Your equation (3) comes from the following steps. First, a dotted index transforms in the complex conjugate representation of an undotted index. For a tensor product, each index transforms according to its own representation. Thus pa˙a↦AabˉA˙a˙bpb˙b=Aabpb˙bA†˙b˙a
where on the left side of the equal sign we have elementwise complex conjugation. Putting the conjugate matrix on the right we have to take a transpose to get order of indices right.
In reasoning about (4) and (5) you are neglecting the transformation of σμa˙a. The correct description of the relation pa˙a=σμa˙apμ is that the 4-vector representation is equivalent to the (12,0)⊗(0,12) representation, by means of the linear transformation σμa˙a:V→(12,0)⊗(0,12)
meaning that
σμa˙a belongs to the space
(12,0)⊗(0,12)⊗V∗, on which the (double cover of the) Lorentz group acts. In fact, it acts like
σμa˙a↦Aa˙bσνb˙bA†˙b˙a(Λ−1)μν
so that
Aμa˙apμ indeed has the correct transformation law.
This post imported from StackExchange Physics at 2015-01-13 11:51 (UTC), posted by SE-user Robin Ekman