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

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

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



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.

Since electron is a massive particle and obeys the dirac equation, we need two weyl spinors of opposite chirality to represent it. So in general for any spin 1/2 massive particle which obeys the dirac equation, we can not express it just as a single-handed weyl spinor.

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  • $\begingroup$ 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/… $\endgroup$ – physics_2015 Jul 29 '17 at 22:18
  • $\begingroup$ "To be more precise and clear about how the Dirac field contains both matter and antimatter parts to it" - but this does not means that one can decompose a dirac spinor into quantities, which represent a matter part and its anti-matter part. $\endgroup$ – physics student Jul 29 '17 at 22:30
  • $\begingroup$ I think the clarification to your confusion is given in the next two paragraphs of that answer. $\endgroup$ – physics student Jul 29 '17 at 23:03
  • $\begingroup$ 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. $\endgroup$ – physics_2015 Jul 29 '17 at 23:29

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