Interpretation of rank 2 spinors While inspecting the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group and defining a right-handed spinor with upper dotted index and a left-handed spinor with lower undotted index and therefore
$$v^{\dot{a}}_b = v^{\nu} \sigma_{b \nu}^{\dot{a}}= \begin{pmatrix}
  v_1+iv_2&-(v_0+v_3) \\v_0-v_3&-(v_1-iv_2)
\end{pmatrix}$$
and 
$$v^{ \dot{a} b} = \epsilon^{bc} v_c^{\dot{a}}$$
with spinor metric 
$$\epsilon^{bc}= \begin{pmatrix}
   0&1 \\ -1&0
\end{pmatrix}$$
it can be seen that the rank 2 spinor  $v^{ \dot{a} b}$  has exactly the same transformation properties as a 4-vector $v_{\mu}$. But in the same way it can seen that for example $v^{\dot{a}}_b$ transforms differently. Are there any physical interpretations of those objects? Do they describe specific particles (not vector particles?)? Any help or reading suggestion would be much appreciated.
 A: The Lorentz group is the set of matrices that preserve the four-vector dot product,
$v^2 = v_0^2 - v_1^2 - v _2 ^2 - v _3 ^2 $
Both $ v ^\mu $ and $ v ^{ \alpha \dot{\alpha} } \equiv v ^ {\mu} (\sigma_\mu) ^{ \alpha \dot{\alpha} } $ transform under the Lorentz group. They just transform under a different representation (but still under the same transformation!). 
Furthermore, there is a one-to-one correspondence between every vector $ v ^\mu $ and $ v ^{ \alpha \dot{\alpha} } $. So you can always choose to work either in one representation or the other.
Note that the only important distinction between the two representations is that for every boost matrix for $ v ^\mu $, $ \Lambda ^{ \mu } _\nu $, there exist two equivalent boost matrices for $ v ^{ \alpha \dot{\alpha} } $, $N _{\alpha \beta}$. This turns out to fix some the inconveniences associated with the fundamental (four-vector) Lorentz group representation.
For example if you have four-vector and you want to boost it in a given direction. You can find the matrix $\Lambda ^\mu_\nu $ and multiply $v ^\mu $ as 
$
v' ^\mu = \Lambda _\nu ^\mu v ^\nu
$
or you could find the matrix $N _{\alpha \beta } $ and apply it to the rank 2 spinor:
$
v' _{\alpha \dot{\alpha}} = N _\alpha ^ \beta v _{\beta \dot{\gamma}} N ^{\ast \, \dot{\gamma}} _{\dot{\alpha}}
$
Both methods are equivalent.
