About six months ago, I was able to show qualitatively that the twist field had more than one stable solution, which implies that it could represent more than just the photon and electron variants. I was easily able to show that any set of closed contours (twist paths) were topologically equivalent as long as no contour crosses, and the unitary field twist theory meets this constraint because twist paths are central force attractive (1/r^2 magnitude) but are repulsive by 1/r^3, so the sum is asymptotically repulsive as a twist path approaches another twist path. This was a big breakthrough because now any interlinked loops or knots become unique and stable solutions, opening the door for representing the particle zoo.
I thought, great, now all I have to do is get some quantitative solutions and determine the relative mass to the twist field ring, and that would prove (or disprove, perhaps) the whole twist field concept.
Turns out, that is an extraordinarily difficult problem, and I’ve spent the last six months trying to figure out how to do it. I finally figured out a crude iterative way to do it.
You would think this is a simple LaGrange mechanics problem, but my in-depth study seems to show this isn’t a workable approach. The contour potential energy must be computed at every point, and is the integral of all other points of the entire contour set. In fact, this problem has a stunning similarity to Feynman path integrals, with the complication that everything (all contour points) can move in 3D+T. It cannot be assumed that the contours are symmetric, in fact if this indeed does model real particles, it’s easy to show that most solutions are not symmetric (contours are identical but displaced or rotated). Worse, it’s likely many solutions are not stable in time, so methodologies invoking gauge invariance can’t be used here.
It was almost immediately obvious that trying to find a minimum path for the contour in the 1/r^2 – 1/r^3 field wouldn’t work (the field is an integral of all contours, but the contour path changes in each delta time), hence the LaGrange mechanics couldn’t be used in practice, the resulting differential equations would be phenomenally complex. Simple iterative methods don’t work because there are constraints that are not really workable in an array simulation–the energy of a loop must remain constant, so its length may not vary–but assuming constant spacing of simulation nodes doesn’t work for several reasons. First, the solution loop length is not known, and fixing it defeats the goal of quantitatively finding that length. Second, applying iterations to a chain of segments means that moving one segment means that a large set of adjacent segments also has to move instantaneously–not impossible, but each segment also has its own movement directives, which then would recursively affect the original movement directive. I thought, well, let’s just make the segments stretchable, but adding that into the vector field complicates the computation significantly and appears to destroy the actual force balance between contour elements. It’s a mess, believe me, I tried.
The approach I came up with is to just find any topologically equivalent set of contours and just start with that. Compute the vector field neighborhood around each contour node and then adjust each contour at each contour point until the vector field has minimal magnitude on each contour point. Yes, there is considerable danger that doing this method of iteration of contours will not be stable and converge, but I can see several outcomes that should yield valuable information anyway. First, if nearly all vector field magnitudes point outwards (or all inwards), this means that the contour energies (and hence loop length) should be adjusted, so closure to a stable mass value should be possible regardless of the stability of the contour path shape iteration. Second, there are many topologically unique solutions–that is already trivial to see. If one contour set isn’t timewise stable or does not converge, either a different contour set could be tried or data from the iteration could be used to find a better starting point for the contour path.
I will put together a new sim (technically no longer a sim, but a generator) that does this contour vector field neighborhood and makes it easy to adjust the contour paths. I have no doubt that over time I will come up with better and faster methods to arrive at solutions.
Agemoz
UPDATE: Some additional thinking showed that taking a vector field derivative will yield the contour normal, and the direction will directly give the desired expansion directive. It would be nice if the normal magnitude would also give the minima that would establish the optimized twist path, but it won’t–it will only give the minimum for that point given that the rest of the contour paths are unchanged. As soon as any other portion of the paths change, this minimum will also change. Perhaps there is a LaGrange multiplier scheme that will work to find the minima for all points on all paths. I’m quickly sensing that there are a number of mathematical tools that can be brought to bear on this problem.
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