Discrete quantum gravity the Lorentz invariant weight for th(2)

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

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