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 -bundle. The intersections of these normal circles presumably have an effect on all aspects of electromagnetism. We don’t really have a local -gauge invariance.
Archive for February, 2014
In orthodox electromagnetic theory, it is considered to be a gauge theory. In particular local gauge invariance is couched in terms of a 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 . The complexity of the extrinsic geometry of 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’ for rather than a proper -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.
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 and the time taken for light to traverse the dimeter is . This term appears directly in the STANDARD gauge transformation of electromagnetism in quantum mechanics.
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.
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.
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.