# How is chirality defined for row vectors?

When working with Dirac spinors, the chirality of a spinor field is determined by its $\gamma_5$ eigenvalue, so if $\psi_L$ is left-handed then $$\gamma_5 \psi_L = - \psi_L.$$ Some sources define a left-handed adjoint spinor as $$\bar{\psi}_L = \bar{\psi} P_R$$ and assert that $\bar{\psi}_L$ is left-handed. I'm confused about how the handedness of a row vector is defined, because it cannot be an eigenvector of $\gamma_5$, the matrix multiplication doesn't make sense. Looking at other sources, I think that a row vector like $\bar{\psi}$ transforms in a so-called 'dual representation' but I'm not sure what that means or how it interacts with chirality.

How is chirality defined for row vectors, and what's the mathematical motivation behind it?

The dual spinor representation restricts Lorentz transformations acting on those spinors to unitary ones, via the restriction $$\gamma^0S^\dagger\gamma^0 = S^{-1}$$ Such that the product $\overline{\psi}\psi$ is Lorentz invariant under an $S$ transformation. As for chirality, the easiest way to see that row vectors transform as you stated is to consider the action of the projection transformations $$\overline{\psi} = (\gamma^0\psi)^\dagger$$ Now consider transforming this via a projection, $\psi \longrightarrow \psi_L = P_L\psi$ $$\overline{\psi} \longrightarrow \overline{\psi}_L = (\gamma^0\psi_L)^\dagger = (\gamma^0P_L\psi)^\dagger = (P_R\gamma^0\psi)^\dagger = \overline{\psi}P_R$$ Additionally, the projection transformed between one another via $$\gamma^0P_L\gamma^0 = P_R$$