- 72

- 4

I have a doubt in using the chiral projection operators. In principle, it should be ##P_L \psi = \psi_L##.

$$ P_L = \frac{1-\gamma^5}{2} = \frac{1}{2} \begin{pmatrix} \mathbb{I} & -\mathbb{I} \\ -\mathbb{I} & \mathbb{I} \end{pmatrix} $$

If I consider ##\psi = \begin{pmatrix} \phi \\ \chi \end{pmatrix}##, with ##\phi## and ##\chi## Weyl spinors.

$$ P_L \psi = \frac{1}{2} \begin{pmatrix} \mathbb{I} & -\mathbb{I} \\ -\mathbb{I} & \mathbb{I} \end{pmatrix} \begin{pmatrix} \phi \\ \chi \end{pmatrix} = \frac{1}{2} \begin{pmatrix} \phi-\chi \\ \chi - \phi \end{pmatrix}$$

I get a spinor which still has the RH and the LH component. Shouldn't I get something with only the LH component and the RH's equal to zero?