# expanding a Dirac spinor in Weyl basis

For a massless electron Dirac spinor in Weyl basis (where $\chi$ is the left-handed spinor and $\eta$ is the right-handed spinor):

$$\begin{pmatrix} \chi \\ \eta \end{pmatrix}$$

How can one decompose this into the electron and positron components?

Thanks!

A single Dirac Spinor, that is $$\Psi= \begin{Bmatrix} \chi_{\alpha} \\ \eta^{\dot{\alpha}} \end{Bmatrix}$$ represents an electron, whereas its charge conjugate $(\bar{\psi} C)^{T}$ could represent its anti-particle, where C is a $4\times4$ matrix.
• Thanks. But from my other question, I had the impression that $\Psi$ could be Fourier expanded in terms of electron and positron components. Please see the responses to the question below: physics.stackexchange.com/questions/348958/… – physics_2015 Jul 29 '17 at 22:18
• Thanks. I'm not sure what are you referring to. Those paragraphs refer how $\Psi$ can be expanded in terms of operators which correspond to positive and negative energies (both electron and positron) and not just either electron or positron. – physics_2015 Jul 29 '17 at 23:29