Apology accepted. If I can clarify though, it's a stress flow. An energy-momentum flow if you like. You could say it's the motion of a wave or field-variation or pulse of potential. This then results in a standing electromagnetic field which IMHO can best be thought of as chiral frame-dragged space.

Alright, stress flow it is.

The latter. But we start with the Möbius strip because you're familiar with that from Dirac's belt (see

Mathspages and spinors (see

Wikipedia). We inflate the flat twisted strip to the ring torus, then inflate that to the spindle-sphere torus.

And it's on the spindle sphere. This post is off to a good start!

The electron and the positron have the opposite chirality, which means the toroidal motion twists either clockwise or anticlockwise like your right-handed and left-handed Mobius strips.

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I haven't dodged or changed the model, you're making a mountain out a molehill that just isn't there. Go and google on

Farsight positron chiral and you can see I've been consistent.

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It isn't. You're claiming that when you haven't even read this thread from the beginning, and when I went through page after page refuting your assertion. See post

#41 where I said the positron has the

opposite chirality to the electron. I've said previously that the spiral was a simplified flat 2D picture, rpenner knows this full well.

Well, yes. I have in fact read this thread from the beginning (even if I didn't start posting for several pages), and you have indeed

*stated* that the electron and positron have opposite chirality. But before page 8, I find two 3D depictions of the electron vs. the positron: the pair of spindle spheres in post 108, and the leftmost toruses in post 125. In both cases, the positron is a time-reversed electron (and you said so verbatim in post 108), but they have the same chirality and are isomorphic to each other. rpenner and I pressed you on this issue, because the electron/positron pairs you were showing did not have the chirality properties you claimed they did. As of page 8, you switched to diagrams in which the electrons and positrons

*do* have opposite chirality and are

*not* time-reversals of each other. In particular, the animated toruses in your latest post share the same format as the ones in post 125, but the positron is clearly animated differently, and they do not inflate into the same spindle spheres as in post 108. From now on, I'm holding you to the latest set of diagrams.

You have to have two loops because the photon has to be interacting with itself displacing itself. Think about a Mobius strip: the photon path is represented by a line drawn twice round the strip, not by the paper.

We're on a roll! I think that in the notation I was using, the Möbius strip has half a twist rather than two; just to be unambiguous, we're looking at a loop that cycles around the major axis twice for each loop around the minor axis.

This, combined with the above details, gives a much clearer picture of the model we're looking at. So clear, in fact, that we can describe it with equations(?!). We can define the stress flow associated with the electromagnetic field throughout all space as a function of spherical coordinates, $$F(\theta,\phi,r)$$ where $$\phi$$ tracks the major axis ("around the equator" coordinate) and $$\theta$$ tracks the minor axis ("around the cross section" coordinate). Although Farsight hasn't said so explicitly, I'm going to assume that the radial coordinate is separable; that is, the full 3D function can be built up from a bunch of concentric spheres with different amplitudes but the same angular properties. My guess is that the radial function $$R$$ would have to asymptote to $$r^{-2}$$ for normalization reasons, but for now let's keep it general:

$$F(\theta,\phi,r)=R(r)f(\theta,\phi)$$

In this form, $$f(\theta,\phi)$$ describes the spherical surface that Farsight has been plotting. So, what is this function? In general it could be time-dependent, but Farsight has told us that it's a steady state, so the function itself should be independent of time even if the "flow" it describes is dynamic in nature. We can further say that the function must be a vector function (because flow has direction) and that the $$\hat{r}$$ component of said vector must always be zero, because steady state flow into (out of) the electron would make it a source (sink) of the field, rather than a conservative circulation. Since Farsight has described his model as bispinor rotation, and just based on the animations provided, the circulation rates about the two axes must be independent of both each other and the location of the sphere. According to a recent reply, the circulation of $$\phi$$ is more specifically twice that of $$\theta$$. And without loss of generality, we'll use the recent pair of toruses to specify that the electron has right-handed chirality and therefore the the same sign on the circulation direction of both variables. All this together gives:

$$F(\theta,\phi,r)=R(r)(2\hat{\phi}+\hat{\theta})$$

Of course, all of this is unitless, and needs to be multiplied by an overall coefficient - probably involving $$\hbar$$ - to make it physically relevant. But really, was that so hard? Now there's only one thorny issue left: this function is at least doubly valued everywhere. Spherical functions often restrict their domains to $$\phi\in[0,\pi]$$ to preserve one-to-one mapping with Cartesian coordinates, but because this sphere was inflated from a torus, it cannot do so and every Cartesian coordinate ends up double-counted. I leave it as an exercise for the reader to show that if this equation is transformed into Cartesian coordinates (Wikipedia has the formulas if you're stuck), the function's values for a given spatial coordinate are always two opposite vectors. (Except at the north and south poles, $$\theta=n\pi$$, where its values are an infinite ring of vectors.) So Farsight: what does this mean, physically? No matter what is "really there" - be it electromagnetic field amplitude, spatial stress, or something else - surely its value must be a well-defined function of spatial coordinates. Simply summing the multiple values gives a value of zero everywhere, so some other mapping must be going on, and I'd ask you to tell me what that is. I assert that for any answer you can give, the resulting model will immediately predict nonsensical behavior.

You are clinging to ignorance here Fednis. Stop doing it.

You said your spindle sphere was isotropic, and I explained why it wasn't. That's not clinging to ignorance, it's just stubbornly insisting that you show me the logical coherence of your model before I accept it.

And do not demand a full-blown theory before you'll listen.

A full-blown theory? Of course not - a theory requires extensive study and comparison with experiment, which takes a long time. But I am going to demand a working hypothesis.