Math

Overview

We will show the following equivalences from established physics to the RQM view

(1) Coulomb

• The coulomb $1/r^2$ forces are a residuals of unestablished resonance potentials
• Unestablished resonance potentials are statistically and rotationally randomly distributed over the distributed sphere (prop. 1/r^2)

(2) Entanglement coupling force

• We calculate the constant distance independent force between entangled/coupled electrons

(3) Energy dissipation through photon emission

• Isolated systems that experience acceleration might distribute this torque internally to their subrotonal states. This works well if the internal structure can integrate this energy.
• If it can not, the lowest level Sonon within the electrons will decouple a Sonon/Anti-Sonon pair (photon) to carry away the needed momentum to change the trajectory.
• The initial Sonon-Pair has some energy and relative speed with which they will find a matching orbital path. So the extraction will be into a special direction “nearly” tangential to the original electron path. Which might be arbitrary and not necessarily in sync with the overall momentum - but most likely it will. So if it is, the Photon polarization will be in direction of the electron rotation but the absorbed direction of the photon opposite to the objects change of trajectory.

(4) Not isolated systems Systems which can distribute the applied torque internally or to neighboring resonance-channels and temporarily available resonance potentials will not eject photons. In this sense photons are not used within compound structures to mediate energy between electrons. Energy mainly dissolves vie the available resonance channels if the systems are able to take them.

(5) Bremsstrahlung

Expectation: Photons are mainly ejected in direction of original electron trajectory, BUT slightly opposite to the trajectory. Yes, this effectively holds for electrons slower than $«c$, then the radiation pattern is mostly symmetric. Sideways relative to instantaneous acceleration. Could even go forward or backward, can be broad and not strongly directional. An electron at c will emit the photon in straight forward direction of the electron (which makes sense, as this can take up any potential energy) When an electron at high speed approaches a nucleus it will mostly continue on its trajectory. As soon as it leaves the nearest point, it well get attracted back and will slow down. This re-enables it to change direction. So at that point in time it is not fulfilling the phase condition of the protonal e+ locked in space anymore so missing the ability of fast phase change, it needs to distribute the change onto its other internal degrees of freedom (resonance channels). This will slightly bend the curve. If these can not take the momentum and the Base-Sonon reach some maximal rotation speed (or maybe at a higher curve than the Reson/electron size) a photon will be ejected.

Hypothesis: What might limit the torque/acceleration energy an electron can take before ejecting a photon? An electron as a particle has no fixed size - as far as measurable. So it might have different sizes it can distribute energy to. And potentially be limited in speed of changing size.

We might think of the coupling constant as a coupling reservoir $\frac{e^2}{\varepsilon_0}$ which relates to “energy * characteristic length”. With this in mind we could say, that the electron needs to size down to a characteristic length of its distance to the next or torque imposing object. So if r decreases, the coupling reservoir decreases too. So when acceleration hit’s the electron at the given distance, it can not take up the coupling change rate anymore and might with it’s overloading angular-speed raise a further Soliton-Pair into existence.

Research results: Emission of a photon depends on: interaction time, impact parameter, energy. So it depends on local acceleration history (change of acceleration per time = accumulated ability to absorb acceleration). It does not depend on absolute distance once the field strength is fixed. Relevant: Same field, same curvature. (Comment: curvature has to do with distance)

Photons are emitted more in random periods with different energies, but some well defined average rate. Curvature/acceleration feeds an emission reservoir, once enough energy is available a photon is emitted. This might actually simply match to the prediction, that once internal torque absorption ability is reached a photon is emitted. But this might be stochastic depending on how good the different internal phase-coupling channels distribute (remember the differential gear).

(6) Radio antenna Prediction: What radio antennas basically do, is, the lead electrons into a curved path creating photons and now it gets interesting:

• either of the size depending on the energy (speed of acceleration times curvature)
• or do they couple with Sonons on the oposite symmetrical side of the antenna? What is the emition direction of a radio wave (photons)? if we see it as a Sonon-Pair, then the creation plane will be most likely within the curvature plane of the electron. But, a photon can not accelerate in it’s planar direction at the speed of c (reference other chapter: Photon travels at speed of light, because it evades it’s energy density). So it will most likely and nevertheless decouple and leave the system into orthogonal direction. (Check: What does that mean for the momentum? Does light have one?)

Check: Radio antennas accelerate electrons linearly with a given frequency. This gives photons in direction symetrically in planar direction perpendicular to the antenna-length. But circular polarization you can get with two antennas and phase shifted frequency. That sounds, nice exactly what we need. But what is the linear one?

(7) Linear Antenna An electron accelerating vertically in one z-direction well create a sonon-pair travalling into z-direction but with a slightly outward horizontal drift. The two Sonons then also are sent out into opposite directions perpendicular to the antennae in arbitrary sideways (horizontal) direction. Now during the next radio frequency period, another electron accelerates downwards with the same Sonon-axis (sideways). These two Sonons, will then re-combined into a photon with an exactly vertical polarization being sent sideways.

Problem: Not all created Sonons will recombine, so there are linear photons with increasing wavelength being sent out vertically but with increasing radius in x/y direction. What is that? Where does this missing energy go to? The distance the Sonons travel before they combine int a photon (if at all) is actually not relevant. The energy they carry will match to the frequency, and they will create vertical circles as soon as they are detached.

(8) Near field energy An antenna is not optimal, so what happens to the remaining Sonons? All Sonons that do not create Photons will most likely couple linearly back with other Sonons. So they will flow back into the antenna or recombine into the antenna wire. But what does that mean for the electrons? Will they get slowed down again while recoupling to the Sonons torques? Most likely.

So the question basically remains: what happens to stray sonons and where do they appear in the energy list? In general they will slow down the electrons coming from the other direction and prevent the energy from leaving the antenna. Or the power has to be increased.

(9) Virtual electrical fields (imposed by magnetic fields) What does the EM Near-Field actually consist of? There are electrically attracting potentials in planar direction sideways, so that is the electrical field. And there will be the dynamical aspect of it with a coming and going. in direction and presence. Creating some magnetic effects not creating a photon-like wave. So this basically means there is more electric field present, than electrons. electric field potential imposed by the magnetic field.

In this sense, the magnetic field is extra torque in the span of an electron so it is actually the electrons resonance coupling field. The electric field is a field imposed by the presence of electrons AND the presence of additional Sonons (1-Tier Electrons) which can also couple to electrons in an inertial way. And create EM-fields.

(10) Findings

Photons and electromagnetic fields are something different (of course). Electromagnetic waves emerge from a electromagnetic field potentially. Why do antennas create electric fields in the first place? Because oszillations of electrons induce the creation of Sonon/Anti-Sonon pairs which will re-combine. With the electrons in the antenna in all different ways. The eventually leave the EM-Field as photons.

Coulomb vs. realized resonance channels

Core proposition

Coulomb interaction primarily describes open or not yet fully coupled systems.
In such cases, interaction energy appears as a free radial potential:

$$ F \propto \frac{1}{r^2} $$

This expresses available coupling energy that has not yet been internally organized.

Once stable resonance channels are realized between constituents, the system no longer behaves as a set of pairwise Coulomb interactions. Instead:

• a collective oscillatory eigenstructure forms
• available Coulomb binding energy becomes internally distributed over the available resonance channels
• energy flows through established resonance channels
• interaction no longer follows simple distance laws

Coulomb then becomes mainly:

a measure of available coupling energy $e^2/\varepsilon_0$,
not a direct descriptor of internal energy distribution.

Open vs closed coupling regimes

Open / not fully resonant systems

• interaction expressed as Coulomb $1/r^2$
• energy exchange radial and external
• pairwise interaction picture valid

Resonantly coupled systems

• multiple resonance channels established
• energy redistributed internally
• system behaves as unified oscillatory structure
• pairwise Coulomb picture loses explanatory relevance
• Coulomb reflects total available binding energy only

Resonance-channel interpretation

When several resonance channels are open:

$$ \frac{e^2}{\varepsilon} $$

is no longer localized between particle pairs but distributed across:

• shared orbitals
• collective oscillation modes
• multi-center coupling geometries

Binding energy becomes a global property of the eigenmode, not a function of individual separations.

Conceptual consequence

Coulomb laws remain valid for:

• uncoupled charges
• partially coupled systems
• external interactions

But internally, once resonance is realized:

binding energy follows resonance topology rather than distance.

Residual non-resonant components may appear externally as weak long-range effects.

Distance–independent resonance coupling term

Starting point: Coulomb force

Between two elementary charges:

$$ F(r) = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2} $$

This describes the open, non-resonant case, where interaction energy is distributed radially over space.

Removing distance dependence via spherical integration

The same interaction can be expressed as the total coupling passing through a spherical shell of radius $r$.

Sphere surface: $$ A(r)=4\pi r^2 $$

Multiply force by sphere area:

$$ F(r)\cdot A(r) = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2}\cdot 4\pi r^2 = \frac{e^2}{\varepsilon_0} $$

All distance dependence cancels.

$$ \boxed{F\cdot A=\frac{e^2}{\varepsilon_0}} $$

Interpretation

This term represents:

total available coupling or resonance capacity
independent of separation distance

It is not a force anymore but a global coupling quantity.

Coulomb $1/r^2$ behavior appears only when this coupling is
distributed freely over a sphere (open system).

Once resonance channels are established, the same total coupling:

$$ \frac{e^2}{\varepsilon_0} $$

can be internally redistributed independent of distance.

Energy times length

How does the $F*A$ calculation of the constant coupling relate to energy? The coupling constant has the unit “energy length”.

So Energy is total coupling capacity divided by characteristic length. And the coupling capacity is energy times the characteristic length.

We can think of $\frac{e^2}{\varepsilon_0}$ as a total available coupling reservoir with dimension energy x characteristic length.

Units and meaning

Force: $$ [F]=\text{N} $$

Area: $$ [A]=\text{m}^2 $$

So: $$ F\cdot A = \text{N·m}^2 $$

Using: $$ 1\ \text{N}=\frac{\text{J}}{\text{m}} $$

gives: $$ \text{N·m}^2=\text{J·m} $$

$$ \boxed{F\cdot A\ \text{has units of energy × length}} $$

This can be interpreted as:

spatially distributable binding or resonance capacity.

Random orientation vs realized coupling

If two systems are not yet aligned:

• resonance potentials are randomly oriented
• only ~½ of the total coupling can effectively interact

$$ C_\text{random}\approx\frac{1}{2}\frac{e^2}{\varepsilon_0} $$

Once a stable resonance channel is realized:

• alignment becomes coherent
• full bidirectional coupling available

Effective usable coupling:

$$ \boxed{C_\text{realized}\approx\frac{e^2}{\varepsilon_0}} $$

Relative to random orientation this appears as:

$$ \boxed{\text{effective factor } \approx 2} $$

because previously only half the potential coupling contributed.

Conceptual summary

Open systems:

• coupling distributed radially
• appears as Coulomb $1/r^2$

Resonantly coupled systems:

• total coupling given by $e^2/\varepsilon_0$
• redistributed internally through resonance channels
• no longer governed by simple distance law