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## S4 MODEL IMPLIES ERGODIC GEODESIC FLOW

### zulfikar.ahmed@gmail.com<zulfikar.ahmed@gmail.com>

4:16 PM (0 minutes ago)

 to jfrenkel, jhansen, jharrington, jharris, jhp, jhricko_4, jhs, jhutasoi, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira

Although the round four sphere itself has totally integrable geodesic flow, generic hypersurfaces of the sphere have positive topological entropy and therefore ergodic geodesic flow.  This is a relatively easy corollary of a theorem that says that for compact manifolds an open dense set of generic metrics have positive topological entropy and therefore ergodic geodesic flow, so some analysis shows that generic embedding metrics into a four sphere (which should be for radius 1/h) which can be characterized by additional Gauss-Codazzi relations also have ergodic geodesic flow and should therefore fall under the quantum ergodicity theorems.  The attached paper has the generic metrics result.

The key issue is how to squeeze the solution of a constant coefficient Riccati equation.  For the Heston model the key point for getting an explicit solution for the probability density is by solving a Riccati equation $y' = ay^2 + by + c$.  For the time-fractional case you want to solve instead $(y/S)' = a y^2 + by + c$ where $S= S_{\alpha}(t)$ is the waiting time of the fractional Poisson process.  The trick is to replace the time variable $t$ with $\int_0^t S_{\alpha}(s) ds$.  So you have to insert this expression into the explicit formula for the Heston probability density in DragulescuYakovenko2002 eq (18) say.
Einstein never liked quantum mechanics for very good reason.  It’s an insane description of the world.  I guess I am chasing a ‘hidden variable’ theory which I am sure is right because otherwise I would not be able to explain the redshift so accurately.  The physical three-dimensional universe is a hypersurface (including the electromagnetic field) of a single fixed-radius four-sphere whose fixed radius $1/h$ gives rise to all quantization.  I am confused about why Kaluza-Klein’s extra dimensions should be small.  It makes little sense to me because quantization in the scale of $h$ would require a large circle of radius $1/h$ and not small circles.  Quantum mechanics is some sort of mathematical gimmick with a Hilbert space and noncommuting operator observables that capture the small scale by localizing and linearizing at the tangent space of the physical universe in the small and then going berserk with eigenfunction expansion basis.  On the other hand, it is frustrating to work in a vacuum on my interpretation of what makes sense.  Kaluza-Klein is a guide but there is more going on because there is no $U(1)$-invariance in an S4 universe because there is a normal bundle but no principal $U(1)$-bundle.  This should lead to experimental predictions correcting MAXWELL’S EQUATIONS which I no longer believe could be describing the universe exactly.  Maxwell’s equations would be true if the extra structure were a principal circle-bundle with fibers of length $2\pi/h$ but I cannot believe that this can be right because it should be an embedding $M\i S^4(1/h)$ where there are circles of the right length along the normal vector but they can intersect.  It is the curvature of the ambient 4-sphere that should produce a ‘cosmological constant’ and also produce the correct level for redshift (not expansion) which are both the right order of magnitude.  This is frustrating, because I am sure I am right but I am also living in the wrong age for this because a century of physics went the wrong way and got itself so confused with so much technical work that recovery seems impossible almost.