Discrete quantum gravity the Lorentz invariant weight for th(5)

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

6.ASO(3,1)invariantforthestatesumofspinfoammodel

AsinthecaseofeuclideanSO(4)invariantmodel,wetakeanondegenerate nitetriangulationofa4-manifold. Weconsiderthe4-simplicesinthehomogeneousspaceSO(3,1)/SO(3)∽H3,

thehyperboloidx|x·x=1,x0>0andde nethebivectorsboneachfaceofthe4-simplex,thatcanbespace-like,nullortimelike( b,b >0,=0,<0,respectively).

Inordertoquantizethebivectors,wetaketheisomorphismb= Lbab=1

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