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
<br /> \gamma_5\psi=+\psi,<br />
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,
<br /> (\gamma_5\psi)^\dagger i\gamma^0=\psi^\dagger\gamma^\dagger_5i\gamma^0\\<br /> =\psi^\dagger(-\gamma^0\gamma^0)\gamma_5i\gamma^0\\<br /> =\bar{\psi}\gamma^0\gamma^0\\<br /> =-\bar{\psi}\gamma_5,<br />
so the chirality condition seems to be rewritten as
<br /> \bar{\psi}\gamma_5=-\bar{\psi}.<br />

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.