Krockner delta is an invariant symbol

[RedX]

A representation of SU(2) is "pseudo-real". Can one form the product [tex]\phi^{\dagger i}\rho_{i} [/tex], where [tex]\phi_i [/tex] and [tex]\rho_i [/tex] 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 [tex]\epsilon_{ij}\phi_{i}\rho_{j}=\phi_{i} \rho^{i}[/tex] with the levi-civita symbol.

However, I've seen the product [tex]\phi^{\dagger i}\rho_{i} [/tex] 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).

 

[Evalesco]

So it seems to work in certain cases, but I'm not sure if it's generally valid.

 

[Entropia]

In this case, the product is allowed because phi and rho are not transforming in the fundamental representation of SU(2), but rather in a pseudo-real representation. This means that the representation is not complex, but it is equivalent to a complex representation after a similarity transformation. This allows for the formation of the product \phi^{\dagger i}\rho_{i} as it is equivalent to \phi^{\dagger i}\rho_{i} = \epsilon_{ij}\phi_{i}\rho_{j}.

In summary, while the product \phi^{\dagger i}\rho_{i} may not be allowed for all representations of SU(2), it is allowed for pseudo-real representations such as those used in the context of giving mass to up-quarks. It is important to consider the specific representation and its properties when determining the validity of such products.