LQG propagator from the new spin foams

The two-point correlation function G^abcd_nm can be written as a ratio of group integrals with insertions q^ab_n q^cd_m e^S.

“Substituting ( 44)-( 45) in (36) we obtain a new expression for the propagator in terms of group integrals:”

There is a unique stationary point of the total action for which Re S_tot vanishes.

“Now we study the stationary points of the total action and show that there is a unique stationary point for which Re S tot vanishes.”

Given that the matrix α in the boundary state has positive definite real part, the real part of the total action is negative or vanishing, Re S_tot ≤ 0.

“Therefore, having assumed that the matrix α in the boundary state has positive definite real part, we have that the real part of the total action is negative or vanishing, Re S tot 0.”

• AssumptionThe authors state this as an assumption: The 10×10 matrix α is complex with positive definite real part and has the symmetries of a regular 4-simplex.

At the lowest order in the vertex expansion, the boundary amplitude of a spin network state with graph Γ5 is μ(j_ab) W_v(j_ab, i_a), involving a single spin foam vertex.

“To this order, the boundary amplitude of a spin network state with graph Γ 5 is given by W |Γ 5 , j ab , i a = μ(j ab )W v (j ab , i a ), (33) i.e. it involves a single spin foam vertex.”

In the large λ limit, the sums over spins in expression (61) can be approximated with integrals over continuous spin variables, with error O(λ^{-N}) for all N > 0.

“In the large λ limit, the sums over spins in expression (61) can be approximated with integrals over continuous spin variables 9 :”

The function P_ab(g+,g-) factorizes as P_ab+(g+) P_ab-(g-).

“we have that the function P ab (g + , g -) factorizes as P ab (g + , g -) = P ab+ (g + )P ab-(g -) (40”

The total action S_tot = log ψ + S is homogeneous of degree one in the spins, so rescaling the spins by λ gives S_tot → λ S_tot.

“Notice that the action S + + S -is a homogeneous function of the spins j ab therefore, rescaling the spins j 0ab and j ab by an integer λ so that j 0ab → λj 0ab and j ab → λj ab , we have that the total action goes to S tot → λS tot .”

The boundary amplitude of a coherent spin network |j, Φ(n)> admits an integral representation over five copies of SU(2)×SU(2) with action S = S+ + S-.

“The boundary amplitude of a coherent spin network |j, Φ( n) admits an integral representation [32,34,35].”

• Also stated in: § 4.1. Integral formula for the amplitude of a coherent spin network · plain

  “W |j, Φ( n) = μ(j ) 5 a=1 dg + a dg - a e S (42”

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The expectation value of the metric operator E^a_c • E^b_c on a coherent spin network is approximately γ^2 j_ca j_cb n_ca • n_cb in the large spin limit.

“Γ 5 , j ab , Φ a |E a c • E b c |hΓ 5 , j ab , Φ a γ 2 j ca j cb n ca • n cb (15”

• Also stated in: § 2. Semiclassical boundary state and the metric operator · plain

  “in the large spin limit.”

The diagonal metric insertion is q^aa_n(g+,g-) = γ^2 j_na(j_na+1).

“q aa n (g + , g -) = γ 2 j na (j na + 1). (60”

For normals n_ab which define non-degenerate tetrahedra and satisfy the continuity condition (16), the equation R_a n_ab = -R_b n_ba admits two distinct sets of solutions, up to global rotations, related by parity.

“For normals n ab which define non-degenerate tetrahedra and satisfy the continuity condition (16), the equation R a n ab = -R b n ba admits two distinct sets of solutions, up to global rotations.These two sets are related by parity.”

At leading order the expectation value of the off-diagonal components is given by the dihedral angle between two faces of a tetrahedron: E_a^n • E_b^n = γ^2 j0na j0nb n_na(j0) • n_nb(j0) + O(j0).

“Therefore we have that at the leading order the expectation value of the off-diagonal components is given by the dihedral angle between two faces of a tetrahedron E a n • E b n = E a ncl • E b ncl + O(j 0 ) = γ 2 j 0na j 0nb n na (j 0 ) • n nb (j 0 ) + O(j 0 ).”

The evaluation of the action on the four classes of critical points gives S = +S_Regge, -S_Regge, +γ^-1 S_Regge, -γ^-1 S_Regge.

“S j ab , ḡ+ a , ḡ- a = +S Regge (j ab ), (76 ) S j ab , ḡ- a , ḡ+ a = -S Regge (j ab ), (77 ) S j ab , ḡ+ a , ḡ+ a = +γ -1 S Regge (j ab ), (78 ) S j ab , ḡ- a , ḡ- a = -γ -1 S Regge (j ab ), (79 )”

Despite 9×9 tensorial components, only six of them are independent as the others are related by symmetries of the configuration.

“Despite the fact that we have 9 × 9 tensorial components, only six of them are independent as the others are related by symmetries of the configuration we are considering.”

The LQG propagator has the correct scaling behaviour.

“We find that the LQG propagator has the correct scaling behaviour.”

All eigenvalues are positive except one, λ1.

“We notice that all the eigenvalues are positive except one, λ 1 .”

The expectation value of the metric operator at leading order is given by the evaluation of the insertion q_ab^n(g+,g-) at the critical point plus O(j0).

“Using the technique developed in the previous sections we can compute it at the leading order in the large spin expansion.We use the integral formula for the metric operator ( 44)-( 45) and the stationary phase analysis of Section 5 and find that the expectation value of the metric operator is simply given by the evaluation of the insertion q ab n (g + , g -) at the critical point E a n • E b n =”

In this limit there is a unique solution for α_k given by (119)-(121).

“In this limit we find a unique solution for α k given by”

The γ-terms contribution to angle-angle correlations is non-trivial.

“On the other hand, their contribution to angle-angle correlations is non-trivial.”

Expression (61) can be rewritten as an integral over d^10 j and d^4 g±, as in equation (64).

“As a result we can re-write expression (61) in the following integral form”

Besides being a stationary point, j_ab = j0_ab is also a critical point of the total action as stated in (72).

“Notice that, besides being a stationary point, this is also a critical point of the total action as stated in (72).”

For diagonal components a=b, the insertion is q_aa^n = (γ j_na)^2, so its evaluation at the critical point gives the area square of the triangle f_na.

“For the diagonal components a = b we have that the insertion is simply given by q aa n = (γj na ) 2 so that its evaluation at the critical point gives the area square of the triangle f na .”

The LQG propagator has the structure G = (γj0)^3 R + γ X + γ^2 Y + O(j0^2).

“In terms of these quantities we have that the LQG propagator has the following structure G abcd nm (α) = (γj 0 ) 3 R abcd nm (α) + γ X abcd nm + γ 2 Y abcd nm + O j 2 0 , (101”

Some off-diagonal components of the metric operator at a node do not commute.

“In particular, from property (ii), we have that some off-diagonal components of the metric operator at a node do not commute [39”

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The γ-terms come from SU(2)×SU(2) group fluctuations.

“They come from SU(2) × SU(2)”

In the limit γ→0 and j0→∞ with γj0 = const = A0, only the Regge contribution survives.

“In the limit γ → 0 and j 0 → ∞ with γj 0 = const = A 0 , only the Regge contribution survives.”

The eigenvalues of the real part of the matrix α are λ5=9216√3 (deg 5), λ4=4608√3/5 (deg 4), λ1=-1024√3/5 (deg 1).

“Its eigenvalues (with the associated degeneracy) are λ 5 = 9216 √ 3, deg = 5,(122) λ 4 = 4608 √ 3 5 , deg = 4,(123) λ 1 = - 1024 √ 3 5 , deg = 1. (124”

Only the matrix R depends on the three parameters α0, α1, α2 appearing on the boundary state.

“Only the matrix R abcd nm depends on the three parameters α 0 , α 1 , α 2 appearing on the boundary state.”

The semiclassical boundary state |Ψ0 used in the LQG calculation has only three free parameters, α0, α1, α2.

“On the other hand, the semiclassical boundary state |Ψ 0 we used in the LQG calculation has only three free parameters, α 0 , α 1 , α 2 .”

The three parameters α_k appearing in the semiclassical boundary state can be chosen so that the tensorial structure of the LQG propagator matches with the one of the graviton propagator.

“The three parameters α k appearing in the semiclassical boundary state can be chosen so that the tensorial structure of the LQG propagator matches with the one of the graviton propagator.”

The leading order of the two-point function is E_a•E_b E_c•E_d = classical product + O(j0^3).

“Using the same technique we can evaluate the leading order of the two-point function.We have that E a n • E b n E c m • E d m = E a n cl • E b n cl E c mcl • E d mcl + O j 3 0 .”

The first two classes of solutions in (75) glue five tetrahedra into two Euclidean non-degenerate 4-simplices related by a reflection.

“The first two classes of solutions in (75) glue five tetrahedra into two Euclidean non-degenerate 4-simplices related by a reflection”

Out of them, four classes of solutions for the couple (g+_a, g-_a) can be found, given by (75).

“Out of them, four classes of solutions for the couple (g + a , g - a ) can be found.They are given by”

The contribution R in (101) matches exactly with the matrix of correlations of areas and angles computed in perturbative quantum Regge calculus with a boundary state as done in [10].

“The contribution R abcd nm in (101) matches exactly with the matrix of correlations of areas and angles computed in perturbative quantum Regge calculus with a boundary state as done in [10].”

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The LQG propagator is given by Eq. (97) in terms of the inverse Hessian of the total action and derivatives of the metric operator insertions at the critical point.

“Using the integral formula ( 44)-( 45) and the result (69) for the asymptotics of connected two-point functions, we can compute the LQG propagator in terms of (the inverse of) the Hessian of the total action and of the derivative of the metric operator insertions at the critical point.”

Requiring the variation of the total action with respect to the group variables g± to vanish leads to two sets of equations, for the real and imaginary parts of the variation.

“The requirement that the variation of the total action with respect to the group variables g ± a vanishes, δ g S tot = 0, leads to the two sets of equations (respectively for the real and the imaginary part of the variation)”

A solution in terms of the parameters α_k can be found only in the limit of vanishing Immirzi parameter, keeping constant the product γj0 = A0.

“We find that a solution in terms of the parameters α k can be found only in the limit of vanishing Immirzi parameter, keeping constant the product γj 0 = A 0 .”

The same quantities computed with the Barrett-Crane spinfoam dynamics do not show the right behavior when the off-diagonal components of the metric operator are considered.

“We observe that the same quantities computed with the Barrett-Crane spinfoam dynamics do not show the right behavior when the off-diagonal components of the metric operator are considered.”

For the first class, Eq. (81) vanishes for j_ab = j0_ab.

“As the phase φ ab 0 in the boundary state is chosen to be exactly γ times the extrinsic curvature, we have that Eq. ( 81) vanishes for j ab = j 0ab .”

Besides the expected Regge contribution, correlations of LQG metric operators have a non-Regge contribution which is proper of the spin foam model.

“Another feature identified in this paper which appears to be general is that, besides the expected Regge contribution, correlations of LQG metric operators have a non-Regge contribution which is proper of the spin foam model.”

The matching is obtained in the limit γ→0 with γj0 fixed.

“The matching is obtained in the limit γ → 0 with γj 0 fixed.”

The γ-terms in (101) are new and proper of the spin foam model.

“On the other hand, the 'γ -terms' in (101), γ X abcd nm + γ 2 Y abcd nm , are new and proper of the spin foam model.”

The second two classes correspond to degenerate configurations with the 4-simplex living in the three-dimensional plane orthogonal to the chosen time direction.

“while the second two classes correspond to degenerate configurations with the 4-simplex living in the three-dimensional plane orthogonal to the chosen”

The Regge term coincides with the correlations of areas and angles computed in Regge calculus with a boundary state [10].

“The Regge term coincides with the correlations of areas and angles computed in Regge calculus with a boundary state [10].”

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The action, measure and insertions in expression (61) are invariant under an SO(4) symmetry that makes an integration dg+ dg- redundant.

“Moreover, notice that the action, the measure and the insertions in (61) are invariant under a SO(4) symmetry that makes an integration dg + dg -redundant.”

The leading-order asymptotic formula for the connected 2-point function is G = (1/λ) H^{-1}_{ij} g_i(x0) h_j(x0) + O(1/λ^2).

“Using (66) it is straightforward to show that the (leading order) asymptotic formula for the connected 2-point function is simply”

The LQG propagator has exactly the form of the connected 2-point function G built from integrals of type (65).

“The reason we are considering the quantity G, built from integrals of the type (65), is that the LQG propagator has exactly this form.”

Normals in different tetrahedra cannot be chosen independently if one wants to peak on a Regge geometry.

“Normals in different tetrahedra cannot be chosen independently if we want to peak on a Regge geometry [44].”

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The spin foam functional W|Γ5,j,i> can be expanded in a formal perturbative vertex expansion in the parameter λ of a Group Field Theory.

“The sum over two-complexes can be implemented in terms of a formal perturbative expansion in the parameter λ of a Group Field Theory [46]:”

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A semiclassical state cannot be an eigenstate of the area because it would not be peaked on a given extrinsic curvature.

“As the extrinsic curvature is the momentum conjugate to the intrinsic geometry, we have that a semiclassical state cannot be an eigenstate of the area as it would not be peaked on a given extrinsic curvature.”

A state with generic normals satisfying the closure condition is peaked on a discontinuous geometry.

“In fact, a state with generic normals (satisfying the closure condition (12)) is peaked on a discontinuous geometry.”

At the leading order in large j0, the LQG propagator is independent of the choice of face amplitude, namely of the function μ(j).

“We will show in the following that, at the leading order in large j 0 , the LQG propagator (3) is in fact independent from the choice of face amplitude, namely from the function μ(j ).”

• AssumptionThe authors state this as an assumption: The function μ(λj_ab) scales as λ^p for some p for large λ.

The Barrett-Crane model's failure to give the correct propagator scaling can be traced back to a missing coherent cancellation of phases between the intertwiner wave function of the semiclassical boundary state and the intertwiner dependence of the model.

“The problem can be traced back to a missing coherent cancellation of phases between the intertwiner wave function of the semiclassical boundary state and the intertwiner dependence of the model.”

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The assignment n_ab = n_ab(j_cd) guarantees that the geometry being peaked on is Regge-like.

“This assignment of normals guarantees that the geometry we are peaking on is Regge-like.”

The Barrett-Crane spin foam model fails to give the correct scaling behavior for off-diagonal components of the graviton propagator.

“The analysis of [12,13] shows that the Barrett-Crane model fails to give the correct scaling behavior for off-diagonal components of the graviton propagator.”

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Under the stated hypotheses, an asymptotic expansion of the integral F(λ) for large λ is available as an extension of the stationary phase approximation that accounts for a complex action.

“Under these hypothesis an asymptotic expansion of the integral F for large λ is available: it is an extension of the standard stationary phase approximation that takes into account the fact that the action S is complex [48].”

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• AssumptionThe authors state this as an assumption: There is a unique critical point in the asymptotic analysis.
• AssumptionThe authors state this as an assumption: The stationary points x0 of S are isolated and the Hessian at a stationary point is non-singular.
• AssumptionThe authors state this as an assumption: The real part of S is negative or vanishing, Re S ≤ 0.

For Euclidean signature and Immirzi parameter 0 < γ < 1, the EPRL γ and FK γ spin foam models coincide.

“We restrict attention to Euclidean signature and Immirzi parameter smaller than one: 0 < γ < 1.In this case the two models coincide.”

• Also stated in: § 3. The new spin foam dynamics · plain

  “We restrict attention to 0 < γ < 1; in this case the two models coincide.”

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Therefore the matching condition (118) can be satisfied, at least in the specific limit considered.

“Therefore the matching condition (118) can be satisfied, at least in the specific limit considered.”

From the non-commutativity of some off-diagonal metric operator components, a Heisenberg inequality for dispersions of metric operators follows.

“From this non-commutativity an Heisenberg inequality for dispersions of metric operators follows.”

The two group elements g+_a and g-_a of the action (43) define an SO(4) element which rotates the initial tetrahedron.

“The two group elements g + a and g - a of the action (43) define an SO(4) element which rotates the”

The system (73) is a gluing condition between tetrahedra.

“The system (73) is a gluing condition between tetrahedra.”

Because of the closure condition (12), for a fixed n the four bivectors (j na n na, j na n na) define an embedding of a tetrahedron in R^4.

“Because of the closure condition (12), for a fixed n the four bivectors (j na n na , j na n na ) define an embedding of a tetrahedron in R 4 .”

As γj0 can be identified with area A0, the LQG propagator and the perturbative graviton propagator have the same scaling.

“As we can identify γj 0 with the area A 0 , we have that the two have the same scaling.”

In the recoupling basis for intertwiner space, the off-diagonal components of the metric operator generally have non-trivial matrix elements.

“Using the recoupling basis for intertwiner space, we have that in general the off-diagonal components of the metric operator have non-trivial matrix elements”

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The eigenvector corresponding to λ1 represents conformal rescalings of the boundary state, j0ab → λj0ab.

“The corresponding eigenvector represents conformal rescalings of the boundary state, j 0ab → λj 0ab .”