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