SU(2) Spinor Question

[div curl F= 0]

Main Question or Discussion Point

I'm having a memory blank on this particular area of field theory. Is the product of two spinors a scalar or scalar type entity and if so, can I treat it like a scalar? (i.e. move it around without worrying about order etc)

i.e.

is $$\Phi_1^{\dagger} \Phi_1$$ a scalar?

and if so does:

$$\Phi_2 \left(\Phi_1^{\dagger} \Phi_1\right) = \left(\Phi_1^{\dagger} \Phi_1\right) \Phi_2$$

where both phi's are SU(2) complex spinors.

Thanks

 

[StatusX]

Yes, if you think back to the definition of SU(2), it is precisely the group under which quantities like $\psi^\dagger \psi$ are scalars (namely, it preserves a hermitian inner product). Also, whether you can commute those quantities is not related to whether they are scalars, but does depend on if they are fermionic or bosonic (though, in this case, anything squared is bosonic, so commutes with everything). Also, if they are operators rather than classical fields you need to be careful.