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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?

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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 $$ \gamma^0P_L\gamma^0 = P_R $$

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