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## Main Question or Discussion Point

I don't quite understand the leptonic SU(2) isospin doublet (electron, neutrino):

As far as I understand, the masses in any multiplet have to be the same always; assuming the neutrino here is massless, so should the electron in this formalism, right? Does this mean that we take the electron in this doublet to be massless (at the Lagrangian level) and then we say that it acquires mass through spontaneous symmetry breaking? Is this the correct way to look at it (i.e. can the electron sit in the same doublet as the neutrino if it acquires a mass "later")?

Also, when looking at the quantum numbers, or charges, under SU(2)x[hypercharge U(1)] breaking down to [electromagnetic U(1)], one says that we know the electric charges for (electron, neutrino) to be (-1,0) and the 3-component of the SU(2) charges, coming from the generator [itex]T^{3}=\frac{1}{2}\sigma^{3}[/itex], to be (-1/2,1/2).

Can someone explain how one works out the eigenvalue of [itex]T^{3}[/itex] for the eigenvectors (electron,neutrino)? Does one have to act with the Pauli matrices onto a Weyl spinor?

As far as I understand, the masses in any multiplet have to be the same always; assuming the neutrino here is massless, so should the electron in this formalism, right? Does this mean that we take the electron in this doublet to be massless (at the Lagrangian level) and then we say that it acquires mass through spontaneous symmetry breaking? Is this the correct way to look at it (i.e. can the electron sit in the same doublet as the neutrino if it acquires a mass "later")?

Also, when looking at the quantum numbers, or charges, under SU(2)x[hypercharge U(1)] breaking down to [electromagnetic U(1)], one says that we know the electric charges for (electron, neutrino) to be (-1,0) and the 3-component of the SU(2) charges, coming from the generator [itex]T^{3}=\frac{1}{2}\sigma^{3}[/itex], to be (-1/2,1/2).

Can someone explain how one works out the eigenvalue of [itex]T^{3}[/itex] for the eigenvectors (electron,neutrino)? Does one have to act with the Pauli matrices onto a Weyl spinor?