= 0$$ This leads us to identify the momentum 4-vector with the $2\times2$ Hermitian matrix $${{\bf p}}= p^\alpha \sigma_\alpha
 = \pmatrix{p^t+p^z& p^x-\ell p^y\cr \noalign{\smallskip} p^x+\ell p^y& p^t-p^z}$$ where we have set $\sigma_t=I$, which reduces (\[DiracII\]) to the two equations $$\begin{aligned}
  -{\widetilde}{{\bf p}}\,\theta - m\eta &=& 0 \label{DiracIIIa}\\
  -m\theta + {{\bf p}}\,\eta &=& 0 \label{DiracIIIb}\end{aligned}$$ where the tilde denotes trace-reversal. Explicitly, $${\widetilde}{{\bf p}}= {{\bf p}}- {{\rm tr\,}}({{\bf p}}) I$$ which reverses the sign of $p^t$, so that $-{\widetilde}{{\bf p}}$ can be identified with the 1-form dual to ${{\bf p}}$. This interpretation is strengthened by noting that $$-{\widetilde}{{\bf p}}{{\bf p}}= \det({{\bf p}}) I = p_\alpha p^\alpha I = m^2 I$$ where the identification of the norm of $p^\alpha$ with $m$ is just the compatibility condition between (\[DiracIIIa\]) and (\[DiracIIIb\]).

**QUATERNIONIC FORMALISM** {#QUATERNIONIC}
==========================

The [*quaternions*]{} ${{\mathbb H}}$ are the associative, noncommutative, normed division algebra over the reals. The quaternions are spanned by the identity element $1$ and three imaginary units, usually denoted $i$, $j$, $k:=ij$. Quaternionic conjugation, denoted with a bar, is given by reversing the sign of each imaginary unit. Each imaginary unit squares to $-1$, and they anticommute with each other; the full multiplication table then follows using associativity.[^3] However, 