m^{-1}\  p^{\mu}\basis{\mu}, \eqno(3.5b)$$ $${\phat}^2 = -1, \eqno(3.5c)$$ where $\Lambda$ is an arbitrary geometric factor (the “spin” degrees of freedom). The unit four-velocity $\phat$ plays the role of the usual $i$ as the generator of quantum phase (when multiplied sinistrally, i.e. on the left). However, each momentum eigenfunction has its own [*unique*]{} generator $\phat$. Note that eq. (3.5a) is invariant under the replacement of ${\bf p} \rightarrow {\bf -p}$, hence we can restrict eq. (3.4b) to $p^4=+E$ without any loss of generality, hence: $p_{\mu}x^{\mu}=\vector{P}\cdot\vector{X}-Et$.

Consider a solution which is a linear combination of eigenfunctions of eq. (3.5a), restricted to $\Lambda=1$. This is a reasonable construct to consider because of the [*superposition principle*]{} of quantum theory: [*the sum of any two physically interpretable solutions will be a reasonably interpretable solution*]{}. Although each eigenfunction separately will have a unit quadratic form, $\overline{\Psi}\Psi=1$, the new solution will display non-scalar “interference” terms due to the differing generators $\phat$. These have been interpreted as a possible useful description as a source for mesonic interactions[@Pezz9302].

Multivectorial Hilbert Space
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In order to have a probabilistic interpretation of quantum theory, we need a positive definite norm. A restricted choice of factor $\Lambda$ in eq. (3.5a) will make the 