A Possible Analytic Solution?

Back from a nice photography trip–somewhat strenuous hiking, but it gave me a chance to figure out a possible Twist Ring solution. I’ve been working on simulation work in an effort to work out the math for the unitary field twist, but the sim was not behaving as expected. Just before I left on the trip, I had a pretty good idea why, but I was able to transform the twist solution to phase space (U2) and from there was able to show that the twist ring that is symmetric axially could not be a valid solution–there would always be a an epsilon region with a phase discontinuity. This helped me visualize that the solution had to be axially dependent–a phase discontinuity simply was not acceptable because then I no longer have a unitary field.
But that really complicates the search for a solution, because if I permit each field vector two degrees of freedom, I am working in SU3 (representing a complex vector field in R3 where each vector is unit magnitude). However, after some thought, I thought of a way to think about it that eventually led to a very exciting solution. The original Twist Ring theory as described in my Paradoxes of the Point Source Electron paper describes the work where I found a unique soliton solution for photons and electrons based on a unitary field twist. However, I did not describe it analytically (mathematically) because I hadn’t yet found it.
I had originally proposed that twists reside in a unitary vector field version of an EM field because quantized particles behaving as E=hv imply a missing degree of freedom, which I hypothesized would be a EM field with constant magnitude. This looked very promising because this provides the appropriate quantization at a given energy without constraining valid frequencies (hence energy) of independent particles. And, it preserved the ability to twist, thus permitting the wonderful results described in the Paradox paper.
Up to now, however, I hadn’t been able to describe the twist without having the solution melt away in time or without a field discontinuity. I tried a variety of twist solutions before finally proving that I could not do it in the simple form in U2 (projected variance along the axis of the twist). I then realized that a solution would have to map onto SU3 with the real and imaginary parts representing the unitary vector angles in polar form. Unable to visualize this much more complex solution, I had a great Aha! moment, and thought, what if I just ignore the region between the twist and the rest of the field? I’ll put a 2Pi twist in the field and surround it with the default field (the field outside the particle, which in our little test case points always to a default direction), separated by a sheath which I just say “I don’t know what’s in there”. But–after thinking for a while, all of a sudden I figured out what could be in there! The twist contains vectors that lie in the radial plane, and so does the default field vectors–but in S3, we have one more degree of freedom, vectors that point tangent to the axis of the twist. All I have to do is make sure that the sheath separating the twist from the default field contains these tangent vectors, and I will have a solution which has no discontinuities, yet can’t melt away without introducing an (impossible) discontinuity. The reason this works when Maxwell field solutions cannot is because the SU3 field is unitary (to enforce quantization) and cannot have field vectors go to zero.
What’s even better–I know there are more twist solutions possible. I’ve found two–the straight line twist (photons) and the ring twist (electrons and positrons, both spin up and spin down). But I can geometrically visualize other twist solutions, which hopefully will yield other particle solutions.

Stay tuned, this is an incredibly exciting result! First I have to confirm it with some analytic and sim work (the sheath is almost certainly not going to confine itself to a cylinder around the twist but will probably mold itself somehow onto the twist). I need to verify that the twist will preserve itself, that there is no way it can dissolve analytically, that is, without requiring a phase discontinuity. Then I need to work out a sim that shows how two twists can form in a field without introducing a phase discontinuity. This is necessary because otherwise spontaneous pair production of particles would be impossible.


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