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I wanted to prove that:

$$ P_L \bar{ \psi } = \bar{ \psi } P_R $$

And I want to know if I did it correctly.

$$ --- $$

Here is what I did:

$$ P_L \bar{ \psi } = \frac{ ( 1 - \gamma_5 ) }{2} \psi^{ \dagger } \gamma_0, $$

$$ = \frac{ 1 }{2} \psi^{ \dagger } \gamma_0 - \frac{ 1 }{2} \gamma_5 \psi^{ \dagger } \gamma_0 $$

$$ = \frac{ 1 }{2} \psi^{ \dagger } \gamma_0 - \frac{ 1 }{2} \psi^{ \dagger } \gamma_5 \gamma_0 $$

$$ = \frac{ 1 }{2} \psi^{ \dagger } \gamma_0 + \frac{ 1 }{2} \psi^{ \dagger } \gamma_0 \gamma_5 $$

$$ = \psi^{ \dagger } \gamma_0 \frac{ ( 1 + \gamma_5 ) }{2} $$

$$ = \bar { \psi } P_R. $$

$$ --- $$

Where

$$ \psi =

\left(

\begin{smallmatrix}

\psi_1 \\

\psi_2 \\

\psi_3 \\

\psi_4

\end{smallmatrix}

\right) $$

And

$$ P_L = \frac{ ( 1 - \gamma_5 ) }{2} \qquad P_R = \frac{ ( 1 + \gamma_5 ) }{2} $$

And

$$ \bar{ \psi } = \psi^{ \dagger } \gamma_0 $$

And

$$ \gamma_0 \gamma_5 + \gamma_5 \gamma_0 = 0 $$

$$ \mbox{ Note: } \gamma_0 \mbox{ and } \gamma_5 \mbox{ are Dirac's matrices. }$$

$$ --- $$

Is there anything wrong with this?

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# Little issue in Relativistic Quantum Physics

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