Dirac spinor question

[Jim Kata]

In Qed they replace the current vector J^{\alpha} by ie\overline{\Psi}\gamma^{\alpha}\Psi. I don't understand how this is done. I understand that
J^{A\dot{A}}=J^{\alpha}{\sigma^{A\dot{A}}_\alpha} but if J^{A\dot{A}} is a rank two matrix then J^{A\dot{A}}=\psi^{A}\psi^{*\dot{A}}+\phi^{A}\phi^ {*\dot{A}}. So shouldn't J^{\alpha} be written as something like
ie(\bar{\Psi}\gamma^{\alpha}\Psi +\bar{\Phi}\gamma^{\alpha}\Phi)?

[Simon_Tyler]

The two 2-component Weyl spinors (and their complex conjugates) need to be combined into a single Dirac spinor (and its Dirac conjugate). And the Dirac matrices are constructed as an off-block-diagonal combination of the Pauli matrices - the Weyl/chiral representation (http://en.wikipedia.org/wiki/Gamma_matrices#Weyl_basis).

This is explained in almost any textbook on supersymmetry (where the 2-component formalism is very common) or, e.g., in QFT text by Srednicki (http://physics.stackexchange.com/questions/6157/list-of-freely-available-physics-books/6167#6167). Also see http://arxiv.org/abs/0812.1594