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.
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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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Claims the paper makes
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- Findingcitation§ II. Bivector geometry of a 4-simplex · p. 5
Condition (6) follows from Tr b1[b2,b3] = 9/8 V^2, where V is the volume of the tetrahedron.
“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].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [23] A. Barbieri , “Quantum tetrahedra and simplicial spin networks,” gr-qc/9707010.
Not checked in the free run.
- Findingnone given§ III. Relativistic spin networks · p. 8
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).
“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).”
Quote in the paperNot a substantial claim - Findingnone given§ III. Relativistic spin networks · p. 8
The sum of the bivector operators for a pair is equal to the bivector operator for the middle edge in the vertex.
“The sum of the bivector operators for a pair is equal to the bivector operator for the middle edge in the vertex.”
Quote in the paperSubstantial claim - Findingreasoning§ III. Relativistic spin networks · p. 8
The vertex is given by the same formula if the top two and bottom two pairs are chosen.
“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.”
Quote in the paperNot a substantial claim - Findingreasoning§ III. Relativistic spin networks · p. 7
The space of vertices is non-empty in many cases, by exhibiting a canonical element.
“However we can show that the space is non-empty in many cases, by exhibiting a canonical element.”
Quote in the paperSubstantial claim
- Findingreasoning§ II. Bivector geometry of a 4-simplex · p. 4
Two simple bivectors b1, b2 determine two planes lying in a hyperplane if and only if b1 + b2 is simple.
“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.”
Quote in the paperSubstantial claim - Interpretationreasoning§ IV. The New Model · p. 9
The new state sum model is a 'constrained' doubling of the Crane-Yetter model.
“Thus the model is a 'constrained' doubling of the Crane-Yetter model.”
Quote in the paperSubstantial claim - Interpretationnone given§ I. Introduction · p. 3
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.
“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.”
Quote in the paperNot a substantial claim - Interpretationcitation§ I. Introduction · p. 3
The new state sum is already very close to a topological theory, in that the deviation from the topological theory [4] is only in the imposed constraints.
“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.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [4] 1997, State sum invariants of 4-manifolds, J. Knot Theory Ram. source
Not checked in the free run.
- Premisecitation§ I. Introduction · p. 2
The Plebanski Lagrangian for general relativity has the form of BF theory with gauge group SL(2,C), with an added term which is a constraint.
“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.”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [8] 1977, On the separation of Einsteinian substructures, J. Math. Phys. source
Not checked in the free run.
- cites [9] 1991, Self-dual 2-forms and gravity, Class. Quantum Grav. source
Not checked in the free run.
- cites [10] 1989, Exactly soluble diffeomorphism invariant theories, Commun. Math. Phys. source
Not checked in the free run.
- Study design and conductreasoning§ IV. The New Model · p. 9
The state sum does not depend on the choice of basis for the space of vertices on tetrahedra in the interior.
“This ensures that the state sum does not depend on the choice of basis for the space of vertices on tetrahedra in the interior.”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. Introduction · p. 3
The new state sum model gives a quantum mechanical version of these constraints.
“The new state sum model gives a quantum mechanical version of these constraints.”
Quote in the paperSubstantial claim - Interpretationreasoning§ III. Relativistic spin networks · p. 8
The corresponding classical phase space is the cotangent space T*S^3.
“The corresponding classical phase space is therefore the cotangent space T ∗ S 3 .”
Quote in the paperNot a substantial claim - Interpretationreasoning§ I. Introduction · p. 2
The topological home for this spin is now on triangles.
“The topological home for this spin is now on triangles.”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. Introduction · p. 3
The essential constraint which marks Einstein gravity from BF theory is the simplicity of bivectors.
“Thus the essential constraint which marks Einstein gravity from BF theory is the simplicity of bivectors.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The frame field gives a linear embedding of the triangle in R^4.
- May combine two claims. 11
- Classified as probably not a substantial claim. 3
May combine two claims.
Findingnone given§ II. Bivector geometry of a 4-simplex · p. 4Each geometric 4-simplex determines a set of bivectors satisfying conditions (1)-(6), and each set of bivectors satisfying these conditions determines a geometric 4-simplex unique up to parallel translation and inversion through the origin.
“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.”
Quote in the paperSubstantial claimMay combine two claims.
Findingnone given§ III. Relativistic spin networks · p. 8Each term in the sum satisfies (2) and (4), and satisfies (3) for the two preferred pairs.
“Each term in the sum obviously satisfies (2) and (4), and satisfies (3) for the two preferred pairs.”
Quote in the paperSubstantial claimMay combine two claims.
Findingnone given§ II. Bivector geometry of a 4-simplex · p. 6A geometric simplex has a metric geometry determined by the lengths of its 10 edges; two geometric 4-simplexes have the same edge lengths if and only if they are related by an isometry of R^4.
“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 .”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ III. Relativistic spin networks · p. 8The 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.
“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.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ II. Bivector geometry of a 4-simplex · p. 6The 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.
“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.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S4 (based).
Interpretationnone given§ I. Introduction · p. 2To pass from 3 dimensions to 4 dimensions one has to replace SO(3) with SO(4), or its spin covering, SU(2) × SU(2).
“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).”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ III. Relativistic spin networks · p. 8Intrinsic spin occurs for non-simple representations, which would arise from sections of spinor bundles over S^3.
“In our picture, intrinsic spin occurs for non-simple representations, which would arise from sections of spinor bundles over S 3 .”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ III. Relativistic spin networks · p. 7In 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.
“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.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Interpretationcitation§ V. Conclusions · p. 10The tensor categories in the proposed state sum are similar to those invented in constructing string theories; terms in the state sum can be interpreted as diagrams in string perturbation theory.
“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.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [29] 1989, Classical and quantum conformal field theory, Commun. Math. Phys. source
Not checked in the free run.
May combine two claims.
Premisenone given§ I. Introduction · p. 2In Einstein gravity, 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.
“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.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ I. Introduction · p. 2In three dimensions, quantum gravity is constructed by the Ponzano-Regge state sum model in which representations of SU(2) label edges.
“In three dimensions, quantum gravity is constructed by the Ponzano-Regge state sum model in which representations of SU(2) label edges [12].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [12]
Not checked in the free run.
May combine two claims.
Premisecitation§ II. Bivector geometry of a 4-simplex · p. 6Giving 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.
“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].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [25] J. W. Barrett , M. Rocek, and R. M. Williams, “A note on area variables in Regge calculus,” gr-qc/9710056.
Not checked in the free run.
May combine two claims.
Premisecitation§ II. Bivector geometry of a 4-simplex · p. 6Specifying the 10 areas does not specify the metric geometry uniquely, although it does for almost all configurations.
“By contrast, specifying the 10 areas does not specify the metric geometry uniquely [24], although it does for almost all configurations.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [24] 1994, First order Regge calculus, Class. Quantum Grav. source
Not checked in the free run.
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 Case B, faces are colored by simple representations N_f of SO(4); each edge is colored by a single fixed normalized Barrett-Crane intertwiner i_BC; the face amplitude is A_f(N_f)=dim N_f=(N+1)^2; and the vertex amplitude is the 15j Wigner symbol constructed from ten simple representations and five Barrett-Crane intertwiners.
“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]
The Barrett-Crane model has been obtained by 'quantizing a geometric 4-simplex'.
“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 quantum theory, the condition on bivectors that they describe planes in R^4 translates into the simplicity condition (A1) on the representation.
“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]
The Barrett-Crane model is a simple model obtained as a systematic quantization of the simplicial SO(4) Plebanski action.
“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]
The second possibility, working in the sector where the constraint condition holds, corresponds to the Barrett-Crane model.
“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]
The Barrett-Crane models are obtained as the γ → ∞ limit of the previous constructions.
“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]
The amplitude of a single vertex bounded by ten SU(2) spins and five SU(2) intertwiners is given by the square of the SU(2) Wigner 15j symbol.
“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]
The new vertex amplitude is essentially given by the square of an SU(2) Wigner 15j symbol.
“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]
The original model of [2,3] and the variant pointed out in [11] appear as limiting γ→0 and γ→∞ cases, respectively.
“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]
The vertex was defined in [2,3] only for the euclidean case and in the absence of an Immirzi parameter γ.
“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]
All these models are defined by the same vertex, the square of the SU(2) Wigner 15j symbol; they differ for the class of boundary states considered and their measure in the spinfoam sum.
“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]
The 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.”
From Spin-foams for all loop quantum gravity (2010) · cites this paper as [12]
Most derivations of spinfoam models involve peculiar procedures or intuitive and ad hoc steps.
“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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Graviton Propagator from Background-Independent Quantum Gravity
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<i>so</i> (4) Plebanski action and relativistic spin-foam model
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Discrete Structures in Gravity
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Ponzano–Regge model revisited: I. Gauge fixing, observables and interacting spinning particles
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Loop quantum gravity: an outside view
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Graviton propagator in loop quantum gravity
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Diffeomorphisms and spin foam models
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Complete LQG propagator: Difficulties with the Barrett-Crane vertex
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“by the Barrett-Crane (BC) vertex [11], where the dependence on the intertwiners is trivial.”
BackgroundDiscrete structures in gravity
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Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles
Freidel, L., & Louapre, D. (2004) · CERN Document Server (European Organization for Nuclear Research) · Cited by 143 · see citing works
Alexandrov, S., & Livine, E. R. (2003) · Physical Review D · doi:10.1103/physrevd.67.044009 · Cited by 141 · see citing works
Loop Quantum Gravity: An Inside View
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Group field theory and simplicial gravity path integrals: A model for Holst-Plebanski gravity
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Spin-foams for all loop quantum gravity
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LQG propagator from the new spin foams
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“Previous attempts to derive the graviton propagator from LQG adopted the Barrett-Crane spin foam vertex [21] as model for the dynamics [7, 8, 9, 10, 11, 12, 13, 14, 15] (see also [22, 23, 24] for investigations in the three-dimensional case).”
MethodAsymptotics of relativistic spin networks
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Self-citationAsymptotics of 6 <i>j</i> and 10 <i>j</i> symbols
Freidel, L., & Louapre, D. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/7/303 · Cited by 113 · see citing works
A spin foam model without bubble divergences
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Casting loop quantum cosmology in the spin foam paradigm
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Spin foam models for quantum gravity from lattice path integrals
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Ten questions on Group Field Theory (and their tentative answers)
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Path integral representation of spin foam models of 4d gravity
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Feynman diagrams of generalized matrix models and the associated manifolds in dimension four
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Quantum geometry from phase space reduction
Conrady, F., & Freidel, L. (2009) · Journal of Mathematical Physics · doi:10.1063/1.3257109 · Cited by 97 · see citing works
Projector on physical states in loop quantum gravity
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Carlo Rovelli, C. (2013) · doi:10.22323/1.140.0003 · Cited by 109 · see citing works
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Markopoulou, F. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/5/301 · Cited by 92 · see citing works
Loop quantum gravity: the first 25 years
Carlo Rovelli, C. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/15/153002 · Cited by 124 · see citing works
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New variables for classical and quantum gravity in all dimensions: I. Hamiltonian analysis
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Group integral techniques for the spinfoam graviton propagator
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Spin Foams and Canonical Quantization
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A note on area variables in Regge calculus
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Asymptotics of 10 <i>j</i> symbols
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Group field theories for all loop quantum gravity
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Lorentzian loop quantum gravity vertex amplitude
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Holonomy spin foam models: Definition and coarse graining
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Effective Spin Foam Models for Four-Dimensional Quantum Gravity
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Regularization and finiteness of the Lorentzian loop quantum gravity vertices
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QUANTUM GRAVITY ASYMPTOTICS FROM THE SU(2) 15j-SYMBOL
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Holonomy spin foam models: asymptotic geometry of the partition function
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- 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 › 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
- quantum gravity, 4-simplex, Ponzano-Regge model, state sum models, spin networks
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