Chiral boost weights in two spacetime dimensions (${\psi }_R^{\prime }(x^{\prime })=e^{\alpha /2}{\psi }_R(x),{\psi }_L^{\prime }(x^{\prime })=e^{-\alpha /2}{\psi }_L(x)$)

For the coordinate Lorentz boost $t^{\prime }=t\cosh \alpha +x\sinh \alpha$, $x^{\prime }=t\sinh \alpha +x\cosh \alpha$, the spinor convention ${\psi }^{\prime }(x^{\prime })=S\psi (x)$ gives $S=\exp (\alpha {\gamma }^5/2)$. In the Dirac field in two spacetime dimensions basis, this yields the displayed weights. The derivatives ${\partial }_t+{\partial }_x$ and ${\partial }_t-{\partial }_x$ scale by $e^{-\alpha }$ and $e^{\alpha }$, respectively, so the two kinetic terms are invariant. Opposite weights cancel in the Dirac mass term. Reversing the coordinate boost convention reverses both exponents.