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Archive for February, 2014

For weak and strong forces, the picture of the potential as a connection on a principal bundle whose curvature is the field strength makes a great deal of sense but the ‘original’ gauge theory for electromagnetism is probably not right.  There is something more subtle going on with electromagnetism — the ‘circles’ of the circle bundle lie in the S4(1/h) universe and intersect each other depending on the extrinsic geometry of the physical universe.  In particular we are not quite dealing with a principal U(1)-bundle.  The intersections of these normal circles presumably have an effect on all aspects of electromagnetism.  We don’t really have a local U(1)-gauge invariance.

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In orthodox electromagnetic theory, it is considered to be a U(1) gauge theory.  In particular local gauge invariance is couched in terms of a U(1) principal bundle.  In the S4 theory, it is clear that the right ‘circle bundle’ for electromagnetism are the normal geodesic circles over any point of the three dimensional submanifold of S^4(1/h).  The complexity of the extrinsic geometry of M leads to many possibilities that these geodesics corresponding to multiple points intersect each other.  In particular, the natural ‘invariance’ that one might expect should be based on ‘potentials’ \phi(z) for z \in S^4(1/h) rather than a proper U(1)-bundle which would take intersections into account.  This is an issue which would be very interesting to check by empirical tests.

The ‘phase’ invariance of electromagnetism is ultimately a claim about some sort of invariance of the embedding of the physical universe in S4(1/h). I simply do not believe in arbitrary invariance of variable translates of M in S4(1/h) for electromagnetism.

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I have been incrementally making progress on thinking about complete geometrization of physics via S4 theory.  It is fairly clear to me now that electromagnetism can be described via the normal geodesic circles of a three-dimensional submanifold M in S4(1/h).  Note that the length of these geodesics is 1/\hbar and the time taken for light to traverse the dimeter is 1/\hbar c.  This term appears directly in the STANDARD gauge transformation of electromagnetism in quantum mechanics.

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Chinese thought-reader

In a metaphysical experience last night, I experienced engaging with a Chinese man who seemed to be interrogating me.  A certain woman, S., with whom I had some contact in the past was somehow connected with my appearance before this man who sneered and sought to unravel my intentions.  A computer screen seemed to read off a long sequence of my thoughts in choppy sentences.  Revealed within was my recent (metaphysical) capture of the seat of emperor although I have doubts about its efficacy in the material world, but as with my previous experiences, I find it never useful to retain power but to abandon it with the sword of forgiveness.

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On S4 it is known by the work of Atiyah-Hitchin-Singer that Yang-Mills equations can be solved for any compact gauge group. For a three-dimensional submanifold, one could pull back these solutions. But the interesting issue is unification of electromagnetism and gravity. This works out beautifully with the normal circles, and the submanifold geometry not only gives us the right cosmological term but gives us an interpretation for the term as arising from ambient curvature.
Quantization is purely an effect of living in a 4-sphere universe. The global geometry of the universe forces quantization even with classical deterministic physics.
These issues are not very difficult for an orthodox physicist to check; the mathematical tools had been available from the late 1970s.  In terms of technical complexity, S4 physics is radically simple compared to other efforts at grand unification and it works.  They might even consider it trivial.  However, the real problem they face with this approach is not technical.  It is philosophical.  The resistance of the scientific community to a fourth spatial dimension has been strong since 1900 but the resistance is deeper in the empiricist foundations of science dating back to the original division between magic and science.  Remember that Newton was a practicing alchemist besides being the founder of the mathematical physics that survived for 300 years before revolutions by Einstein and the quantum theory in twentieth century.  It will not be hard to verify that S4 physics is extremely competitive in terms of parsimony.  It is much simpler than other approaches that retain the established interpretation of quantum mechanics.  From a parsimony standpoint, S4 physics is unified and simple (and fits the facts) compared to monumentally complex pieces forcing three space dimensions not fitting together.  The complexity is partly due to the empiricist constraints (such as three spatial dimensions).  It is not very surprising thus that S4 physics (which allows us to open the door to metaphysics) would be simpler.  That, I claim, is because metaphysical reality is objectively real and can only be evaded by highly complex incoherent physics.  We should be grateful for the advances made under the banner of sharp empiricism and yet be able to return to the divide rather than dogmatically refusing a simple unified physics which lets in all the ‘woo woo’ which had been fought for a few centuries.  Let the parsimony of a unified physics speak for Nature rather than being dictated by died in the wool empiricists.

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Imagine a curve on a 2-sphere.  Now consider the normal lines to the curve, which are all great circles.  Now imagine that the curve evolves by moving along the normal lines at some steady pace.  The convex portions of the curve will move closer to each other and the concave portions will move further away.  My picture of the physical universe replacing the curve but in a 4-sphere.  Analogous reasoning will yield that the convex portions will move closer to each other.  This is the picture of gravity via extrinsic geometry that I would like to establish.  Now Einstein has other things happening but I think even without relativity, the picture of gravity translates to a very simple geometric picture.  We also have a second role for the normal circles, which is to act as the principal U(1) bundle for electromagnetism.  Thus we have an extremely geometric and simple unification of electromagnetism and gravity in this picture.  Does the actual universe follow this simple picture?  We have more than sufficient evidence that the universe is a four-sphere — four dimensional from the observed 5, 8, 10, and 12 fold rotational symmetry of crystals, compact by heat equation arguments, but also because of the perfect match of cosmological constant and quantization of energy, by the close match of the theoretical and measured slope of the redshift.  So this geometric picture has some validity established already.  The above unification of gravity and electromagnetism is natural.  Now gauge theory on the 4-sphere is an established subject generally and the Standard Model itself can be reproduced on a 4-sphere.

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My general view is that electromagnetism is probably an SU(2) gauge theory on S4(1/h).  But I just realized that for a three-dimensional submanifold of a scaled 4-sphere, there is a very special circle bundle, perfect of a U(1) gauge theory that does not occur for any other four-manifold.  It is that the normal vector is tangent to the great circle geodesic.  I have high hopes of being able to produce a U(1) gauge theory using this circle bundle which could describe actual electromagnetism in the actual universe.

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