Spin-foams for all loop quantum gravity
Open accessAbstract
The simplicial framework of Engle–Pereira–Rovelli–Livine spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity. The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams. The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations (or cubulations). In particular, the notion of embedded spin-foam we use allows us to consider knotting or linking spin-foam histories. Also the main tools, the vertex structure and the vertex amplitude, are naturally generalized to an arbitrary valency case. The correspondence between all the SU (2) intertwiners and the SU (2)× SU (2) EPRL intertwiners is proved to be 1–1 in the case of the Barbero–Immirzi parameter |γ| ⩾ 1.
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TL;DR
The paper generalizes EPRL spin-foams from simplicial complexes to arbitrary linear 2-cell complexes and embedded spin-foams, so that all LQG spin-network states—including knotted/linked graphs and arbitrary valency—can have spin-foam histories. It also proves a 1-1 correspondence between SU(2) intertwiners and SU(2) × SU(2) EPRL intertwiners for |γ| ≥ 1, except when the EPRL co-domain is trivial and the domain is non-trivial.
Key findings
- The simplicial EPRL spin-foam framework is generalized to arbitrary linear 2-cell spin-foams and embedded spin-foams, admitting all spin-network states of loop quantum gravity, not only those defined by triangulations or cubulations; knotting or linking spin-foam histories are allowed.
- The vertex structure and vertex amplitude are generalized to arbitrary valency; a vertex is encoded by the spin-network induced on the boundary of its neighbourhood.
- The correspondence between all SU(2) intertwiners and SU(2) × SU(2) EPRL intertwiners is proved to be 1-1 for |γ| ≥ 1, unless the co-domain of the EPRL map is trivial and the domain is non-trivial.
- An n-valent Barrett-Crane intertwiner is defined equivalently to earlier definitions, and the EPRL intertwiner is generalized; in the limit γ → ±∞ the EPRL intertwiner becomes the BC intertwiner.
Methods
- Theoretical/mathematical construction within loop quantum gravity and spin-foam models; no empirical data or sample are used.
- Foams are defined as oriented linear 2-cell complexes with boundary; faces are coloured by irreducible representations and internal edges by intertwiners, inducing a boundary spin-network; vertex traces, spin-foam traces, and embedded spin-foams are then defined.
- The injectivity proof for the map I ↦ ι_EPRL(I) uses the spin composition map and mathematical induction.
Limitations
- Stated: “We do not study the FK intertwiners in the γ > 1 in this paper.”
- Stated: the discussion notes that the limits γ → 0, ±1, ±∞ are mathematically well defined but the model “seems to be losing its key properties”; for example, γ → ±∞ gives the BC intertwiner with 0 or 1 dimension per vertex, γ → ±1 gives an SU(2) BF amplitude, and γ → 0 does not resemble a topological theory.
- Evident from the text: no numerical, observational, or semiclassical/convergence validation of the generalized amplitudes is presented; the work is a formal mathematical construction.
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Claims the paper makes
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- Findingnone given§ Spin-Foams for All Loop Quantum Gravity · p. 1
The correspondence between all SU(2) intertwiners and the SU(2) × SU(2) EPRL intertwiners is 1-1 in the case of the Barbero-Immirzi parameter |γ| ≥ 1, unless the co-domain of the EPRL map is trivial and the domain is non-trivial.
“The correspondence between all the SU(2) intertwiners and the SU(2) × SU(2) EPRL intertwiners is proved to be 1-1 in the case of the Barbero-Immirzi parameter | γ | ≥ 1, unless the co-domain of the EPRL map is trivial and the domain is non-trivial.”
- Also stated in: § C. The goal of the paper · p. 2 · boosted
“We show that the correspondence is 1-1 in the case of the Barbero-Immirzi parameter γ ≥ 1, unless the co-domain of the EPRL map is trivial and the domain is non-trivial.”
Quote in the paperSubstantial claim - Also stated in: § C. The goal of the paper · p. 2 · boosted
- Findingnone given§ Spin-Foams for All Loop Quantum Gravity · p. 1
The simplicial framework of EPRL spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity.
“The simplicial framework of Engle-Pereira-Rovelli-Livine spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity.”
Quote in the paperSubstantial claim - Findingnone given§ Spin-Foams for All Loop Quantum Gravity · p. 1
The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams.
“The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams.”
Quote in the paperSubstantial claim - Findingreasoning§ 3. The injectivity of I ↦→ ι EPRL ( I ) in the case γ ≥ 1 · p. 20
The map I ↦ ι_EPRL(I) is injective in the case |γ| ≥ 1.
“3. The injectivity of I ↦→ ι EPRL ( I ) in the case γ ≥ 1”
Quote in the paperSubstantial claim - Findingnone given§ B. The limits γ = ±∞ , ± 1 , 0 · p. 23
The EPRL theory does not resemble a topological theory at all in the limit γ → 0.
“but, quite surprisingly, the EPRL theory does not resemble a topological theory at all.”
Quote in the paperNot a substantial claim
- Findingreasoning§ D. Spin-foam of a spin-network function · p. 13
The spin-network s induced on the boundary Σ×{0} is obtained from s_in by flipping the orientations of the edges and taking the hermitian adjoint of its intertwiners.
“The spin-network s induced on the boundary Σ ×{ 0 } is obtained from s in by flipping the orientations of the edges and taking the hermitian adjoint of its intertwiners, hence,”
Quote in the paperSubstantial claim - Findingnone given§ B. The limits γ = ±∞ , ± 1 , 0 · p. 22
The EPRL intertwiner has a limit in the case γ → ±1.
“However the EPRL intertwiner has a limit in that case:”
Quote in the paperNot a substantial claim - Findingnone given§ A. Summary · p. 22
There is a 1-1 correspondence between the EPRL SU(2) × SU(2) intertwiners and all the SU(2) intertwiners in the case of the Immirzi-Barbero parameter |γ| ≥ 1 and non-trivial co-domain of the EPRL map.
“We have also proved the 1-1 correspondence between the EPRL SU(2) × SU(2) intertwiners and all the SU(2) intertwiners in the case of the Immirzi-Barbero parameter | γ | ≥ 1 and non-trivial co-domain of EPRL map.”
Quote in the paperNot a substantial claim - Findingcitation§ B. The limits γ = ±∞ , ± 1 , 0 · p. 22
The BC intertwiner space has 0 or 1 dimension per vertex.
“The BC intertwiner space has 0 or 1 dimension per vertex and has been proven to insufficiently accommodate all the gravitational degrees of freedom.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [9] Perez A, 2003, Class. Quantum. Grav. source
Not checked in the free run.
- Findingnone given§ 3. The injectivity of I ↦→ ι EPRL ( I ) in the case γ ≥ 1 · p. 20
The spin composition map cannot annihilate a nonzero simple tensor K_BC = I_B W_C.
“The key property is that the map can not annihilate a nonzero simple tensor K BC = I B W C”
Quote in the paperNot a substantial claim - Findingnone given§ 2. The vertex spin-network · p. 11
The spin-foam vertex trace at a vertex equals the evaluation of the conjugate spin-network induced on the boundary of the foam neighbourhood of that vertex.
“The vertex trace (3.11) of the spin-foam equals the evaluation of the conjugate spin-network”
Quote in the paperSubstantial claim - Findingreasoning§ 3. The injectivity of I ↦→ ι EPRL ( I ) in the case γ ≥ 1 · p. 21
The projection P+ in (4.42) acts as the identity under the conditions of the injectivity proof.
“Hence the projection P + in (4.42) acts as the identity,”
Quote in the paperSubstantial claim - Findingnone given§ A. Summary · p. 22
The structure of each vertex of a generalized spin-foam can be completely encoded in a spin-network induced locally on the boundary of the vertex neighbourhood.
“The most important result is the characterization of a vertex of a generalized spin-foam. The structure of each vertex can be completely encoded in a spin-network induced locally on the boundary of the vertex neighbourhood.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The spin-foams are histories of the embedded spin-network functions defined within the framework of the diffeomorphism invariant quantization of the theory of connections.
- Findingreasoning§ 2. The EPRL intertwiners · p. 19
In the limit γ → ±∞, the EPRL intertwiner becomes the BC intertwiner ι_BC.
“That is, the EPRL intertwiner becomes the BC intertwiner ι BC .”
Quote in the paperNot a substantial claim - Findingreasoning§ 1. The BC intertwiners · p. 18
The non-vanishing of ι_BC whenever the SU(2) intertwiner space Inv(ρ j1 ⊗ ... ⊗ ρ jn) is not empty itself follows from the nondegeneracy of the invariant antisymmetric bilinear form ǫ12, which implies the nondegeneracy of each ǫj, as well as the nondegeneracy of the product ǫ.
“The non-vanishing of ι BC whenever the SU(2) intertwiner space Inv ( ρ j 1 ⊗ ... ⊗ ρ j n ) is not empty itself, follows from the nondegeneracy of the invariant antisymmetric bilinear form ǫ 1 2 which implies the nondegeneracy of each ǫ j , as well as the nondegeneracy of the product ǫ .”
Quote in the paperSubstantial claim - Premisecitation§ A. Recent spin-foam models of gravitational field · p. 2
The Barrett-Crane model was shown not to have sufficiently many degrees of freedom to ensure the correct classical limit.
“Nonetheless, it was shown not to have sufficiently many degrees of freedom to ensure the correct classical limit [9].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [9] Perez A, 2003, Class. Quantum. Grav. source
Not checked in the free run.
- Premisecitation§ A. Recent spin-foam models of gravitational field · p. 2
The EPRL and Freidel-Krasnov derivations produced a new, consistent spin-foam model valid for the Barbero-Immirzi parameter in the interval -1 ≤ γ ≤ 1.
“The result of their effort is a new, consistent spin-foam model valid for the Barbero-Immirzi parameter belonging to the interval -1 ≤ γ ≤ 1, a promising candidate for a path integral formulation of LQG, and two different models for | γ | > 1 [14-16]”
Quote in the paperNot a substantial claim0 of 3 sources checked - Interpretationreasoning§ Spin-Foams for All Loop Quantum Gravity · p. 1
The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations (or cubulations).
“The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations (or cubulations).”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. Summary · p. 22
Our approach is basis independent.
“Our approach is basis independent.”
Quote in the paperSubstantial claim - Premisecitation§ A. Recent spin-foam models of gravitational field · p. 2
There are two different spin-foam models for |γ| > 1.
“The result of their effort is a new, consistent spin-foam model valid for the Barbero-Immirzi parameter belonging to the interval -1 ≤ γ ≤ 1, a promising candidate for a path integral formulation of LQG, and two different models for | γ | > 1 [14-16]”
Quote in the paperNot a substantial claim0 of 3 sources checked - Interpretationreasoning§ B. The incompatibilities between LQG and some SFMs · p. 2
The EPRL model does not define a spin-foam history of a generic spin-network state of loop quantum gravity.
“Therefore it does not define a spin-foam history of a generic spin-network state of LQG.”
Quote in the paperSubstantial claim- Rests on an assumption: In the LQG canonical framework, there is no justified way to restrict the graphs to those dual to triangulations of the underlying 3-manifold.
- Premisecitation§ C. Simple intertwiners · p. 16
In the limit γ → ∞, the FK model is no longer the BC model.
“In particular γ →∞ is no longer the BC model [21].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [21] Freidel L, 2008, Class. Quantum Grav. source
Not checked in the free run.
- Interpretationreasoning§ C. Simple intertwiners · p. 16
The real difference between the EPRL and BC approaches is in replacing the Palatini action by that of Holst.
“Hence, the real difference is in replacing the Palatini action by that of Holst.”
Quote in the paperSubstantial claim - Premisecitation§ C. Simple intertwiners · p. 16
The FK quantum simplicity constraint is encoded in a construction of suitable kinematical semiclassical states.
“The quantum simplicity constraint is encoded in a construction of suitable kinematical semiclassical states.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [15] Barrett J W, 1998, Adv. Theor. Math. Phys. source
Not checked in the free run.
- Interpretationcitation§ C. Simple intertwiners · p. 16
Rewriting the EPRL paper replacing the Holst action by that of Palatini and repeating each argument and every calculation yields the BC intertwiners.
“One can just rewrite the EPRL paper [6] replacing the Holst action by that of Palatini and repeating each argument and every calculation. The result will be the BC intertwiners.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [6] source
Not checked in the free run.
- Interpretationreasoning§ B. The incompatibilities between LQG and some SFMs · p. 2
There are still several incompatibilities between loop quantum gravity and spin-foam models.
“Although the correspondence between LQG and SFMs was settled down recently due to the EPRL model, there are still several incompatibilities between these frameworks.”
Quote in the paperSubstantial claim- Rests on an assumption: Loop quantum gravity is a quantum theory of gravitational field with all its local degrees of freedom and is not a discretized theory.
- Rests on an assumption: The kinematical observables labelled by curves and 2-surfaces form a complete set for the continuum theory.
- Rests on an assumption: Loop quantum gravity is a diffeomorphism invariant theory of fields on manifolds.
- Rests on an assumption: The diffeomorphisms play a crucial role in loop quantum gravity and are responsible for the diversity of embedded graphs and spin-networks labelling the quantum states.
- Rests on an assumption: In the LQG canonical framework, there is no justified way to restrict the graphs to those dual to triangulations of the underlying 3-manifold.
- Interpretationcitation§ 1. The BC intertwiners · p. 18
Rewriting the EPRL paper for the Palatini action yields, for every intertwiner space Inv(ρ j1 ⊗ ... ⊗ ρ jn) ⊗ Inv(ρ j1 ⊗ ... ⊗ ρ jn), a unique solution, exactly ι_BC.
“Remarkably, if we rewrite the EPRL paper [6] for the Palatini action (rather the that of Holst considered by EPRL), we will derive for every intertwiner space Inv ( ρ j 1 ⊗ ... ⊗ ρ j n ) ⊗ Inv ( ρ j 1 ⊗ ... ⊗ ρ j n ) a unique solution. This solution will be exactly our very ι BC (see also below).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [6] source
Not checked in the free run.
- Interpretationreasoning§ C. Simple intertwiners · p. 16
The failure of the BC approach does not consist in the too strong formulation of the quantum constraints.
“It is often stated that the failure of the BC approach consists in the too strong formulation of the quantum constraints. This is in fact not true.”
Quote in the paperSubstantial claim - Premisecitation§ B. The incompatibilities between LQG and some SFMs · p. 2
The discreteness of loop quantum gravity is a property of the quantum representation.
“The discreteness of LQG is a property of the quantum representation [18].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [18] Reisenberger M P, 1997, Class. Quantum Grav. source
Not checked in the free run.
- Premisecitation§ C. Simple intertwiners · p. 16
The BC states are insufficient to define physical semi-classical states of quantum gravity.
“A few years ago it was noticed [9], that the BC states are insufficient to define physical semi-classical states of quantum gravity.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [9] Perez A, 2003, Class. Quantum. Grav. source
Not checked in the free run.
- Premisecitation§ A. Recent spin-foam models of gravitational field · p. 2
The values of the Barbero-Immirzi parameter predicted by various black hole models belong to the interval -1 ≤ γ ≤ 1.
“The values of the Barbero-Immirzi parameter predicted by various black hole models belong to the former interval [17].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [17] source
Not checked in the free run.
- Interpretationreasoning§ A. Summary · p. 22
LQG does not have to be reformulated in terms of the piecewise linear category and triangulations to match it with the EPRL SFM.
“The conclusion coming from our work is that we do not have to reformulate LQG in terms of the piecewise linear category and triangulations to match it with the EPRL SFM.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The spin-foams are histories of the embedded spin-network functions defined within the framework of the diffeomorphism invariant quantization of the theory of connections.
- Premisenone given§ B. The scheme for embedded spin-foam models · p. 15
The recent spin-foam models of Barrett-Crane, Engle-Pereira-Rovelli-Livine and Freidel-Krasnov fall into a common scheme.
“The recent spin-foam models of Barrett-Crane [8], Engle-Pereira-Rovelli-Livine [6] and FreidelKrasnov [15] fall into a common scheme.”
Quote in the paperSubstantial claim0 of 3 sources checked - Premisecitation§ B. The incompatibilities between LQG and some SFMs · p. 2
The EPRL model uses only simplicial complexes (or cubulations) and the spin-networks defined on their boundaries.
“The EPRL model on the other hand, uses only simplicial complexes (or, recently cubulations [28]) and the spin-networks defined on their boundaries.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [28] source
Not checked in the free run.
- Premisecitation§ C. Simple intertwiners · p. 16
The FK model uses the Holst action.
“The FK model also uses the Holst action.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [15] Barrett J W, 1998, Adv. Theor. Math. Phys. source
Not checked in the free run.
- Premisecitation§ B. The incompatibilities between LQG and some SFMs · p. 2
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].”
Quote in the paperSubstantial claim0 of 4 sources checked
- May combine two claims. 8
- Classified as probably not a substantial claim. 6
May combine two claims.
Findingnone given§ Spin-Foams for All Loop Quantum Gravity · p. 1The vertex structure and the vertex amplitude are naturally generalized to arbitrary valency case.
“Also the main tools as the vertex structure and the vertex amplitude are naturally generalized to arbitrary valency case.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Findingnone given§ 2. The vertex spin-network · p. 11The conjugate spin-network is the spin-network induced on the boundary of the spin-foam obtained from (κ,ρ,ι) by removing the neighbourhood of the vertex v.
“The conjugate spin-network itself is the spin-network induced on the boundary of the spin-foam obtained from ( κ, ρ, ι ) by removing the neighbourhood of the vertex v .”
Quote in the paperSubstantial claimMay combine two claims.
Findingnone given§ B. The limits γ = ±∞ , ± 1 , 0 · p. 22The limit γ → ±∞ exists at each level: the Holst action converges to the Palatini action, j+ = j-, k = 0, the EPRL intertwiner converges to the BC intertwiner, and all the EPRL derivation converges to a finite limit.
“That limit exists at each level: the Holst action converges to the Palatini action, j + = j -, k = 0, the EPRL intertwiner converges to the BC intertwiner, all the EPRL derivation converges to a finite limit.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingnone given§ B. The limits γ = ±∞ , ± 1 , 0 · p. 22The limit γ → ±1 cannot be extended to the entire derivation although the Holst action does converge perfectly well to the self dual action.
“The limit that can not be extended to the entire derivation although the Holst action does converge perfectly well to the self dual action is”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Interpretationnone given§ Spin-Foams for All Loop Quantum Gravity · p. 1The notion of embedded spin-foam used allows to consider knotting or linking spin-foam histories.
“In particular the notion of embedded spin-foam we use allows to consider knotting or linking spin-foam histories.”
- Also stated in: § C. The goal of the paper · p. 2 · plain
“The notion of embedded spin-foam we use, allows to consider knotting or linking spin-foam histories.”
Quote in the paperSubstantial claim- Also stated in: § C. The goal of the paper · p. 2 · plain
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone given§ 2. The EPRL intertwiners · p. 19The classical Holst action becomes the Palatini action in the limit γ → ±∞.
“This is consistent with the fact that the classical Holst action becomes the Palatini action in the limit γ →±∞ .”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ A. Summary · p. 22The set of all possible vertices is given by the set of the spin-networks, and the set of all spin-foams can be obtained by gluing the vertices and the static spin-foams.
“The set of all the possible vertices is given by the set of the spin-networks and and set of all spin-foams can be obtained by gluing the vertices and the 'static' spin-foams.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The spin-foams are histories of the embedded spin-network functions defined within the framework of the diffeomorphism invariant quantization of the theory of connections.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Interpretationnone given§ A. Summary · p. 22The 3j, 6j, 10j and 15j symbols used in the context of BC and EPRL intertwiners and amplitudes are related to a specific choice of basis in the space of intertwiners valid in the case of simplicial spin-foams and spin-networks.
“In the literature, the 3 j , 6 j , 10 j and 15 j symbols are used extensively in the context of the BC and EPRL intertwiners and amplitudes. From our point of view, they are just related to a specific choice of the basis in the space of intertwiners valid in the case of the simplicial spin-foams and spin-networks.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationnone given§ A. Summary · p. 22The generalized spinfoam framework is compatible with the original LQG framework and accommodates all the spinnetwork states and the diffeomorphism covariance.
“The generalized spinfoam framework is compatible with the original LQG framework and accommodates all the spinnetwork states and the diffeomorphism covariance.”
Quote in the paperNot a substantial claim- The authors state this as an assumption: The spin-foams are histories of the embedded spin-network functions defined within the framework of the diffeomorphism invariant quantization of the theory of connections.
May combine two claims.
Premisenone given§ B. The limits γ = ±∞ , ± 1 , 0 · p. 23The limit γ → 0 is the limit in which the Holst action is no longer equivalent to the Palatini action and, upon the rescaling by γ, defines a topological theory.
“The limit in which the Holst action is no longer equivalent to the Palatini action and (upon the rescaling by γ ) defines a topological theory is”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ C. The goal of the paper · p. 2The definition of the n-valent Barrett-Crane intertwiner is equivalent to that of [10-12].
“That definition is merely equivalent to that of [10-12], however we spell it out in our notation for the clarity and the sake of precision.”
Quote in the paperNot a substantial claim0 of 3 sources checkedMay combine two claims.
Interpretationreasoning§ B. The incompatibilities between LQG and some SFMs · p. 2The simplicial spin-foam models do not allow knotted or linked states and do not allow graphs with vertices more than 4-valent.
“The simplicial SFMs do not allow such states as well as they do not allow graphs with vertices more than 4-valent.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ B. The limits γ = ±∞ , ± 1 , 0 · p. 23In the limit γ → ±1, the amplitude turns into the SU(2) BF amplitude and the limit theory is the SU(2) BF theory.
“where I is an arbitrary SU(2) intertwiner, and the amplitude turns into the SU(2) BF amplitude. So the limit theory is the SU(2) BF theory.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Premisecitation§ C. The goal of the paper · p. 2The converse construction of a vertex from any given embedded spin-network was described by Reisenberger.
“The converse construction of a vertex from any given embedded spin-network was described by Reisenberger [13].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [13] source
Not checked in the free run.
The paper relies on this without support.
Loop quantum gravity is a quantum theory of gravitational field with all its local degrees of freedom and is not a discretized theory.
“Loop quantum gravity is a quantum theory of gravitational field with all its local degrees of freedom. This is not a discretized theory.”
The paper relies on this without support.
The kinematical observables labelled by curves and 2-surfaces form a complete set for the continuum theory.
“The (kinematical) observables labelled by curves and 2-surfaces form a complete set for the continuum theory.”
The paper relies on this without support.
Loop quantum gravity is a diffeomorphism invariant theory of fields on manifolds.
“Secondly, LQG is a diffeomorphism invariant theory of fields on manifolds.”
The paper relies on this without support.
The diffeomorphisms play a crucial role in loop quantum gravity and are responsible for the diversity of embedded graphs and spin-networks labelling the quantum states.
“The diffeomorphisms play a crucial role in LQG. They are responsible for the diversity of embedded graphs and spin-networks labelling the quantum states.”
The paper relies on this without support.
In the LQG canonical framework, there is no justified way to restrict the graphs to those dual to triangulations of the underlying 3-manifold.
“In the LQG canonical framework, whose quantum states are labelled by embedded spin-networks, there is no justified way to restrict the graphs to those dual to triangulations of the underlying 3-manifold.”
The paper relies on this without support.
Loop quantum gravity admits knotted and linked graphs.
“In particular, LQG admits knotted and linked graphs.”
The authors state this as an assumption.
The spin-foams are histories of the embedded spin-network functions defined within the framework of the diffeomorphism invariant quantization of the theory of connections.
“We combine it in below with the assumption that the spin-foams are histories of the embedded spin-network functions defined within the framework of the diffeomorphism invariant quantization of the theory of connections (see Section II).”
The paper relies on this without support.
The self dual action defines the same Einstein's (Euclidean) gravity.
“That is very strange, taking into account that the self dual action still defines the same Einstein's (Euclidean) gravity.”
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 EPRL amplitudes can be generalized to arbitrary complexes with boundaries given by graphs with nodes of arbitrary valence.
“as in BF theory, the EPRL amplitudes can be generalized to arbitrary complexes with boundaries given by graphs with nodes of arbitrary valence. The extension of the model to arbitrary complexes was first studied in [237, 236] and has been revisited in [140].”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [236]
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“The discreteness of LQG is a property of the quantum representation [18].”
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Chirco, G., Carlo Rovelli, C., & Ruggiero, P. (2015) · Classical and Quantum Gravity · doi:10.1088/0264-9381/32/3/035011 · Cited by 15 · see citing works
Volume simplicity constraint in the Engle-Livine-Pereira-Rovelli spin foam model
Bähr, B., & Belov, V. (2018) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.97.086009 · Cited by 14 · see citing works
Spin Foams, Refinement Limit, and Renormalization
Asante, S. K., Dittrich, B., & Steinhaus, S. (2023) · Springer eBooks · doi:10.1007/978-981-19-3079-9_106-1 · Cited by 21 · see citing works
Thermally correlated states in Loop Quantum Gravity
RUGGIERO, P. (2014) · Electronic Theses and Dissertations Repository (University of Pisa) · Cited by 14 · see citing works
Area law from loop quantum gravity
Hamma, A., Hung, L.-Y., Marcianò, A., & Zhang, M. (2018) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.97.064040 · Cited by 12 · see citing works
Magliaro, E., & Perini, C. (2012) · International Journal of Modern Physics D · doi:10.1142/s0218271812500903 · Cited by 12 · see citing works
Rennert, J. (2017) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.95.026002 · Cited by 13 · see citing works
Commuting Simplicity and Closure Constraints for 4D Spin Foam Models
Han, M., & Thiemann, T. (2013) · Max Planck Digital Library · Cited by 13 · see citing works
- Authors
- Kamiński, W., Kisielowski, M., & Lewandowski, J.
- Affiliations
- University of Warsaw (PL)
- Pennsylvania State University (US)
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2010-03-19
- Volume
- 27
- Issue
- 9
- Pages
- 095006
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2096508020
- Semantic Scholar
- 197267ae67f3e95833ebfaf598932227e87dceb1
- Cited by
- 191
- References
- 40
- Citations by year
- 2026: 4 · 2025: 1 · 2024: 8 · 2023: 13 · 2022: 8 · 2021: 9 · 2020: 7 · 2019: 13 · 2018: 17 · 2017: 7 · 2016: 12 · 2015: 7 · 2014: 16 · 2013: 24 · 2012: 21
- FWCI
- 15.20
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- HAL (submittedVersion, other-oa)
- Trinity's Access to Research Output (TARA) (Trinity College Dublin) (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
- spin foam models, loop quantum gravity, EPRL model, spin network states, vertex amplitude, Barbero-Immirzi parameter, diffeomorphism invariance
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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