The lorentz group consists of rotations + boosts.
L+ = proper orthodronous lorentz group
= (det(T)=1,T00=>1 for T an element in L+)
General T:
B o s t
o
s R
t
Where R is a rotation.
L+ is not compact.
universal covering group of L+ = SL(2,C)
Generators:
(Jps)nu=-i(nnp*nsm-nns*npu
=> [Jun,Jps]=i(nupJns-nnpJus+nsuJpn-nsnJpu)
Define J1=J23, Ki=Ji0.
=> [Ji,Jj]=ieijkJk, [Ki,Jj]=ieijkKk, [Ki,Kj]=-ieijkJk
Define Mi=.5*(Ji+iKi), Ni=.5*(Ji-iKi)
=> [Mi,Nj]=0,[Mi,Mj]=ieijkMk,[Ni,Nj]=ieijkNk
=> = SU(2) x SU(2) (nearly, -since it's not compact, it is not unitary)
Action on eigenstates:
Let |ma>|nb> with a= -m,...,m and b= -n,...,n => dim of irrep - (2m+1)(2n+1)
M3|ma>|nb>=a|ma>|nb>
M1+-iM2|ma>|nb>=(m(m-1)-a(a+-1))^.5 |m+-a>|nb>
Defining representation: (1/2,1/2)