How Does Chirality Affect the Dirac Adjoint in Quantum Mechanics?

[synoe]

I have a question about chirality.
When a spinor $\psi$ have plus chirality, namely
$$\gamma_5\psi=+\psi,$$
how can I write this condition for the Dirac adjoint $\bar{\psi}=\psi^\dagger i\gamma^0$?

Let me choose the signature as $\eta_{\mu\nu}=\mathrm{diag}(-,+,+,+)$ and define $\gamma_5\equiv i\gamma^0\gamma^1\gamma^2\gamma^3$. Taking the Dirac adjoint of the left hand side of the above equation,
$$(\gamma_5\psi)^\dagger i\gamma^0=\psi^\dagger\gamma^\dagger_5i\gamma^0\\ =\psi^\dagger(-\gamma^0\gamma^0)\gamma_5i\gamma^0\\ =\bar{\psi}\gamma^0\gamma^0\\ =-\bar{\psi}\gamma_5,$$
so the chirality condition seems to be rewritten as
$$\bar{\psi}\gamma_5=-\bar{\psi}.$$

However, for example, considering a quantity $\bar{\psi}\gamma_5\psi$, this result is inconsistent. Where is wrong in the above calculation?

 

[fzero]

It's not inconsistent, since $\bar{\psi}\gamma_5 \psi =0$ if $\psi$ is chiral. Essentially you have already proved it, but you can also verify it in terms of the chiral components.