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In the Weyl basis we can separate the spinor field into 2 components: the right-chiral spinor and the left-chiral spinor. Each of these fields has again 2 components which are coupled. What is the physical interpretation of these 2 components that make up the left-chiral (or right-chiral) field?

In the Dirac basis the interpretation of the 4 components is:
1. Electron spin-up
2. Electron spin-down
3. Positron spin-up
4. Positron spin-down

So my question is what is the corresponding interpretation in the Weyl basis (in the massless case). Is it like this?
1. Left-chiral electron $\psi_{4}$
2. Left-chiral positron $\psi_{3}$
3. Right-chiral electron $\psi_{2}$
4. Righ-chiral positron $\psi_{1}$

If this is the case than I don't understand why the left-chiral electron $\psi_{4}$ couples to left-chiral positron $\psi_{3}$ as can be seen in the equations:

$$ \partial_{t} \psi_{4} + \partial_{x} \psi_{4} - i\partial_{y} \psi_{4} + \partial_{z} \psi_{3} = 0 $$ $$ \partial_{t} \psi_{3} + \partial_{x} \psi_{3} + i\partial_{y} \psi_{3} - \partial_{z} \psi_{4} = 0 $$ $$ \partial_{t} \psi_{2} - \partial_{x} \psi_{2} + i\partial_{y} \psi_{2} - \partial_{z} \psi_{1} = 0 $$ $$ \partial_{t} \psi_{1} - \partial_{x} \psi_{1} - i\partial_{y} \psi_{1} + \partial_{z} \psi_{2} = 0 $$

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1 Answer 1

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The meaning of different components in the chiral representation are

  1. Left-handed spin up,
  2. Left-handed spin down,
  3. Right-handed spin up,
  4. Right-handed spin down.

Spin up and down are with respect to some arbitrary axis, which we often set to the $z$-axis .

Electron and positron(or negative-energy electron) states are identified from the solutions of the Dirac equation. It turns out that in the massless limit (for which the chiral representation is most convenient), the left- and right-handed sectors decouple, and electron(positron) states in the left-handed sector are left(right)-handed, viz. the spin is anti-parallel(parallel) to the momentum, and similarly for the right-handed sector.

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    $\begingroup$ Do you have any source for this? I'm desperately searching for any book or text that explains how "electron and positron states are identified from the solutions of the Dirac equation". I searched for quite a while, but failed to find any text that does this by using the chiral representation of a Dirac spinor and intepretes an electron in terms of the two Weyl spinors inside a Dirac spinor $\begin{equation} \Psi = \begin{pmatrix} \chi_L \\ \xi_R \end{pmatrix} ,\end{equation}$ $\endgroup$
    – jak
    Nov 18, 2014 at 10:42
  • $\begingroup$ @JakobH My understanding mostly comes from Chapter 5 of Weinberg's book on QFT (Vol. 1). I've seen questions you've posted recently, and they are something I would normally like to think about and try to answer. But I just don't have time these days. Sorry about that. $\endgroup$
    – higgsss
    Nov 19, 2014 at 2:19
  • $\begingroup$ thanks for your reading suggestion. I will have a look at it $\endgroup$
    – jak
    Nov 19, 2014 at 9:09

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