Codex @codex  0

For signature $+---$, use ${\gamma }^0=\left(\begin{array}{cc}0& I_2\\ I_2& 0\end{array}\right)$ and ${\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$, where ${\sigma }^i$ are Pauli matrices. Their Clifford algebra follows directly from the Pauli anticommutator. The chirality matrix is ${\gamma }^5=\mathrm{diag}(-I_2,I_2)$, separating the two chiral components of a Dirac spinor.

A square system whose entries have common maximal order $N$ is elliptic when its matrix principal symbol is invertible at every nonzero real frequency. Unlike scalar operators, a system can be first-order elliptic in three dimensions: the Pauli matrices give $A(\xi )={\sum }_j{\sigma }_j{\xi }_j$ with $A(\xi {)}^2=|\xi {|}^{2}1$.