The category proposed should be useful for studying smoothly or real-analytically embedded spin foams, as well as piecewise-linearly embedded ones.

“The category we propose should be useful for studying smoothly or real-analytically embedded spin foams, as well as piecewise-linearly embedded ones.”

May combine two claims.

Any spin foam F in M determines a state F|S in L2(A_S/G_S), so spin foams in M determine quantum 3-geometries for S.

“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.”

Classified as probably not a substantial claim.

This gives an 'abstract' spin foam model, where there is no picture of spacetime other than that provided by the spin foam itself.

“This gives an 'abstract' spin foam model, where there is no picture of spacetime other than that provided by the spin foam itself.”

Classified as probably not a substantial claim. May combine two claims.

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 Λ.

“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 .”

• cites [12] Baez J, 1996, Lett. Math. Phys. sourceNot checked yet

  Not checked in the free run.

May combine two claims.

Crane and Yetter [26] constructed a q-deformed version of the Ooguri model and showed that with suitable normalizations the partition function is convergent and independent of the choice of triangulation of M.

“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.”

• cites [26] Crane L, 1993Not checked yet

  Not checked in the free run.

May combine two claims.

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.

“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.”

• cites [42] Penrose R, 1971Not checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

For SU(2) spin networks on the 3-sphere, evaluation on the flat connection is the same as the Penrose evaluation, up to certain signs.

“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.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The first spin foam model is just another way of looking at the state sum model of Barrett and Crane [19].

“The first model is just another way of looking at the state sum model of Barrett and Crane [19].”

• cites [19] Barrett JNot checked yet

  Not checked in the free run.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In three dimensions, the Ponzano-Regge model is based on SU(2) and the Turaev-Viro model is based on the quantum group SUq(2).

“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).”

• cites [46] Regge T, 1961, Nuovo Cimento sourceNot checked yet

  Not checked in the free run.

• cites [62] sourceNot checked yet

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May combine two claims.

The quantized version of the constraints 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.

“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.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Barbieri's work suggests changing the definition of E_i in terms of the cotriad so that the volume quantity is negative when the cotriad is left-handed.

“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.”

• cites [18] Barbieri A Quantum tetrahedraNot checked yet

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May combine two claims.

The Ponzano-Regge model corresponds to Euclidean quantum gravity with Λ=0 but is divergent, while the Turaev-Viro model is convergent and corresponds to Euclidean quantum gravity with Λ=0.

“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.”

May combine two claims.

Loll's area and volume operators for 6-valent spin networks match the above up to constant factors except she omits an absolute value in the volume operator.

“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.”

• cites [34][35][36][37]Not checked yet

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