Discrete quantum gravity the Lorentz invariant weight for th(3)

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

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