I am working through some course notes where the aim is to derive the equations of motion satisfied by the left handed and right handed components of the Dirac spinor ##\psi##. From the Dirac lagrangian, we have $$\mathcal L = \bar \psi (i \not \partial P_L - m P_L)\psi_L + \bar \psi (i \not \partial P_R - m P_R)\psi_R$$ The next line is $$\mathcal L = \bar \psi_L i \not \partial \psi_L - \bar \psi_R m \psi_L + \bar \psi_R i \not \partial \psi_R - \bar \psi_L m \psi_R$$ I understand where the terms involving m come from but I am not sure about the other two. Can anyone help?(adsbygoogle = window.adsbygoogle || []).push({});

Another question is to do with the acting of the projection operators onto Dirac spinors. Since $$P_L \psi = P_L \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix} = \frac{1}{2}(1-\gamma^5)\begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix} = \frac{1}{2} \left( \begin{pmatrix} 1 & 0 \\0 & 1 \end{pmatrix} - \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \right) \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}$$ which does not project out the left handed component. If I use the Weyl representation of ##\gamma^5##, it works but I am trying to understand why not in the Dirac representation of the ##\gamma^5##.

Thanks!

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# Projection operators and Weyl spinors

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