language is strikingly similar, our approach differs fundamentally from theirs in its description of momentum. Ultimately, this hinges on our interpretation of the obvious $SU(2)$ as spin, whereas Schücking and coworkers interpret it as isospin.

**COMPLEX FORMALISM** {#COMPLEX}
=====================

The standard Weyl representation of the gamma matrices in signature is $$\label{CGamma}
\gamma_t = \pmatrix{0& I\cr \noalign{\smallskip} I& 0\cr}
\qquad\qquad
  \gamma_a
  = \pmatrix{0& \sigma_a\cr \noalign{\smallskip} -\sigma_a& 0\cr}$$ where $\sigma_a$ for $a=x,y,z$ denote the usual Pauli matrices [^2] $$\sigma_x = \pmatrix{0& 1\cr \noalign{\smallskip} 1& 0\cr} \qquad
  \sigma_y = \pmatrix{0& -\ell\cr \noalign{\smallskip} \ell& 0\cr} \qquad
  \sigma_z = \pmatrix{1& 0\cr \noalign{\smallskip} 0& -1\cr}$$ and $I$ is the $2\times2$ identity matrix.

The original formulation of the Dirac equation involves the even part of the Clifford algebra, historically written in terms of the matrices $\alpha_a=\gamma_t\gamma_a$ and $\beta=\gamma_t$. Explicitly, the momentum-space Dirac equation in this signature can be written as $$\label{DiracI}
(\gamma_t\gamma_\alpha \, p^\alpha - m \, \gamma_t) \, \Psi = 0$$ where $\alpha=t,x,y,z$ and $\Psi$ is a 4-component complex (Dirac) spinor.

Writing $\Psi$ in terms of two 2-component complex Weyl (or Penrose) spinors $\theta$ and $\eta$ as $$\Psi = \pmatrix{\theta\cr \noalign{\smallskip} \eta}$$ and expanding (\[DiracI\]) leads to $$\label{DiracII}
\pmatrix{p^tI-p^a\sigma_a& -m\cr \noalign{\smallskip} -m& p^tI+p^a\sigma_a}
  \pmatrix{\theta\cr \noalign{\smallskip} \eta} 