- #1
StephvsEinst
- 41
- 1
Hey!
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?
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?