Hermitian square-root construction of Dirac plane waves ($u=\left(\genfrac{}{}{0ex}{}{\sqrt{p\cdot \sigma }\xi }{\sqrt{p\cdot \bar{\sigma }}\xi }\right)$)

In mostly-minus signature and the Weyl representation, take $p^0=E=\sqrt{|\mathbf{p}{|}^2+m^2}>0$. The Hermitian matrices $A=p\cdot \sigma =E-\mathbf{p}\cdot \mathbf{\sigma }$ and $B=p\cdot \bar{\sigma }=E+\mathbf{p}\cdot \mathbf{\sigma }$ commute and obey $AB=m^{2}I$. Their positive square roots of a matrix therefore obey $\sqrt{A}\sqrt{B}=mI$. It follows that $u=(\sqrt{A}\xi ,\sqrt{B}\xi {)}^T$ solves $(/p-m)u=0$ and $v=(\sqrt{A}\zeta ,-\sqrt{B}\zeta {)}^T$ solves $(/p+m)v=0$. They multiply $e^{-ipx}$ and $e^{ipx}$ respectively. For normalized two-spinors, $u^{†}u=v^{†}v=2E$. The formula extends by positive-semidefinite square roots to the massless limit.