The bifurcation of sub-waves in space and time
• Is the bifurcation of sub-waves a process in space or in time, or both?
• How are elementary matter- and force particles formed?
• What is the role of the potential well?


In sub-wave / microvita theory, the bifurcation of sub-waves in space and in time is closely interwoven, but each is treated differently.

The bifurcation in space, (B) subjective > (B) objective, constitutes a spectral analysis of the source wave numbers (cycles per unit of distance) and is initially manifested through the formation of matter-waves.

3 source sub-waves
4 bifurcated sub-waves
sub-wave #
wave number
angular velocity
1
Phi^0 = 1
1
2
Phi^1 = Phi
1
3
Phi^2
1
sub-wave # wave number angular velocity
0 - electron Phi^2 - Phi - 1 = 0 1 - 1 - 1 = -1
1 - photon Phi^2 - Phi + 1 = 2 · 1 1 - 1 + 1 = 1
2 - gluon Phi^2 + Phi - 1 = 2 · Phi 1 + 1 - 1 = 1
3 - quarks Phi^2 + Phi + 1 = 2 · Phi^2 1 + 1 + 1 = 3

Integral bifurcation table, with emphasis on spacial bifurcation. Only non-negative wave numbers are considered.
Note the factor 2 each time - this is the wave number bifurcation, only occuring at Phi (1.618) synchronized scaling.


The common angular velocity (or phase or frequency) of the source sub-waves bifurcates in time: (A) subjective > (A) objective. It manifests through the formation of a spherical potential well in which the matter-wave or -particle moves in a closed trajectory.

Increasing curvature of the matter wave, while space-time is transformed into a potential well with a nucleus.
Note that unlike in philosophy, the straight flow ("nada") also has an inner oscillation - this is the sub-wave.


The matter-wave is 1-dimensional, implying a bound sub-wave with unit angular velocity (1 electron). The nucleus (incl. its surrounding medium), is 3-dimensional, implying a bound sub-wave with triple angular velocity (3 quarks).


3 source sub-waves
4 bifurcated sub-waves
sub-wave #
wave number
angular velocity
1
Phi^0 = 1
1
2
Phi^1 = Phi
1
3
Phi^2
1
sub-wave # wave number angular velocity
0 - electron Phi^2 - Phi - 1 = 0 1 - 1 - 1 = -1
1 - photon Phi^2 - Phi + 1 = 2 · 1 1 - 1 + 1 = 1
2 - gluon Phi^2 + Phi - 1 = 2 · Phi 1 + 1 - 1 = 1
3 - quarks Phi^2 + Phi + 1 = 2 · Phi^2 1 + 1 + 1 = 3

Integral bifurcation table, with emphasis on angular velocities.

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Electron- and quark (Dirac) spinors, modeled as resp. (2, -1) and (2, -3) toruses.
The dashed tracks are the electric resp. accurate "color" charges, both a (2, 1) torus.
The white nodes show the interdigitation between the matter- and force- spinors.
In the current gauge the charge is cycling whereas the particles are still.


The potential-well is 2-dimensional, it is a topological- or surface phenomenon which constitutes the charge. The associated particles are the photon resp. gluon, force-carriers. The spherical potential well is nothing but the (space and time) bifurcation itself in dimensional form.

The spherical shell or surface-area, in a general sense, accounts for the source sub-waves' common phase cycle, bifurcating in self-interactive, quadratic form, and therefore with dual angular velocity. The source phases and the topologically bifurcated phases together form a resonant potential well.


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Squaring a source sub-wave's phase angle (blue), bifurcates its angular velocity (red).
The two components together form a quadratic potential well based on self-interaction,
relating the sub-waves to a particle's mass (bifurcated wave's magnitude).


If the bifurcation is synchronized (which is the case for perceptual states and for harmonic entities), the quadratic part of a potential well is formed specifically by distinct, positively bifurcated sub-wave phase pairs of the resp. constituent waves. A phase pair couples the charge with a resp. Dirac spinor.

The temporal bifurcation is effectively a gauge phenomenon, indicating that the ever present, implicit (principal or axiomatic) bifurcation of the sub-stratum (also, the "swabhava" in philosophy) is now explicitly being realized as a known resp. perceived state. This known state, in its most rudimental form, is a hydrogen atom.


3 source sub-waves
4 bifurcated sub-waves, with potential wells and Dirac spinors
sub-wave #
wave number
angular velocity
1
Phi^0 = 1
1
2
Phi^1 = Phi
1
3
Phi^2
1
sub-wave # wave number angular velocity
0 - electron Phi^2 - Phi - 1 = 0 +1 - 1 - 1 = {-1}
1 - photon (Phi^2) - Phi + (1) = 2 · 1 (+1) - 1 (+ 1) = [{2} - 1] = 1
     
2 - gluon (Phi^2 + Phi) - 1 = 2 · Phi (1 + 1) - 1 = [{2} - 1] = 1
3 - quarks Phi^2 + Phi + 1 = 2 · Phi^2 1 + 1 + 1 = {3}

Integral bifurcation table, with emphasis on temporal bifurcation and potential well. The paired sub-waves
constituting the quadratic, charged sphere are in round brackets. The factors 2 in the angular velocities show the
phase bifurcation. The potential wells are in square brackets, while {2,3} and {2,-1} are the resp. Dirac spinors.



The spherical potential well and bifurcation scheme (both simplified)
Note that it should not too literally be viewed as an atom model.



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Bifurcation tree, scaling to synchronization


In summary, it was shown that the bifurcation of sub-waves consists of a spectrum analysis of the source wave numbers in space, and a topological analysis of the common phase cycle in time. The space- and time bifurcation together constitutes a dimensional analysis, in the form of a spherical potential well.

 


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