How to Make Direct Product of Representations for the Lorentz Group?

[Marco_84]

[SOLVED] Lorentz Representations

I am reading about the Lorentz group on Schweber.
My problem is the following:
I don't really understand how to make the direct product of Representations for this Group.
I know that we need only 2 mubers since the invarints of the gropu are 2.
I know the general reciepie for the SU(n) case is it the same?

Let me ask an example explicity: I've read that Weyl is (1/2,0) Repr and its conjugate is obviously (0,1/2) left handed and right handed; while Dirac is (1/2,1/2).
How can i show that (1/2,0)x(0,1/2)=(1/2,1/2).

The book doesn't go so deep. i think i have to read somewhere else.

thanks in advance
marco

 

[Marco_84]

[Never mind I've figured out by myself, it wws just necessary how works complexification for sl(2,C)..

marco

 

[dextercioby]

Did you figure out how (1/2,0) \otimes (0,1/2) =(1/2,1/2) ?

 

[Marco_84]

Did you figure out how (1/2,0) \otimes (0,1/2) =(1/2,1/2) ?

yeah it is something like this:
(j1,j1')x(j2,j2')=Sum all possible combination of( j,j'). where

j=j1+j2,j1+j2-1,...,|j1-j2|.
j'=j1'+j2',j1'+j2'-1,...,|j1'-j2'|.

because after complexification you have that so(1,3) is isomorphic to su(2)xsu(2)...

bye bye