address the more general [*meta-monogenic*]{} or “hyper-holomorphic” situation, in the -Euclidean spacetime metric case, which so far has not received much attention in Clifford analysis.

Relativistic Quantum Wave Equations
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The massive [*meta-harmonic*]{} equation over the spacetime domain ${\bf R}^{3,1}$ is known as the [Klein-Gordon equation]{}[@Drell65], $$(\Box^2 - m^2)\ \phi({\bf x})=0, \eqno(3.1)$$ where $\Box^2=\ddd{x}+\ddd{y}+\ddd{z}-\ddd{t}$ is the [d’Alembertian]{} (the generalization of the [Laplacian]{}, but in non-Euclidean spacetime). Note that some authors use the symbol $\Box$ without the square for the d’Alembertian, which makes the use of the same symbol for the factored Dirac operator ambiguous. Physically, eq. (3.1) is interpreted to be the relativistic generalization of Schrödinger’s quantum wave equation. Historically, $\phi$ was a complex scalar function describing a charged spinless particle. If however the function $\phi$ is a vector, then eq. (3.1) is called the [Proca equation]{}, describing a spin-one particle (the “vector boson”equivalently electromagnetism with mass).

Factoring the Klein-Gordon operator minimally requires four anticommuting algebraic entities $\basis{\mu}$, $$(\Box - m)\ \Psi({\bf x})=0, \eqno(3.2a)$$ $$\Psi({\bf x})=(\Box + m)\ \phi({\bf x})=
(\basisu{\mu}\partial_{\mu}+m)\ \phi({\bf x}), \eqno(3.2b)$$ $$\{\basis{\mu},\basis{\nu}\}=2g_{\mu \nu}, \eqno(3.2c)$$ where $\basis{\mu}=g_{\mu\nu}\basisu{\nu}$, and $\Box=\basisu{\mu}\partial_{\mu}$ is the spacetime gradient (we hesitate to call it the “Dirac operator”, because this term is sometimes used by 