Understanding Dirac Spinor Question in QED

[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 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 http://physics.stackexchange.com/questions/6157/list-of-freely-available-physics-books/6167#6167"