Discrete quantum gravity the Lorentz invariant weight for th

In a recent paper [1] we have constructed the spin and tensor representations of SO(4) from which the invariant weight can be derived for the Barrett-Crane model in quantum gravity. By analogy with the SO(4) group, we present the complexified Clebsch-Gorda

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002 voN 41 1v8601140/cq-r:gviXraDiscretequantumgravity:TheLorentzinvariantweightfortheBarrett-Cranemodel

MiguelLorente

DepartamentodeF´ sica,Univ.deOviedo,33007Oviedo,SpainInst.f¨urtheor.Physik,Univ.Tuebingen,72076Tuebingen,Germany

Abstract.Inarecentpaper[1]wehaveconstructedthespinandtensorrep-resentationsofSO(4)fromwhichtheinvariantweightcanbederivedfortheBarrett-Cranemodelinquantumgravity.ByanalogywiththeSO(4)group,wepresentthecomplexi edClebsch-Gordancoef cientsinordertoconstructtheBiedenharn-DolginovfunctionfortheSO(3,1)groupandthesphericalfunctionastheLorentzinvariantweightofthemodel.

1.ReviewoftheEuclidianmodel

Givenatriangulationofa4-dimensionalRiemannianmanifold,weassignebivectorstothefacessatisfyingappropiateconstraints[2].ThenweidentifythebivectorswithLiealgebraelementsandassociatearepresentationofSO(4)toeachtriangleandatensortoeachtetrahe-drum,invariantunderSO(4).Therepresentationchosenissimple,i.e.j1=j2.

Nowitiseasytoconstructanamplitudeforthequantum4-simplex.Thegraphforaspinfoamisthe1-complex,dualtotheboundaryofthe4-simplexhaving ve4-valentvertices(correspondingtothe vetetrahedra)witheachofthetenedgesconnectingtwodifferentvertices(correspondingtothetentrianglesofthe4-simplexeachsharedbytwotetrahedra).Nowweassociatetoeachtriangle(thedualofwhichisanedge)asimplerepresentationofthealgebraofSO(4)andtoeachtetrahedra(thedualofwhichisavertex)anintertwiner;andtoa4-simplextheproductofthe veintertwinersandthesumforallpossiblerepresentations.Theproposedstatesumis:

Z=∑

Atr

Jtriang∏

.

tetrahedra

∏

Atetr.

simp.

4 simplex

∏

AForthesimplerepresentations(j1=j2)attachedtoeveryfaceofthetetrahedrum,weusedtheelementarysphericalfunctions,thatcanbecalculatedfromtheBiedenharn-Dolginovfunctioninthecasej=m=0,namely,

d(j,0)

sin(2j1+1)τ

0001(τ)=

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