Spacetime as a Feynman diagram: the connection formulation

The quote only partly states this claim.

The formal perturbation series of the expectation value of Θ* equals the sum over the set A_Γ of admissible 2-complexes J bounded by Γ of the overlap of the state θ with the Hartle-Hawking vacuum of the spin foam model on J, with symmetry factors.

“The formal perturbation series of the expectation value of Θ ∗ ,”

• Also stated in: § Theorem · p. 11 · plain

  “the sum over the set A Γ of admissible 2-complexes J bounded by Γ of the overlap of the state θ with the Hartle-Hawking vacuum of the spin foam model on J .”

May combine two claims.

The interaction term (41) is A times the result of contracting together the indices of five c* in the pattern of a 4-simplex; it can be obtained from (37) by replacing intertwiners W_I by [c_I]* and setting all parallel transporters to 1.

“It's just A times the result of contracting together the indices of five c ∗ in the pattern of a 4-simplex. More precisely it can be obtained from (37) by replacing the intertwiners W I by [ c I ] ∗ and setting all the parallel transporters to 1 .”

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

The only flat connections on Γ5 are the trivial connection and its gauge transforms.

“The only such connections are the trivial connection and its gauge transforms”

May combine two claims.

For the simplest analog integral f(λ), the coefficients in its formal power series expansion about λ=0 diverge as (n!)^{3/2} for large n, and the integral converges for purely imaginary λ.

“The coefficients in the formal power series expansion of f about λ = 0 also diverge as ( n !) 3 / 2 for large n and, like (44) the integral converges for purely imaginary λ .”

May combine two claims.

Adopting r_ij = r*_ji for i>j, the lower indices of intertwiners in (25) can be raised using ε; the resulting tensors W^{m_k...}_{...I_k} are invariant and thus intertwiners for incident representations r_ki, and they form an orthonormal basis of such intertwiners.

“The lower indices of the intertwiners appearing in (25) have been raised using ε . The invariance of ε implies that the resulting tensors W m k. I k are also invariant and thus intertwiners for the incident representations r ki . It is easy to verify that they form an orthonormal basis of such intertwiners.”

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

Other closed surfaces can also contribute singular factors.

“Other closed surfaces can also contribute singular factors 17”

The quote only partly states this claim.

The corresponding propagator is given by formula (33) with δ^b_a = ∏_{t=1}^4 δ^{b_t}_{a_t}.

“The corresponding propagator is 19”

• Also stated in: § 3.3 The field theory in terms of spin foams · p. 21 · plain

  “with δ b a = ∏ 4 t =1 δ b t a t .”

May combine two claims.

The factor dim r_y results from the factors of ε(r_ij)_{x_ij x_ji} in (38) and (41) associated to each wedge; when orientations are changed to match 2-cells these are replaced by Kronecker deltas, and when propagators for b are substituted, the Kronecker deltas associated to wedges in a given 2-cell contract in a chain around the 2-cell, contributing a factor dim r_y.

“This last factor results from the factors of ε ( r ij ) x ij x ji that appear in (38) and (41), associated to each wedge. When the orientations of the edges are changed to match those of the 2-cells these are replaced by Kronecker deltas. Then, when the propagators for b are substituted in, the Kronecker deltas associated to wedges in a given 2-cell end up being contracted in a chain around around”

May combine two claims.

In BF theory, the weight w of a connection has redundant delta distributions and is infinite whenever J contains topological 2-spheres.

“For instance, in the model of BF theory that we have discussed the weight w of a connection has redundant delta distributions, and is thus infinite, whenever J contains topological 2-spheres.”

May combine two claims.

The free action can be expressed as I0[ψ] = 1/2·4! ∫ G^4 ψ* ψ d^4 h, valid because ψ is real; using (24) yields formula (31).

“The free action is easily expressed in terms of these coefficients if we write it as I 0 [ ψ ] = 1 2 · 4! ∫ G 4 ψ ∗ ψ d 4 h . This is valid because ψ is real. We find, using (24), that”

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

Using the reality of ψ to replace ψ by ψ* in (4), the integrals can be carried out using (24) and yield formula (38) for V.

“Evaluating V is now straightforward. If we use the reality of ψ to replace ψ by ψ ∗ in (4) the integrals can be carried out using (24) and we obtain”

May combine two claims.

With V(g) real, V_{n1...n5} is real and the integral (44) is convergent when λ is pure imaginary; it is a Gaussian times a function of modulus O(|x|^m).

“For simplicity we will suppose that V ( g ) is real, as is the case in BF theory, and in the models of [6] and [27], then also V n 1 ...n 5 is real and we see that the integral is convergent when λ is pure imaginary - the integral is a Gaussian times a function of modulus O ( | x | m ).”

• cites [6] Reisenberger M, 1994Not checked yet

  Not checked in the free run.

• cites [27] Rovelli C, 1995Not checked yet

  Not checked in the free run.

• AssumptionThe authors state this as an assumption: V(g) is real, as is the case in BF theory and in the models of [6] and [27].

May combine two claims.

(34) determines ε(r) up to phase; consequently ε(r*)_{mn} equals ε(r)_{nm} up to a phase factor, and when r=r*, ε(r) is either purely symmetric or antisymmetric.

“(34) determines ε ( r ) up to phase. Consequently ε ( r ∗ ) mn equals ε ( r ) nm up to a phase factor, and thus when r = r ∗ , ε ( r ) is either purely symmetric or antisymmetric.”

May combine two claims.

The number of admissible 2-complexes increases very rapidly with the number of atoms; the number of complexes of n atoms with a given boundary Γ having m 4-valent nodes is given approximately by formula (43).

“The number of admissible 2-complexes increases very rapidly with the number of atoms. In fact the number of complexes of n atoms with a given boundary Γ having m 4-valent nodes is given approximately by”

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

There exists a unitary matrix ε(r)_{mn} such that relation (34) holds, and ε(r)^{mn} = [ε(r)_{mn}]^* is its inverse in the sense ε(r)_{mx} ε(r)^{nx} = δ_n^m.

“We therefore introduce ε ( r ) mn , a unitary matrix such that”

• Also stated in: § 3.3 The field theory in terms of spin foams · p. 22 · plain

  “and let ε ( r ) mn = [ ε ( r ) mn ] ∗ , which is the inverse of ε ( r ) mn in the sense that ε ( r ) mx ε ( r ) nx = δ n m .”

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

ψ*=ψ and (34) imply the reality conditions satisfied by coefficients b, given by formula (42) with ε(r)^{mn}=∏_{t=1}^4 ε(r_t)^{m_t n_t}.

“ε also allows us to express the reality conditions satisfied by the coefficients b : ψ ∗ = ψ and (34) implies that”

• Also stated in: § 3.3 The field theory in terms of spin foams · p. 24 · plain

  “(with ε ( r ) mn = ∏ 4 t =1 ε ( r t ) m t n t ).”

May combine two claims.

(37) is a spin network expansion of V with bivalent nodes assigned the intertwiners ε(r_ij)_{x_ij x_ji}, which is sqrt(dim r_ij) times the unique normalized bivalent intertwiner; orientations of edges manifest only in the ordering of indices in these εs.

“(37) is thus a spin network expansion of V with the bivalent nodes assigned the intertwiners ε ( r ij ) x ij x ji (which is √ dimr ij times the unique (mod phase) normalized bivalent intertwiner). In (37) the orientations of the edges manifest themselves only in the ordering of the indices in these ε s.”

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

For BF theory on a triangulated manifold, this means dividing out a factor δ(0)_regulated = [∑_r (dim r)^2]_regulated for each independent 2-sphere in J.

“For BF theory on a triangulated manifold this means dividing out a factor δ (0) regulated = [ ∑ r ( dimr ) 2 ] regulated for each independent 2-sphere in J .”

May combine two claims.

The first step in the argument for f can be repeated for the regulated functional integral (44); the resulting integral converges for Rλ≤0.

“The first step in this argument can be repeated for the regulated functional integral (44).”

• Also stated in: § 4.1 On the possible finiteness of the field theory corresponding to regulated spin foam models · p. 27 · plain

  “Again we find that the integral converges for ℜ λ ≤ 0.”

May combine two claims.

V is simpler when expressed in terms of spin network coefficients in an expansion of the gauge invariant field ψbar(h)=∫G ψ(gh)dg; inserting projector (39) onto invariant r1⊗...⊗r4 tensors gives expansion (40) with c(r)_I n = b(r)_m n [W(r)^m_I]^*, and in terms of c the interaction term is (41).

“V is even simpler when expressed in terms of the spin network coefficients in an expansion of the 'gauge invariant' field ¯ ψ ( h ) = ∫ G ψ ( g h ) dg . A spin network type expansion of this field can be obtained by inserting into the expansion (30) of ψ the projector”

• Also stated in: § 3.3 The field theory in terms of spin foams · p. 24 · plain

  “with c ( r ) I n = b ( r ) m n [ W ( r ) m I ] ∗ . In terms of the coefficients c the interaction term is”

May combine two claims.

The triangular matrix model is recovered by choosing the vertex function V(g_ij)= A (dim r)^{3}/3! tr U(r)(g12 g23 g31) so that only one representation r appears in its spin network expansion; then V = A/3! c(r)_{n1 n2} c(r)_{n2 n3} c(r)_{n3 n1}.

“The triangular matrix model is recovered by choosing the vertex function to be V ( g ij ) = A ( dimr ) 3 / 3! tr U ( r ) ( g 12 g 23 g 31 ) so that only one representation r appears in its spin network expansion. Then V = A/ 3! c ( r ) n 1 n 2 c ( r ) n 2 n 3 c ( r ) n 3 n 1 .”

May combine two claims.

Even with the above generalization all atoms have spokes of valence four and all bounding graphs of atoms and complexes have non-trivial nodes of valence four only.

“Even with the above generalization all atoms have spokes of valence four and thus all bounding graphs of atoms and of complexes have non-trivial nodes of valence four only.”

May combine two claims.

When the formalism is applied to two-dimensional models with atoms dual to triangles, the analog of c(r)_I n is c(r1)_{n1 n2} ≡ b(r1,r2)_{m1 m2 n1 n2} (1/√dim r1) δ_{r1 r*2} ε(r1)_{m1 m2}, with r1 required to equal r*2 because r1⊗r2 contains the trivial representation only in this case; ε(r)/√dim r is the only normalized bivalent intertwiner.

“When our formalism is applied to two dimensional models with atoms dual to triangles we find that the analog of c ( r ) I n is c ( r 1 ) n 1 n 2 ≡ b ( r 1 ,r 2 ) m 1 m 2 n 1 n 2 1 √ dimr 1 δ r 1 r ∗ 2 ε ( r 1 ) m 1 m 2 ( r 1 has to equal r ∗ 2 because r 1 ⊗ r 2 contains the trivial representation only in this case. ε ( r ) / √ dimr is the only normalized bivalent intertwiner).”

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

Such a field theory was recently found for a particular gravity model.

“Such a field theory was recently found for a particular gravity model [1].”

• cites [1] sourceNot checked yet

  Not checked in the free run.

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

Several spin foam models of gravity have been proposed, most of which live on finite simplicial lattice spacetime.

“In the last few years several spin foam models of gravity have been proposed, most of which live on finite simplicial lattice spacetime.”

• cites [5] Iwasaki J, 1994Not checked yet

  Not checked in the free run.

• cites [6] Reisenberger M, 1994Not checked yet

  Not checked in the free run.

• cites [26] sourceNot checked yet

  Not checked in the free run.

Classified as probably not a substantial claim.

A model in this connection formalism is nothing other than a lattice gauge theory defined on an unusual lattice in order to bring boundary data and locality to the fore.

“A model in this connection formalism is nothing other than a lattice gauge theory defined on an unusual lattice in order to bring boundary data and locality to the fore.”

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

The lattice truncates the presumably infinite set of gravitational degrees of freedom down to a finite set.

“The lattice truncates the presumably infinite set of gravitational degrees of freedom down to a finite set.”

• AssumptionRests on an assumption: Four dimensional gravity models have infinitely many degrees of freedom in any given spacetime topology.

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

Three dimensional general relativity as well as four dimensional BF theory can also be given a spin foam formulation.

“Three dimensional general relativity [15, 3, 25] as well as four dimensional BF theory [17, 18, 4] can also be given a spin foam formulation.”

• cites [15] Gambini R, 1997Not checked yet

  Not checked in the free run.

• cites [3] Reisenberger M, 2000Not checked yet

  Not checked in the free run.

• cites [25] Zapata J A, 1997Not checked yet

  Not checked in the free run.

May combine two claims.

The work generalizes the result for a particular gravity model, Boulatov's and Ooguri's models of three and four dimensional topological field theories, and the old matrix models of two dimensional systems with dynamical topology.

“Our work generalizes this result as well as Boulatov's and Ooguri's models of three and four dimensional topological field theories, and ultimately the old matrix models of two dimensional systems with dynamical topology.”

• cites [1] sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The atomic boundary is a graph in the boundary of the simplex so the connection on it is a discrete approximation to the continuum connection on the boundary of the 4-simplex.

“The atomic boundary is a graph in the boundary of the simplex so the connection on it is a discrete approximation to the continuum connection on the boundary of the 4-simplex.”

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

If the model is a simplicial approximation to a spacetime manifold field theory, then V is an approximation to exp(i [Effective action in 4-simplex]) in terms of the connection on the boundary of the atom in the 4-simplex.

“If the model is a simplicial approximation to a spacetime manifold field theory, as all the gravity models aim to be, then V is an approximation to exp( i [Effective action in 4-simplex]) in terms of the connection on the boundary of the atom in the 4-simplex.”

Classified as probably not a substantial claim.

Simplicial spin foams live on the 2-skeleton of the cellular complex dual to the simplicial complex.

“Simplicial spin foams live on the 2-skeleton of the cellular complex dual to the simplicial complex.”

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

Many of the underlying ideas go back to the work of Boulatov on three dimensional gravity and to matrix models of two dimensional theories.

“Many of the underlying ideas go back to the work of Boulatov [33], on three dimensional gravity, and to matrix models of two dimensional theories [34].”

• cites [33] Crane L, 1993Not checked yet

  Not checked in the free run.

• cites [34] Crane L, 1993Not checked yet

  Not checked in the free run.

Classified as probably not a substantial claim.

The dual 2-skeleton is spacetime as far as spin foams are concerned.

“The dual 2-skeleton, the two dimensional complex formed by these cells (together with the dual 0cells), thus is spacetime as far as spin foams are concerned.”

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

In a spin network basis, the representations U(r) placed on edges are particular representatives chosen one out of each unitary equivalence class, and the same set of representatives is used on all edges.

“Now recall that in a spin network basis the representations U ( r ) that may be placed on edges are particular representatives chosen one out of each unitary equivalence class. Moreover, in order to keep the definition of the spin network basis simple we shall use the same set of representatives on all edges.”

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

One can also simply cut off the sum over representations, leaving the model otherwise unchanged.

“One can also simply cut off the sum, leaving the model otherwise unchanged.”

May combine two claims.

The spin network basis is not quite unique; conventions have to be introduced to define a particular basis.

“The difficulty has to do with the fact that the spin network basis is not quite unique. Some conventions have to be introduced to define a particular basis.”

May combine two claims.

To express the field theory in spin foam terms, one must choose a particular spin network basis on the prototypical atomic boundary Γ5, which is then the basis on every atomic boundary in complexes generated by the field theory; this inflexibility complicates calculation of Feynman diagram amplitudes.

“On the other hand, in order to express our field theory in spin foam terms, which boils down to expressing it in terms of the coefficients in spin network expansions of V , θ , and ψ , we have to choose a particular spin network basis on the prototypical atomic boundary Γ 5 . This is then the basis on every atomic boundary in the complexes generated by the field theory. The resulting inflexibility”

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

The parameters can be chosen as real linear combinations {x_p} such that I0[ψ]=x^T x.

“These paramenters can be the b ( r ) m n , but we will use a set of linear combinations { x p } N p =1 of these chosen so that they are real and I 0 [ ψ ] = x T x .”

May combine two claims.

In the spin foam sum, infinities take the form of a divergence of the sum over representations and can be regulated by cutting off this sum.

“In the spin foam sum the infinities take the form of a divergence of the sum over representations and can thus be regulated by somehow cutting off this sum.”

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

All representations of SU(2) are equivalent to their conjugates, but unitary irreducible representations of non-integer spin are not real and so are not equal to their conjugates.

“For instance, all representations of SU (2) are equivalent to their conjugates, but unitary irreducible representations of non-integer spin are not real and so are not equal to their conjugates.”

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

When translating from an integral over connections to a sum over spin foams on a particular given 2-complex, one can choose spin network bases on the atomic boundaries so as to simplify calculations on that particular 2-complex.

“When translating from an integral over connections to a sum over spin foams on a particular, given, 2-complex one can choose the spin network bases on the atomic boundaries so as to simplify the calculations on that particular 2-complex.”

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

The spin foam formulation is conjugate to the connection formulation in the sense that if a connection is a history of the configuration variables of the model a spin foam is essentially a history of the momentum variables.

“The spin foam formulation is conjugate to the connection formulation in the sense that if a connection is a history of the configuration variables of the model a spin foam is essentially a history of the momentum variables”

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

Any local lattice gauge theory with compact gauge group G can be given a spin foam formulation.

“Any local lattice gauge theory with compact gauge group G can be given a spin foam formulation.”

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

Once the spin foam model is regulated by cutting off sums over representations, it is not unreasonable to hope that integral (6) can be assigned a finite value by analytic continuation.

“In fact it is not unreasonable to hope that once the spin foam model is regulated by cutting off the sums over representations (as we discussed in the context of BF theory) that the integral (6) can be assigned a finite value by analytic continuation.”

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

When r=r*, one is free to choose the representative of the equivalence class of the conjugate representation, and therefore free to set ε(r)=ε(r*) for all r.

“When r = r ∗ one is of course free to choose the representative of the equivalence class of the conjugate representation as one likes. One could simply choose the conjugate representation itself and use ε ( r ) mn = δ mn . We are therefore free to set ε ( r ) = ε ( r ∗ ) for all r .”

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

This is the origin of the factor δ(0)^{N0-N1}_regulated in the weight of a history in simplicial BF theory.

“This is the origin of the factor δ (0) N 0 -N 1 regulated (with N 0 the number of 0simplices and N 1 the number of 1simplices in the triangulation) in the weight of a history in simplicial BF theory [18].”

• cites [18] Rovelli C, 1998, Living Rev. Rel.Not checked yet

  Not checked in the free run.

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

In lattice models of topological theories or Yang-Mills theories, the continuum limit theory is usually defined by limiting values of observables as the lattice is refined, not by a sum over lattices.

“In lattice models of topological theories, or Yang-Mills theories for that matter, the continuum limit theory is usually defined not by a sum over lattices but by the limiting values of the observables of the theory as the lattice is refined.”

Classified as probably not a substantial claim.

A representation matrix of the parallel transporter along an edge is a two-point tensor with one index living at each end of the edge.

“Notice that a representation matrix of the parallel transporter along an edge is a two point tensor, with one index living at each end of the edge.”

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

Renormalization is more subtle in the context of a general admissible 2-complex because the spheres do not necessarily span all the closed 2-surfaces, for instance a 2-complex can have a torus as its only closed surface.

“Renormalization is more subtle in the context of a general admissible 2-complex because the spheres do not necessarily span all the closed 2-surfaces (for instance one can make a 2-complex having a torus as it's only closed surface).”

May combine two claims.

Each spin network on the boundary of an atom defines a spin foam on the body of the atom, with the intertwiner at a 4-valent node becoming the spin foam intertwiner on the spoke to the central 0-cell and the representation on the chain between nodes defining the representation on the wedge bounded by the chain.

“Each spin network on the boundary of an atom defines a spin foam on the body of the atom: The intertwiner I i on the 4-valent node i is the spin foam intertwiner on the spoke connecting the node with the 0-cell at the center of the atom; The representation r ij on the chain connecting nodes i and j defines the representation on the wedge of the atom bounded by the chain.”

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

The same prescription can be applied to θ to express Θ in terms of the coefficients c*.

“This prescription can also be applied to θ to express Θ in terms of the coefficiants c ∗ .”

May combine two claims.

This number increases as (n!)^{3/2} for large n, so unless the vertex function V is very special indeed the radius of convergence of series (7) is zero.

“This increases as ( n !) 3 / 2 for large n , so unless the vertex function V is very special indeed the radius of convergence of the series (7) is zero.”

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

Any model having a finite set of types of atoms can be handled by these means, including models with spacetimes of any dimensionality to the extent that this dimensionality can be captured in the combinatorial structure of the atoms.

“Any model having a finite set of types of atoms can be handled by these means, including of course models with spacetimes of any dimensionality to the extent that this dimensionality can be captured in the combinatorial structure of the atoms.”

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

The theory may be generalized to accommodate other valences q by introducing additional fields depending on q group elements instead of four like ψ does, and adding a quadratic free action to the total action for each such field.

“The theory may be generalized to acomodate other valences q by introducing additional fields depending on q group elements instead of four like ψ does, and adding a quadratic free action to the total action for each such field.”

Classified as probably not a substantial claim.

If F(λ) is continuable then it will be defined and finite on an open region in the complex plane, possibly including λ=1, the value corresponding most closely to a simple sum of the spin foam model over all admissible 2-complexes.

“If it is continuable then F ( λ ) will be defined and finite on an open region in the complex plane, possibly including λ = 1 which is the value corresponding most closely to a simple sum of the spin foam model over all admissible 2-complexes.”