RedX
Apr5-09, 03:06 PM
A representation of SU(2) is "pseudo-real". Can one form the product \phi^{\dagger i}\rho_{i} , where \phi_i and \rho_i transform in the fundamental representation?
If a representation is complex, Krockner delta is an invariant symbol, so you can form such a product.
SU(2) is not complex, so the only product you should be able to form is \epsilon_{ij}\phi_{i}\rho_{j}=\phi_{i} \rho^{i} with the levi-civita symbol.
However, I've seen the product \phi^{\dagger i}\rho_{i} before, applied to SU(2) doublets phi and rho, in the context of giving mass to up-quarks (phi would be a Higgs doublet and rho would be a lepton doublet).
If a representation is complex, Krockner delta is an invariant symbol, so you can form such a product.
SU(2) is not complex, so the only product you should be able to form is \epsilon_{ij}\phi_{i}\rho_{j}=\phi_{i} \rho^{i} with the levi-civita symbol.
However, I've seen the product \phi^{\dagger i}\rho_{i} before, applied to SU(2) doublets phi and rho, in the context of giving mass to up-quarks (phi would be a Higgs doublet and rho would be a lepton doublet).