Dirac Gamma Matrices: Is Invariance Under Lorentz Transformation?

[eoghan]

Hi!
I can define
$\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$
I know that the four gamma matrices $\gamma^i\:\:,\;i=0...3$ are invariant under a Lorentz transformation. So I can say that also $\gamma ^5$ is invariant, because it is a product of invariant matrices.
But this equality holds:
$$\gamma ^5=\frac{i}{4!}\epsilon_{\mu\nu\rho\sigma}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}$$
and this expression is not invariant!
So, is $\gamma^5$ invariant or isn't it?

 

[fzero]

$\gamma^\mu$ transforms like a 4-vector.

 

[eoghan]

But $\gamma^5$ transforms like a pseudo-scalar because of $\epsilon_{\mu\nu\rho\sigma}$