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Cited by 241Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryBlack Holes and Theoretical Physics

Flipped spinfoam vertex and loop gravity

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Engle, Jonathan, Pereira, Roberto, Rovelli, Carlo·2008·Nuclear Physics B
DOI 10.1016/j.nuclphysb.2008.02.002
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Abstract

We introduce a vertex amplitude for 4d loop quantum gravity. We derive it from a conventional quantization of a Regge discretization of euclidean general relativity. This yields a spinfoam sum that corrects some difficulties of the Barrett–Crane theory. The second class simplicity constraints are imposed weakly , and not strongly as in Barrett–Crane theory. Thanks to a flip in the quantum algebra, the boundary states turn out to match those of SO ( 3 ) loop quantum gravity—the two can be identified as eigenstates of the same physical quantities—providing a solution to the problem of connecting the covariant SO ( 4 ) spinfoam formalism with the canonical SO ( 3 ) spin-network one. The vertex amplitude is SO ( 3 ) and SO ( 4 ) -covariant. It rectifies the triviality of the intertwiner dependence of the Barrett–Crane vertex, which is responsible for its failure to yield the correct propagator tensorial structure. The construction provides also an independent derivation of the kinematics of loop quantum gravity and of the result that geometry is quantized.

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Written from the paper’s full text.

TL;DR

The paper introduces a 4d spinfoam vertex amplitude for loop quantum gravity by quantizing a Regge discretization of Euclidean general relativity in Plebanski form, imposing the second-class simplicity constraints weakly and using a flipped symplectic structure. The resulting boundary state space matches SO(3) loop quantum gravity, providing a link between covariant SO(4) spinfoam and canonical SO(3) spin-network formalisms.

Key findings

  • Imposing the off-diagonal simplicity constraints weakly, rather than strongly as in Barrett-Crane theory, yields a nontrivial intertwiner space; the paper reports that the constraint count for a 4-simplex leaves \(60-(24+10+10)=16\) degrees of freedom of the tetrad \(e(v)\).
  • The physical boundary Hilbert space is isomorphic to the SO(3) LQG state space on the graph dual to the boundary triangulation; the vertex amplitude is both SO(3)- and SO(4)-covariant.
  • The construction corrects the trivial intertwiner dependence of the Barrett-Crane vertex and is argued to recover the degrees of freedom needed for correct low-energy \(n\)-point functions.
  • The flipped symplectic structure is selected over the unflipped one because the latter would lead to a macroscopic area spectrum; the flipped choice leads to the nontrivial intertwiner space, while the opposite choice leads to the Barrett-Crane intertwiner space.

Methods

  • The authors discretize Euclidean general relativity via Regge calculus on a fixed triangulation and re-express it using Plebanski two-form variables, holonomies, and bivectors associated with triangles.
  • They quantize the boundary phase space as an SO(4) lattice Yang-Mills theory on \(L^2[SO(4)^L]\), impose diagonal simplicity, closure, and off-diagonal simplicity constraints, with the latter imposed weakly.
  • They derive the vertex amplitude for a single 4-simplex and the spinfoam partition function for boundary SO(3) spin networks, using the flipped Poisson structure and a topological term related to the Immirzi parameter.

Limitations

  • The paper states that it works in Euclidean signature and on a fixed triangulation; issues of recovering triangulation independence and relation to Lorentzian signature are deferred to elsewhere.
  • The paper states that whether the model is non-trivial and/or reproduces general relativity in an appropriate limit is unknown, and that study of its semi-classical limit and \(n\)-point functions is needed.
  • Evident from the text: the work is a theoretical construction without numerical or observational validation, and the low-energy correctness of the vertex remains an open question.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

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Crossref · 10.1016/j.nuclphysb.2008.02.002

Cites

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Statistics

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Claims the paper makes

0 of 37 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § 1 Introduction · p. 1

    The vertex is both SO(3) and SO(4) covariant.

    “Third, the vertex can be seen as a vertex over SO (3) spin networks or SO (4) spin networks, and is both SO (3) and SO (4) covariant.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 1 Introduction · p. 1

    The simplicity constraints can be imposed weakly, as 〈φC_n ψ〉 = 0.

    “In this paper we show that the simplicity constraints can be imposed weakly ( 〈 φC n ψ 〉 = 0), and that the resulting theory has remarkable features.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 3.6 Boundary terms · p. 15

    The boundary action gives additivity to first order in the algebra.

    “This gives additivity to first order in the algebra.”
    Quote in the paperNot a substantial claim
  • Findingreasoning
    § 3.3 Dynamics · p. 13

    To lowest order in θ_f, the bulk term in (59) is precisely the bulk Regge action.

    “To lowest order in θ f , this is precisely the bulk Regge action.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: For any metric g, one can construct a corresponding Regge geometry ∆_λ approximating g with typical lattice spacing λ, such that as λ→0, ∆_λ approaches g.
  • Findingreasoning
    § 3.6 Boundary terms · p. 15

    If R_f ≡ 1, the discrete boundary action reduces precisely to the continuum boundary action.

    “If we choose R f ≡ 1, the above discrete boundary action becomes”
    • Also stated in: § 3.6 Boundary terms · p. 15 · plain
      “which reduces precisely to the continuum boundary action.”
    Quote in the paperNot a substantial claim
  • May combine two claims. 37
  • Classified as probably not a substantial claim. 28
  • The quote only partly states this claim. 2

  • May combine two claims.

    Findingreasoning
    § 1 Introduction · p. 1

    The resulting theory's boundary quantum state space matches exactly the one of SO(3) loop quantum gravity, with no degrees of freedom lost.

    “First, its boundary quantum state space matches exactly the one of SO (3) LQG: no degrees of freedom are lost.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § 1 Introduction · p. 1

    The vertex can be seen as a vertex over SO(3) spin networks or SO(4) spin networks.

    “Third, the vertex can be seen as a vertex over SO (3) spin networks or SO (4) spin networks, and is both SO (3) and SO (4) covariant.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § 4.1 Kinematical Hilbert space and operators · p. 19

    After (92) and (88) are satisfied, only two of the three off-diagonal simplicity constraints acting on each tetrahedron are independent; these constraints form a second class system.

    “After (92) and (88) are satisfied, only two of the three off-diagonal simplicity constraints acting on each tetrahedron are independent. These constraints form a second class system.”
    Quote in the paperSubstantial claim

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

    Findingnone given
    § 3.2 Relation with geometry · p. 11

    The area is also the norm of its associated selfdual or antiselfdual bivector, and these equalities follow from the simplicity constraints.

    “It is important to notice that, as mentioned in the previous section, the area is also (the square root of) the norm of its associated selfdual , or, equivalently, antiselfdual bivector:”
    Quote in the paperNot a substantial claim

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

    Findingnone given
    § 3.7 Boundary variables · p. 16

    Unlike in the bulk, each boundary tetrahedron bounds just a single simplex of the triangulation, and each boundary triangle bounds just two boundary tetrahedra.

    “Notice that, unlike what happens in the bulk, each boundary tetrahedron bounds just a single simplex of the triangulation; and each boundary triangle bounds just two boundary tetrahedra.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 1 Introduction · p. 2

    Using this, the paper finds a nontrivial subspace of the SO(4) intertwiner space, which corresponds to closed tetrahedra and maps naturally to an SO(3) intertwiner space.

    “Using this, we find a nontrivial subspace of the SO (4) intertwiner space, which corresponds to closed tetrahedra and maps naturally to an SO (3) intertwiner space.”
    Quote in the paperSubstantial claim

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

    Findingnone given
    § 3.7 Boundary variables · p. 16

    This cotangent bundle has a natural symplectic structure, which defines Poisson brackets.

    “This cotangent bundle has a natural symplectic structure, which defines Poisson brackets.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given
    § 3.8 Summary of the classical theory · p. 17

    The off-diagonal simplicity constraint can be replaced by the stronger condition that there is an n^I for each t such that (91) holds, equivalently there is a gauge in which (91) holds.

    “The second can be replaced by the (stronger) condition that there is an n I for each t such that”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingnone given
    § 3.8 Summary of the classical theory · p. 17

    Discretized GR can be defined by the action (59) with appropriate boundary terms (80) as a function of the variables [Σ_f(t), V_vt, U_tt'(v), λ_vf].

    “Summarizing, discretized GR can be defined by the action (59) with the appropriate boundary terms (80) now defined as a function of the variables [Σ f ( t ) , V vt , U tt ′ ( v ) , λ vf ]).”
    Quote in the paperSubstantial claim

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

    Findingnone given
    § 3 Regge discretization · p. 9

    The quantities Σ^IJ_f(t) and V_vt can be taken as a discretization of the continuous gravitational fields Σ^IJ and ω^IJ that define general relativity in the Plebanski formalism.

    “The quantities Σ IJ f ( t ) and V vt can be taken as a discretization of the continuous gravitational fields Σ IJ and ω IJ that define GR in the Plebanski formalism.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 3.1 Constraints on Σ · p. 10

    This stronger requirement implies the off-diagonal simplicity constraints and is implied by the physical solution (51) of these constraints.

    “It is immediate to see that this implies the off-diagonal simplicity constraints, and that it is implied by the physical solution (51) of these constraints.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given
    § 3.7 Boundary variables · p. 16

    The space of boundary variables (Σ^L_l, Σ^R_l, U_l) codes the phase space of discretized GR and is precisely the same as the phase space of a Yang-Mills SO(4) lattice theory, identifiable as the cotangent bundle of C = SO(4)^L.

    “Since we are in a first order formalism, the space of these variables (Σ L l , Σ R l , U l ) code the phase space of discretized GR. In fact, this space is precisely the same as the phase space of a Yang-Mills SO (4) lattice theory, and it can be identified as the cotangent bundle of the configuration space C = SO (4) L , where L is the number of links on Γ.”
    Quote in the paperSubstantial claim

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

    Findingnone given
    § 3.6 Boundary terms · p. 15

    The boundary action is not gauge-invariant, as it is not in the continuum theory.

    “The boundary action is not gauge-invariant, as it is not in the continuum theory.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 3.7 Boundary variables · p. 16

    Of the constraints (46,47,48,43,31), only (46,47,48,43) act separately at each tetrahedron and impose direct restrictions on boundary data; constraint (31) necessarily involves bulk variables.

    “Finally, of the constraints (46, 47, 48, 43, 31), only (46, 47, 48, 43) act separately at each tetrahedron, and thus impose direct restrictions on boundary data. We call (46, 47, 48, 43) the kinematical constraints. The constraint (31) on the other hand necessarily involves bulk variables; we thus call it the dynamical constraint.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 4.2 The physical intertwiner state space K ph · p. 20

    The off-diagonal simplicity constraints are odd under exchange of i^+ and i^-; states in K_ph are symmetric in i^+ and i^-; hence <φ|C_ll'|ψ>=0 for all φ,ψ∈K_ph, so K_ph is a possible solution of the constraint equations.

    “The off-diagonal simplicity constraints are odd under the exchange of i + and i -, namely under exchange of self dual and antiselfdual sectors. But the states in K ph are symmetric in i + and i -. Hence 〈 φ | C ll ′ | ψ 〉 = 0 , ∀ φ, ψ ∈ K ph , that is, K ph can be considered as one possible solution of the constraint equations.”
    Quote in the paperSubstantial claim

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

    Findingnone given
    § Proof. · p. 29

    For each i, V_i and U_i are orthogonal complements.

    “the fact that for each i , V i and U i are orthogonal complements (which one can also show).”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Findingreasoning
    § 4.2 The physical intertwiner state space K ph · p. 22

    The embedding f : H_SO(3) → H_ph is the hermitian conjugate of π.

    “Let us now describe the conjugate embedding f : H SO (3) →H ph . This is defined as the hermitian conjugate of π , using the fact that π is a linear map between Hilbert spaces.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 4.3 Dynamics · p. 24

    For a spin-network state ψ = ψ_{j_ab, i_a}, the amplitude is given explicitly by the formula and extends by linearity to more general states Ψ = f(ψ).

    “Let us consider the specific case when ψ is a spin-network state ψ = ψ { j ab } , { i a } . The amplitude is then given explicitly by”
    • Also stated in: § 4.3 Dynamics · p. 25 · plain
      “This amplitude extends by linearity to more general states Ψ = f ( ψ ).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Proof. · p. 29

    Exactly one of span{f1, f2, f3} and span{ ̃f1, ̃f2, ̃f3} is three dimensional, and the other is one dimensional.

    “One can prove that exactly one of span { f 1 , f 2 , f 3 } and span { ˜ f 1 , ˜ f 2 , ˜ f 3 } is three dimensional, and the other is one dimensional.”
    Quote in the paperSubstantial claim

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

    Findingreasoning
    § C Symplectic structure · p. 33

    In the γ ≫ 1 case, one obtains the more standard symplectic structure (86), and this symplectic structure along with constraint (105) leads to the Barrett-Crane model.

    “As a side remark, consider the case γ ≫ 1. In this case, one obtains the more standard symplectic structure (86). As noted in § 4.2, this symplectic structure, along with the constraint (105), imposed as discussed above, leads to the Barrett-Crane model.”
    Quote in the paperNot a substantial claim

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

    Findingreasoning
    § 4.3 Dynamics · p. 25

    Adding the topological term (78) incorporating γ does not change the quantum dynamics defined above; the final vertex amplitude differs at most by a constant.

    “Observe that the quantum dynamics defined above does not change if we add the topological term (78) incorporating γ .”
    • Also stated in: § C Symplectic structure · p. 33 · plain
      “neither does the dynamics change with the introduction of the γ -term in the action.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingnone given
    § C Symplectic structure · p. 32

    The symplectic structure obtained from the action when fixing the U's on the boundary is the one determined with U's fixed.

    “Thus we obtain the symplectic structure from the action when we fix the U 's on the boundary.”
    • Also stated in: § C Symplectic structure · p. 32 · plain
      “we simply do not address this problem, and take the symplectic structure to be the one determined with U 's fixed.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The unconstrained BF action can be used to derive the symplectic structure because general relativity is quantized as BF theory constrained at the quantum level.

  • Classified as probably not a substantial claim.

    Findingnone given
    § 4.3 Dynamics · p. 24

    As Ψ satisfies all kinematical constraints, it is of the form Ψ = f(ψ) for some ψ ∈ H_LQG.

    “As Ψ satisfies all the kinematical constraints, it is of the form Ψ = f ( ψ ) for some ψ ∈ H LQG .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § C Symplectic structure · p. 33

    For γ ≪ 1 and γ ≫ 1, one recovers the same constraint *Π_f · Π_f ∼ 0, and in these two limits the diagonal simplicity constraints again imply simplicity of the representations of the edges (j+ = j-).

    “Note that for γ ≪ 1 and γ ≫ 1 one recovers the same constraint ∗ Π f · Π f ∼ 0 being used before, whence, in these two limits, the diagonal simplicity constraints again imply simplicity of the representations of the edges ( j + = j -).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 4.2 The physical intertwiner state space K ph · p. 21

    The projection π maps simple SO(4) spin-network states to SO(3) spin-network states; the spin (j,j) on each edge is mapped to spin 2j in the SO(3) spin-network; each SO(4) intertwiner is mapped to an SO(3) intertwiner via orthogonal projection.

    “Thus π so-defined maps simple SO (4) spin-network states to SO (3) spin-network states. The spin ( j, j ) on each edge of the SO (4) spin-network is mapped to spin 2 j in the SO (3) spin-network. Each SO (4) intertwiner I v ∈ Inv SO (4) ( H ( j 1 ,j 1 ) ⊗ ... ⊗ H ( j 4 ,j 4 ) ) is mapped to an SO (3) intertwiner simply via orthogonal projection onto the subspace Inv SU (2) ( H (2 j 1 ) ⊗ ... ⊗H (2”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § B Existence of tetrad · p. 28

    If e^I_μ is such that B^IJ_i = B^μν_i e^I_μ e^J_ν for i ∈ {1,2,3}, and the closure constraint holds, then B^IJ_4 = B^μν_4 e^I_μ e^J_ν as well; an analogous statement holds with B replaced by *B.

    “then B IJ 4 = B µν 4 e I µ e J ν as well. An analogous statement holds with B IJ i replaced by ∗ B IJ i .”
    Quote in the paperSubstantial claim

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

    Premisenone given
    § 1 Introduction · p. 1

    The Barrett-Crane theory's boundary state space is similar to but does not exactly match that of loop quantum gravity; in particular the volume operator is ill-defined.

    “Its boundary state space is similar to, but does not exactly match, that of LQG; in particular the volume operator is ill-defined.”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § 1 Introduction · p. 1

    The kinematics of loop quantum gravity is rather well understood, while its dynamics is not understood as cleanly.

    “While the kinematics of loop quantum gravity (LQG) [1] is rather well understood [2, 3], its dynamics is not understood as cleanly.”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [1] source

      Not checked in the free run.

    • cites [2] 2. A Ashtekar, J Lewandowski, 'Quantum Theory of Geometry II : Volume Operators', Math. Phys. 1 (1997) 388-429. - (1990) 80-152. A Ashtekar, C Rovelli, L Smolin, 'Weaving a classical geometry with quantum threads', Phys. Rev. Lett. 69 , 237 (1992). C Rovelli, L Smolin, 'Discreteness of Area and Volume in Quantum Gravity', Nucl. Phys. B442 (1995) 593-619; Erratum: Nucl. Phys. B456 , 734 (1995). A Ashtekar, J Lewandowski, 'Quantum Theory of Geometry I: Area Operators' Class. Quant. Grav. 14 (1997) A55-A82. Adv. Th. - [2] AAshtekar, J Lewandowski, 'Background independent quantum gravity: A status source

      Not checked in the free run.

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  • May combine two claims.

    Premisecitation
    § 1 Introduction · p. 1

    In three dimensions, the Ponzano-Regge model's boundary states match those of loop quantum gravity and its vertex amplitude can be obtained as a matrix element of the hamiltonian of 3d loop quantum gravity.

    “This picture is nicely implemented in three dimensions (3d) by the Ponzano-Regge model [10], whose boundary states match those of LQG [11] and whose vertex amplitude can be obtained as a matrix element of the hamiltonian of 3d LQG [12].”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [10]

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    • cites [11]

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    • cites [12]

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

    Premisecitation
    § 1 Introduction · p. 1

    The Barrett-Crane theory is the best studied model in the 4d euclidean context, based on the vertex amplitude introduced by Barrett and Crane.

    “The best studied model in the 4d euclidean context is the Barrett-Crane (BC) theory [8], which is based on the vertex amplitude introduced by Barrett and Crane [7].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [8] source

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

    Premisereasoning
    § 1 Introduction · p. 2

    It is hard to find a proper derivation of a spinfoam model from the classical field theory that follows well-tested quantization procedures.

    “It is hard to find a proper derivation of a spinfoam model from the classical field theory, which follows well-tested quantization procedures.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given
    § 1 Introduction · p. 2

    The vertex of the theory is similar to, but different from, the Barrett-Crane vertex, leading to a dynamics that might be better behaved in the low-energy limit.

    “Finally, (iv) the vertex of the theory is similar to, but different from, the BC vertex, leading to a dynamics that might be better behaved in the low-energy limit.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 4.1 Kinematical Hilbert space and operators · p. 19

    The closure constraint (92) does not need to be imposed at this stage, since it is not an independent constraint, but it is implemented by the dynamics.

    “Strictly speaking, the closure constraint (92) does not need to be imposed at this stage, since it is not an independent constraint, but it is implemented by the dynamics, as shown in the last section.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 3 Regge discretization · p. 7

    Because the quantum theory has no degrees of freedom below the Planck scale, a Regge approximation is good for small L and is a low-energy approximation for large L.

    “Since (as we shall see, and as implied by LQG) the quantum theory has no degrees of freedom below the Planck scale, it follows that a Regge approximation is good for L small, and it is a low-energy approximation for L large.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisereasoning
    § 2.2 Selfdual structure of SO (4) · p. 5

    A simple representation (j,j) of SO(4) is reducible under the subgroup SO(3)_n and transforms as j ⊗ j.

    “Clearly, it transforms in the representation j ⊗ j , where j indicates the usual spin j representation of SU (2).”
    Quote in the paperSubstantial claim

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

    Premisenone given
    § 3 Regge discretization · p. 7

    The quantum theory has no degrees of freedom below the Planck scale.

    “Since (as we shall see, and as implied by LQG) the quantum theory has no degrees of freedom below the Planck scale, it follows that a Regge approximation is good for L small, and it is a low-energy approximation for L large.”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § 1 Introduction · p. 2

    Quantization of Regge calculus has been considered in the past and its relation to spinfoam theory has been pointed out.

    “Quantization of Regge calculus has been considered in the past [24] and its relation to spinfoam theory has been pointed out (see [25] and references therein).”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [24] source

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    • cites [25] source

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  • May combine two claims.

    Interpretationreasoning
    § 3.7 Boundary variables · p. 17

    Choosing the flipped symplectic structure (85) leads to a nontrivial intertwiner space, while the opposite choice leads to the Barrett-Crane trivial intertwiner space.

    “Thus, we choose the fl ipped symplectic structure (85) as a basis for the quantization of theory. This choice leads to a nontrivial intertwiner space, while the opposite choice leads to the Barrett-Crane trivial intertwiner space, as we shall see below.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: In order to arrive at the Ashtekar formalism and LQG, the topological term is needed.

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

    Premisecitation
    § 1 Introduction · p. 2

    Montesinos has discussed the classical indeterminacy of the symplectic structure in detail.

    “Montesinos, in particular, has discussed the classical indeterminacy of the symplectic structure in detail.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [29] source

      Not checked in the free run.

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

    Premisereasoning
    § 3 Regge discretization · p. 7

    In Regge calculus, curvature is concentrated on the d-2 simplices; in dimension 4, curvature is concentrated on the triangles.

    “In Regge calculus, curvature is concentrated on the d -2 simplices. In dimension 4, curvature is therefore concentrated on the triangles f .”
    Quote in the paperNot a substantial claim

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

    Interpretationnone given
    § 1 Introduction · p. 2

    The flip can be viewed as the equivalent of the Ashtekar trick, which yields a connection as phase space variable.

    “The flip can be viewed as the equivalent of the Ashtekar 'trick', which yields a connection as phase space variable.”
    Quote in the paperNot a substantial claim

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

    Premisenone given
    § 3 Regge discretization · p. 7

    In a 4d triangulation, the geometry induced on a given tetrahedron from the geometry of the two adjacent 4-simplices is the same.

    “The geometry induced on a given tetrahedron from the geometry of the two adjacent 4-simplices is the same.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation
    § 3.7 Boundary variables · p. 17

    Both symplectic structures are equivalent to the lattice Yang-Mills theory Poisson brackets; the difference is only whether the electric field is identified with B or with Σ.

    “Both structures are equivalent to the lattice Yang Mills theory Poisson brackets [32], the difference is only whether we identify the electric field with B or with Σ.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [32]

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    Premisenone given
    § 1 Introduction · p. 2

    Loop quantum gravity predicts an actual physical discretization of spacetime.

    “This persuasion is reinforced by the LQG prediction of an actual physical discretization of spacetime.”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § 1 Introduction · p. 2

    The possibility and relevance of a flipped symplectic structure was noticed by Baez and Barrett and by Montesinos.

    “The possibility and the relevance of a 'flipped' symplectic structure was noticed by Baez and Barrett in [26] (where they attribute the observation to Jos´ e-Antonio Zapata) and by Montesinos [29].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [26] source

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  • May combine two claims.

    Premisenone given
    § 2.2 Selfdual structure of SO (4) · p. 5

    The split so(4) = su(2)+ ⊕ su(2)- is canonical, but there is no canonical isomorphism between su(2)+ and su(2)- or between SU(2)+ and SU(2)-.

    “In fact, the split so ( 4 ) = su ( 2 ) + ⊕ su ( 2 ) -is canonical, but there is no canonical isomorphism between su ( 2 ) + and su ( 2 ) -or between SU (2) + and SU (2) -.”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § 3.6 Boundary terms · p. 15

    The additivity condition for boundary terms is S[S] = S[S'] + S[S''].

    “The additivity condition is then that S [ S ] = S [ S ′ ] + S [ S ′′ ] [35].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [35]

      Not checked in the free run.

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

    Premisecitation
    § 1 Introduction · p. 2

    From experience with QCD, one can derive the persuasion that a nontrivial quantum field theory should be related to a natural lattice discretization of the corresponding classical field theory.

    “From the experience with QCD, one can derive the persuasion that a nontrivial quantum field theory should be related to a natural lattice discretization of the corresponding classical field theory [22].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [22] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

    Interpretationnone given
    § 1 Introduction · p. 2

    The geometrical interpretation for all the variables becomes fully transparent.

    “In particular: (i) the geometrical interpretation for all the variables becomes fully transparent;”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 1 Introduction · p. 2

    It also provides a novel independent derivation of the loop quantum gravity kinematics, and in particular of the quantization of area and volume.

    “It also provides a novel independent derivation of the LQG kinematics, and, in particular, of the quantization of area and volume.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 4.2 The physical intertwiner state space K ph · p. 21

    If the flipped symplectic structure had not been chosen, one would have obtained J_ij=0 instead of J_0i=0, i.e. vanishing of the SO(3) Casimir, and the resulting space is precisely the one-dimensional Barrett-Crane intertwiner space.

    “Notice that if we hadn't chosen the flipped symplectic structure, then we would have obtained J ij = 0, instead of J 0 i = 0, that is, the vanishing of the SO (3) Casimir. Following the same procedure as above, we would have defined the space”
    • Also stated in: § 4.2 The physical intertwiner state space K ph · p. 21 · plain
      “and this is precisely the one dimensional Barrett-Crane intertwiner space.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 5 Conclusions · p. 25

    The derivation of loop quantum gravity given here starts from a lattice discretization and later finds that in the quantum theory the area of the triangles of the triangulation is discrete, so the regularization receives a physical meaning.

    “The derivation of loop quantum gravity given here is somewhat intermediate: it starts from a lattice discretization, and later finds out that in the quantum theory, say, the area of the triangles of the triangulation is discrete and therefore the regularization receives a physical meaning.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 5 Conclusions · p. 26

    The theory obtained can be seen as an independent derivation of loop quantum gravity kinematics, and in particular of the fact that geometry is quantized.

    “This can be seen as an independent derivation of the loop quantum gravity kinematics, and, in particular, of the fact that geometry is quantized.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 4.2 The physical intertwiner state space K ph · p. 21

    K_ph is isomorphic to the SO(3) intertwiner space, and the constrained boundary space H_ph is precisely the SO(3) LQG state space H_SO(3) associated to the graph dual to the boundary of the triangulation.

    “Third, we have the remarkable result that K ph is isomorphic to the SO (3) intertwiner space, and therefore the constrained boundary space H ph is precisely the SO (3) LQG state space H SO (3) associated to the graph which is dual to the boundary of the triangulation, namely the space of the SO (3) spin networks on this graph.”
    Quote in the paperSubstantial claim

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

    Premisenone given
    § C Symplectic structure · p. 32

    The topological term in the GR action does not change the equations of motion.

    “Recall, however, that in the LQG approach the action that is quantized is the GR one with a topological term that doesn't change the equations of motion:”
    Quote in the paperNot a substantial claim

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

    Premisenone given
    § C Symplectic structure · p. 30

    The BF equations of motion dictate that all holonomies around closed loops are trivial.

    “The BF equations of motion dictate that all holonomies around closed loops are trivial.”
    Quote in the paperNot a substantial claim
56 of 56

  • The authors state this as an assumption.

    For any metric g, one can construct a corresponding Regge geometry ∆_λ approximating g with typical lattice spacing λ, such that as λ→0, ∆_λ approaches g.

    “suppose one has a metric g which one wishes to approximate. Then, for each real number λ , one can construct a corresponding Regge geometry ∆ λ approximating g , such that the typical lattice spacing is λ , as measured w.r.t. g . As λ → 0, ∆ λ approaches g , and one can consider expansions in λ .”

  • The paper relies on this without support.

    In order to arrive at the Ashtekar formalism and LQG, the topological term is needed.

    “But in order to arrive at the Ashtekar formalism and LQG, we know that the topological term is needed.”

  • The authors state this as an assumption.

    The unconstrained BF action can be used to derive the symplectic structure because general relativity is quantized as BF theory constrained at the quantum level.

    “We use the unconstrained action in deriving the symplectic structure because our general philosophy is that we are quantizing GR as a BF theory constrained at the quantum level .”
3 of 3

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The basic idea of the EPRL model was to relax the imposition of the Plebanski constraints that reduce BF theory to general relativity in the quantum theory.

    “The basic idea was to relax the imposition of the Plebanski constraints that reduce BF theory to general relativity in the quantum theory.”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [169]

  • Relaxing the constraints naturally leads to a simple model that has a clearer relationship with the canonical picture provided by LQG.

    “There is a natural way to relax the constraints and this leads to a simple model that has a more clear relationship with the canonical picture provided by LQG.”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [169]

  • Imposing the constraints strongly, as in the Barrett-Crane model, implies additional conditions that are not present in the classical theory.

    “Imposing the constraints strongly, as in the Barrett-Crane model, implies the imposition of additional conditions that are not present in the classical theory.”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [169]

  • The EPRL model is a modification of the Barrett-Crane model that was recently introduced and extended to arbitrary Immirzi parameter.

    “A modification of the Barrett-Crane model was recently introduced in [168, 169] and extended for arbitrary Immirzi parameter in [164].”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [169]

  • Spin-foam models are derived as quantizations of discretized classical theories.

    “The SFMs on the other hand, are derived as quantizations of discretized classical theories [6, 13, 19, 20].”

    From Spin-foams for all loop quantum gravity (2010) · cites this paper as [19]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Doubled particle variables have the wrong Poisson bracket

    Minor

    With the stated canonical brackets and the definitions of q− and p−, their bracket is (1+1)/4 = 1/2, not 1. The stated value is used to characterize the example's second-class constraints.

  • Both vector triples are defined from the same intersections

    Minor

    The proof defines f̃1 from V2 ∩ V3, just as it defines f1, and says to proceed cyclically for both triples. One may therefore choose the triples identically, contradicting the subsequent claim that exactly one spans three dimensions and the other one dimension.

2 of 2

Cites

Flagged first, then by citations

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Cited by

Sorted by citations

Authors
Engle, J., Pereira, R., & Rovelli, C.
Affiliations
Centre National de la Recherche Scientifique (FR)
ORCID
Engle, Jonathan
Venue
Nuclear Physics B
Publisher
Elsevier BV
Published
2008-07
Volume
798
Issue
1-2
Pages
251-290
ISSN
0550-3213 (ISSN-L)
DOI
10.1016/j.nuclphysb.2008.02.002
OpenAlex
W2088552537
Semantic Scholar
febbee02f9755d54b9dc0a050a0aae22d8ae578f
Cited by
241
References
20
Citations by year
2025: 2 · 2024: 7 · 2023: 4 · 2022: 5 · 2021: 4 · 2020: 4 · 2019: 6 · 2018: 6 · 2017: 8 · 2016: 11 · 2015: 9 · 2014: 12 · 2013: 27 · 2012: 27
FWCI
26.80
Type
journal article
Language
en
Open access
Licence
Locations
  • Nuclear Physics B (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • HAL AMU (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Keywords
loop quantum gravity, Barrett-Crane model, Regge calculus, simplicity constraints, spin networks

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