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I'm having some problems in understanding the direct products of representation in group theory.

For example, take two right weyl spinors.

We can then write[tex]\tau_{0\frac{1}{2}}\otimes\tau_{0\frac{1}{2}}=\tau_{00}\oplus\tau_{01}[/tex]

Now they make me see that [itex](\sigma_2\psi_R^*)^+\sigma_2\psi_L=-\psi_R^+\psi_L[/itex] (where σ_2 is the second Pauli matrices, + indicates the adjoint and ψ_R is a right weyl spinor (and so is [itex]\sigma_2\psi_R^*[/itex])) and since this is invariant they say that this is the [itex]\tau_{00}[/itex].

Then since [tex]\Delta(\psi_L^+\sigma^\mu\psi_L)=\Lambda^\mu{}_\nu (\psi_L^+\sigma^\mu\psi_L)[/tex] (where [itex]\sigma^\mu=(1,-\sigma_k)[/itex] with 1 as the identity 2x2 matrix and σ as the pauli matrices, and Δ is the total variation of the field) transforms as a vector (with the Lorentz matrix) [itex]\tau_{11}[/itex] is a vector.

Now, there are some things i miss from the discussion above.

First of all, the [itex]\tau_{mn}[/itex] shouldn't indicate the matrices that act on the spinors? Here I'm treating those as the spinors themselves!

In second place, I cannot figure out why [itex]\psi_L^+\sigma^\mu\psi_L[/itex] should itself be a 4vector, since a [itex]\tau_{01}[/itex] acts on (or IS, i don't know) on 3 vectors.

To close, let me make another example:

In an exercise there was told that a second rank tensor [itex]t_{\mu\nu}[/itex] transforms according to the reducible representation [itex]T=\tau_{\frac{1}{2}\frac{1}{2}}\times\tau_{\frac{1}{2}\frac{1}{2}}[/itex] of the Lorentz group O(1,3).

It was asked to find the representation into the sum of irriducible representation.

It's said that the decomposition is [tex]T=\tau_{00}\otimes\tau_{10}\otimes\tau_{01}\otimes\tau_{11}\otimes[/tex]

where the scalar is the trace of the tensor, the [itex]\tau_{10}\otimes\tau_{01}[/itex] os the antisymmetric tensor and the last one is the traceless symmetric tensor.

This is ok, since i guess that this is the only interpretation that make the dimension match.

But here again is the interpretation of the [itex]\tau_{\frac{1}{2}\frac{1}{2}}[/itex] that messes me up: if i treat them as the elements on which the matrices acts upon they are 4-vectors [itex]a^\mu[/itex], and this is ok since the tensor product of two 4vectors is a matrices which can be decomposed into its trace, symmetric and antisymmetric part.

But if I see as matrices I lose all the sense of the exercise

Thanks a lot for the attention!