I know that I promised to finish with the posts on the gauge principle, but in the meantime, I’ve been distracted by a couple of things. They are related. One is the discussion going on at the ISUS discussion forums, and the other is, as a result of the ISUS discussion, I’ve started to write the book on all this stuff.
Now, however, it’s almost midnight and I can’t sleep for the thoughts about spin that are running through my head. With the new understanding of the discrete and continuous number systems that are the result of the two groups of RNs, one group under addition, isomorphic to the integers, and the other group under multiplication, isomorphic to the non-zero rationals, the explanation of quantum spin, which is so enigmatic to the LST community, turns out to be just as simple as it can be.
During the course of the ISUS discussion, I tried to illustrate the connection between the scalar vibrations of the SUDRs and TUDRs and rotation in the complex plane, using the equations of GA, implemented as a simulation that I borrowed from a GA website. I showed 135 degrees of bivector rotation, in the graphic below.
Figure 1. Expanding/Contracting Sphere in Geometric Algebra
At first, I thought this was really a convincing demonstration of the equivalency of SHM and the spherical expansion/contraction cycle, but it turns out that it isn’t, not at all.
However, when I realized why it wasn’t, everything fell into place, and the long mystery of what quantum spin actually is was solved at last. You see, the problem with using GA equations, where the inner and outer products represent the sine and cosine of the angle of rotation, is that the trivector goes through the contract/expand cycle four times as the bivector rotates through 360 degrees.
When I first realized this, I was really embarrassed, and hoped that nobody would catch it on the ISUS site. I was really puzzled though, what is wrong here? I thought I knew that the SUDR cycle had to be one cycle per revolution, synchronized with the bivector rotation; that is, one complete expansion over the first 180 degrees of rotation, and one contraction over the second 180 degree rotation, or one cycle per 360 degrees of bivector rotation, not four cycles!
Then it dawned on me. The GA equations are equivalent to the complex rotations, but the SUDR cycle isn’t. It’s equivalent to one half of SHM! You can easily see this, when you compare the SUDR expansion, from a point to the unit sphere, to the extension of a spring mounted weight. As the SUDR expands from the point to the sphere, the spring goes from fully compressed, to fully extended, and as the SUDR contracts from the sphere to the point, the spring goes from fully extended, to fully compressed. When the SUDR and TUDR are combined as an S|T unit, the two, inverse, oscillations constitute SHM, just as two springs do, when their oscillations are 180 degrees out of phase.
One complete, expansion/contraction, cycle, then, is equivalent to one 360 degree rotation, and it takes two, inverse, rotations, coupled together, to constitute one cycle of SHM, just as it does for the extension/compression cycle. The reason this is so is now very clear as well: The 720 degree spin is the equivalent to 360 degree rotation, but it is continuous motion, not discrete motion. Recall that discrete motion has two components, each passing through zero, between positive and negative one, while continuous motion has two components, but each passes through one between positive and negative zero. Duh! One is the inverse of the other, as we’ve been discussing.
The set of meshed, counter-rotating, gears is the perfect representation of this, as I’ve already noted, but I didn’t make the mathematical connection with rotation in the complex plane, until this came up, but look how easy and plain it is to see:
- 1 complex rotation of unit circle: -1 <——> 0 <——> 1 (sine) -1 <——> 0 <——> 1 (cosine)
- 1 SUDR|TUDR exp/cont cycle:: - .5 <——> 1 <——> 2 (SUDR) -2 <——> 1 <——> .5 (TUDR)
- 1 spring1|spring2 ext/comp cycle: - .5 <——> 1 <——> 2 (sprg1) -2 <——> 1 <——> .5 (sprg2)
The expansions of the spheres, like the extensions of the springs, never pass through 0, at 90 degrees, as they do in the graphic of figure 1 above. Instead, at the half-way point, they have doubled their initial value (their “0”), from .5 to 1, and the contractions of the spheres, like the compressions of the springs, never pass through 0, at 90 degrees. Instead, at the half-way point, they have halved their initial value (their “0”), from 2 to 1. Meanwhile, the sine and cosine have both passed through 0 twice, once each for every 180 degrees of rotation.
So, unlike the trivector/bivector relationship. or the sine/cosine relationship, the expansion of the SUDR to a spatial sphere is equivalent to 180 degrees of rotation, not 90 degrees, and the contraction of the TUDR to a temporal point, at the same time, is also equivalent to a 180 degrees of rotation. Thus, quantum spin is not two 360 degree vector rotations around the unit circle, but two simultaneous, inverse, scalar expansions/contractions, totaling 720 degrees of equivalent rotation, in every cycle.
Moreover, now that we can understand quantum spin, in such simple terms, we should be able to cut through quantum mechanics like a hot knife through butter. Not only this, but we can also see that comparing amplitude modulation to frequency modulation of a carrier signal, in terms of octave changes, now provides us the same type of association between them. I suspect that this will enable us to tie M2 and M4 motion together, through M3 motion, just as the Chart of Motion suggests we should be able to do; that is, we can get one integrated picture of classical and quantum mechanics.
Man this stuff is exciting. No wonder I can’t sleep.