Vector under Chiral transformation

[PhyAmateur]

Was reading how do vectors transform under chiral transformation and found the following:

If $$V^\mu$$ is a vector; set $$ V^\mu = \bar{\psi} \gamma^\mu \psi= $$

$$\bar{\psi}\gamma^\mu e^{-i\alpha\gamma^5}e^{i\alpha\gamma^5}\psi =$$
$$\bar{\psi}\gamma^\mu\psi = V^\mu $$

My questions are why is it that vector takes the form $$V^\mu = \bar{\psi}\gamma^\mu\psi$$ and does the same thing apply to $$\partial_\mu$$ I mean is $$\partial_\mu$$ written as $$\bar{\psi}\gamma^\mu\psi$$ ?

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?