representations of the Lorentz group in 4 dimensions

2012-07-18T13:05:29Z

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.

User - MathOverflow

representations of the Lorentz group in 4 dimensions