A while ago, I discovered a showstopper–a contradiction between two parts of the theory I’ve been working on that proposes an underlying unitary rotation field for the particle zoo. The theory is based, in part, on two discoveries: that any Fourier construction of particles (a sum of waves that results in a group wave delta function) will appear to move at constant speed regardless of observer frame of reference, thus providing a basis for special relativity, and secondly, that quantized energy states can emerge from an R3+I unitary rotation field. Lots of work has resulted from this basic model of existence, including the quantized formation of stable solitons.
However, the showstopper problem needed to be resolved, and I think I have done so, although I’ve not proven it yet. The problem is this: how can a sum of waves exist in a unitary single valued vector field? There is no magnitude component in such a field, so the only way to “sum” a Fourier composition of waves is to sum the rotations at any given point. This doesn’t really work when you try to classically doppler shift the resulting field, there’s no wave components present in the resulting field and the special relativity behavior can’t emerge. I’ve looked at abandoning the doppler shift approach, but there are only a few other ways that special relativity could emerge and so far they all seem unworkable as an underlying field for particles.
Coming back to the original premise, I can resolve the paradox if doppler shifting can occur on a single wave cycle (rather than requiring a sum of waves). I believe that this should be true for this reason–when generating a Fourier sum of a delta function, normally waves of infinite span are used. However, in the limiting case, all parts of the sum cancel out except in the region of the delta function, so the constant speed derivation is just as valid if you only use the sum of waves in the immediate region of the delta function. A single cycle of oscillation will still doppler shift, and the apparent constant velocity of the resulting delta function is valid whether the infinite waves are summed or the region bounded (single cycle) waves are summed. If there is only a single cycle wave present, its shape and velocity are still defined by the math of the original theorem with a different set of limits (described in this paper: group_wave_constant_speed) and now the contradiction is resolved.
There’s more work to do, I think it would be pretty easy to blow holes in this framework as it is. Nevertheless, it’s the first time I’ve been able to work out a promising answer to the showstopper contradiction.
Agemoz
Tags: particle zoo, physics, quantum, special relativity, vector field
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