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Hypersphere sizes as given by , and hypersphere vorticitation rates as given by do not appear to equate with such phenomena as the Compton wavelength of a particle or with its ‘Compton’ frequency .
Fundamental quanta can appear as either point particles or as wave like phenomena depending on how we choose to measure them. To a simple approximation quanta fly like waves but take off and land like discrete particles. This wave-particle duality lies at the heart of quantum theory and arises because of the quantisation of phenomena down at the Planck scale and the Heisenberg indeterminacy relationships. Plus we conventionally ascribe zero mass to lightspeed particles like photons to avoid conflicts with the Special Relativity model, even though photons hit targets with a measurable momentum.
Now the Heisenberg uncertainty/indeterminacy relationships usually appear as complimentary pairs such as: -
Where the indeterminacy of position and momentum, or the indeterminacy of energy and time multiply up to about Planck’s constant reduced (ђ= ). However in dimensional terms, and there seems no reason not to decompose it into complementary trialities of qualities such as: -
Where in the first case we have a triality of indeterminacies of mass, cross sectional area, and time, and in the second case we have a triality of indeterminacies of mass, length, and velocity. There seems little reason to regard mass as somehow more fundamental and inviolate than any of the other characteristics that remain indeterminate within Planck limits.
Now the Compton wavelength gives a measure of the effective apparent ‘size’ of a quantum wave-particle, and the sort of minimum sized aperture through which it can pass:
The corresponding hypersphere length for a quantum of the same mass comes out at: -
and for most quantum scale objects this comes out at a much smaller length.
However quantisation at the Planck scale, which implies quantisation of spacetime itself, has the effect of giving that mass a larger apparent length: -
Where the Planck length,
Similarly the Hypersphere frequencies, masses, and lengths have the following relationships to the Compton characteristics that some choices of measurement make apparent: -
Furthermore, as the information content H, of the entire universe probably corresponds to its surface area in Planck units, as in the Beckenstein-Hawking Conjecture, then: -
(Surface area/Planck area)
(or (Hypersurface area/Planck volume))
Then those bits of information have to suffice for all the Planck volumes in the universe. Thus the universe has a serious information deficit with only 1 bit per Planck volumes. This number, , corresponds to the ‘Ubiquity Constant’ U, where and it means that only one bit of information seems available to specify the state of every Planck units of length, or, by similar argument, for every Planck units of time, and that this may well represent the effective quantisation or ‘grain size’ or ‘pixilation’ scale of spacetime.
We note that the universe does not appear to actually exhibit any behaviour at scales of less than or , twenty orders of magnitude above the Planck length and the Planck time.
Addendum of 12/3/15
The product of the Hypersphere Length and the Compton Wavelength for any particle, always equals the Planck Area, which equates to one bit of information in the Beckenstein-Hawking Conjecture.
This may have considerably more significance than at first appears, for it implies that these two ‘sizes’ of a fundamental particle represent its complementary aspects in the wave-particle duality.
The Hypersphere Length would correspond to the particle like nature of a fundamental quanta, and it never has a zero value, just an extraordinarily small value, way below the Planck Length, but it does not exist as a singularity like ‘point particle’. We can readily observe quanta behaving like very small particles in experiments to measure their particle like nature but we cannot confirm from such experiments that they exist as zero dimensional point particles.
The Compton Wavelength corresponds to the wave like nature of a fundamental quanta and it always exceeds the Planck length, as we can readily observe if we set up an experiment to measure its wavelike properties.
Research and Azathothian musings continue……………….
Addendum of 19/3/15
The meaning of two times or frequencies for a particle seemed rather strange until I stumbled upon this: -
Herein lies the problem. For the charge and size of electrons in particular, their magnetic field is way too high. They’d need to be spinning faster than the speed of light in order to produce the fields we see. As fans of the physics are no doubt already aware: faster-than-light = no. And yet, they definitely have the angular momentum necessary to create their fields.
Aha! Calculations to follow, and then to the question of two different masses. (Could the apparently extreem weakness of gravity compared to the other interactions come in here somehow?)
14/4/15 Hmm, mass has two manifestations, inertial mass and gravitational mass. These remain indentical (the strong equivalence principle) in the sense that they have the same proportion for any body, but with a huge conversion factor G. Inertial mass in effect has a strength 10^11 times stronger than the corresponding gravitational mass........
Perhaps all 'forces/fields' have a strong and a weak forms arising from their hypersphere particle and quantum wave manifestations.
Mass - Inertia and gravity. (and gravitomagnetism from ordinary spin)
Electroweak - Electrostatic and magnetic from ordinary spin.
Strong - dunno, but quark properties don't summate at all well to the properties of the hadrons they supposedly constitute.
A Speculation: - Hypersphere Cosmology depends on spacetime curvature. The hypothesis of Three Dimensional Time depends on torsion (spin). Einstein-Cartan theory includes both curvature and torsion. EC theory remains unfalsified and in play, though few theorists reference it today, except perhaps to remove the nonsensical spacetime singularities that arise in the conventional big bang theory and in the black holes that straight general relativity predicts. Instead EC theory proposes that mater particles have a minimum size rather than a point like nature and that they resist compression beyond this, so instead of a big bang singularity we may have a universe which bounces back and forth between a very small and a rather large size, yet the theory does not usually get used to eliminate ‘ordinary’ black holes, although it does eliminate possible pesky little black holes of less than 1016kg, the mass of a substantial asteroid.
In EC theory, particles have a spin or torsion component which gives them a minimum spatial displacement, the Cartan Length lCA.
Where lCA3 = Gh2/mc4 where h = Planck’s constant.
Interestingly, we can decompose the Cartan ‘volume’ lCA3 above into: -
Cartan volume = Planck area (Gh/c3) x Compton length (h/mc)
And/or/or possibly both…….
Cartan volume = Compton area (h/mc)2 x Hypersphere length (Gm/c2)
(Note that hypersphere length lH differs from hypersphere external radius r,( lH = pi r)
These components may in some way correspond to the wave/particle duality and can confirm that particles do indeed have some kind of hypersphere properties.