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There is a very nice account of Schroedinger’s approach to his equation and wave mechanics here.  One of the reasons I have been very excited by the idea that a scaled 4-sphere is our actual universe is that especially on a sphere waves occur naturally.  This is the content of Fourier’s original discovery that all reasonable functions on a circle can be approximated by sinusoids and the intuition persists on higher dimensional sphere.  The radius of $1/h$ furthermore makes the 4-sphere naturally produce the scaling constant $h^2$ from the Laplacian on a unit sphere which produces natural Schroedinger operators.  But this is a formality only.  The more fundamental physical intuition that led Schroedinger to wave mechanics is precisely the analogy of geometric optics refined to wave optics; so Hamiltonian geometrical mechanics refines to the wave mechanics.  The opportunity with S4 physics is to dig even deeper and produce an EXPLANATION for why nature operates by wave-particle duality and wavelike behavior more generally — it is enforced by the shape of the universe.  I have claimed many times before that quantum mechanics is an approximation of a classical physics but classical physics in the sense of the INTERPRETATION of the wave-function with modulus squared being a probability distribution.  Deterministic waves do exist without interpretations in terms of probability distributions.  Thus far my approach has been very sweeping rather than detailed technical work but has succeeded in showing that the universe must be a scaled 4-sphere.  This approach has already led to the idea that electromagnetism is in fact not a $U(1)$ gauge theory.  I will proceed now with poring through Schroedinger’s papers on wave mechanics to understand the natural formulation of wave mechanics on a 4-sphere.  I suspect that the geometry and function theory on hypersurfaces of a four-sphere can produce a complete coherent physics explaining the phenomena we have known to date.