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Can anyone explain to me the math behind this simple step:

$$ P_L \psi = \psi P_R $$ where $$ P_L = \frac{1}{2} ( 1 + \gamma_5 ) $$ and $$P_R = \frac{1}{2} ( 1 - \gamma_5 )$$.

And why is $$ \bar{\psi }P_R \gamma^{\mu } \psi = \bar{\psi } \gamma^{\mu } P_L \psi ,$$

where $$ \gamma_5$$ and $$\gamma_\mu $$ are Dirac matrices.

Can anyone help??

Edit: The psi's $$ \psi $$ represent Dirac spinors.

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# Simple chirality question

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