 Solution

In the chirality basis of a, the two rows of the Dirac Hamiltonian equation giveThe Dirac mass term thus couples the two opposite chirality components. In particular, the Lagrangian density isFor $m=0$, the coupling disappears and the massless Dirac equation splits into the independent transport equationsThese 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.