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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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Written from the paper’s full text.
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
20 No basis given · 16 Not checked yet · 13 Own interpretation · 1 Couldn’t confirm · accuracy not yet measured
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“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.”
What the sources say
- Could not verifycites [6] sourceExperimental — accuracy not yet measured
The Conclusion (p. 19) describes a sum-over-surfaces formulation of quantum gravity, and Appendix B (pp. 21–23) discusses an Ooguri–Crane–Yetter model. Neither passage addresses replacing the Holst action with the Palatini action in an EPRL calculation or obtaining BC intertwiners. Searches for those terms found no passage in our copy that makes the claimed connection. (LLM, conf=0.78)
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- Could not verify
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In the paper
“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.”
boostedstates an associationSingle case- Asserted: passes — The authors put it forward as true.
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- Also stated in: § C. The goal of the paper · p. 2 · boosted
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The simplicial framework of Engle-Pereira-Rovelli-Livine spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity.”
plainstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams.”
plainstates a cause- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“3. The injectivity of I ↦→ ι EPRL ( I ) in the case γ ≥ 1”
plainstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“but, quite surprisingly, the EPRL theory does not resemble a topological theory at all.”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“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,”
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“However the EPRL intertwiner has a limit in that case:”
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
boostedstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“The BC intertwiner space has 0 or 1 dimension per vertex and has been proven to insufficiently accommodate all the gravitational degrees of freedom.”
What the sources say
- Not checked yetcites [9] Perez A, 2003, Class. Quantum. Grav. source
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- It may state two separate claims as one. 8
- Probably not a substantial claim. 6
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Also the main tools as the vertex structure and the vertex amplitude are naturally generalized to arbitrary valency case.”
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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 .”
plainstates an association- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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”
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It may state two separate claims as one.
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In the paper
“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.”
plainstates a cause- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
- Also stated in: § C. The goal of the paper · p. 2 · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This is consistent with the fact that the classical Holst action becomes the Palatini action in the limit γ →±∞ .”
plainstates a cause- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- 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.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plainstates an association- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The generalized spinfoam framework is compatible with the original LQG framework and accommodates all the spinnetwork states and the diffeomorphism covariance.”
plainstates an association- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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”
plainstates a cause- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
What the sources say
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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In the paper
“The simplicial SFMs do not allow such states as well as they do not allow graphs with vertices more than 4-valent.”
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“The converse construction of a vertex from any given embedded spin-network was described by Reisenberger [13].”
What the sources say
- Not checked yetcites [13] source
plainstates an association- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
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?
In the citing paper
“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]
Citations
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Metadata
- 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 › Nuclear and High Energy 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
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