(on the [*right*]{}) by an arbitrary complex number.

There is also a single [*complex*]{} massless [spin-${1\over2}$]{} particle, with the opposite helicity, which is given in momentum space by $$\label{Oz}
\hbox{\O}_z=\pmatrix{0\cr 1\cr}$$ As with the other massless momentum space states, this describes both a particle and an antiparticle. Alone among the particles, this one does not contain $k$, and hence does not depend on the choice of identification of a particular quaternionic subalgebra ${{\mathbb H}}$ satisfying ${{\mathbb C}}\subset{{\mathbb H}}\subset{{\mathbb O}}$.

Spin Operators {#SPINOP}
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We saw in the previous section that the spin up particle state ${e_{\scriptscriptstyle\uparrow}}$ is a simultaneous eigenvector of the spin operator and particle projections, that is $$2{\widehat}{L}_z {e_{\scriptscriptstyle\uparrow}}= {e_{\scriptscriptstyle\uparrow}}= \Pi_+ {e_{\scriptscriptstyle\uparrow}}$$ Remarkably, ${e_{\scriptscriptstyle\uparrow}}$ is also an eigenvector of the remaining spin operators, namely $$2{\widehat}{L}_x {e_{\scriptscriptstyle\uparrow}}= -{e_{\scriptscriptstyle\uparrow}}\,k \qquad
  2{\widehat}{L}_y {e_{\scriptscriptstyle\uparrow}}= -{e_{\scriptscriptstyle\uparrow}}\,k\ell$$ although the eigenvalues are not real. Similar statements hold for the corresponding spin down and antiparticle states, although with different eigenvalues.

We find it illuminating to consider the equivalent right eigenvalue problem $$L_z \psi = \psi \lambda$$ for the non-Hermitian operator $L_z$. The operator $L_z$ admits imaginary eigenvalues $\pm \ell/2$, which correspond to the usual spin eigenstates. But $L_z$ also admits other imaginary eigenvalues! These correspond precisely 