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The fundamental equation of S4 physics is just the wave equation for $\mathcal{D}^2$ where $\mathcal{D}$ is the Dirac equation on $S^4(1/h)$.  Consider the physical universe $M$ that is a smooth (or at worst some $C^{\alpha}$ closed hypersurface that evolves with the wave equation
$(\frac{\partial^2}{\partial t^2} - \mathcal{D}_{S^4})u =0$
for spinor fields which are defined in detail in the great book Spin Geometry of Lawson and Michelson.  Now consider the restriction of this equation to $M$ and linearize at a tangent space.  This gives you a linear Dirac equation, which is I claim what Dirac got here.  Details will have to be worked out but intuitively it’s clear because $\mathcal{D_M} - \mathcal{D_{S^4(1/h}}$ is the potential which in the case of the Dirac operator has been worked out rigorously by many people such as Hijazi et. al.