Gamma 5 Dirac Matrice: Understanding Pseudo-Scalar

[IRobot]

Hi, i wonder somethin'

when I see that definition of the matrice:
$$\gamma^5=\gamma^0\gamma^1\gamma^2\gamma^3$$
I don't see why it is a pseudo-scalar.

 

[Rus Almighty]

If you inspect $$\gamma^5$$,you can see that it anti commutes with $$\gamma^0$$ (while 0 stands for the time coordinate).
so under space inversion the expression will change it's sign
$$\psi^{\dag}\gamma^{0}\gamma^5\psi\rightarrow-\psi^{\dag}\gamma^0\gamma^5\psi$$
and that is a property of a Pseudoscalar

 

[IRobot]

Hum i see, thanks :).