= Hermitian square-root construction of Dirac plane waves
{c}
{title2=$u=\binom{\sqrt{p\cdot\sigma}\,\xi}{\sqrt{p\cdot\bar\sigma}\,\xi}$}
In mostly-minus signature and the <Weyl representation>, take $p^0=E=\sqrt{|\mathbf p|^2+m^2}>0$. The <Hermitian matrices> $A=p\cdot\sigma=E-\mathbf p\cdot\boldsymbol\sigma$ and $B=p\cdot\bar\sigma=E+\mathbf p\cdot\boldsymbol\sigma$ commute and obey $AB=m^2I$. 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 $(\not p-m)u=0$ and $v=(\sqrt A\zeta,-\sqrt B\zeta)^T$ solves $(\not p+m)v=0$. They multiply $e^{-ipx}$ and $e^{ipx}$ respectively. For normalized two-spinors, $u^\dagger u=v^\dagger v=2E$. The formula extends by positive-semidefinite square roots to the <massless limit>.
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