Similarly, in this language, the chiral projection operator is constructed from $$\Upsilon^5
  = \sigma^t \sigma^x \sigma^y \sigma^z
  = - \pmatrix{\ell& 0\cr 0& \ell\cr}$$ However, as with spin, we must multiply by $\ell$ in the correct place, obtaining $${\widehat}\Upsilon^5 = \ell_R \circ \Upsilon^5$$ As a result, even though $\Upsilon^5$ is a multiple of the identity, ${\widehat}\Upsilon^5$ is not, and the operators ${1\over2}(\sigma_t\pm{\widehat}\Upsilon^5)$ project ${{\mathbb H}}^2$ into the Weyl subspaces ${{\mathbb C}}^2\oplus{{\mathbb C}}^2 k$ as desired.

Combining the spin and particle information, over the quaternionic subalgebra ${{\mathbb H}}\subset{{\mathbb O}}$ determined by $k$ and $\ell$, we thus find 1 massive [spin-${1\over2}$]{} particle at rest, with 2 spin states, namely $${e_{\scriptscriptstyle\uparrow}}= \pmatrix{1\cr k\cr}
\qquad
  {e_{\scriptscriptstyle\downarrow}}= \pmatrix{-k\cr ~~1\cr}$$ whose antiparticle is obtained by replacing $k$ by $-k$ (and changing the sign in (\[Proportional\])). We also find 1 massless [spin-${1\over2}$]{} particle involving $k$ moving in the $z$-direction, with a single helicity state, $${\nu_z}= \pmatrix{0\cr k\cr}$$ which corresponds, as usual, to both a particle and its antiparticle. It is important to note that $${\nu_{{\scriptscriptstyle-}z}}= \pmatrix{k\cr 0\cr}$$ corresponds to a massless particle with the same helicity moving in the opposite direction, not to a different particle with the opposite helicity. Each of the above states may be multiplied 