Representation of the Lorentz group and correspondence with the $SL(2,\mathbb{C})$ group

We can find a correspondence between the restricted Lorentz group and the group $$SL(2,\mathbb{C})$$ if to each coordenate $$x^{\mu}$$ we associate a $$2\times 2$$ hermitian matrix $$X$$ given by $$X = x^{\mu} \sigma_{\mu},$$ where $$\sigma_{0} = 1$$ (the identity matrix) and $$\sigma_{i}$$, $$i = 1,2,3$$, are the Pauli matrices. It's easy to see that $$det X = (x^{0})^{2} - (x^{1})^{2} - (x^{2})^{2} - (x^{3})^{2},$$ which is a invariant. Now, if I do a Lorentz transformation $$x^{\mu} \rightarrow x'^{\mu} = \Lambda^{\mu}_{\ \ \ \nu}\ x^{\nu}$$, the transformation in the matrix $$X$$ is $$X \rightarrow X' = A X A^{\dagger} = x'^{\mu} \sigma_{\mu}.$$ Since $$det X = det X'$$, $$|det A | = 1$$ and I can choose $$det A = 1 \Rightarrow A \in SL(2,\mathbb{C})$$. So, for each Lorentz transformation $$\Lambda$$ I can associate two elements $$\pm A$$ of the group $$SL(2,\mathbb{C})$$. My first question is: do the matrices $$A(\Lambda)$$ forms a spinorial representation of the Lorentz group? How can I show it?

The book of Wu-Ki Tung says that these matrices satisfy the relation $$A \epsilon A^{T} = \epsilon$$, where

$$\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$

is a kind of metric tensor for the matrices $$A$$. I can see that this equation is analogous to the equation $$\Lambda^{T} g \Lambda = g$$, where $$g$$ is the metric tensor of the Minkowsk space. But how can I deduce this relation? (mainly the form of the matrix $$\epsilon$$) the book does'nt explain it.

• – Qmechanic May 31 at 2:50