“Sum over surfaces” form of loop quantum gravity

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

U_{N,N⃗} can be written as a limit of products of small time propagators.

“Let us write U N, ⃗ N as a limit of products of small time propagators.”

May combine two claims.

For every two s-knots si and sf in H_diff, the matrix element between them is well defined, independent of the chosen representatives S_i and S_f.

“For every two s-knots s i and s f in H diff , we can arbitrarily pick S i and S f such that S i ∈ s i and S f ∈ s f , and we have the key result that”

May combine two claims.

The integration over the Shift amounts to imposing the Diff constraint and is equivalent to an integration over the diffeomorphism group.

“The integration over the Shift amounts to imposing the Diff constraint. Indeed, it turns out to be equivalent to an integration over the diffeomorphism group, as in the group integration technique for solving the Diff constraint.”

May combine two claims.

The operator U(T) is well defined in H_diff and is diff-invariant.

“The operator U ( T ) is now well defined in H diff ! Indeed, it is immediate to see that it is diff-invariant.”

May combine two claims.

The perturbation expansion of the proper time propagator U(T) is finite and computable order by order.

“We show that the perturbation expansion of this operator is finite and computable order by order.”

May combine two claims.

The sum over the intermediate state S1 in the expansion is finite because both D and D† yield a finite number of terms when acting on a spin network state.

“The first point to be noticed in this expression is that the sum over the intermediate state S 1 is finite. This is because both D and D † yield a finite number of terms only, when acting on a spin network state.”

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

The explicit expression for U ( T ) that we obtain is simple and well defined order by order for any T.

“The explicit expression for U ( T ) that we obtain is simple and well defined order by order for any T .”

• AssumptionThe authors state this as an assumption: The potential singularities in the proper time slicing can be ignored for the purposes of this paper.

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

Each individual term in the expansion (55) is finite.

“Each individual term in the expansion (55) is finite.”

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

The expansion (64) is explicitly computable.

“The essential property of the expansion (64) is that it is finite order by order, and explicitly computable.”

May combine two claims.

S_v is a colored graph in the 3-d space ∂B, having 4 = n_i + n_f nodes (satisfying Clebsch-Gordan relations), corresponding to the intersections between the 4 edges emerging from p and ∂B.

“A short reflection will convince the reader that S v is a colored graph in the 3-d space ∂B , having 4 = n i + n f nodes (that satisfy Clebsh Gordan relations), corresponding to the intersections between the 4 edges emerging from p and ∂B .”

May combine two claims.

The expansion is in powers of T, or equivalently in inverse powers of the Planck length.

“Notice that the expansion is in power of T , or, equivalently, in inverse powers of the Planck length, because this must divide T in order to recover physical dimensions.”

May combine two claims.

A_v(σ) is non-vanishing only if n_i = 1 and n_f = 3 or if n_i = 3 and n_f = 1.

“A v ( σ ) is non vanishing only if n i = 1 and n f = 3 or if n i = 3 and n f = 1.”

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

The vertex factor in the OCY partition function is the 15j symbol of the colorings adjacent to the vertex.

“the vertex factor is the 15j symbol of the colorings adjacent to the vertex.”

May combine two claims.

The vertex factor is different in the two theories: in BF theory it is a 15j symbol, while in GR it is a combination of 9j and 6j symbols.

“The vertex factor is different in the two theories: in BF theory it is a 15j symbol, while in GR it is a combination of 9j and 6j symbols.”

May combine two claims.

The vertices of BF have always five edges and ten faces, while vertices of GR have (at least with the ordering considered so far) four edges and an arbitrary number of faces.

“The vertices of BF have always five edges and ten faces, while vertices of GR have (at least with the ordering considered so far) four edges and an arbitrary number of faces.”

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

The matrix element of the hamiltonian between the two partial spin networks is the 15j symbol of the 15 colors associated to the four links and the one node of the incoming spin network, and the six other links and four other nodes of the newly created tetrahedron.

“The matrix element of the hamiltonian between these two (partial) spin networks is the 15j symbol of the 15 colors associated to the four links and the one node of the incoming spin network, and the six other links and four other nodes of the newly created tetrahedron.”

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

The operator U is a key quantity for the theory; computing it virtually amounts to solving the quantum constraints, including the hamiltonian constraint, which codes all the dynamics of the theory.

“The operator U is a key quantity for the theory. Computing it virtually amounts to solving the quantum constraints, including the hamiltonian constraint, which codes all the dynamics of the theory.”

• AssumptionThe authors state this as an assumption: The potential singularities in the proper time slicing can be ignored for the purposes of this paper.

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

The scalar product 〈 s | U | s ′ 〉 defines the physical scalar product of the theory.

“Second, the scalar product 〈 s | U | s ′ 〉 defines the physical scalar product of the theory.”

• AssumptionThe authors state this as an assumption: The potential singularities in the proper time slicing can be ignored for the purposes of this paper.

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

U is the projector on the physical state space of the theory.

“First of all, it is the projector on the the physical state space of the theory.”

• AssumptionThe authors state this as an assumption: The potential singularities in the proper time slicing can be ignored for the purposes of this paper.

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

U is just the integral over proper time of the multifingered proper time evolution operator (37).

“Thus, U is just the integral over proper time of the multifingered proper time evolution operator (37).”

• AssumptionThe authors state this as an assumption: The potential singularities in the proper time slicing can be ignored for the purposes of this paper.

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

The proper time propagator P ( s f , s i ; T ) codes the dynamics of the theory, but it has no direct physical meaning: only after integration over proper time we obtain a quantity that we can, in principle, compare with experiments.

“A simple answer is that P ( s f , s i ; T ) codes the dynamics of the theory, but it has no direct physical meaning: only after integration over proper time we obtain a quantity that we can, in principle, campare with experiments.”

• AssumptionThe authors state this as an assumption: The potential singularities in the proper time slicing can be ignored for the purposes of this paper.

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

The construction generalizes to a multifingered proper time.

“The construction generalizes to a multifingered proper time.”

• AssumptionThe authors state this as an assumption: The potential singularities in the proper time slicing can be ignored for the purposes of this paper.

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

The proper time propagator is the first step toward the computation of the physical evolution operator U, in the same sense as the Feynman-Nambu proper time propagator.

“First of all, the proper time propagator is the first step toward the computation of the physical evolution operator U , in the same sense as the Feynman-Nambu proper time propagator.”

• AssumptionThe authors state this as an assumption: The potential singularities in the proper time slicing can be ignored for the purposes of this paper.

May combine two claims.

The discreteness of the spectra of area and volume is a central result in loop quantum gravity, first obtained by Rovelli and Smolin in [20], and later confirmed and clarified by a number of authors [21-25].

“The discreteness of the spectra of area and volume -and the 'quantized' structure of space that these spectra suggests- is a central result in loop quantum gravity, first obtained by Rovelli and Smolin in [20], and later confirmed and clarified by a number of authors [21-25].”

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

Matrix elements of the hamiltonian constraint operator have been computed explicitly in [34], using the methods developed in [23].

“Matrix elements of this operator have been computed explicitly in [34], using the methods developed in [23].”

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May combine two claims.

Slicing by equal proper time hypersurfaces generally develops singularities because the geodesics that define the proper time intersect, causing the canonical evolution to break down.

“Generally, slicing by equal proper time hypersurfaces develops singularities, because the geodesics that define the proper time intersect. This causes the canonical evolution to break down”

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

For trivalent nodes, S_v is a tetrahedron: the function A() must respect tetrahedral invariance.

“For instance, for trivalent nodes, S v is a tetrahedron: the function A ( ) must respect tetrahedral invariance.”

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

One can discretize general relativity over a simplicial lattice, and express the gravitational degrees of freedom as colored branched surfaces over the (dual) twoskeleton.

“It is shown in [9] that one can discretize general relativity over a simplicial lattice, and express the gravitational degrees of freedom as colored branched surfaces over the (dual) twoskeleton.”

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

The PRTV model can be seen as a quantization of 3d GR, or a quantization of 3d Chern-Simon theory.

“The OCY model is a 4d generalization of the Ponzano-Regge-Turaev-Viro (PRTV) model [12,13], which, in turn, can be seen as a quantization of 3d GR, or a quantization of 3d Chern-Simon theory.”

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Classified as probably not a substantial claim.

The orderings of the hamiltonian constraint studied so far generate a dynamical evolution that appears excessively 'local'; they preserve the general structure of the network on which they act, simply 'dressing' nodes.

“It has been observed [55] that the orderings of the hamiltonian constraint studied so far generate a dynamical evolution that appear to be excessively 'local'. They preserve the general structure of the network on which they act, simply 'dressing' nodes.”

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

The PRTV model is a theory of the dynamics of spin networks having the same physical interpretation as the spin network basis states in continuum 3d GR.

“In [14] it was shown that the PRTV model is a theory of the dynamics of spin networks (loops in the terminology of [14]) having the same physical interpretation as the spin network basis states in continuum 3d GR.”

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May combine two claims.

One consequence of adding the new vertices is that the faces of the surfaces in the sum do not need anymore to be topologically two-disks, and there is no sense in the requirement of the surfaces being well-ordered.

“Also, we note that one of the consequences of adding the new vertices is that the faces of the surfaces in the sum do not need anymore to be (topologically) two-disks, as followed from the original Hamiltonian expansion, and there is no sense in the requirement of the surfaces being well-ordered.”

May combine two claims.

This function is universal, and characterizes general relativity, in the same manner in which the Feynman vertex factor characterizes a QFT.

“This function is universal, and characterizes general relativity, in the same manner in which the Feynman vertex factor characterizes a QFT.”

May combine two claims.

In string theory, the contribution of each surface to the sum is given by the area of the surface, and therefore it depends on a fixed background metric on the manifold.

“In string theory, the contribution of each surface to the sum is given by the area of the surface, and therefore it depends on a fixed background metric on the manifold.”

• cites [7] sourceNot checked yet

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May combine two claims.

BF theory and simplicial GR are theories with crossing symmetric vertices.

“BF theory [11] and simplicial GR [8,9] are theories of this kind, and have crossing symmetric vertices.”

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Probably not a substantial claim: may not pass S1 (asserted).

No long range interaction is likely to emerge from a hamiltonian with these features.

“These difficulties have been recently detailed in Ref. [6], where it is argued that no long range interaction is likely to emerge from a hamiltonian with these features.”

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

Divergences can arise in summing the series.

“Divergences can arise in summing the series, and in integrating over proper time.”

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

The (2,2) term gives a rearranging of two nodes.

“The (2,2) term (third line of Figure 8) gives a rearranging of two nodes.”

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

Divergences can arise in integrating over proper time.

“Divergences can arise in summing the series, and in integrating over proper time.”

May combine two claims.

The integral over 4-geometries is replaced here by a sum, and explicit computation can be performed.

“The integral is replaced here by a sum, and explicit computation can be performed.”

May combine two claims.

The simplicial construction corresponds to a cut of the sum (56) in two respects: the maximum number of vertices is fixed by the triangulation, and vertices have 10 faces at most.

“Thus, the simplicial construction corresponds to a cut of the sum (56) in two respects: the maximum number of vertices is fixed by the triangulation, and vertices have 10 faces at most.”

May combine two claims.

The EEF ordering of the hamiltonian constraint has non-vanishing action on the vacuum and can create a small tetrahedron.

“Reverse this order, choosing EEF , and have the small loop being inserted first. Then the hamiltonian constraint has non-vanishing action on the vacuum as well, because the grasping term can grasp the 'small loop'. In particular, this may create a 'small' tetrahedron.”

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

Our expression for the proper time propagator U(T) of GR as a sum over worldsheets resembles in many ways the worldsheet sum for the projector on physical states, U_OCY, of the OCY model.

“More specifically, our expression for the proper time propagator U ( T ) of GR as a sum over worldsheets resembles in many ways the worldsheet sum [8,56] 14 for the projector on physical states, U OCY , of the OCY model.”

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Classified as probably not a substantial claim.

In canonical loop quantum gravity, the area of a surface S pierced by a single link with color p = 2j is given by the formula (59).

“recall that according to canonical loop quantum gravity the colors of the spin networks are quanta of area: the area of a surface S pierced by a single link with color p = 2 j is [20]”

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

The OCY model is a 4d generalization of the Ponzano-Regge-Turaev-Viro (PRTV) model.

“The OCY model is a 4d generalization of the Ponzano-Regge-Turaev-Viro (PRTV) model [12,13], which, in turn, can be seen as a quantization of 3d GR, or a quantization of 3d Chern-Simon theory.”

• cites [12] sourceNot checked yet

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May combine two claims.

The action of GR is local and 4-d diff invariant, and therefore the action of the 4-geometry of a small region ('the vertex') is independent of how this region is sliced by equal time slices.

“The action of GR is local and 4-d diff invariant, and therefore the action of the 4-geometry of a small region ('the vertex') is independent of how this region is sliced by equal time slices.”

• AssumptionRests on an assumption: The action of general relativity is local and 4-d diffeomorphism invariant.

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

The structure of quantum general relativity in the form presented in this paper is surprisingly similar to the Ooguri-Crane-Yetter (OCY) four dimensional topological quantum field theory.

“The structure of quantum general relativity in the form presented in this paper is surprisingly similar to the OoguriCrane-Yetter (OCY) four dimensional topological quantum field theory [10,11] a rigorously defined simplicial lattice version of four dimensional SU (2) BF theory.”

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

An arbitrary crossing symmetric vertex is obtained by replacing (60) with a function A(S_v) of the spin network S_v associated to the vertex.

“An arbitrary crossing symmetric vertex is obtained by replacing (60) with a function A ( S v ) of the spin network S v associated to the vertex”

• AssumptionThe authors state this as an assumption: Crossing symmetry is a valid requirement for the theory.

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

The 4d OCY model, like the 3d PRTV model, but unlike 4d GR, has no local degrees of freedom.

“On the other hand, the 4d OCY model like the 3d PRTV model, but unlike 4d GR has no local degrees of freedom, so one also expects large differences between the theories.”

Classified as probably not a substantial claim.

If the results of the canonical theory can be applied, then a surface σ assigns a (possibly vanishing) area to each triangle of the triangulation T.

“If we make this assumption, then we can say that the area of S is A ( S ) given in (59). Therefore, a surface σ assigns a (possibly vanishing) area to each triangle of the triangulation T .”

• AssumptionThe authors state this as an assumption: The results of the canonical theory can be applied to the intersection between a 3d hypersurface Σ and the colored surface σ to determine area.

Classified as probably not a substantial claim.

In BF theory, triangulation independence holds: refining the triangulation does not change the overall sum.

“In the case of BF theory there is a crucial theorem holding: triangulation independence. Refining the triangulation does not change the overall sum.”

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

Both quantum GR and quantum BF theory are invariant under diffeomorphism.

“Both theories are invariant under diffeomorphism.”

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

The worldsheet amplitudes in the BF theory considered are the amplitudes in the projector on physical states, while the GR worldsheet amplitudes of the present paper are from the sum for the proper time propagator U(T).

“First of all, the worldsheet amplitudes in the BF theory that we are considering are the amplitudes in the projector on physical states, while the GR worldsheet amplitudes of the present paper are from the sum for the proper time propagator U ( T ), so we might be comparing apples and oranges.”

May combine two claims.

Diff invariance of the sum is implemented in two different ways in BF and in GR: in BF, invariance is obtained thanks to triangulation independence; in GR, invariance is obtained by summing over arbitrarily fine triangulations.

“Therefore: Diff invariance of the sum is implemented in two different ways in BF and in GR, corresponding to the fact that BF is topological, while GR is not: in BF, invariance is obtained thanks to triangulation independence; in GR invariance is obtained by summing over arbitrarily fine triangulations.”

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

The elementary vertex of BF theory 'opens up' a four-valent intersection of a spin-network into a 'small' tetrahedron.

“A moment of reflection shows that the elementary vertex of BF theory 'opens up' a four-valent intersection of a spin-network into a 'small' tetrahedron:”

May combine two claims.

In GR, one must sum over arbitrarily complicated surfaces, or equivalently sum over arbitrarily fine triangulations of the manifold as well.

“In the case of GR we have to sum over arbitrarily complicated surfaces, or, equivalently, sum over arbitrarily fine triangulations of the manifold as well.”

May combine two claims.

In both the OCY/BF construction and the GR construction, there is a sum over the same kind of branched colored surfaces, and the weight for each surface is the product of vertex factors, where vertex factors are simple SU(2) invariant functions of the adjacent colorings.

“In both cases, we have a sum over the same kind of branched colored surfaces, and the weight for each surface is the product of vertex factors, where vertex factors are simple SU (2) invariant functions of the adjacent colorings.”

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

GR is expected to have an infinite number of degrees of freedom.

“We expect GR to have an infinite number of degrees of freedom.”