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
Withthehelpofequations(1),(2)and(3)wecanconstructtherepresentationfortheboostoperator,ortheBiedenharn-Dolginovfunction,namely,
iτK (µ,γ)3 djj′m(τ)=ψjmeψj′m=
=
∞
∞
1d j mpj m
(m µ,m+µ)
1
(λ,γ)e iτλd j′ mpj′ m
(m µ,m+µ)
(λ,γ)ρ(λ)dλ
5.SphericalfunctionforthegroupSO(3,1)
GivenalocallycompactgroupGwithcompletelyirrep.Tg,andacompactsubgroupK G,
( nitedimensional),wede nethesphericalfunction
f(g)=TrEkTg=Tr{Tk}
whereEkistheprojectorEk:Tg→Tk.
Thesphericalfucntions[4]arefunctionsonthehomogeneousspaceK\Gandinvariantonrightcosets:f(kg)=f(g)
Weapplythisde nitiontotherepresentationofSO(3,1)givenbyBiedenharnfunction,withµ=0,projectedintotheidentityrepresentationofSO(3),withj=m=0(theelementarysphericalfunction).Wehave
0,γ
f(τ)=Trd000
∞
=
2 iλτd0e
=
e iλτ
4πshπγ
chπ
=
∞
∞ ∞
e iλτ
Γ1
λ+γ
(0,0)2p0ρ(λ)dλ
2
=
1Γ
2
dλ= 2
chπ
λ γ
γ
e
τ
n=0
∑
∞
fortheprincipalserieswithl0=0,l1=iγ.Weobtainbythesameway
f(τ)=
1shτ
e2τ
n
=
1shτ
forthecomplementaryseries,withl0=0,l1=σ,|σ|<1
4