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Relativistic spin networks and quantum gravity
Open accessAbstract
Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2)×SU(2). Relativistic quantum spins are related to the geometry of the two-dimensional faces of a 4-simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3-simplex. This leads us to suggest that there may be a four-dimensional state sum model for quantum gravity based on relativistic spin networks that parallels the construction of three-dimensional quantum gravity from ordinary spin networks.
Checked 2026-10-06 · 9 references not checked yet
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TL;DR
The paper defines relativistic spin networks using the spin covering SU(2) × SU(2) of SO(4), relating quantum spins to bivectors on the 2-faces of a 4-simplex. It proposes a 4-dimensional state sum model for quantum gravity as a constrained doubling of the Crane-Yetter topological state sum.
Key findings
- A geometric 4-simplex is characterized by 10 bivectors on its triangles satisfying six conditions; the theorem states that any set satisfying these conditions determines a geometric 4-simplex unique up to parallel translation and inversion through the origin. Condition (6) is supported by the calculation \(\mathrm{Tr}\, b_1[b_2,b_3] = \frac{9}{8} V^2\), where \(V\) is the volume of the tetrahedron.
- Relativistic spin networks label faces with irreducible representations of SU(2) × SU(2); the classical simplicity condition for bivectors is transcribed quantum-mechanically by requiring \(j=k\), called simple representations.
- The proposed state sum is a “constrained” doubling of the Crane-Yetter model; for each 4-simplex, weights are linear combinations of products of two 15j-q symbols with identical representations but opposite orientation.
- A canonical vertex is exhibited as the usual SU(2) vertex doubled, with the middle edge labelled by \((j,j)\), showing the vertex space is non-empty in many cases.
Methods
- Classical Euclidean 4-dimensional geometry: an abstract 4-simplex \((01234)\) is embedded in \(\mathbb{R}^4\), and oriented triangles determine bivectors \(b = \frac{1}{2} e_{01} \wedge e_{02} \in \mathbb{R}^4 \wedge \mathbb{R}^4\); the paper proves that six listed properties characterize the embedding.
- Quantum construction: bivectors are replaced by bivector operators in representations of SU(2) × SU(2), with simplicity imposed by \(j=k\); a state sum on a triangulated 4-manifold is defined by summing weights over labellings of 2-simplexes and tetrahedra.
- The analysis is mathematical and axiomatic; no empirical data or numerical simulations are used.
Limitations
- The paper states it has an axiomatic description of the space of vertices but not an explicit construction; if the only allowable vertex is the one explicitly constructed, the state sum does not seem to give a topological invariant of manifolds, so its relationship to topology is unclear.
- The authors state they have not analysed the quantum analogues of inequalities (5) and (6), do not yet have an analogue of the Ponzano-Regge asymptotic formula linking the model to the Einstein action, and lack a complete argument that it contains local degrees of freedom or propagating plane waves.
- The paper states that the key remaining issue is how probability amplitudes behave under refinement of the triangulation, analogous to ultraviolet problems; possible routes to a finite theory are suggested but not established.
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In the paper
“Condition (6) follows from the calculation Tr b 1 [ b 2 , b 3 ] = 9 8 V 2 , where V is the volume of the tetrahedron [23].”
What the sources say
- Not checked yetcites [23] A. Barbieri , “Quantum tetrahedra and simplicial spin networks,” gr-qc/9707010.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe 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 simple representations of SU(2) × SU(2) occur as the summands of the Hilbert space of functions on S 3 , which has the obvious action of SO(4).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes? Based: unclearThe 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 sum of the bivector operators for a pair is equal to the bivector operator for the middle edge in the vertex.”
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In the paper
“The orthogonality relation of the 6j-symbol shows that the vertex is given by the same formula if the top two and bottom two pairs are chosen.”
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In the paper
“However we can show that the space is non-empty in many cases, by exhibiting a canonical element.”
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In the paper
“Two simple bivectors b 1 , b 2 determine two planes lying in a hyperplane (of dimension 3) if and only if b 1 + b 2 is simple.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA 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
“Thus the model is a 'constrained' doubling of the Crane-Yetter model.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA 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 new state sum has a very natural relationship with Regge calculus in a new formulation, in that the classical version of the theory has the metric as its variable in the same way as Regge calculus.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes? Based: unclearIn the paper
“However, it is already very close to a topological theory, in that the deviation from the topological theory [4] is only in the imposed constraints.”
What the sources say
- Not checked yetcites [4] 1997, State sum invariants of 4-manifolds, J. Knot Theory Ram. source
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In the paper
“In fact, the Plebanski Lagrangian [8,9] for general relativity has the form of BF theory [10] with gauge group SL(2 , C ), with an added term which is a constraint.”
What the sources say
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- It may state two separate claims as one. 11
- Probably not a substantial claim. 3
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
“Theorem. Each geometric 4-simplex determines a set of bivectors satisfying these conditions, and each set of bivectors satisfying these conditions determines a geometric 4-simplex unique up to parallel translation and inversion through the origin.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt 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
“Each term in the sum obviously satisfies (2) and (4), and satisfies (3) for the two preferred pairs.”
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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
“A geometric simplex has a metric geometry, which is determined by the lengths of its 10 edges. Thus two geometric 4-simplexes have the same edge lengths if and only if they are related by an isometry of R 4 .”
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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
“The corresponding momentum in the classical case is l µν x µ p ν , for a point ( x, p ) ∈ T ∗ S 3 . This is the dual pairing of so(4) with the simple bivector x ∧ p in R 4 , which explains how simple bivectors are associated to triangles.”
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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
“Therefore the bivector data determines a set of edge lengths. Two sets of bivector data determine the same edge lengths if and only if the bivectors are all transformed simultaneously by an orthogonal rotation.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt 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
“The idea is that to pass from 3 dimensions to 4 dimensions one has to replace SO(3) with SO(4), or its spin covering, SU(2) × SU(2).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. 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
“In our picture, intrinsic spin occurs for non-simple representations, which would arise from sections of spinor bundles over S 3 .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt 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
“Now in the relativistic case, the theory is for four-dimensional manifolds and the relativistic spin network vertices reside in a tetrahedron. There will be one bivector operator for each face of a tetrahedron, and so this will be a four-valent vertex.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“It is interesting that the sort of tensor categories which go into the state sum we are proposing are so similar to the ones invented in constructing string theories [29]. Terms in our state sum can be interpreted as diagrams in string perturbation theory, by connecting together the diagrams we are associating to the 4-simplexes, and interpreting elements in the representations as string states.”
What the sources say
- Not checked yetcites [29] 1989, Classical and quantum conformal field theory, Commun. Math. Phys. source
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. 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
“However, in Einstein gravity it is important that so(4) ∗ is identified with R 4 ∧ R 4 , as B is constrained to be of the form B = ∗ e ∧ e , where e is a vector valued one-form.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“In three dimensions, quantum gravity is constructed by the Ponzano-Regge state sum model in which representations of SU(2) label edges [12].”
What the sources say
- Not checked yetcites [12]
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“However giving the areas of the four faces of a tetrahedron does not specify the geometry of the tetrahedron uniquely, so that two 4-simplexes cannot be glued together by specifying the areas of the 16 triangles in their union [25].”
What the sources say
- Not checked yetcites [25] J. W. Barrett , M. Rocek, and R. M. Williams, “A note on area variables in Regge calculus,” gr-qc/9710056.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“By contrast, specifying the 10 areas does not specify the metric geometry uniquely [24], although it does for almost all configurations.”
What the sources say
- Not checked yetcites [24] 1994, First order Regge calculus, Class. Quantum Grav. source
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The authors state this as an assumption.
The frame field gives a linear embedding of the triangle in R^4.
“Assuming the frame field gives a linear embedding of the triangle in R 4 , the integral over this triangle,”
The paper relies on this without support.
In the classical limit, when the representation is large, vector operators behave as ordinary vectors in 3-dimensional space with probability near to 1.
“The intuition is that in the classical limit, when the representation is 'large', these behave as ordinary vectors in 3-dimensional space, with probability near to 1.”
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
“Case B. In this case, as in case A, faces are colored by simple representation N f of SO (4). Each edge, however, is colored by a single, fixed, intertwiner i BC , the ( normalized ) Barrett-Crane intertwiner [1].”
From Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space (2000) · cites this paper as [1]
In the citing paper
“The Barrett-Crane model has been obtained by 'quantizing a geometric 4-simplex' [1].”
From Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space (2000) · cites this paper as [1]
In the citing paper
“Simple representations emerge in this process: in the quantum theory, the condition on bivectors that they describe planes in R 4 translates into the the simplicity condition (A1) on the representation.”
From Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space (2000) · cites this paper as [1]
In the citing paper
“However, there is a simple model that can be obtained as a systematic quantization of the simplicial SO(4) Plebanski’s action.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [59]
In the citing paper
“The second possibility is to work in the sector where . We will show later that this corresponds to the famous Barret-Crane model [59] with the various limitations discussed in Section 10.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [59]
In the citing paper
“The Barrett-Crane models [60, 59] can easily be obtained in the present framework as a suitable limiting case γ → ∞ of the previous constructions.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [59]
In the citing paper
“Following [2, 3, 7], the amplitude of a single vertex bounded by ten SU (2) spins j ab , a, b = 1 , ..., 5 and five SU (2) intertwiners i a is given by”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [2]
In the citing paper
“A different vertex has been recently introduced in [2, 3]; its amplitude essentially given by the square of an SU (2) Wigner 15 j symbol.”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [2]
In the citing paper
“The original model of [2, 3] and the variant pointed out in [11] appear as limiting γ → 0 and γ → ∞ cases, respectively.”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [2]
In the citing paper
“The vertex was defined in [2, 3] only for the euclidean case, and in the absence of an Immirzi parameter γ .”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [2]
In the citing paper
“All these models ([2, 3, 11, 12],) are defined by the same vertex, namely the square of the SU (2) Wigner 15 j symbol; they differ for the class of boundary states considered and their measure in the spinfoam sum.”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [2]
In the citing 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.”
From Spin-foams for all loop quantum gravity (2010) · cites this paper as [12]
In the citing paper
“Indeed, although spinfoam models have been derived in a number of different manners [6, 7, 8, 21], most derivations involve peculiar procedures or intuitive and ad hoc steps.”
From Flipped spinfoam vertex and loop gravity (2008) · cites this paper as [21]
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Metadata
- Authors
- Barrett, J. W., & Crane, L.
- Affiliations
- University of Nottingham (GB)
- Kansas State University (US)
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 1998-06-01
- Volume
- 39
- Issue
- 6
- Pages
- 3296-3302
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W2069771760
- Semantic Scholar
- de1b77a7e917bd8c77d8f612728f6d4d4b3d470b
- Cited by
- 404
- References
- 25
- Citations by year
- 2026: 3 · 2025: 4 · 2024: 11 · 2023: 11 · 2022: 13 · 2021: 7 · 2020: 10 · 2019: 15 · 2018: 13 · 2017: 14 · 2016: 7 · 2015: 9 · 2014: 12 · 2013: 29 · 2012: 22
- FWCI
- 13.39
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (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
- quantum gravity, 4-simplex, Ponzano-Regge model, state sum models, spin networks
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