I'm trying to follow Schwichtenberg's (Physics from Symmetry, Springer, 2nd edition, page 80) method to derive the representation $(1/2,1/2)$ of the Lorentz Group. A brief overview follows:
We find representations of SU(2), given by an integer or half integer j.
Through complexification, we see that the Lorentz Group SO(1,3) is equivalent to two copies of SU(2).
Before studying the (1/2,1/2) representation, we study the (1/2,0) and (0,1/2) representations (the two 2D representations) that correspond to a left-chiral spinor and a right-chiral spinor, respectively. Furthermore, these spinors are denoted with a lower undotted index and an upper dotted index, respectively (this is known as Van der Waerden notation). Where $\chi $ denotes a spinor: $$\chi _L=\chi _a \;\;\;\;\Big|\;\;\;\; \chi _R=\chi ^{\dot{a}}\tag{1}$$
In his proof, he mentions the fact that the representation has the following property $$(1/2,1/2)=(1/2,0)\otimes (0,1/2) \tag{2}$$
We start by acknowledging that the object (let's call it $v$) that's going to be transformed by this representation (we will act on this object) will transform differently, and without interference, under each of the two copies of SU(2) in this representation.
Then "the object will have 2 indices, each transforming under a separate two-dimensional copy of $\mathfrak{su}2$" and he writes $$v^{\dot{b}}_a\tag{3}$$
Why do we pick the indices to be that way? (By this I mean that I would have naturally picked two letters without much thought, rather than letters that may/may not be dotted and that may be an upper or lower index) Does eq. (2) have to do with it?
I have been reading on building representations through tensor products but I can't seem to connect these two ideas.
Furthermore, he says "Instead of examining $v^{\dot{b}}_a$, let's examine $v_{a\dot{b}}$", and object that we get from lowering an index using the spinor metric. We do this because the basis will then match with the Pauli Matrices. How do we know that the lower indices have a basis given by the Pauli Matrices? And thus why do upper indices have a different basis?
The rest of his method is clear. I just haven't been able to understand these few lines that motivate the rest. Thanks!