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:

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


1 Answer 1


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.

  • 1
    $\begingroup$ 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
    Commented 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
    Commented Nov 19, 2014 at 2:19
  • $\begingroup$ thanks for your reading suggestion. I will have a look at it $\endgroup$
    – jak
    Commented Nov 19, 2014 at 9:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.