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## GEOMETRIC QUANTIZATION FOR S4 PHYSICS

I happen to have a firm conviction that quantum mechanics is incorrect as a description of the actual universe regardless of its empirical accuracy.  On the other hand, it is probably a theory that is a correct approximate linear model.  Now geometric quantization is an approach to link classical to quantum mechanics.  In the axiomatic geometric quantization, functions $f\in C^{\infty}(M)$ are mapped to operators $\hat{f}$ satisfying some axioms: (a) linearity, (b) constant functions get mapped to constant times identity, (c) Poisson brackets map as $\{ f_1, f_2 \} = f_3$ go to $[\hat{f_1},\hat{f_2}] = -i\hbar \hat{f_3}$.  Now in geometric quantization on a line bundle $B\rightarrow \Simga$ for a symplectic $\Sigma$ with $\hat{f} s = -i\hbar\nabla_{X_f} s + f s$.  Let’s consider the physical space $M\i S^4(1/h)$ and $\Sigma = T^*M$ which is symplectic.  There’s a natural line bunlde for $M$ which is the normal bundle of the embedding, so we can get the geometric quantization ‘prequantum bundle’ here canonically.

See Ritter for the geometric quantization fundamentals:  RitterGeometricQuantization.

The key issue in geometric quantization is the need for polarizations.  Now uncertainty principles exist for compact riemannian manifolds as you can find in the monumental thesis of Erb (UncertaintyPrinciplesRiemannianManifolds).  So the basic idea is that geometric quantization specialized to a hypersurface of the 4-sphere might allow us to dispense with quantum mechanical interpretation altogether and recognize that all quantum mechanical features arise from the geometry of the three dimensional physical universe embedded in an S4 universe.