I see it now. Reviewing the field structure of an electron moving close to the speed of light, which has a field discontinuity, showed the way (my quandary about how a field discontinuity has to exist for E field twists to be possible, but I couldn’t resolve that with the quantizing behavior of a unitary field. I worked for quite a while trying to find a way that static discontinuities could exist, but finally came to the point that they couldn’t, hence no static twists are possible. Then I guessed that special relativity would provide a context where twists would be stable, but didn’t see a viable way until now.

In special relativity, there is the concept of light cones defined for every point in spacetime. For each point in R3+T, there are two 4D conic shaped regions that define other time-separated points and other space-separated points. The time-forward looking cone defines those points which could interact with the originating point, and the time-backwards-looking cone defines those points that could affect the behavior of the originating point. The key is this–no spatially separated point can influence or be influenced by the originating point. The light cone itself defines a boundary where no connection can be made, and thus is a perfect candidate for defining (for that point) a permissible field discontinuity. The trouble is, the discontinuity must be permissible for all points at a given time, which means a static point of discontinuity isn’t possible, nor is any point moving on any timelike trajectory (spacelike trajectories would be faster than the speed of light and can’t happen). Only a point moving along a world line path **on the light** cone can provide a valid solution. A world line on the cone means that the point is moving at the speed of light. **A discontinuity can only exist if it moves with this point!** But–a twist isn’t just a point, it has to have a length defined by the wavelength of the photon? No problem, as long as this “length” only lies on a path on the light cone. A string of worldline points, (lying on a single world line on a light cone) can all sustain a twisting field discontinuity somewhere on the light cone for each of the points. The twist field vector never has to match the default background field vector until the discontinuity vanishes (matches the background field vector) which can only happen with integral rotations–our quantization of a twist in a background field vector direction.

Thus, the photon cannot have a spherical shell (or topological equivalent) for its discontinuity–the sheath of the twist mentioned several posts prior. The discontinuity in its entirety for a particular set of points must lie on the 3D light cone of every point where a discontinuity exists. As time passes, the location of the possible loci of the discontinuity will move–but if the discontinuity is more than a point at a given point in time, then the allowable region of the future (and past) discontinuity loci must be the intersection of lightcones from all points–a severe constraint on the discontinuity. A point discontinuity can exist as a path on a light cone and could sustain a twist. A (spacelike) line or volume discontinuity, straight or not, moving at c cannot work because no light cone points at one end of the line/volume will lie on light cone points at the other end of the line, and thus we will have a reachable discontinuity in space–already shown to be an impossibility. Only a point moving at c (in 3D+T this will be a line on the light cone) will sustain a discontinuity, and thus cannot enclose a region, and thus a photon cannot have a finite volume radius.

But what I think does work is a twist about a point moving at speed c. This twist of a point, or rather, in our field case, a field vector at that point, can do whatever it wants as long as it moves at speed c–if any slower, the discontinuity becomes spacelike and we get a contradiction–a spatial discontinuity in the field, that cannot work. Note that the point twist does not have to move at c in a straight line, but, for example, could move at speed c in a circle or other path (a spiral in 3D + T, this is legal because the light cone moves as the point moves. It can be generalized that any path will work as long as the point’s speed is c). A twist ring still seems to be a workable solution and is a soliton that has the characteristics (mass/energy) of an electron.

More to come…

Agemoz

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