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
thatleadstothecommutationrelationsoftwoindependentangularmomenta:
Ap,Aq=iεpqrAr
Bp,Bq=iεpqrBr Ap,Bq=0
J3φm1m2=mφm1m2,K3φm1m2=λφm1m2,henceA3φm1m2=1
2(m iλ)φm1m2≡m2φm1m2
SinceJ3andK3commuteweconstructtherepresentationsoftheseoperatorsinthebasiswhereJ3andK3arediagonal,[3]
Noticethatλisarealcontinuousparameter,butm1andm2arecomplexconjugateandm¯1=m2
Inbothbasisthelabelsoftherepresentations(l0,l1)takesthevaluesl0=µ(integerorhalfinteger),l1=iγ(γ∈R)fortheprincipalseries,andl0=0,l1=σ,|σ|<1,σ∈R,forthecomplementaryseries.FortheCasimiroperatorswehave
2 22 2¯J Kψjm=l0+l1 1ψjm
¯)ψjm=l0l1ψjm(J¯·K
plexi edClebsch-Gordancoef cientsandtherepresentationoftheboostopera-torInordertoconnectthebasisψjmandφm1m2wecanusethecomplexi edClebsch-Gordan
coef cients:∞
ψjm=
∞
dλ m1m2|jm φm1m2
(1)
Wehaveusedintegrationbecauseλisacontinuousparameter.Itcanbeprovedthatthesecoef cientesarerelatedtotheHahnpolynomialsofimaginaryargument[3]
(m µ,m+µ)pj m(2) m1m2|jm =f(λ,γ),ff¯=1,dj m
fortheprincipalseries,
m1m2|jm =f
m+µ+1Γ
2 2
and
2
dn=
Γm µ+14π
2
(4)
Γ(m µ+n+1)Γ(m+µ+n+1)|Γ(m+iγ+n+1)|2