Chiral transport equations in two spacetime dimensions ($({\partial }_t+{\partial }_x){\psi }_R=({\partial }_t-{\partial }_x){\psi }_L=0$)

The massless Dirac equation in one spatial dimension separates into opposite chirality sectors. Its initial profiles evolve as ${\psi }_R(t,x)=f(x-t)$ and ${\psi }_L(t,x)=h(x+t)$, by the chain rule along the straight characteristics. Thus wave packets translate without distortion at the speed of light. A nonzero Dirac mass term couples the two sectors, so these separate rigid-translation formulas do not apply to a massive field.