From the interpretative perspective, my opinion of right way to look at quantum mechanics is through Schroedinger’s project of producing a ‘wave mechanics’ to replace classical point masses in analogy with wave optics replacing geometric optics. This wave mechanics is deterministic but operates on objects that are inherently wavelike — the solutions of Schroedinger equation or Dirac equations. The stochasticity in the *interpretation* of quantum mechanics is essentially misguided as there exist examples where the probability interpretation of solutions of Dirac equation produces negative probabilities (I will return later with a reference). If one then asks, why does our universe contain these wavelike objects, the S4 approach provides a very simple answer: the geometry of our universe is formally very similar to the circle where the waves are the Fourier basis with ‘quantized frequencies’. Objects on a circle universe are the sinusoids and by analogy in a 4-sphere universe are eigenspinors of the natural Dirac operator on the canonical spin bundle. Quantum mechanics in an S4 universe thus can be thought of as describing these wave-like objects interacting with each other which are all eigenspinors of the Dirac operator. It is always worthwhile recalling that in an S4 universe, since gravity is naturally described via Euler-Lagrange equations of an Einstein-Hilbert action to choose hypersurfaces, there is never any explicit conflict between gravity and quantum phenomena; furthermore, the magic of the radius with the Planck constant is that it not only reproduces the right scale for quantum phenomena but it also produces a cosmological constant of which is close to the measured cosmological constant.

There is then a natural question of what the relation between gravity and these spinors on are and using the Lawn-Roth analysis one would identify the effect of spinors on the hypersurfaces to an effect on the *mean curvature*. This simple picture then must be modified when the objects are not only spinor fields but spinor fields twisted by sections of a vector bundle (carrying a representation of a gauge group ).

This picture is sufficiently different from the orthodox picture that a great deal more work might be required to understand it fully. It is easier to understand the space-only picture because we can consider adding an absolute time or consider a Lorentzian metric on so that eigenspinors of the Dirac operator are multiplied by for appropriate (the eigenvalue of Dirac) and similarly for twisted Dirac operators.

The above picture, or any other picture that is based on an background space gains it’s claim to describing the actual universe first and foremost from the empirical evidences that this model of the universe predicts cleanly the slope of the observed redshift.

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