Hi,
First of all I should say I am quite uneducated in group theory, so my question can be very naive. Sorry about that.
I'm reading Srednicki's "Quantum Field Theory" and I have a bit of trouble understanding how one can label Lorentz representations as two different su(2) algebras.
In particular, with the definitions:
$J_i = \frac{1}{2} \epsilon_{ijk} M^{jk}$ ; $K_i = M^{i0}$
where $M$ are the generators of the Lorentz group, one can build:
$N_i = \frac{1}{2}(J_i - iK_{i})$
$N^{\dagger}_i = \frac{1}{2}(J_i + iK_{i})$
so that, in terms of the Ns:
$[N_i,N_j] = i\epsilon_{ijk}N_k$
$[N_i^{\dagger},N_j^{\dagger}] = i\epsilon_{ijk}N_k^{\dagger}$
$[N_i,N_j^{\dagger}] = 0$
now there are two su(2) representations that do not mix with each other.
However, I cannot see how one can have, simultaneously, different dimensions for the two representations since they are obtained from the very same matrices. For example:
$J_i = N_i + N_i^{\dagger}$,
which seems to imply that both Ns have to be matrices of the same dimension.
Thanks a lot.