Asymptotics of 6j and 10j symbols

David Louapre (ENS-Lyon); Laurent Freidel (PI)

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Asymptotics of 6j and 10j symbols

Laurent Freidel (PI), David Louapre (ENS-Lyon)

High Energy Physics - Theory · Submitted 17 Sep 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Finally we compute the asymptotic expansion of the 10j symbol which is shown to be non-oscillating in agreement with a recent result of Baez et al. We discuss the physical origin of this behavior and a way to modify the Barrett-Crane model in order to cure this disease.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.87 · mentions code or data availability 0.02.

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arXiv listing abstract v4. Results and the main claim are underlined; show every role.

It is well known that the building blocks for state sum models of quantum gravity are given by 6j and 10j symbols. In this work we study the asymptotics of these symbols by using their expressions as group integrals. We carefully describe the measure involved in terms of invariant variables and develop new technics in order to study their asymptotics. Using these technics we compute the asymptotics of the various Euclidean and Lorentzian 6j-symbols. Finally we compute the asymptotic expansion of the 10j symbol which is shown to be non-oscillating in agreement with a recent result of Baez et al. We discuss the physical origin of this behavior and a way to modify the Barrett-Crane model in order to cure this disease.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	It is well known that the building blocks for state sum models of quantum gravity are given by 6j and 10j symbols.	background 1.00	0.03	0.02	asserted 1.00
s1	In this work we study the asymptotics of these symbols by using their expressions as group integrals.	contribution 0.81	0.90	0.05	asserted 0.97
s2	We carefully describe the measure involved in terms of invariant variables and develop new technics in order to study their asymptotics.	method 0.87	0.95	0.04	asserted 1.00
s3	Using these technics we compute the asymptotics of the various Euclidean and Lorentzian 6j-symbols.	result 0.74	0.96	0.14	asserted 1.00
s4	Finally we compute the asymptotic expansion of the 10j symbol which is shown to be non-oscillating in agreement with a recent result of Baez et al. We discuss the physical origin of this behavior and a way to modify the Barrett-Crane model in order to cure this disease.	result 0.88	0.92	0.70	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from proof or derivation 1.00.

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Original abstract

It is well known that the building blocks for state sum models of quantum gravity are given by 6j and 10j symbols. In this work we study the asymptotics of these symbols by using their expressions as group integrals. We carefully describe the measure involved in terms of invariant variables and develop new technics in order to study their asymptotics. Using these technics we compute the asymptotics of the various Euclidean and Lorentzian 6j-symbols. Finally we compute the asymptotic expansion of the 10j symbol which is shown to be non-oscillating in agreement with a recent result of Baez et al. We discuss the physical origin of this behavior and a way to modify the Barrett-Crane model in order to cure this disease.

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References

Forward citations

Cited by 9 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A New Spin Foam Model for 4d Gravity
gr-qc 2007-08 conditional novelty 7.0 of 10

A new spin foam vertex is constructed that solders neighbouring tetrahedra, is argued to quantize the gravitational rather than the topological sector of Plebanski gravity, and admits a quantized Immirzi parameter.
Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory
hep-th 2005-02 conditional novelty 7.0 of 10

The Ponzano-Regge spin foam amplitude with particle insertions reduces, as the Newton constant goes to zero, to ordinary Feynman diagram evaluation, and its full gravitational expansion resums into a braided non-commu...
Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles
hep-th 2004-01 conditional novelty 7.0 of 10

Gauge-fixing both symmetries of the Ponzano-Regge model yields finite transition amplitudes and a single operator formalism that unifies gauge fixing, time insertion, and the coupling of interacting spinning particles.
On the semiclassical limit of 4d spin foam models
gr-qc 2008-09 conditional novelty 6.0 of 10

In the large-spin limit the FK-gamma spin foam amplitude on an arbitrary triangulation is exponentially suppressed unless the face spins come from a discrete geometry, in which case it reduces to exp(i times the Regge...
Area-angle variables for general relativity
gr-qc 2008-02 conditional novelty 6.0 of 10

Replacing edge lengths with triangle areas and 3d dihedral angles, plus local closure and gluing constraints, reproduces Regge calculus on a 4-simplex and removes the non-locality that blocked area-only formulations.

Show 4 more citing papers

The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
gr-qc 2007-08 conditional novelty 6.0 of 10

The Barrett-Crane spinfoam vertex does not reproduce the correct long-distance graviton propagator in loop quantum gravity, and the failure traces to its independence from intertwiners.
A new spinfoam vertex for quantum gravity
gr-qc 2007-05 conditional novelty 6.0 of 10

Rewriting the SU(2) BF spin foam vertex in a coherent state basis makes triangle normals explicit, and the closure-satisfying (semiclassical) configurations are shown to dominate the intertwiner norm exponentially in ...
Spin Foam Models for Quantum Gravity
gr-qc 2003-01 accept novelty 3.0 of 10

Spin foam models give a finite, background-independent path integral for quantum gravity on a fixed 2-complex, with the Barrett-Crane model derivable from a simplicial Plebanski action and convergent in both Riemannia...
Les Houches lectures on Spinfoam Path Integrals
gr-qc 2026-07 accept novelty 2.0 of 10

A pedagogical review of spinfoam path integrals, from 1d quantum mechanics and 2d BF theory through Ponzano-Regge/Turaev-Viro to the 4d EPRL model, accurate but with no new results.

How the literature uses it

31 arXiv papers cite this in 68 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 55 sentences without a confident label

Uses its method or tool 2 papers

Starting from [ref], we use the character formula ($θ$ is the $SU(2)$ angle) [equation] to divide the integral into several parts (the sines from [ref] cancel with the sines from the integral) [equation] to which stationary point method can be applied – boundary terms contribute non geometric asymptotics as shown in [ref] and will be omitted.

The Barrett-Crane model: asymptotic measure factor · Stationary point conditions · 0.96
The above formula can be used to extract the asymptotic behavior of the 6j-symbol at large spin through a saddle point analysis [ref] .

A New Recursion Relation for the 6j-Symbol · 6j-Symbol as a Group Integral · 0.83

Cites it as support 2 papers

Accordingly, we investigate the consequences of replacing ([ref]) by a similar expression that is conjectured to result from the large spin limit of the spin foam vertex amplitude corresponding to each simplex: [equation] where the term $D_σ$ is dominant but slowly varying; this ansatz is motivated by the results on the asymptotics of the Barrett-Crane model [ref] .

Semiclassical regime of Regge calculus and spin foams · Introduction · 0.96
The suggested problems with the models were of two main types: 1) the absence or very limited nature of correlation between amplitudes of neighboring simplices glued together (the so-called ultra-locality problem); 2) analytical and numerical evidence (in the case of a single simplex) that the simplicial geometries dominating the partition function of the model in the limit of large representations are the degenerate ones [ref] , [ref] , [ref] .

A New Spin Foam Model for 4d Gravity · Introduction · 0.97

Cites it as background 7 papers

The asymptotics of the 6j-symbol has been studied and proved in [ref] , and next-to-leading orders have been obtained in [ref] .

A New Recursion Relation for the 6j-Symbol · Introduction · 0.92
Its asymptotic limit, relevant to the semi-classical behaviour of the model, has been well-studied analytically [ref] .

Sub-leading asymptotic behaviour of area correlations in the Barrett-Crane model · Introduction · 0.82
The most studied Wigner symbol is the 6j-symbol, whose asymptotics is known when all spins are large (the Ponzano-Regge formula, [ref] ), or with one being small (the Edmonds' formula [ref] ).

Asymptotics of Wigner 3nj-symbols with Small and Large Angular Momenta: an Elementary Method · Introduction · 0.96

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