Back after dealing with some unrelated stuff. I had started work on a new simulator that would test the Twist Theory idea, and in so doing ran into the realization that the mathematical premise could not be based on any sort of electrostatic field. To back up a bit, the problem I’m trying to solve is a geometrical basis for quantization of an EM field. Yeah, old problem, long since dealt with in QFT–but the nice advantage of being an amateur physicist is you can explore alternative ideas, as long as you don’t try to convince anyone else. That’s where crackpots go bad, and I just want to try some fun ideas and see where they go, not win a Nobel. I’ll let the university types do the serious work.
OK, back to the problem–can an EM field create a quantized particle? No. No messing with a linear system like Maxwell’s equations will yield stable solitons even when constrained by special relativity. Some rule has to be added, and I looked at the old wave in a loop (de Broglie’s idea) and modified it to be a single EM twist of infinitesimal width in the loop. This still isn’t enough, it is necessary that there be a background state for a twist where a partial twist is metastable, it either reverts to the background state, or in the case of a loop, continues the twist to the background state. In this system–now only integer numbers of twists are possible in the EM field and stable particles can exist in this field. In addition, special relativity allows the twist to be stable in Minkowski space, so linear twists propagating at the speed of light are also stable but cannot stop, a good candidate for photons.
If you have some experience with EM fields, you’ll spot a number of issues which I, as a good working crackpot, have chosen to gloss over. First, a precise description of a twist involves a field discontinuity along the twist. I’ve discussed this at length in previous posts, but this remains a major issue for this scheme. Second, stable particles are going to have a physical dimension that is too big for most physicists to accept. A single loop, a candidate for the electron/positron particle, has a Compton radius way out of range with current attempts to determine electron size. I’ve chosen to put this problem aside by saying that the loop asymptotically approaches an oval, or even a line of infinitesimal width as it is accelerated. Tests that measure the size of an electron generally accelerate it (or bounce-off angle impact particles) to close to light speed. Note that an infinitely small electron of standard theory has a problem that suggests that a loop of Compton size might be a better answer–Heisenberg’s uncertainty theorem says that the minimum measurable size of the electron is constrained by its momentum, and doing the math gets you to the Compton radius and no smaller. (Note that the Standard Model gets around this by talking about “naked electrons” surrounded by the constant formation of particle-antiparticle pairs. The naked electron is tiny but cannot exist without a shell of virtual particles. You could argue the twist model is the same thing except that only the shell exists, because in this model there is a way for the shell to be stable).
Anyway, if you put aside these objections, then the question becomes why would a continuous field with twists have a stable loop state? If the loop elements have forces acting to keep the loop twist from dissipating, the loop will be stable. Let’s zoom in on the twist loop (ignoring the linear twist of photons for now). I think of the EM twist as a sea of freely rotating balls that have a white side and a black side, thus making them orientable in a background state. There has to be an imaginary dimension (perhaps the bulk 5th dimension of some current theories). Twist rotation is in a plane that must include this imaginary dimension. A twist loop then will have two rotations, one about the loop circumference, and the twist itself, which will rotate about the axis that is tangent to the loop. The latter can easily be shown to induce a B field that varies as 1/r^3 (formula for far field of a current ring, which in this case follows the width of the twist). The former case can be computed as the integral of dl/r^2 where dl is a delta chunk of the loop path. This path has an approximately constant r^2, so the integral will also vary as r^2. The solution to the sum of 1/r^2 – 1/r^3 yields a soliton in R3, a stable state. Doing the math yields a Compton radius. Yes, you are right, another objection to this idea is that quantum theory has a factor of 2, once again I need to put that aside for now.
So, it turns out (see many previous posts on this) that there are many good reasons to use this as a basis for electrons and positrons, two of the best are how special relativity and the speed of light can be geometrically derived from this construct, and also that the various spin states are all there, they emerge from this twist model. Another great result is how quantum entanglement and resolution of the causality paradox can come from this model–the group wave construction of particles assumes that wave phase and hence interference is instantaneous–non-causal–but moving a particle requires changing the phase of the wave group components, it is sufficient to limit the rate of change of phase to get both relativistic causality and quantum instantaneous interference or coherence without resorting to multiple dimensions or histories. So lots of good reasons, in my mind, to put aside some of the objections to this approach and see what else can be derived.
What is especially nice about the 1/r^2 – 1/r^3 situation is that many loop combinations are not only quantized but topologically stable, because the 1/r^3 force causes twist sections to repel each other. Thus links and knots are clearly possible and stable. This has motivated me to attempt a simulation of the field forces and see if I can get quantitative measurements of loops other than the single ring. There will be an infinite number of these, and I’m betting the resulting mass measurements will correlate to mass ratios in the particle zoo. The simulation work is underway and I will post results hopefully soon.
Agemoz
PS: an update, I realized I hadn’t finished the train of thought I started this post with–the discovery that electrostatic forces cannot be used in this model. The original attempts to construct particle models, back in the early 1900s, such as variations of the DeBroglie wave model of particles, needed forces to confine the particle material. Attempts using electrostatic and magnetic fields were common back then, but even for photons the problem with electrostatic fields was the knowledge that you can’t bend or confine an EM wave with either electric or magnetic fields. With the discovery and success of quantum mechanics and then QFT, geometrical solutions fell out of favor–“shut up and calculate”, but I always felt like that line of inquiry closed off too soon, hence my development of the twist theory. It adds a couple of constraints to Maxwell’s equations (twist field discontinuities and orientability to a background state) to make stable solitons possible in an EM field.
Unfortunately, trying to model twist field particles in a sim has always been hampered by what I call the renormalization problem–at what point do you cut off the evaluation of the field 1/r^n strength to prevent infinities that make evaluation unworkable. I’ve tried many variations of this sim in the past and always ran into this intractable problem–the definition of the renormalization limit point overpowered the computed behavior of the system.
My breakthrough was realizing that that problem occurs only with electrostatic fields and not magnetic fields, and finding the previously mentioned balancing magnetic forces in the twist loop. The magnetic fields, like electrostatic fields, also have an inverse r strength, causing infinities–but it applies force according to the cross-product of the direction of the loop. This means that no renormalization cutoff point (an arbitrary point where you just decide not to apply the force to the system if it is too close to the source) is needed. Instead, this force merely constrains the maximum curvature of the twist. As long as it is less that the 1/r^n of the resulting force, infinities wont happen, and the curve simulation forces will work to enforce that. At last, I can set up the sim without that hokey arbitrary force cutoff mechanism.
And–this should prove that conceptually there is no clean particle model system (without a renormalization hack) that can be built from an electrostatic field. A corollary might be–not sure, still thinking about this–that magnetic fields are fundamental and electrostatic fields are a consequence of magnetic fields, not a fundamental entity in its own right. The interchangability of B and E fields in special relativity frames of reference calls that idea into question, though, so I have to think more about that one! But anyway, this was a big breakthrough in creating a sim that has some hope of actually representing twist field behavior in particles.
Agemoz
PPS: Update–getting closer. I’ve worked out the equations, hopefully correctly, and am in the process of setting them up in Mathematica. If you want to make your own working sim, the two forces sum to a flux field which can be parametrically integrated around whatever twist paths you create. Then the goal becomes to try to find equipotential curves for the flux field. The two forces are first the result of the axial twist, which generates a plane angle theta offset value Bx = 3 k0 sin theta cos theta/r^3, and Bz = k0 ( 3 cos^2 theta -1)/r^3. The second flux field results from the closed loop as k0 dl/r^2). These will both get a phase factor, and must be rotated to normalize the plane angle theta (some complicated geometry here, hope I don’t screw it up and create some bogus conclusions). The resulting sum must be integrated as a cross product of the resulting B vector and the direction of travel around the proposed twist path for every point.
Agemoz's Physics Blog 