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Elementary scattering theory for Schroedinger equation in flat 3-space can be based on plane-wave solutions.  Since $1/h$ is quite large relative to scales in the domain of the scattering analysis, there is little difficulty in supposing that the flat 3-space phenomena approximate scattering on a hypersurface of S4(1/h) relatively flat.  In particular, scattering theory for a flat 3-space locally described would approximate that on a hypersurface of S4(1/h) and vice versa which justifies modeling the latter situation which we can claim to be the exact theory.  The substantive issue for us is that so long as we exhibit the scattering theory behavior for a well-defined set of objects on $S^4(1/h)$ that cover cases of interest in the quantum field theoretic view directly, we could provide a theory that reproduces some of the important calculations in quantum field theory without accepting the theory itself.
More precisely, let us start with the fixed set of objects the eigenspinors of the Dirac operator on the 4-sphere.  These can be associated with the possible particles or fields.  Add to these the eigenspinors of twisted Dirac operators also on $S^4(1/h)$.  Now consider defining scattering theory for these objects directly.  We want to consider the Dirac equation directly rather than the Schroedinger equation because we have an exact match of the spectrum with integer multiples of $\pm(2+k)h$ which is not quite the same as the Laplace spectrum.  A priori, given that the geoemetry of Dirac wave scattering is very similar to Schroedinger wave scattering and that flat 3-space locally is going to be well-approximated by a piece of $S^3(1/h)$ for example there is no reason to expect that the quantum mechanical scattering theory might be quite close to scattering of these classical fixed objects.  But we can produce theories that have substantially different content than quantum field theory by the latter.