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
2.SpinorrepresentationofSL(2,C)Wede nethecomplexvaluedpolynomials
p(z,z¯)=∑Cαβzαz¯β
asthebasicstatesofthespinorrepresentations:
Tap(z,z¯)=(a10z+a11)k(a¯10z¯+a¯11)np
a00z+a01
a¯10z¯+a¯11
aa
foranya∈SL(2,C,a≡0001.
a10a11
k n
Thisrepresentationwithlabels(l0,l1)=2+1,k,n∈N,isirreducible,and nitedimensional.Ifweenlargethisrepresentationwithcomplexvalues,k,n,weget
a00z+a01
,Taf(z,z¯)=(a10z+a11)l0+l1 1(a¯10z¯+a¯11)l1 l0 1f
a¯10z¯+a¯11therepresentationbecomesin nitedimensional.Forl0integerorhalfintegerandl1com-plextherepresentationisirreducible.Theprincipalseriesofunitaryrepresentationsfor
SL(2,C)arede ned,ontheHilbertspaceofcomplexfunctions,withscalarproduct(f1,f2)=
f1(z)f2(z)dz,as
a00z+a01
Taf(z)=(a10z+a11)µ 1+iγ(a¯10z¯+a¯11) µ 1+iγf
a10z+a11
withl0=0,l1=σ,σ∈R,|σ|<1.3.RepresentationsofthealgebraofSL(2,C)
weobtaintheunitaryrepresentationsofthealgebraofSL(2,C)inthebasiswheretheoperators
J3andJ¯2arediagonal,namely:J3ψjm=mψjm,J¯2ψjm=j(j+1)ψjm
Itispossiblealsotoconstructacomplexi edoperators
¯=1A
22¯),(J¯ iK
¯+=B¯A
Giventhegeneratorsofrotations(J1,J2,J3)=J¯andofpureLorentztransformations
¯satisfyingthecommutationrelations:(K1,K2,K3)=K
¯+=J¯J,J,p,q,r=1,2,3εJ,J=ipqrrpq
¯+=K¯,K Jp,Kq =iεpqrKr
Kp,Kq= iεpqrJr