A quick spin on spinors

“No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors.” From Graham Farmelo. The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius

The concept of spinors is obviously a tricky one to people in the fields of math and physics. So how can I be of any help? Well, I could copy paste some statements about spinors and see how that relates to my fundamental unit. Who knows, they might turn out to be the same ”thing”.

  1. “…spinors appear when we imagine that instead of a single rotation, the coordinate system is gradually (continuously) rotated between some initial and final configuration.”
    Good fit! My unit oscillates back and forth between two distinct configurations. Those are the momentary states when/where the unit experiences phase shifts to and from projection/extension and conjection/contraction. I usually describe this as the unit being what defines the coordinate system values of X,Y,Z.
  2. “…sensitive to how the gradual rotation of the coordinates arrived there: they exhibit path-dependence
    Good fit! The gradual change is of opposite directions vertically, but same direction horizontally. When vertical values decrease, horizontal values increase and vice versa. So one path is “positive” in the sense that the real plane extends, and the other is “negative” since the projected values are then conjected. This is probably the cause of quantization. It seems reasonable to assume path-dependence of value X as either increasing or decreasing.
  3. Spinors actually exhibit a sign-reversal..” Perfect fit! As explained above, all values are likely to be reversed/inversed as the unit phase shifts at the relevant end configurations. The different topologies of rotation can perhaps be linked to the perpendicular relation of pole/axis to charge/radius. While my unit rotates around the vertical axis, that axis itself is assumed to be non-rotational. Instead, it is compressed when unit extends horizontally, and it elongates when compression is relaxed i.e. when axis function as a tensor. The topologies of circular and linear would be close to opposite as far as I can see. But if this is what is meant by “inequivalent gradual (continuous) rotations of the coordinate system”, I don’t know.
  4. “..a spinor must belong to a representation of the double cover of the rotation group SO(n, R), or more generally of double cover of the generalized special orthogonal group SO+(p, q, R) on spaces with metric signature (p, q).”
    You got me there Buddy! When reading “double cover” I think of chocolate wrapping, but that makes no sense in this context, so I Google it up. It turns out that what is covered is a continuous function p which is mapped from a topological space C to a topological space X. Seems like p-function translates “open” values to “definite” values. Then I stumble on the concept of “balls” in math, and things start to get really complicated. But however complex and diverse, the balls of math seems to resonate fairly well to my extreme simplicity. An Euclidean plane ball is a disc, an Euclidean 3-space ball is a volume bounded by a 2-D spherical shell and in 1-D space it is a line segment. All of those are present in my unit so, at a glance, it seems possible that this oscillator covers, if not everything so at least lots of it.All in all, I believe someone knowing Lorentz groups and spin matrices would have a field day with this unit/oscillator. And even with my limited understanding of formal physics and math, it seems possible that this is indeed what causes the effect of geometry. The problem might be that this unit cannot be detected empirically as a particular pure state. That requires a parity of such units, and such a pair is, at least in my mind, forced to counter each others values as to obey von Neumann entropy zero. That is, if A is forced by observation to express “contraction”, then B is forced by parity to express “extension”. That would seem an obvious function of parity, to have A and B oscillate in transverse directions. If both were to expand equally, I see them bounce off each other and separate. On the other hand, if they both contract, I fear they would condense to a single, coherent state. Opposition would be what keeps the oscillator oscillate, whether it is a single, pair or a triplet.

    Science seems bound to appearances, and in my image, appearance is a quality of space extensions and that is only half the truth. How space hides itself will be revealed in time, but “time” is not an observable. Time is linear momentum which enables spatial extensions to be observed “at a distance”. Without axial compression/relaxation, Alice and Bob would be blind to system AB. A spinning surface is going nowhere but in circles, but a pole on the other hand…can be understood as an “arrow of time”.
    In short: pole is messenger and surface is the message. In my image, the fundamental unit oscillates between both configurations. Not local, not nonlocal.

    Does this make any sense? Most certainly not!