I kept thinking that a continuous twist solution (with no discontinuities) was a good thing, but it’s not, actually. My previous post on electron size (two posts ago) is why–one of the ways to discover whether a particle is point-like or not is to bombard it with high energy particles. If there is any diffuse (distributed) component to the particle, higher energy collisions will eventually plow straight through the muck. But if the particle is truly pointlike, high energy collisions will bounce back even as the energy is increased, which is what we see with electron/positron collisions.

If the twist ring is to model the electron, it cannot be a continuous function. I noted previously that if there were discontinuities in the unitary vector field twist, this would introduce potential energy infinities–but this more accurately models what happens with the electron anyway. It vastly complicates my mathematical description of the twist field, but it has to be true–I have just about convinced myself that there is no possible continuous solution. You can transform the twist angle vector field to a two element scalar field of angles (since I propose that the vector field is unitary). When you do this to a field twist, you create a volume of angle pairs. This volume, say a filled sphere, has zero pairs on the entire surface representing the background angles. At the center lies a value (Pi,0) representing the twist. A continuous vector field can be represented this way as a continuous but periodic scalar field (where 2 Pi maps to 0). A diameter path of the twist has a zero on one end, and 2 Pi on the other–still zero–but with a full range from 0 to 2 Pi along the axis. Even though 2 Pi is the same as zero, continuity requires (by inductive analysis) that any path on any sub-surface of the sphere that includes the twist and the 0 and 2 Pi points, must include the twist as well. There is no “sheath” that can enclose the twist that won’t have the twist on its surface as well–but this is a contradiction since there must be some sphere big enough that the surface has all zero values–no twist. The only way out is if there is some sheath that is discontinous from its internal composition.

This is a major shift in my thinking–probably a good thing as I mentioned, since the point-like electron requires infinite potentials due to its point-like collision behavior.

So how do I model this thing now? I still want to create a simulatable model, but when infinite potentials are allowed, anything goes–I can make up whatever I want. The trouble with that is the same kind of trouble we have with string theory. Add a bunch of dimensions, and glory be!! We can accomodate both the math of general relativity and quantum field theory, but so what, you haven’t really described anything. Good models show basic intrinsics that separate them from models that don’t match reality. Stuffing all possibilities into a model does not give any predictive behavior–and I have to give careful thought to introducing infinities into the twist model so that we still have something useful.

Agemoz

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