= Solution
A <projection operator> satisfies $P^2=P$; an orthogonal <projection operator> also has $P^\dagger=P$. It extracts a subspace while leaving already projected states unchanged. Since $(\gamma^5)^2=1$ and $(\gamma^5)^\dagger=\gamma^5$, the <chiral projectors> are
$$
\boxed{P_L=\frac{1-\gamma^5}{2},\qquad P_R=\frac{1+\gamma^5}{2}},\qquad P_LP_R=0,\quad P_L+P_R=1.
$$
They separate the <Dirac spinor> into opposite <fermion chirality> components. The <Standard Model> weak <charged current> contains $\bar\psi\gamma^\mu P_L\psi$, selecting left-chiral particle fields and the corresponding right-helicity massless <antiparticles>; right-chiral charged <fermions> are weak-isospin singlets. The <weak neutral current> can couple to both <fermion chirality> components with different strengths. <Chirality> and <helicity> should not be equated for a massive <fermion>: a Lorentz boost can reverse its <helicity>, while the field representation defining <chirality> is a different notion.
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