I'm studying about quantum-spin (in a syllabus about non-relativistic quantum-mechanics though), but I have some trouble understanding everything. So I would like to ask some small questions, which may perhaps clear up the picture as a whole.
So they are trying to determine the influence of a spatial rotation, to the state in the two dimensional spin-space. So they try to determine the operator $Û_R$ responsible for this rotation (which acts on the spin-space and thus is represented by a 2x2 matrix).
To do this they first try to determine $Û_{\epsilon}$ which is the infinitesimal unitary operator for rotation about an infinitesimal angle $\epsilon$ and they state that such an operator can always be written as: $$Û_{\epsilon}= 1-i\epsilonÂ$$ with $Â$ hermitic. So $Â$ is also acting on the two dimensional spin-space and can be represented by a 2x2 matrix. So far so good.
But now they state that $Â$ has to be constructed in such a way that $<\chi\vert\chi>$ and $<\chi'\vert\chi'>$ and $<\chi'\vert\chi>$ are rotation-invariant. Where $\vert\chi>$ and $\vert\chi'>$ are states of the two dimensional spin-space and $\vert\chi'> = Û_{\epsilon}\vert\chi>$. What I don't understand is:
1) What is meant by rotation invariance? Because if we rotate $\vert\chi>$ and $\vert\chi'>$ by taking the action of a unitary rotation-operator $Û_{\theta}$ , then these inproducts are necessarily conserved, whatever $Â$ may be. So I guess they mean something else.
2) Why must $Â$ be constructed so that this is true? Perhaps this will be clear if I understand 1) though.
Furthermore they state that this can only be the case if: $$Â = \vec{1}_n\cdot\vec{Â} = \frac{1}{2}\vec{1}_n\cdot\vec{\sigma}$$ With $\vec{Â}$ and $\vec{\sigma}$ hermitical vector operators. So I wonder:
3) Why is this only the case if this is true?
4) Somewhat further they also mention that $Tr(\sigma_z) = 0$ but I don't really see here too why this has to be the case.
I hope that someone can answer one (or more) of these questions, I think it will help a lot in understanding the other claims of the chapter about quantum-spin.