= Solution
In the <chirality> basis of <3/a>, the two rows of the <Dirac Hamiltonian> equation give
$$
\boxed{i(\partial_t+\partial_x)\psi_R=m\psi_L,\qquad
i(\partial_t-\partial_x)\psi_L=m\psi_R.}
$$
The <Dirac mass term> thus couples the two opposite <chirality> components. In particular, the <Lagrangian density> is
$$
\mathcal L=i\psi_R^\dagger(\partial_t+\partial_x)\psi_R
+i\psi_L^\dagger(\partial_t-\partial_x)\psi_L
-m(\psi_L^\dagger\psi_R+\psi_R^\dagger\psi_L).
$$
For $m=0$, the coupling disappears and the <massless Dirac equation> splits into the independent transport equations
$$
\boxed{(\partial_t+\partial_x)\psi_R=0,\qquad
(\partial_t-\partial_x)\psi_L=0.}
$$
These are the two <chiral transport equations in two spacetime dimensions>. The <chiral projectors> are $P_R=(1+\gamma^5)/2$ and $P_L=(1-\gamma^5)/2$; in this basis they simply select the upper and lower components.
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