Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity

May combine two claims.

Integrating over the shift in the proper-time evolution operator imposes the diffeomorphism constraint, replacing spin networks by their diffeomorphism equivalence classes.

“It happens that the integration over the lapse is trivial, since there is no dependence on it in the integral, while the integration over the shift is just the imposition of the diffeomorphism constraint, so its effect is to replace the spin networks in the matrix elements with their diffeomorphism equivalence classes.”

May combine two claims.

In the expansion of the proper-time evolution operator, at each order n the operator D acts n times, with n factors A and a finite number of terms from a sum over vertices, edges and signs; the expansion is finite order by order.

“So, at each order n we have the operator D acting n times, there are n factors A and a finite number of terms coming from a sum over vertices, edges and = ±1.Moreover, since the sum over intermediate states S 1 is finite, the expansion above is finite order by order.”

May combine two claims.

For a 4-simplex, the large-spin limit of the Barrett-Crane amplitude equals the cosine of the Regge action on a triangulated 4-manifold, so it includes the semiclassical discrete-gravity path-integral amplitude.

“the asymptotics of the Barrett-Crane amplitude for a 4-simplex gives the cosine of the Regge action for a triangulated 4-manifold, and so contains the amplitude for a semiclassical path integral formulation of discrete general relativity.”

• cites [67] Gaul M, 1996Not checked yet

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

Performing the integrals, the amplitude for a single boundary tetrahedron is given by eq (78), where the B's are un-normalized Barrett-Crane intertwiners, unique up to scaling.

“Performing explicitly all the integrals, the amplitude for a single tetrahedron on the boundary turns out to be A e = simple I,L 1 J 1 J 2 J 3 J 4 C J 1 J 2 J 3 J 4 I k 1 k 2 k 3 k 4 C J 1 J 2 J 3 J 4 L m 1 m 2 m 3 m 4 = 1 J 1 J 2 J 3 J 4 B J 1 J 2 J 3 J 4 k 1 k 2 k 3 k 4 B J 1 J 2 J 3 J 4 m 1 m 2 m 3 m 4 (78 )”

• cites [15] Ashtekar A, 1991Not checked yet

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• cites [59] sourceNot checked yet

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• cites [60] Hawking S W, 1979Not checked yet

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• cites [79] Roberts J, 1998Not checked yet

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May combine two claims.

The result of the gluing is the insertion of an amplitude for the interior tetrahedra and the disappearance of the boundary terms B.

“We see that the result of the gluing is the insertion of an amplitude for the tetrahedra (dual edges) in the interior of the triangulated manifold, and of course the disappearance of the boundary terms B since the tetrahedron is no longer part of the boundary of the new manifold”

May combine two claims.

At each stationary phase point, the signs ε_kl are all equal; the angles φ are dihedral angles of a geometric 4-simplex whose triangle areas are 2j_kl+1; the integrand is exp(iμS_E), where S_E is the Regge-calculus Einstein action for a 4-simplex and μ=±1.

“The result is that, for each stationary phase point, the kl are all positive or all negative; the φ are the dihedral angles for a geometric 4-simplex with triangle areas A kl = 2j kl + 1; the integrand is exp iµS E , where S E = A kl φ kl is the Regge calculus version of the Einstein action for a 4-simplex, and µ = ±1.”

• cites [67] Gaul M, 1996Not checked yet

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May combine two claims.

The propagator is represented by four parallel lines, each a delta function, and the vertex operator has the structure of a 4-simplex with five vertices and four lines emerging from each.

“The propagator (see figure 14) can be represented by four straight parallel lines, each representing a delta function in (89), while the vertex operator (90) (see figure 15) has the structure of a 4-simplex, with five vertices and four lines coming out of each of them, and connecting it with the others according to the deltas in the expression above.”

May combine two claims.

The W functions must be gauge invariant, which requires all indices α to be paired and summed; each independent choice of indices and pairing defines an independent W function.

“However, the W functions have to be invariant under the gauge group G to which the g belong, and this requires all the indices α to be suitably paired (with the same representations for the paired indices) and summed over.Each independent choice of indices and of their pairing defines an independent W function.”

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

The cycles, edges, and vertices form a 2-complex, and because each face is labelled by a simple SO(4) representation, it is a labelled 2-complex, i.e. a spin foam.

“The abstract set formed by these cycles, the edges and the vertices is a 2-complex.Moreover, since each face (cycle) is labelled by a simple representation of SO(4), this is a labelled 2-complex, i.e. a spin foam.”

May combine two claims.

The sum over 2-complexes, which includes a sum over topologies, is another potential source of divergences, and its convergence has not yet been studied in detail.

“Of course, there is another potential source of divergences in the model, that is the sum over 2-complexes, that includes a sum over topologies, but the possible convergence of this latter sum has not yet been studied in detail (see [79] for a brief discussion).”

• cites [79] Roberts J, 1998Not checked yet

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May combine two claims.

The kinetic term acts as a projector onto gauge invariant fields and is equal to its own inverse, so the propagator is the kinetic operator itself.

“which corresponds to a projector into the space of gauge invariant fields, and is such that it is equal to its inverse, so that the propagator is K itself.”

May combine two claims.

W(s) is a linear functional on the algebra A and is positive definite.

“Moreover, W (s) is a linear functional on this algebra, which turns out to be positive definite.”

May combine two claims.

Imposing the Barrett-Crane constraints appropriately in the Ooguri model produces the Barrett-Crane state sum rather than the Crane-Yetter one, thereby giving four-dimensional quantum gravity instead of a topological theory.

“It could be expected that imposing in a proper way the Barrett-Crane constraints in the Ooguri model would give, in the end, the Barrett-Crane state sum instead of the Crane-Yetter one, so giving quantum gravity in four dimensions instead of a topological theory.This is indeed the case, as we will see.”

• cites [79] Roberts J, 1998Not checked yet

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• cites [60] Hawking S W, 1979Not checked yet

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May combine two claims.

The projector P_g enforces SO(4) invariance on the field's four arguments, corresponding to the closure constraint, while P_h projects onto fields constant on SO(3) orbits in SO(4).

“The first projector P g imposes on the field the invariance under the action of SO(4) on its four arguments (closure constraint of section 4); the second one P h projects the field over the subspace of fields that are constant on the orbits of SO(3) in SO(4).”

• AssumptionThe authors state this as an assumption: The Euclidean scalar field over SO(4)^4 is required to be invariant under any permutation of its four arguments.

May combine two claims.

The field theory defined above is perturbatively finite at all orders, but more work is needed to check that the sum defining the perturbation expansion in Feynman diagrams makes sense.

“Beacuse of this potential difficulty, all we can really say is that the field theory defined above is perturbatively finite at all orders, but more work is needed to check that the sum defining the perturbation expansion in Feynman diagrams makes sense.”

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

The potential term in momentum space contains the Barrett-Crane amplitude B_BC, which is a product of five Barrett-Crane intertwiners.

“where B BC is the Barrett-Crane amplitude given by a product of five Barrett-Crane intertwiners as in section 4.6.”

May combine two claims.

Transition functions between spin networks can be computed perturbatively in Feynman diagrams and are given by a sum over spin foams interpolating between the boundary spin networks.

“The transition functions between spin networks can be easily computed using a perturbative expansion in Feynman diagrams.As we have seen above, this turns out to be given by a sum over spin foams σ interpolating between the n spin networks representing their boundaries, for example”

May combine two claims.

The quantities W(s,s') are fully gauge invariant and represent transition amplitudes between physical states.

“The quantities W (s, s ) ≡ phys s|s phys = s|P |s (98 ) are fully gauge invariant (invariant under the action of all the constraints) objects and represent transition amplitudes between physical states.”

• AssumptionThe authors state this as an assumption: The projection operator P is real and physically represents invariance under exchange of past and future boundaries.

May combine two claims.

The exact relation between sums over spin foams and path integrals over equivalence classes of 4-metrics is not understood, so the path integral parallel is only an analogy at the present stage.

“This parallel with path integrals has, however, to be taken just as an analogy, at the present stage of development of the theory, since the exact relation between sums over spin foams and path integrals over equivalence classes of 4-metrics is not understood.”

May combine two claims.

The operator φ_s is self-adjoint because P is real, and it satisfies [φ_s, φ_s']=0.

“We can introduce in H phys the operator φ s |s phys = |s ∪ s phys (99 ) with the properties of being self-adjoint (because of the reality of P ) and of satisfying [φ s , φ s ] = 0, so that we can define”

• AssumptionThe authors state this as an assumption: The projection operator P is real and physically represents invariance under exchange of past and future boundaries.

May combine two claims.

The general structure of the amplitude is the same spin foam structure as before, with face, edge, and vertex amplitudes and an integral over continuous representations instead of a sum, but the particular form and representations differ for the Q1 and Q-1 cases.

“The general structure of this amplitude is, of course, the same spin foam structure we encountered before, with amplitudes for faces, edges, and vertices of the 2-complex, and an integral over the continuous representations instead of a sum, but of course the particular form of this amplitude and the representations involved are different for the two cases based on Q 1 or Q -1 .”

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

The general expression for evaluating closed spin networks containing only (0,ρ) representations is given by formula (107) from [34].

“The general expression for the evaluation of closed spin networks in which only representations of the type (0, ρ) are present is given by the formula [34]”

• cites [34] Ashtekar A, 1997, Class. Quantum Grav. sourceNot checked yet

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May combine two claims.

In both cases one can give an explicit formula for evaluating spin networks and for the amplitudes appearing in the spin foam partition function.

“In both cases one can give an explicit formula for the evaluation of spin networks, and for the amplitudes appearing in the spin foam partition function, as we will see in the next section.”

May combine two claims.

The field theory over group formalism yields the sum over 2-complexes and supplies a weight for each 2-complex.

“As we have seen, the field theory over group formalism provides a way to obtain this sum over 2-complexes in a straightforward manner, and also gives a prescription for the weight to be assigned to each 2-complex, but the geometrical and physical meaning of this approach has still to be fully investigated and understood.”

May combine two claims.

Simple representations of type (j,0) correspond to timelike bivectors, while those of type (0,ρ) correspond to spacelike bivectors.

“Consequently, simple representations of the type (j, 0) correspond to timelike bivectors, while those of the type (0, ρ) correspond to spacelike ones.”

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

In the Q1 case the amplitude is given by formula (113) from [102,103].

“In the first case (Q 1 ) the amplitude is given by [102,103]”

• cites [102] Barrett J W, 1998, Adv. Theor. Math. Phys. sourceNot checked yet

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• cites [103] Barrett J W, 1998Not checked yet

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May combine two claims.

P_g imposes the Barrett-Crane closure constraint, and P_h± imposes the simplicity constraint, with U+ = SU(2) and U- = SU(1,1)×Z2.

“imposing the Barrett-Crane closure constraint, and P h ± φ(g i ) = U ± . . . U ± dh 1 . . . dh 4 φ(g 1 h 1 , g 2 h 2 , g 3 h 3 , g 4 h 4 ) (110 ) imposing the simplicity constraint, where U ± are two subgroups of SL(2, C); more precisely, U + = SU (2) and U -= SU (1, 1) × Z 2 .”

• AssumptionThe authors state this as an assumption: The Lorentzian field φ(g1,g2,g3,g4) over four copies of SL(2,C) is completely symmetric in its four arguments.

May combine two claims.

Choosing the first sign in the projector leads to the Barrett-Crane model based on Q1 with only spacelike (0,ρ) representations, since SL(2,C)/SU(2) is Q1; choosing the second sign leads to the model on Q-1 with both series, since SL(2,C)/(SU(1,1)×Z2) is isomorphic to Q-1 with opposite points identified.

“Choosing the first sign in the projector appearing in the action leads to the Barrett-Crane model based on Q 1 and containing only spacelike simple representations (0, ρ), since SL(2, C)/SU (2) Q 1 , while the choice of the second sign in the projector leads to the Barrett-Crane model on Q -1 , with both series of simple representations (0, ρ) and (j, 0)”

The quote only partly states this claim.

The partition function can be expanded in terms of the theory's Feynman graphs as a power series in λ, as in the Euclidean case.

“As was shown in section 5 for the Euclidean case, we can expand the partition function in terms of the Feynman graphs of the theory (power series in λ) as in (87).”

May combine two claims.

A spin network in the dual complex of a triangulation, with one 4-valent vertex per tetrahedron and edges intersecting triangles, completely characterizes a state of the quantum tetrahedron and gives volume to it and areas to its faces.

“We then immediately recognize that this spin network completely characterizes a state of the quantum tetrahedron, and is a state in T , and gives volume to it and areas to its faces (also matching the results from loop quantum gravity).”

May combine two claims.

The area operator in loop quantum gravity has a discrete spectrum, with eigenvalues given by sums over spins of edges intersecting the surface.

“This means that the area is quantized and has a discrete spectrum of eigenvalues!”

• Also stated in: § T (α) S (A) = Tr U(A, α) S (A). · plain

  “Â( ) S (A) = i j i (j i + 1) S (A) (8 )”

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

A spin foam formulation exists for lattice Yang-Mills theory, and its partition function equals that of the lattice gauge theory and can be used to compute the strong coupling expansion.

“A spin foam formulation can also be given [28,29] for lattice Yang-Mills theory; here the partition function of the spin foam model is equal to that of the lattice gauge theory and can be used to calculate the strong coupling expansion.”

• cites [28] Rovelli C, 1995, Phys. Rev. sourceNot checked yet

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• cites [29] Rovelli C, 1995Not checked yet

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May combine two claims.

As the initial spin network slides through the manifold, its edges trace 2-surfaces and its vertices trace lines.

“The edges of s i will describe 2-surfaces, while the vertices will describe lines.”

May combine two claims.

Spin foam models for quantum gravity are built by translating geometric data of a triangulated manifold into combinatorics and group theory, so metric concepts are emergent rather than fundamental.

“Spin foam models for quantum gravity are obtained by translating the geometric information about a (usually triangulated) manifold into the language of combinatorics and group theory, so that the usual concepts of a metric and of metric properties somehow emerge from them, and are not regarded as fundamental.”

May combine two claims.

In the covariant path integral approach, topology can change, and the path integral can include a sum over topologies, producing a foam-like structure of spacetime.

“Topology itself is also allowed to change, and the path integral can be completed by a sum over different topologies, giving rise to a foam-like structure of spacetime [39], with quantum fluctuation from one 4-metric to another for a given topology, but also from one topology to another.”

• cites [39] sourceNot checked yet

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May combine two claims.

In canonical quantum gravity, the matrix elements of the proper-time evolution operator between spin network states are transition amplitudes, and computing them is equivalent to solving the theory by imposing the Hamiltonian and diffeomorphism constraints.

“Now we want to calculate the matrix elements of this operator between two spin network states (two s-knots).These are to be interpreted as transition amplitudes between quantum states of the gravitational field, and computing them is equivalent to having solved the theory imposing both the Hamiltonian and diffeomorphism constraints.”

• cites [16] sourceNot checked yet

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Classified as probably not a substantial claim.

Spin networks are purely algebraic and combinatorial objects defined without reference to an embedding.

“They are spin networks, purely algebraic and combinatorial objects, defined regardless of any embedding in the following way”

Classified as probably not a substantial claim.

For a consistent spin foam model, the amplitude Z(F) must satisfy Z(F)Z(F') = Z(F F') under gluing.

“Moreover, this amplitude should satisfy Z(F )Z(F ) = Z(F F ) (12 ) where F F : → is the spin foam obtained by gluing together F : → and F : → along .”

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

A generic slice of a spin foam is a spin network, so every spin foam can be viewed as a composition of smaller spin foams along common spin networks.

“We note also that a generic slice of a spin foam is a spin network so that every spin foam can be considered as a composition of 'smaller' spin foams formed along common spin networks.”

Classified as probably not a substantial claim.

An s-knot is a purely algebraic kinematical quantum state of the gravitational field in which vertices give volume and edges give areas to the space in which it is embedded.

“So we can say that an s-knot is a purely algebraic kinematical quantum state of the gravitational field in which the vertices give volume and the edges give areas to the space in which we embed it.”

May combine two claims.

The 3D diffeomorphism-invariant quantum states of the gravitational field are spin networks (s-knots), and they do not live in space but define spatial location itself.

“These are, as we stress again, the 3D diffeomorphism invariant quantum states of the gravitational field (provided we add to them homotopy-theoretic information of the kind mentioned above) and, accordingly, they do not live anywhere in the space but define 'the where' itself”

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

If the group G is trivial, a spin network is just a one-dimensional complex representing space, and a spin foam is just a two-dimensional complex representing spacetime.

“In fact, if we choose G to be the trivial group, then a spin network is just a one-dimensional complex, thought to represent space, and a spin foam is just a two-dimensional complex representing spacetime.”

May combine two claims.

Area and volume operators are diagonal on the invariant tensor space of a quantum tetrahedron, with area eigenvalues sqrt(j(j+1)).

“Moreover, we can define four area operators Âi = Êi Êi and a volume operator V = | IJK ÊI 1 ÊJ 2 ÊK 3 | and find that all are diagonal on Inv(j 0 ⊗ j 1 ⊗ j 2 ⊗ j 3 ) (the area having eigenvalue √ j i (j i + 1)).”

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

Classically, the geometry of a tetrahedron in a triangulated 3-manifold is characterized by its edge lengths up to rescaling and reflection, and can be studied with Regge calculus.

“Classically, its geometry is characterized by its edge lengths (up to rescaling and reflection), and it can be studied with the discretized version of general relativity given by Regge calculus [49,50].”

• cites [49] Oeckl R, 2000Not checked yet

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• cites [50] Reisenberger M, 1997, Class. Quantum Grav. sourceNot checked yet

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May combine two claims.

There exist spin foam formulations for topological field theories, Abelian and non-Abelian lattice gauge theories, and gravity.

“There exist spin foam formulations for topological field theories [23][24][25], lattice gauge theories, both Abelian and non-Abelian [28,29], and gravity [15,[30][31][32][33][34].”

• cites [23][24][25]Not checked yet

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• cites [28] Rovelli C, 1995, Phys. Rev. sourceNot checked yet

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• cites [29] Rovelli C, 1995Not checked yet

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• cites [15] Ashtekar A, 1991Not checked yet

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May combine two claims.

The topological invariance of the Turaev-Viro theory is reflected in the independence of its partition function from the chosen triangulation, which also implies that summing over triangulations is unnecessary.

“The topological invariance of the theory is reflected in the independence of Z from the particular triangulation chosen (this also implies that it is not necessary to sum over triangulations).”

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

The amplitude structure obtained in this approach is the same as the one outlined in equation (11).

“Moreover, the structure of the amplitude calculated in this approach is the same as the one outlined in (11).”

May combine two claims.

The Turaev-Viro partition function was proved to give, in the asymptotic limit, 3D Euclidean gravity with a cosmological constant, and can be thought of as a quantization of BF theory, which is classically equivalent to 3D gravity and also equivalent to two copies of Chern-Simons theory.

“This partition function was then proved [51,52] to give, in the asymptotic limit, 3D Euclidean gravity with cosmological constant, and can be thought of as giving a quantization of BF theory [53,54], a topological field theory, classically equivalent to 3D gravity”

• cites [51] Reisenberger M, 1996Not checked yet

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• cites [52] Reisenberger M, 1997Not checked yet

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• cites [53] Iwasaki J, 1999Not checked yet

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• cites [54] Iwasaki J, 2000Not checked yet

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• cites [55] Barrett J W, 2000, Class. Quantum Grav. sourceNot checked yet

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

Spin foam faces give areas to the triangles they intersect, and spin foam edges give volume to the tetrahedra they intersect.

“spin foam faces give areas to the triangles they intersect and spin foam edges give volume to the tetrahedra they intersect.”

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

The formalism of relativistic spin networks was developed and generalized to higher-valent vertices in [63].

“We note that the formalism of 'relativistic spin networks' (or 'simple spin networks'), whose evaluation is interesting also from the purely mathematical point of view since it gives an invariant of the isotopy class of labelled embedded graphs, was then developed and generalized to higher-valent vertices in [63].”

• cites [63] Penrose R, 1971Not checked yet

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May combine two claims.

The Hilbert space of a quantum 4-simplex is the infinite-dimensional subspace of L2(A_S/G_S) consisting of states annihilated by the quadratic simplicity constraints for all triangles and tetrahedra, and it was constructed explicitly by Barrett and Crane.

“We conclude that the Hilbert space of a quantum 4-simplex is the (infinite-dimensional) subspace of”

May combine two claims.

For i≠i', a solution is the Barrett-Crane intertwiner, which Reisenberger showed to be unique.

“For the case i = i a solution is the Barrett-Crane intertwiner Inv BC , then shown to be unique by Reisenberger, as we said in section 4.”

May combine two claims.

The quadratic simplicity constraints can be expressed as equality of left-handed and right-handed triangle areas for i=i', and as equality of left-handed and right-handed parallelogram areas for i≠i'.

“These constraints can be expressed in terms of left-handed and right-handed triangle areas as”

May combine two claims.

A spin foam model can be viewed as giving transition amplitudes for the gravitational field and thus a precise formulation of quantum dynamics as a sum over histories of states.

“One is to consider it as giving the transition amplitudes for the gravitational field, and consequently giving a precise formulation of the dynamics of the quantum theory”

May combine two claims.

Barrett and Crane suggested using the quantum deformation of SO(4) at a root of unity to obtain a finite sum over representations, but results in [60,61] show that the sum is finite even in the undeformed case.

“Barrett and Crane also suggest using the quantum deformation of SO(4) at a root of unity in order to obtain a finite sum over representations, but the results of [60,61], to be discussed in section 5, show that the sum above is finite even in the undeformed case.”

• cites [60] Hawking S W, 1979Not checked yet

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• cites [61] sourceNot checked yet

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May combine two claims.

Ponzano and Regge proposed a quantum gravity model in which one triangulates a three-dimensional manifold, assigns a 6j-symbol to each tetrahedron, multiplies all 6j-symbols, and sums over all possible irreducible representations of SU(2), obtaining a discretized path integral for quantum gravity, with the problem that the sum diverges.

“This led Ponzano and Regge to propose a quantum gravity model in which one triangulates a three-dimensional manifold, assigns a 6j -symbol to each tetrahedron in the triangulation, multiplies all the 6j -symbols for all the tetrahedra forming the manifold, and then sums over all the possible irreducible representations of SU (2), obtaining a kind of discretized path integral for quantum gravity,”

Classified as probably not a substantial claim.

The coloured 2-complexes arising from the time evolution of spin networks are spin foams.

“These coloured 2-complexes arising from the evolution in time of spin networks are precisely spin foams!”

May combine two claims.

The construction was generalized to four dimensions by Crane and Yetter, giving a discrete topological field theory corresponding in the continuum to BF theory and an invariant of 4-manifolds.

“This construction was generalized to the four-dimensional case by Crane and Yetter [25,26], giving a discrete topological field theory corresponding in the continuum to BF theory [57] and an invariant of 4-manifolds.”

• cites [25] Reisenberger M, 1996Not checked yet

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• cites [26] Rovelli C, 1999, Phys. Rev. sourceNot checked yet

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• cites [57] sourceNot checked yet

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May combine two claims.

The simplicity constraints imply that all four faces of the tetrahedron lie in a common hyperplane, so after imposing them one is in some sense back to the three-dimensional situation.

“These have a geometrical significance in that they imply that all the four faces of the tetrahedron lie in a common hyperplane, so that, after having imposed them, we are in some sense back to the three-dimensional situation.”

May combine two claims.

At each event where the Hamiltonian constraint acts, the 2-complex branches, and this branching is the elementary interaction vertex of the theory.

“So, at each event in which the Hamiltonian constraint acts, the 2-complex 'branches' and this branching is the elementary interaction vertex of the theory.”

May combine two claims.

The commutation relations of the quantum operators show that the four triangle area operators commute with each other and with the parallelogram area operators, among which only two are independent, while the parallelogram area operators have nonvanishing commutators among themselves.

“Analysing the commutation relations of the quantum operators corresponding to the triangle and parallelogram area operators, it turns out (see [13,58]) that, while the four triangle area operators commute with each other and with the parallelogram areas operators (among which only two are independent), the last ones have nonvanishing commutators among themselves.”

• cites [13] sourceNot checked yet

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• cites [58] Crane L, 1995Not checked yet

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Classified as probably not a substantial claim. The quote only partly states this claim.

The proposed state sum for gravity for a fixed triangulation is given by equation (27).

“The proposed state sum (spin foam model) for gravity (for fixed triangulation ) is then”

May combine two claims.

The Ponzano-Regge-Turaev-Viro model is a state sum model of the kind described earlier and is also interpretable as a spin foam model.

“It is apparent that the Ponzano-Regge-Turaev-Viro model is a state sum model of the kind described in the previous section, but it is also interpretable as a spin foam model.”

May combine two claims.

Using a metric on R4, ∧2 R4 can be identified with so(4)* ≅ so(3) ⊕ so(3), which can be interpreted as a classical phase space with a Poisson structure.

“using a metric on R 4 we can identify 2 R 4 with so(4) * so(3) ⊕ so(3), in turn interpretable as a classical phase space equipped with a Poisson structure.”

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

For a classical 4-simplex, the bivectors assigned to triangles satisfy total antisymmetry, closure constraints for each tetrahedron, and simplicity constraints for triangles sharing at least one edge.

“The conditions that these bivectors satisfy are: total antisymmetry in the indices, the closure constraints”

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

The representation of G labelling a face gives an area to the triangle that face intersects, and the intertwiner labelling an edge gives a volume to the tetrahedron that edge intersects.

“we can think that the representation of G labelling a face gives an area to the triangle this face intersects, and that the intertwiner labelling an edge gives a volume to the tetrahedron the edge intersects.”

May combine two claims.

Turaev and Viro made the Ponzano-Regge idea rigorous using the quantum deformation of SU(2) at a root of unity, obtaining a truncation of the representation sum, a finite model, and a new topological invariant for 3-manifolds given by a partition function.

“This idea was then made rigorous much later by Turaev and Viro [24], who used the same kind of construction but with the quantum deformation of SU (2) at a root of unity, instead, obtaining a truncation of the sum over representation, and thus a finite model, and a new topological invariant for 3-manifolds given by the partition function”

• cites [24] sourceNot checked yet

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May combine two claims.

This result suggests that the correct way to derive a state sum implementing the Barrett-Crane constraints from a generalized matrix model is to impose the constraints on representations only in the interaction term of the field over a group manifold, as in [60], because this leads to the correct edge amplitudes from gluing, which are absent in [79].

“Also, this result suggests that the correct way of deriving a state sum that implements the Barrett-Crane constraints from a generalized matrix model (we will discuss this topic in section 5) is like in [60], i.e. imposing the constraints on the representations only in the interaction term of the field over a group manifold, because this derivation leads to the correct edge amplitudes coming from”

• cites [60] Hawking S W, 1979Not checked yet

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• cites [79] Roberts J, 1998Not checked yet

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May combine two claims.

After the constraints reducing BF to gravity are imposed, the B field becomes a product of tetrad fields, so its geometrical information is encoded in the representation theory of the gauge group.

“the B field turns into a product of tetrad fields after the imposition of the constraints reducing BF theory to gravity, so giving the geometrical information about our manifold, and that this information, in spin foam models, has to be encoded into the algebraic language of representation theory”

May combine two claims.

The discretization of B by values on triangles is exact for a topological field theory like BF but only approximate for a non-topological theory like gravity.

“Note that this gives an exact theory for a topological field theory like the BF one, but it represents only an approximation for a non-topological theory like gravity.”

May combine two claims.

A quantum tetrahedron does not have a unique metric geometry, because some geometrical quantities cannot be determined even when the system is in a well-defined quantum state.

“Consequently, we can say that a quantum tetrahedron does not have a unique metric geometry, since there are geometrical quantities whose value cannot be determined even if the system is in a well-defined quantum state.”

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

The evaluation of a simple spin network in four dimensions for a classical group SO(4) was first given in [64].

“The first one was given [64] for a classical group and in four dimensions (the group is SO(4)).”

• cites [64] Gaul M, 2000Not checked yet

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

The gluing of two 4-simplices along a common tetrahedron was not addressed in the original paper [15] and is necessary to construct a partition function for the whole manifold.

“This important issue was not addressed in the original paper [15] and of course it is necessary to settle it in order to construct properly a partition function for the whole manifold.”

• cites [15] Ashtekar A, 1991Not checked yet

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May combine two claims.

The Barrett-Crane model quantizes the geometric sector and does not see the non-geometric sector.

“Consequently we can say that the Barrett-Crane model gives a quantization of this geometric sector and does not 'see' the non-geometric one.”

May combine two claims.

The Barrett-Crane constraints are essentially two: the simplicity constraint, requiring simple representations on triangles, and the closure constraint, requiring the tensor assigned to each tetrahedron to be SO(4)-invariant.

“The constraints are essentially two: the simplicity constraint, saying that the representations by which we label the triangles are to be chosen from the simple representations of SO(4) (Spin(4)), and the closure constraint, saying that the tensor assigned to each tetrahedron has to be an invariant tensor of SO(4) (Spin(4)).”

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

This is an example of the kind of quantum uncertainty relations expected in a quantum gravity theory.

“This is a very interesting example of the kind of quantum uncertainty relations that we can expect to find in a quantum gravity theory, i.e. in a theory of quantum geometry.”

May combine two claims.

The Barrett-Crane state sum model can be viewed as a quantization of the Plebanski action and is therefore strongly related to gravity.

“Consequently, we can look at the Barrett-Crane state sum model as a quantization of the Plebanski action, and so strongly related (even if somewhat different) to gravity.”

May combine two claims.

This approximation improves as the triangulation is refined or as one sums over all possible triangulations.

“Nevertheless this approximation would get better and better when we refine the triangulation, or sum over all the possible different triangulations”

May combine two claims.

The second constraint is implemented by requiring representation functions to be invariant under SO(3), realizing them as harmonic functions on SO(4)/SO(3)≅S^3, which was proven to completely characterize simple representations of SO(D) for any D.

“We can implement the second constraint at this level by requiring that all the representation functions have to be invariant under the subgroup SO(3) of SO(4), so realizing these representations in the space of harmonic functions over the coset SO(4)/SO(3) S 3 , which was proven in [65,73] to be a complete characterization of the simple representations of SO(D) for any dimension D”

• cites [65] Gaul M, 1999Not checked yet

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• cites [73] sourceNot checked yet

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May combine two claims.

The restriction on B is always possible classically, so the two theories agree classically but differ quantumly because interference between sectors cannot be avoided.

“This restriction on the B field is always possible classically, so the two theories do not differ at the classical level, but they are different at the quantum level, since in quantum theory one cannot avoid interference between different sectors.”

May combine two claims.

The integration over SU(2) at each vertex with normalized Haar measure ensures that the tensor product of representations at each vertex contains the trivial representation and implies restrictions on the edge labels.

“The integration is done with the Haar measure on SU (2), normalized to one, and ensures that the tensor product of representations at each vertex contains the trivial representation, and implies some restrictions on the edge labels.”

May combine two claims.

The Barrett-Crane constraints are quantum constraints on the SO(4) representations assigned to each triangle, so they can be imposed at the quantum level.

“These are quantum constraints on the representations of SO(4) which are assigned to each triangle of the triangulation, so they can be imposed at this 'quantum' level.”

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

This way of gluing is local, depending only on the parameters of the common tetrahedron.

“In this naive sense we could say that this way of gluing is local, because it depends only on the parameters of the common tetrahedron”

May combine two claims.

The four bivectors assigned to the faces of a tetrahedron automatically satisfy a chirality constraint, so they lie in the geometric sector II rather than the non-geometric sector I.

“Moreover, in [58] it was shown that the four bivectors assigned to the faces of a tetrahedron satisfy automatically a 'chirality constraint', implying that they belong to the geometric sector II above and not to the non-geometric sector I.”

• cites [58] Crane L, 1995Not checked yet

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May combine two claims.

The procedure naturally implements a sum over spin foams with weights as in (87), which is necessary to restore full covariance and remove dependence on the chosen triangulation.

“In addition, using this procedure we have also implemented in a natural way a sum over spin foams, with weights as in (87), necessary to restore the full covariance of the theory and get rid of the dependence on the particular triangulation chosen.”

May combine two claims.

Fat graphs representing Feynman diagrams are in direct correspondence with 2-complexes, which can be interpreted as the dual 2-skeleton of a triangulation.

“More precisely [92,93], the fat graphs representing Feynman diagrams of our field theory are in direct correspondence with 2-complexes, as we have seen, that can in turn be interpreted as the dual 2-skeleton of a triangulation”

• cites [92] Baez J, 1999Not checked yet

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• cites [93] sourceNot checked yet

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May combine two claims.

Starting from classical BF theory, quantizing, and imposing the constraints that are the quantum analogue of those giving the Plebanski action from BF at the classical level, yields precisely the Barrett-Crane spin foam model, strongly confirming the conjecture in [68] that this model quantizes the Plebanski action.

“Let us stress that the fact that, starting from classical BF theory, quantizing, and imposing the constraints that are the quantum analogue of those giving the Plebanski action from the BF one at the classical level, as discussed in section 4.5, we obtained precisely the Barrett-Crane spin foam model, representing a strong confirmation of the conjecture in [68] that this model gives a quantization”

• cites [68] Smolin L, 1997Not checked yet

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May combine two claims.

The sum in equation (87) can be interpreted as a sum over spin foams, the amplitude Z( ) as the amplitude for each spin foam, and the partition function itself defines a spin foam model.

“This means that the sum in (87) can be interpreted as a sum over spin foams (labelled 2-complexes), the amplitude Z( ) as an amplitude for each spin foam, and the partition function itself defines a spin foam model.”

May combine two claims.

Matrix models were first created to formulate two-dimensional quantum gravity, then Boulatov extended them to three dimensions where the model yields the Ponzano-Regge-Turaev-Viro spin foam formulation of 3D quantum gravity, and Ooguri extended them to four dimensions where the model yields the Crane-Yetter spin foam formulation of 4D BF theory.

“Matrix models were invented more than 15 years ago to give a precise formulation of 2D quantum gravity (or 'zero-dimensional string theory') [83][84][85][86][87], then generalized to three dimensions by Boulatov [88], whose model gives the Ponzano-Regge-Turaev-Viro (spin foam) formulation of 3D quantum gravity (still a topological theory)”

• cites [88] Reshetikhin N, 1991, Invent. Math. sourceNot checked yet

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• cites [89] sourceNot checked yet

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May combine two claims.

H is just the Hilbert space H_phys of the canonical theory, so the physical state space annihilated by all canonical constraints can be defined without using the projection operator P.

“Clearly, H is just the Hilbert space H phys of the canonical theory, so this means that there is the possibility of defining the physical state space, annihilated by all the canonical constraints, without making use of the projection operator P .”

May combine two claims.

The convergence is due mainly to the edge amplitudes (82), which can be interpreted as arising from gluing 4-simplices along tetrahedra in the partition function (83).

“This is due basically to the presence of the edge amplitudes (82) that we can interpret, following the results of section 4.6 [62], as coming from the gluing of 4-simplices along tetrahedra in the partition function (83).”

• cites [62] Wheeler J A, 1963Not checked yet

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May combine two claims.

Under suitable conditions, a fat graph can be put in correspondence with a four-dimensional triangulated manifold, so the sum over Feynman graphs is a sum over triangulations.

“In fact, under suitable conditions [92,93], a fat graph can be put in correspondence with a four-dimensional triangulated manifold, so that the sum over Feynman graphs is a sum over triangulations.”

• cites [92] Baez J, 1999Not checked yet

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• cites [93] sourceNot checked yet

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May combine two claims.

The vertices can be interpreted as 4-simplices and the propagators as the tetrahedra they share.

“Knowing this, we can interpret the vertices as referring to 4-simplices, and the propagators to the tetrahedra they share.”

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

To obtain a complete spin foam model for gravity, one should implement a sum over triangulations or dual 2-complexes, and both the implementation and its convergence remain open problems.

“Of course, in order to obtain a complete spin foam model for gravity we should implement a sum over triangulations or over dual 2-complexes, in some form, and both the implementation itself and the issue of its convergence are still open problems, not completely solved yet”

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

A way to implement naturally a sum over spin foams, giving also a sum over topologies, is given by generalized matrix models.

“we note that a way to implement naturally a sum over spin foams, giving also a sum over topologies, is given by the generalized matrix models that we will discuss in section 5.”

May combine two claims.

The field components depend on five parameters, corresponding to simple representations on the faces of a tetrahedron and an intertwiner in its bulk, and these determine the quantum geometry of the tetrahedron, so the components are a quantum amplitude for a particular tetrahedron geometry.

“Regarding the physical interpretation of the field theory above, note that the field J 1 J 2 J 3 J 4 depends on five parameters, corresponding to the assignment of simple representations to the faces of a tetrahedron and of an intertwiner to the bulk of it.”

May combine two claims.

The full dynamical content of the theory is given by the W functions, and these can be computed in a fully covariant fashion.

“The full (dynamical) content of the theory is given by the W functions, and these, as we are going to see now, can be computed in a fully covariant fashion.”

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

Similar models can be constructed in higher dimensions by replacing SO(4) with SO(D) and SO(3) with SO(D-1), yielding the higher-dimensional analogue of the Barrett-Crane model.

“Also, it is possible to construct similar models in higher dimensions, replacing SO(4) with SO(D) and SO(3) with SO(D -1), so that the representations used are realized on the homogeneous space SO(D)/SO(D -1) S D-1 , and so obtaining the higher-dimensional analogue of the Barrett-Crane model.”

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

The quantum theory defined by action (86) is the second quantization of a quantum theory of geometry of quanta of space that are created and annihilated, and four-dimensional spacetime is interpretable as the Feynman history of creation, destruction, and interaction of these elementary quanta of space.

“This suggests a very interesting interpretation for the quantum theory defined by the action (86): it is the second quantization (in the sense of quantum field theory) of a quantum theory of geometry of these quanta of space, in which these are created and annihilated.”

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

The sum over colourings is interpretable as a sum over geometries for a fixed triangulation and topology, giving a rigorously defined discrete path integral for Euclidean quantum gravity that is finite at all orders without regularization.

“This is a surprising and striking result since, as we have seen in the previous sections, this sum is interpretable as a sum over geometries for a fixed triangulation of the manifold and a fixed topology, so we have a rigorously defined discrete version of a path integral formulation for Euclidean quantum gravity, or sum over geometries, which moreover is finite at all orders, without any need for”

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

A spin foam model of Lorentzian quantum gravity in 3D, generalizing the Ponzano-Regge-Turaev-Viro state sum, was constructed by Freidel [98].

“Again we concentrate on the Barrett-Crane model in 4D, but we point out that a spin foam model of Lorentzian quantum gravity in 3D, generalizing the Ponzano-Regge-Turaev-Viro state sum, was constructed by Freidel [98], using methods similar to those used in section 4.6.”

• cites [98] sourceNot checked yet

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

The first derivation of the Barrett-Crane model from a field theory over a group manifold appeared in [79], and an alternative derivation was proposed in [60].

“The first derivation of the Barrett-Crane model from a field theory over a group manifold was given in [79], and an alternative one was proposed in [60].”

• cites [79] Roberts J, 1998Not checked yet

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• cites [60] Hawking S W, 1979Not checked yet

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May combine two claims.

The Feynman graphs of the model are interpreted as encoding the geometry of a four-dimensional Euclidean spacetime, although no geometric notion enters their definition.

“Then these Feynman graphs are interpreted as encoding the geometry of a four-dimensional (Euclidean) spacetime, but no geometric notion enters in their definition.”

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

These observables are purely quantum and have no analogue in the classical theory.

“Clearly these are purely quantum observables and have no analogue in the classical theory.”

May combine two claims.

Gravity in four and higher dimensions is not a topological theory, although it is strongly related to topological BF theory.

“gravity in four and higher dimensions is not a topological theory (even if strongly related to topological BF theory)”

May combine two claims.

The model obtained using Q1 involves only tetrahedra lying in spacelike hypersurfaces, hence only spacelike faces and only (0,ρ) representations.

“The model obtained using Q 1 involves only tetrahedra lying in spacelike hypersurfaces, thus having only spacelike faces.Consequently only representations (0, ρ) are present.”

May combine two claims.

Important results on deriving black hole entropy were obtained in canonical loop quantum gravity [117], and a first attempt in a 3D spin foam model is in [118], but a complete satisfactory procedure in four dimensions is still missing.

“Important results were already obtained in the canonical loop quantum gravity approach [117], and a first attempt to perform this derivation in the context of a spin foam model in 3D is in [118], but a complete and satisfactory procedure in the four-dimensional case is still missing and would be of paramount importance.”

• cites [117] Capovilla R, 2001Not checked yet

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• cites [118] Freidel L, 1999, Adv. Theor. Math. Phys. sourceNot checked yet

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May combine two claims.

The relevant Barrett-Crane models use the Q1 realization, involving only spacelike (0,ρ) representations, and the Q-1 realization, involving both timelike (j,0) and spacelike (0,ρ) representations; the first was developed in [34,102] and the second in [103].

“The relevant Barrett-Crane models are those using the realizations on Q 1 , which involves only spacelike representations of the type (0, ρ), and Q -1 , which instead involves both timelike (j, 0) and spacelike (0, ρ) representations.The first case was developed in [34,102], while the second one appeared in [103].”

• cites [34] Ashtekar A, 1997, Class. Quantum Grav. sourceNot checked yet

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• cites [102] Barrett J W, 1998, Adv. Theor. Math. Phys. sourceNot checked yet

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• cites [103] Barrett J W, 1998Not checked yet

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

The causality of spin network evolution was studied in [99][100][101]107], where a general class of globally causal models was outlined.

“The issue of causality in the evolution of spin network was studied in great generality in [99][100][101]107], where a general class of globally causal models for this evolution was outlined.”

• cites [99][100][101]107]Not checked yet

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Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing).

In [114] it was argued that the system describing perturbations of a spin foam model for quantum gravity will be a kind of perturbative string theory.

“it was argued that the system describing the perturbations of a spin foam model for quantum gravity will be given by a kind of perturbative string theory”

• cites [114] Livine R E, 2001Not checked yet

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• cites [106] Vilenkin N J, 1993Not checked yet

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

The canonical space of a spin foam theory is associated with spacetime boundaries and is spanned by spin networks, because a spin foam boundary is a spin network; in the Barrett-Crane model these are SO(4) or SO(3,1) spin networks with simple representations, whereas loop quantum gravity uses SU(2).

“the canonical space of a spin foam theory is associated with boundaries of spacetime, and so is spanned by spin networks, since the boundary of a spin foam is a spin network”

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

The characterization of a quantum 4-simplex embedded in Minkowski space is based on the same analysis reviewed in section 4.1 [34].

“The characterization of a quantum 4-simplex embedded in R 4 endowed with a Minkowski metric is based on the same kind of analysis we reviewed in section 4.1 [34].”

• cites [34] Ashtekar A, 1997, Class. Quantum Grav. sourceNot checked yet

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May combine two claims.

The form of the Lorentzian Barrett-Crane model depends on how the simple representations are realized, and this leads to a different geometric interpretation of the resulting model.

“The actual form of the Barrett-Crane model in the Lorentzian case differ, depending on the way we choose to realize our simple representations, with a consequently different geometrical interpretation of the resulting model.”

May combine two claims.

The model based on Q-1 involves only timelike tetrahedra, so its faces can be timelike or spacelike and both kinds of simple representations appear.

“The model based on Q -1 , on the other hand, involves only timelike tetrahedra, so that its faces can be either timelike or spacelike and both kinds of simple representations appear.”

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

The general idea, interpretation, and construction of the Lorentzian Barrett-Crane model are analogous to those in the Euclidean signature.

“As far as the Barrett-Crane model is concerned, the general idea and interpretation of the model and its construction are analogous to those in the Euclidean signature.”

May combine two claims.

There is no definite proof so far that the Barrett-Crane model has truly physical degrees of freedom and a sensible linearized approximation describing linearized gravity, possibly with additional fields; it is not known whether it contains gravitons.

“This issue is of crucial importance, since there is no definite proof so far that the Barrett-Crane model possesses truly physical degrees of freedom and admits a sensible linearized approximation describing linearized gravity, maybe coupled with additional fields, i.e. we do not know yet if the model contains gravitons, as is compulsory for a good theory of quantum gravity.”

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

Including matter or gauge fields is another step in developing the model, with recent results in [127-129].

“Another step in the development of the model is the inclusion of matter (or gauge fields) in it (recent results on this issue are in [127-129]).”

• cites [127] Freidel L, 1998Not checked yet

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• cites [128] O'Loughlin M, 2000Not checked yet

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• cites [129] sourceNot checked yet

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

The Lorentzian Barrett-Crane model provides a non-perturbative and algebraic definition of local causality.

“It encodes a non-perturbative and algebraic definition of local causality and, of course, presents all the features that make spin foam models interesting models of quantum geometry.”

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

Different spin foam models for gravity were proposed [28,30,32,33], and the Reisenberger model [30,31] is of much interest, but its precise relation to the Barrett-Crane model is not clear.

“In fact, we mentioned that different spin foam models for gravity were proposed [28,30,32,33], and in particular the Reisenberger model [30,31] is of much interest, but the precise relation between them and the Barrett-Crane one is not at all clear.”

• cites [28] Rovelli C, 1995, Phys. Rev. sourceNot checked yet

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• cites [30] Baez J C, 1996Not checked yet

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• cites [32] sourceNot checked yet

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• cites [33] Ashtekar A, 1995Not checked yet

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• cites [31] Baez J C, 1995Not checked yet

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

The construction in [99][100][101]107] is deeply related to the causal set formalism [108] for quantum gravity.

“The construction in [99][100][101]107] is also deeply related to the causal set formalism [108] for quantum gravity, with an interesting additional (see the introduction) convergence of ideas.”

• cites [99][100][101]107]Not checked yet

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• cites [108] Barrett J W, 1998Not checked yet

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May combine two claims.

For any non-degenerate triangulation with finitely many simplices, the Lorentzian amplitude (113) is finite, with the integral over simple continuous representations of the Lorentz group converging absolutely.

“In fact, it was proven [105], using the results of [104], that, for any non-degenerate and finite (finite number of simplices) triangulation, the amplitude (113) is finite, in the sense that the integral over the simple continuous representations of the Lorentz group converges absolutely.”

• cites [105] Freidel L, 1999Not checked yet

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• cites [104] sourceNot checked yet

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