0,1,8,9$ (or equivalently $\alpha=t,x,y,z$) is complex, and corresponds to the 4-dimensional momentum of the particle, with squared mass $$m^2 = p_\alpha p^\alpha = -\det(\pi(P))$$ Inserting (\[PComplex\]) into (\[StringII\]), we recover precisely (\[DiracVI\]), and we see that we have come full circle: Solutions of the [*octonionic*]{} Weyl equation (\[Weyl\]) are described precisely by the [*quaternionic*]{} formalism of Section \[QUATERNIONIC\], and the dimensional reduction scheme determines the mass term.

Particles {#PARTICLES}
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For each solution $\psi$ of (\[SolII\]), the momentum is proportional to $\psi\psi^\dagger$ by (\[Proportional\]). Up to an overall factor, we can therefore read off the components of the 4-dimensional momentum $p_\alpha$ directly from $\pi(\psi\psi^\dagger)$. We can use a Lorentz transformation to bring a massive particle to rest, or to orient the momentum of a massless particle to be in the $z$-direction.

If $m\ne0$, we can distinguish particles from antiparticles by the sign of the term involving $m$, which is the coefficient of $\sigma_k$ in $P$. Equivalently, we have the particle/antiparticle projections (at rest) $$\Pi_\pm
  = {1\over2} \left( \sigma_t \pm \sigma_k \right)$$ If $m=0$, however, we can only distinguish particles from antiparticles in momentum space by the sign of $p^0$, as usual; this is the same as the sign in (\[Proportional\]). 