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}