The three length scales don’t have anything to do with each other. The Planck units are the result of dimensional analysis, whether they have any physical meaning isn’t known.

The Compton wavelength is, similarly, the result of dimensional analysis, with the difference that it is known, thanks to experiment, what happens at that length scale, namely diffraction effects of particle motion, due to quantum fluctuations become visible.

The Schwarzschild radius does have a well-defined meaning, also, in classical gravity; in quantum gravity the corresponding length scale is related to the entropy of a black hole, when the microstates can be resolved.

At Planck lentgh the density of vacuum energy becomes so high than we get classical gravitation. It seemes contradictatory but this the theory of gravitation.

No, the Planck scale doesn’t have anything to do with vacuum energy. If one assumes that classical gravity makes any sense at the Planck scale, any object of mass greater than the Planck mass and of size less than the Planck length is a black hole, therefore a spacetime singularity, hidden behind an event horizon. The problem is that classical probes of such a system don’t make sense, because matter at such scales is known to have quantum, not classical, properties.

Quote: "The three length scales don’t have anything to do with each other. The Planck units are the result of dimensional analysis, whether they have any physical meaning isn’t known. "

Correct. In fact the Planck dimensional analysis was carried out before the Kaufmann experiments with free moving electrons in 1901-1903 and before Millikan experimentally established the invariance of the electron rest mass and its invariant unit charge in 1913.

They are only theoretical mathematical limit values that have no relation with actual physical limits.

Similarly, the Schwarzschild radius is a theoretical mathematical limit that never found any experimental confirmation.

The electron Compton wavelength λc=2.426310215E-12 m is the wavelength of the amount of electromagnetic energy of which the rest mass of the electron is made m0=9.10938188E-31 kg that oscillates at the electron Compton frequency f0= 1.235589976E20 Hz.

The mass of the electron is the smallest elementary mass ever discovered and confirmed to exist.

The masses of protons and neutrons are already systems involving smaller elementary masses of the same order of magnitude as the electron, and so are not elementary themselves.

Actual point-singularities never were observed and seem not to really develop at the subatomic level of magnitude as put in perspective in this analysis:

Article From E=mc2 in normal space to E=m0cIcK in the complex config...

André Michaud , Stam Nicolis , Alaya Kouki ,

Interrelation of Planck length, Schwarzschild radius and Compton wavelength on the Planck scale: https://www.researchgate.net/post/Discussion_Addressed_Convergence_of_Quantum_Mechanics_and_General_Relativity_at_the_Planck_Scale

This section delves into the profound relationship between the Planck length, Schwarzschild radius, and Compton wavelength at the Planck scale, emphasizing their convergence in the context of quantum gravity. It elucidates how a black hole's Schwarzschild radius, derived from its mass, becomes comparable to the Planck length when the mass is equivalent to the Planck mass. The discussion also encompasses the Compton wavelength of particles, particularly photons, which, despite having no rest mass, can be related to their energy. This interconnection suggests that at extremely small scales, traditional boundaries between quantum mechanics and gravity blur, indicating a deep link between matter, energy, and spacetime.

Quantum Gravity's Implications

The convergence of these concepts at the Planck scale implies that our current understanding of physics may need to adapt. The merging of quantum mechanics and general relativity suggests that at this scale, spacetime may not behave in the classical sense, leading to new physics where the effects of both theories are equally significant. This could provide insights into phenomena like black hole thermodynamics and the nature of singularities, offering a potential path toward a unified theory of quantum gravity.

Distinction Between Rest Mass and Energy-Based Mass

It’s crucial to differentiate between rest mass (invariant mass) and energy-based mass, especially in the context of quantum mechanics and relativity. Rest mass is the mass of an object measured when it is at rest relative to the observer and is a fundamental property of particles. In contrast, energy-based mass refers to the concept that mass can be derived from energy through Einstein's equation E = mc². In high-energy physics, particularly for massless particles like photons, their effective mass can be interpreted from their energy, given by E=hc/λ. Thus, while rest mass remains constant, energy-based mass can vary based on the particle's energy, leading to different implications in gravitational and quantum contexts.

The Planck length (ℓP) is the Schwarzschild radius (Rg) of a black hole with energy (E) equal to the Compton wavelength (λ) of a photon (hc/λ):

The relationship can be expressed as:

ℓP = Rg = λ

where: The energy (E) of a black hole with Schwarzschild radius (Rg) is equal to the energy of a photon (hc/λ) with Compton wavelength (λ).

This statement ties together three significant concepts—Planck length, Schwarzschild radius, and Compton wavelength—by demonstrating how they converge in an interesting way when examining extremely small scales, specifically the Planck scale.

This can be explained through the following steps:

Black Hole’s Schwarzschild Radius at the Planck Scale:

The Schwarzschild radius (Rg) of a black hole is determined by its mass. At extremely small mass scales, particularly when the mass is equivalent to the Planck mass, the Schwarzschild radius becomes comparable to the Planck length (ℓP). In other words, a black hole with a mass equal to the Planck mass (mP) would have an event horizon radius approximately equal to the Planck length.

Mathematically:

Rg = 2G·mP/c² ≈ ℓP

​

This is a fundamental length at which quantum gravitational effects are expected to become significant, meaning that general relativity and quantum mechanics both play critical roles.

Compton Wavelength, Photon Energy, and Planck Mass Relationship:

The Compton wavelength of a particle (with rest mass denoted by m) is inversely related to the mass of the particle: as the mass increases, its Compton wavelength decreases. For photons, which have no rest mass (m=0), the Compton wavelength is determined by their energy. In this context, the Compton wavelength is represented as λ = h/mc, which simplifies to λ = h/E when considering photons.

For a photon, the energy associated with the Compton wavelength can be expressed as:

E = hc/λ

This relationship shows that the energy of a photon is inversely proportional to its Compton wavelength. As the wavelength increases, the energy decreases, highlighting the fundamental connection between wavelength and energy in the context of quantum mechanics.

If we associate this photon energy with the energy of a black hole (i.e., the rest energy of a black hole with Planck mass mP), the wavelength of the photon becomes directly comparable to the Planck length.

Note:

The Planck mass is the minimum mass of a classical object (M) that corresponds to its Schwarzschild radius (Rg). The Planck mass (mP) is approximately 21.76 micrograms (µg). It's defined by an equation that uses the speed of light (c), reduced Planck's constant (ℏ), and the gravitational constant (G).

Linking the Two—Photon and Black Hole:

The statement says that at the Planck scale, a photon with the same energy as the rest mass energy of a Planck mass black hole will have a wavelength equal to the Schwarzschild radius of that black hole.

Essentially:

ℓP = Rg = λ

This means that the photon’s wavelength and the black hole’s event horizon are equal in size at this extreme quantum limit, where the energy of the photon corresponds to the energy required to form a black hole with a radius equal to the Planck length.

Implication:

Quantum Gravity Intersection:

This is a profound realization because it implies that at such small scales (the Planck scale), there is a deep connection between quantum mechanics and gravity. The Schwarzschild radius (typically a classical gravitational concept) and the Compton wavelength (a quantum mechanical concept) are equal at this scale. This suggests that the traditional divide between quantum mechanics and general relativity might blur at these extreme conditions.

Planck Mass and Photon’s Energy:

The Planck mass is the smallest possible mass for a black hole to exist. The energy of the photon with a Compton wavelength equal to the Planck length is enormous, and this photon behaves like a black hole. Any photon with such a small wavelength (Planck length) has so much energy that it can be seen as a black hole with mass equal to the Planck mass.

Photons and Black Holes at Planck Scale:

A photon with a wavelength this small can be thought of as a black hole itself. This reveals a fundamental quantum gravitational effect: light (normally massless) can, in extreme conditions, exhibit black hole-like behaviour.

Conclusion:

This statement highlights a profound and theoretically significant relationship: at the Planck scale, where quantum mechanics and general relativity merge, a black hole’s Schwarzschild radius, the energy of a photon, and its Compton wavelength all converge. This suggests that in the realm of quantum gravity, the distinctions between matter, energy, and spacetime become deeply intertwined.

-Soumendra Nath Thakur

29-09-2024

It's possible to write many sentences, that are grammatically and syntactically correct, but don't mean anything. The problem of quantum gravity pertains to describing the quantum fluctuations of spacetime geometry, nothing more, nothing less. Such fluctuations can become relevant at some multiple of the Planck scale, but the coefficient isn't known; just like diffraction effects when dealing with waves become relevant at some multiple of the wavelength, without the coefficient being necessarily equal to 1.

Mr. Stam Nicolis

I noted with interest your remarks: "It is possible to write many sentences that are grammatically and syntactically correct but lack substantive meaning," and "The problem of quantum gravity concerns the description of quantum fluctuations in spacetime geometry—nothing more, nothing less."

However, I would like to respectfully challenge your assertion that my previous comment, titled "Interrelation of Planck length, Schwarzschild radius and Compton wavelength on the Planck scale," lacks meaningful content. I am curious to understand the scientific basis upon which you claim it does not contribute to the very discussion.

Regarding your statement, "The problem of quantum gravity pertains solely to describing quantum fluctuations of spacetime geometry," I invite you to clarify how this assertion aligns with the broader topic of our discussion, "Convergence of Quantum Mechanics and General Relativity at the Planck Scale."

Should you find your comment irrelevant to the discussion topic, I would appreciate your reasoning for not addressing it within this context. Conversely, if you believe your comment is relevant, I invite you to further explain how your assertion aligns with the broader topic.

Regards,

Soumendra Nath Thakur

To whom it may concern,

The Planck length is highly relevant in the context of quantum gravity, as it represents a fundamental length scale crucial for understanding quantum gravitational effects.

The Planck length, one of the Planck units introduced by Max Planck, plays a pivotal role in theoretical physics. For instance, the speed of light is one Planck length per Planck time.

At this scale, the Planck length serves as a lower bound for the smallest possible length in spacetime. It is theoretically impossible to construct a device capable of measuring lengths smaller than this scale. Moreover, at the Planck scale, gravity's strength is expected to become comparable to other fundamental forces, potentially leading to a unification of all forces.

Additionally, the Planck length may represent the approximate lower limit for the formation of black holes. It is at this scale where quantum gravitational effects become significant, allowing for the measurement of the geometry of space and time.

In conclusion, the Planck length may represent a minimal, fundamental length, thereby completing the set of fundamental scales in nature.

Quote: "At this scale, the Planck length serves as a lower bound for the smallest possible length in spacetime. It is theoretically impossible to construct a device capable of measuring lengths smaller than this scale."

Sure it is possible. we can go at will to the very infinite minus limit by integration with any fraction of the Planck length, from 1 x Planck-length to 0.0000....1 x Planck length. We do it regularly with integration.

More than Planck´s scale is Planck´s equation revealing that energy is observed in a discrete form. The only variable is time in an inverse relation. This time/period is the clue for understanding Special Relativity space contraction. The 4th dimension is Ct but what "t?"; the traditional way is Minkowski´s evolution of events or the passage of time. A new understanding can be achieved by Ctau; where tau is Planck´s periodic presence of energy. This Ctau expresses the wavelength of the presence of energy (Lambda). When more energy is present, this Lambda (4thD) gets smaller and so its local space (Lorentz´s space contraction. Poincare expresses this with a simple invariance; (Delta x)2 + (Delta y)2 + (Delta z)2 + (i Delta Ctau)2 = constant. Relativity sharing Quantum time... a joining point, best regards

The Planck units, once more, just express dimensional analysis, not physics, absent further input, just like the wavelength of a wave doesn't imply a sharp scale.

The problem of quantum gravity doesn't have anything to do with dimensional analysis but with the property of classical gravity that it is described by a gauge theory, whose gauge group is noncompact, it's the group of diffeomorphisms. So the exercise is to find a quantum theory, whose classical limit has as gauge group the group of diffeomorphisms.

Dear Stam Nicolis ,

Thank you for your insights regarding the Planck units and their implications for quantum gravity. However, I would like to address some points in your response concerning the interrelation of the Planck length, Schwarzschild radius, and Compton wavelength at the Planck scale.

While I acknowledge that dimensional analysis plays a role in our understanding of physical concepts, I contend that reducing the discussion to mere dimensionality overlooks the profound physical implications of these relationships. The convergence of the Planck length, Schwarzschild radius, and Compton wavelength at the Planck scale is not just a mathematical curiosity but rather a critical insight into how quantum mechanics and general relativity intersect. At these scales, the distinctions between matter, energy, and spacetime begin to blur, suggesting a deeper interconnectedness that demands our attention.

Moreover, the properties of classical gravity described by gauge theories, particularly with noncompact gauge groups, certainly present important challenges. However, these challenges cannot be fully understood without considering the implications of energy and mass at such fundamental levels. The relationships you mention are indeed pivotal in guiding us toward a more unified understanding of quantum gravity.

In summary, I submit that addressing the convergence of these fundamental concepts at the Planck scale offers significant insights into the nature of the universe and should not be dismissed as mere dimensional analysis.

Regards, Soumendra Nath Thakur

Jose Oreste Mazzini

Dear Jose,

You wrote: "Poincare expresses this with a simple invariance; (Delta x)2 + (Delta y)2 + (Delta z)2 + (i Delta Ctau)2 = constant"

Would you clarify what numerical value this constant may have.

Involving the complex plane as Minkowski did with time t in this 4th term involves considerations that I am interested in that seem to define time as being the real axis in the quaternion geometric structure .

Best Regards, André

Dear André, my answer is involved with a new interpretation that I´ve made in 2021. The core idea is that physical presence in 3D is intermittent as Planck´s first quantum equation. The novel point is that the absence in 3D is supported by the transitory existence in the 4th D. So, existence will be in an oscillatory form between 3D and the 4th dimension (Ctau). With this, the quantum system in 3D (Delta x-y-z) is intimately linked with time tau. The constant in Poincare´s equation is zero. On each fluctuation, this quantum system will be observed only with one of its eigenvalues and, randomly new eigenvalues will be present one-by-one, in accordance with the Born/Schrodinger's equation. More than a hidden-variables is a hidden-solutions; only one observed with no QM measurement problem or collapsed wavefunction. I realize that Minkowski´s view is the general understanding but considering Ctau is also valid. The important time is Planck´s tau and it will be one joining point between relativity and QM. It also considers that energy is the one involved with space contraction and time dilation. Einstein in his General Relativity, clearly shows space on one side of the equation and the energy-tensor matrix at the other side. It´s energy the main actor over space and time. For example, when a traditional spacetime grid is shown with two massive objects, how is the distance in the axis Ct understood? Meanwhile, a 3D grid considering the scale information given by the scalar Ctau is more precise. A quaternion of the energy as scalar, plus the three vectors of 3D space. Hope this unusual form will inspire you and don´t perturb you; best regards

General relativity provides a classical description of gravitational phenomena. Quantum fluctuations don't enter Einstein's equations anywhere.

Quantum mechanics is the framework for describing the quantum fluctuations of any physical system, the framework doesn't know anything about gravity, electrodynamics and so on.

Within the formalism of quantum mechanics, the calculations that make sense are those that involve the transition probabilities from one state of the system in question to another; which assumes knowledge of what is the space of states. For an electromagnetic system this is known; for a gravitational system it isn't, because gravitational systems have spacetime singularities, that are absent from other interactions. Nowhere in this discussion do the Compton wavelength or the Planck scale enter the picture. The Schwarzschild radius depends on the system studied and is a way of distinguishing those spacetime observers that have that particular singularity in their future, from those that don't. It makes sense in that way in classical gravity. In the part of quantum gravity that is understood, the Schwarzschild radius enters in the expression for the dimension of the space of states that a quantum observer can explore: It is exp(A/4), in units where G=1,hbar=1,c=1 and A=4πR^2, where R is the Schwarzschild radius.

Jose Oreste Mazzini

Dear Jose,

Wow! Jose, far from perturbing me, your take at the same time connects in major ways with my own analysis and gets my mind racing in directions that I did not suspect.

So much stuff to address!

Without being overly extensive not to become boring, I will address some of the issues that clearly stand out for me.

First, you wrote: "The core idea is that physical presence in 3D is intermittent as Planck´s first quantum equation. The novel point is that the absence in 3D is supported by the transitory existence in the 4th D. So, existence will be in an oscillatory form between 3D and the 4th dimension (Ctau). "

This is the first time I meet someone aware that fundamental energy in the universe might not entirely reside within what we map as Cartesian 3D-space. Your seeing it as physically oscillating between 3D space and the 4th dimension Ctau plane is highly interesting to me, because from my own analysis, generally described, half of the energy of the electron rest mass (the half that oscillates at the electron Compton frequency 1.235589976E20 Hz) oscillates exactly in this manner in and out of 3D space, that I named X-space.

This is represented, if interested, with Figure 16 in the following paper.

What is more, within the related trispatial geometry, all oscillatory motion of stabilized energy quanta except for the case above occur entirely outside of 3D X-space, with only non-oscillating unidirectional momentum energy residing continuously within 3D X-space. Represented with Figure 17 in the same paper.

Article From E=mc2 in normal space to E=m0cIcK in the complex config...

You wrote: "The constant in Poincare´s equation is zero. On each fluctuation, this quantum system will be observed only with one of its eigenvalues and, randomly new eigenvalues will be present one-by-one, in accordance with the Born/Schrodinger's equation."

Thanks for clarifying the value of the constant. With regard to Schrödinger's equation, you may be interested in the fact that the following analysis reveals that the de Broglie phase wave velocity on which Schrödinger grounded the development of his equation was erroneous in a manner that the researchers of the 1920's could not be aware of. Published in the following article:

https://www.scirp.org/pdf/jmp2024156_17505289.pdf

You wrote: "Meanwhile, a 3D grid considering the scale information given by the scalar Ctau is more precise. A quaternion of the energy as scalar, plus the three vectors of 3D space."

With regard to the quaternion orthogonal structure, note that the trispatial geometry mentioned above, that underlies my own general analysis, is an expanded geometry that emerged from the combination of the quaternion 3D orthogonal coordinates system with the triple orthogonal electromagnetic relation that establishes that the vectorial direction of motion in X-space of an electromagnetic quantum results from the vectorial cross-product of its local E-field and B-field, which is the 3-way orthogonal relation so fundamental in electromagnetism.

If interested, this geometry is explained in this article:

Article Evolution From the Complex Plane to the Quaternion Coordinat...

Your summary description definitely inspires me and gets my mind racing. I will certainly investigate further.

Best Regards, André

Thanks André for having a positive response to this novel approach. Just a little more about this fluctuation... The energy involved is what Einstein found in his great equation mC2. Note that this energy is added in a vectorial form with kinetic energy [E2 = (mC2)2 + (pC)2]. Meanwhile, the energy, from different origin, is added in a scalar way. This indicates that mC2 is the dynamic energy between 3D and the 4th D. This is why they are added in a Pythagoras way; both energies are from the same family. With the same thinking, the angular momentum of this fluctuation (cw and ccw) is what is understood as spin "S" that again has a vectorial addition with 3D angular momentum "L" [|J|2 = |S|2 + |L|2]. With great interest, I will read the papers mentioned. You can also find my papers at Academia.edu and ResearchGate for further analysis. Have a great weekend. Regards

Sorry André, an additional idea for the weekend. Space contraction and time dilation are presented by me as a scale change of the quantum system space and time. For example, an engineer or architect ruler that has six scales, the rules have a unique dimension but the "value" of it changes in accordance with the scale selected. So, the quantum space occupied is the same, the great difference is its scale that follows the presence of energy. Taking this into account, the classical ladder or train paradox is solved. Black holes reveal this, they are so huge that they are seen. Their internal dimensional value is the same all-over, not a singularity point, It´s a singular sphere! (engineer ruler with the same value from the beginning to the end). With the same thought, the example of time dilation is imagining a clock with its dial changing values in all-over. Not changing the speed of its moving handle because the handle will indicate the common passage of time and is the way to link it from the other reference point. For example, when no energy is present, the dial goes from zero to twelve. with a Lorentz factor of two, the dial goes from zero to six. At the speed of light, the dial has zero all-over. I think that is the way to imagine "being on top of a light beam"; not a shrink space but traveling through a space with the same value. Idem with time, the clock handle moving linked by "the passage of time" with the other quantum systems. See my article " A new interpretation of the special theory of relativity" DOI 10.21275/SR239118031232.

Jose Oreste Mazzini

Dear Jose,

Note that the energy of the rest mass of the electron that I refer to is also the same energy referred to as E=m0c2 referred to by Einstein, particularly in his explicit reference of the electron at rest in his 1910 paper.

You wrote: "This indicates that mC2 is the dynamic energy between 3D and the 4th D."

Yes. This is the oscillating half of this energy that I also was referring to, represented with Figure 16, and also with Equation (18) and related Figure 9 in the first paper referred.

For the Spin of the electron, instead of a rotation, in the trispatial context, it is not an intrinsic property as in the classical conception of the Spin, but a relative magnetic property related to parallel magnetic repelling spin orientation and anti-parallel attractive magnetic spin orientation between 2 electrons, too difficult to summarize, here, but completely analyzed in Section 10 of this article:

Article Electromagnetic and Kinematic Mechanics Synchronized in thei...

This perspective was established as an outcome of this physically carried out experiment:

Article On The Magnetostatic Inverse Cube Law and Magnetic Monopoles...

I have already downloaded your papers available on RG for printing and reading at leisure in the coming days. Note that all my papers are also available in Spanish if more convenient.

Have a nice weekend too.

Best Regards, André

Dear Jose, your point about space contraction and time dilation noted.

Looking forward to see what you will think of my stuff.

Hi Andre, very interesting Figure 16 and, congratulations for your excellent work. The next step is to imagine what is going on in the atom. Following my proposal, the electron is in-and-out of the 3D assuming each time only one of its eigenstates, The electron is in one instance in 3D with a horizontal orbit and almost together in its vertical solution; doing a perfect electric shield to the proton. With this model, a tiny and compact elemental particle can distribute around the nucleus meanwhile when observed, it will be only in one place (no collapse thoughts nor diffuse presence at 3D). Another observation is that some physical parameters (like total energy and total momentum) are at the 4th dimension meanwhile other physical parameters are present in 3D. This is also a way that my proposal understand Heisenberg´s Uncertainty Principle; no simultaneity presence of these parameters. Also, there is no commutative multiplication between them because action h is involved... hope you have a great day and looking forward to see what you think of my strange model... Regards

Jose Oreste Mazzini

Dear Jose,

I don't find your model strange at all. In fact, I think it is the first progress in Quantum Mechanics since Feynman introduced his path integral, but a progress that brings at long last a logical explanation leading to understanding actual localization of the electron at any given moment within the possible volume within which it is oscillating, in context of QM.

Overall, I think that your work repatriates QM within the relm of logical theories.

You may have by now noticed that I am dealing with the subatomic level of magnitude from an entirely electromagnetic perspective, but a perspective that allows harmonizing the electromagnetic equations set with the kinematic equation set by incorporating in the latter set the missing magnetic energy that Kaufmann's data allowed identifying as being induced simultaneously with the momentum enegy by the E-field during his experiments.

I am off to a medical exam in a few minutes. Later in the day, I will complete my message.

See your later.

Soumendra Nath Thakur, André Michaud, Jose Oreste Mazzini

A recently published article proves that an electron’s gravitational properties can be derived from quantum mechanical wave properties. For example, at distance r from an electron, the slope of an electron's gravitational curvature is Lp2/rcr . In this, Lp is Planck length and rc is an electron's Compton radius. For another example, an electron’s gravitational radius (rG) is rG = Lp2/rc = 6.76x10-58 m). This is half an electron's Schwarzschild radius.

The forces between two electrons have also been calculated from quantum mechanical wave properties. For example, if the gravitational force between two electrons is designated FGe and the electrostatic force between two electrons is designated Fqe, then from quantum mechanical wave proprties, the ratio of these forces is FGe/Fqe = Lp2/rc2α = 2.4×10‑43 where α is the fine structure constant. Usually, this force ratio is calculated using G, mass me, charge e, in FGe/Fqe = 4πεoGme2/e2≈ 2.4×10-43. The article that achieves this is:

Article Oscillating Spacetime: The Foundation of the Universe

Jose Oreste Mazzini

Dear Jose,

OK, I am back in service. Some more remarks.

While the complex plane turns out to be ambient by structure in the whole universe at the subatomic level in your model, which is what is needed from the QM perspective that you analyzed, you may have figured out already that the 3D coordinate systems of both the Y and the Z complex spaces of the trispatial structure are local to every fundamental energy quantum in the universe, with the "real" X-space being ambient in the whole universe.

The electric charges of each quantum that oscillate (the oscillating charges of photons) or remain stable (the charge energy of the electron) within electrostatic Y-space interact with the energy of all other charges of all other quanta in the universe while they reside permanently or fleetingly within their own Y-space (The Coulomb law), and the energy of the oscillating magnetic component of each quantum while present in their local Z-space interact with the energy of all other magnetic components while they are present within their own local Z-space.

In the latter case this is what is related to parallel and antiparalles magnetic spin alignment between a pair of electrons for example.

I clearly undertand the hydrogen atom as described from your model and it makes total sense in harmony with the Schrödinger's equation representation.

You may have noticed in the same paper where you found Figure 16 the last figure before the conclusion (Figure 21). This figure illustrates how the hydrogen atom can remain stable and why the electron cannot crash on the proton, from the electromagnetic perspective.

If interested, this stability as analyzed and explained from the electromagnetic perspective is available in the following paper originally published in 2018:

Article The Hydrogen Atom Fundamental Resonance States

But this version, republished as a book chapter upon invitation in 2020 is a little more complete, as it also provides the mechanics of photon absorption and emission by an electron in the hydrogen atom:

Article An Overview of the Hydrogen Atom Fundamental Resonance State...

Your equation term iCtau was a revelation for me because I directly understand how to use it in context of my trispatial geometry. This is what got my mind racing the first time I saw it.

About your rhetorical question in the intro of your book: "What is oscillating in the extraordinary Schrödinger equation Psi?", In the trispatial model, the Ψ*Ψ function delimits the volume within which the permanently localized electron oscillates within its ground state orbital in the hydrogen atom. This volume is described in relation with Equation (4) in this paper:

Article Evolution From the Complex Plane to the Quaternion Coordinat...

So much more to discuss, but I have a question about a specific speed that you mention in the intro of your book: "a speed of 86.6% of the speed of light"

Why this specific speed?

I am very curious about it because it also appears in the analyses of Buckminster Fuller and also in one of my papers.

Thanks André for your interest and questions. The speed was just a value so Lorentz´s gamma factor is equal to two (a round value). In my proposal, Schrodinger´s wavefunction deals with the oscillating presence; that is the way I give a physical meaning. Now, another daring one is to think if the quantum system goes in-and-out 3D, what happens with its volumetric space? Does it goes away and be filled by the surrounding space? If so, then the space will be in acceleration towards the energetic system at the rate of its frequency... something like the equivalent principle that made Einstein so happy. The model embraces the relation between gravity and space and not with the graviton idea. Many thoughts that with an open mind will challenge our knowledge (or at least have fun thinking different). Best regards Oreste

John A. Macken

Dear John,

I have begun reading your quoted paper.

I note on page 4 that you quote Einstein's postulates on which he grounded SR as "1) The speed of light in a vacuum is invariant in all inertial frames of reference." and "2) All the physical laws are the same in all inertial frames of reference."

I just want to mention that these are not the exact formulation of this postulates in his original third 1905 paper. Particularly different for the speed of light postulate:

The original postulates of SRT quoted directly from his 1905 paper are, as you will be able to verify yourself from his original paper, a link to which I give after the quote:

"für alle Koordinatensysteme, für welche die mechanischen Gleichungen gelten, auch die gleichen elektrodynamischen und optischen Gesetze gelten, wie dies für die Großen erster Ordnung bereits erwiesen ist. Wir wollen diese Vermutung (deren Inhalt im folgenden ,,Prinzip der Relativität" genannt werden wird) zur Voraussetzung erheben und außerdem die mit ihm nur scheinbar unverträgliche Voraussetzung einführen, daß sich das Licht im leeren Raume stets mit einer bestimmten, vom Bewegungszustande des emittierenden Körpers unabhängigen Geschwindigkeit V fortpflanze."

"for all coordinate systems for which the mechanical equations apply, the same electrodynamic and optical laws also apply, as has already been proven for the first-order magnitudes. We want to make this assumption (the content of which will be called the “principle of relativity” in the following) a prerequisite and also introduce the only apparently incompatible assumption that light always propagates in empty space at a certain speed V independent of the state of motion of the emitting body."

Einstein, A. (1905c) Zur Elektrodynamik bewegter Körper. Annalen der Physik, vol. 17, no 10,‎ 30 Juni 1905, p. 891–921

(DOI 10.1002/andp.19053221004,

https://onlinelibrary.wiley.com/doi/epdf/10.1002/andp.19053221004

On a separate issue, is it possible from the Planck dimensions to derive the exact rest mass oft he electron. I would need this to be able to connect with your derivations.

Best Regards, André

André Michaud Thank you for the exact quote from Einstein's 1905 paper. I believe the two postulates I used are substantially the same as the two parts you highlighted in a bold font.

You also asked the question, "Is it possible from the Planck dimensions to derive the exact rest mass of the electron?"

The referenced paper starts with the concept that to explain Lorentz transformations, spacetime must be an oscillating medium that propagates waves at the speed of light. The enormous impedance of this medium is calculated to be c3/G = 4x1035 kg/s. A wave-based model of an electron is developed and tested. The model of the electron and the wave propagation medium is sufficiently detailed that it is possible to calculate the electron's approximate energy, spin, electric field, gravitational field and gravitational force. The calculation of an electron's energy is equation 10 in this paper.

Most importantly, the wave-based model of an electron also generates an electron's electrostatic and gravitational properties using quantum mechanical wave properties. I believe this is the first time an electron model is complete enough for these calculations to be made. Since a single model generates both the electrostatic and gravitational forces, the model also reveals how these forces are closely related.

Article Oscillating Spacetime: The Foundation of the Universe

Jose Oreste Mazzini

Dear Jose,

The reason I was asking why you specifically referred to 86.6% of the speed of light is that this is the velocity at which the effective mass of an electron reaches twice its rest mass.

You answered: " The speed was just a value so Lorentz´s gamma factor is equal to two (a round value). "

You may be interested to know that I also mention this velocity in Section (R. The Fusion Reactor) of this paper published in 2013:

Article Inside Planets and Stars Masses

It is at this velocity of 259 627 884 m/s (86.6% of the speed of light) that the carrying energy of an electron reaches the amount of .637420828E-13 joules (1.022 MeV), of which half (0.511 MeV) accounts for the mass increase of the electron (0.511 being the energy making up the initial rest mass plus this additional 0.511) and the remaining half (1.022/2=0.511) is the momentum energy that propells this total increased mass of the electron at this velocity of 86.6% of the speed of light.

Pair production from destabilizing photons of 1.022 MeV energy is well understood in the engineering community, as witnessed by this article published on this issue in the American Journal of Physics:

https://pubs.aip.org/aapt/ajp/article-abstract/74/5/457/1044262/A-simple-electron-positron-pair-production?redirectedFrom=fulltext

You may be interested to know also that according to electromagnetic mechanics, if this carrying energy in the exact amount of 1.022 MeV happens to be destabilized when the electron reaches this exact velocity of 86.6% of the speed of light, it is also likely to immediately destabilized into a pair of electron and positron, momentarily establishing the following triad with the electron that was being accelerated, of 2 electrons plus 1 positron with insufficient energy to escape from each other's interaction, that can result in them irreversibly accelerating to become a neutron as analyzed in this paper:

Article The Mechanics of Neutron and Proton Creation in the 3- Spaces Model

This process has been described and is now in the public domain following a patent application represented with Figure 18 in Section (X. Beyond the fission and fusion of atomic nuclei) of this paper:

Article Evolution From the Complex Plane to the Quaternion Coordinat...

Composite fermions such as neutrons and protons are well established as could resulting from electron-positron collisions (e+e-) as described in this paper from the Nuclear Physics B – Proceedings supplement, Volume 66, Issues 1-3, July 1998, pages 91-95:

https://www.sciencedirect.com/science/article/abs/pii/S0920563298000176

Full description of the conversion process is available at each of these 3 links:

https://patentscope.wipo.int/search/en/detail.jsf?docId=WO2017041162&recNum=1&maxRec=&office=&prevFilter=&sortOption=&queryString=&tab=PCT+Biblio

https://www.ic.gc.ca/opic-cipo/cpd/eng/patent/2997508/summary.html?type=number_search&tabs1Index=tabs1_1

https://ppubs.uspto.gov/dirsearch-public/print/downloadPdf/20180261348

I understand well your development and I conclude that it is a clear breakthrough in Quantum Mechanics that will help the upcoming generation to move beyond the Heisenberg uncertainty principle and to at long last resume fundamental research to finish exploring the subatomic level of magnitude.

Thank you Andrè for your well sustained comment, best regards

Soumendra Nath Thakur ,

if you like, dive with me into a fundamental scale world:

Imagine a symmetric 3D-grid:

• All neighbored locations have distance unit space-step.
• Each location has a value. (Its unit will appear later...)
• Each of these locations has six immediate neighbors: left, right, up, down, front and rear.
• The delay for transporting the value between two locations is one time-step.
• There is one single process, which is permanently performed by each location: minimize within one time-step the sum of all value differences to the immediate neighbors

Initial test situation:

All values are zero, except that of exactly one single location.

Now start this model within a simulation.

Preprint One Unique Possible Universe

• You will see, how a spherical front wave is initiated, which propagates with a constant velocity.
• You will see that at the origin of this sphere, at its focus, oscillation does not stop. Instead, it keeps the period time of a constant number of time-steps: the smallest wave period time.
• You will see that the propagating waves' period length increase, the less, the longer they are travelling.

Now, we make a jump into our physical world:

• we associate the constant propagation velocity with our measured c.
• we associate the smallest wave period time with our (hypothetical) PLANCK-time.
• we associate the unit of the values with our unit Volt -> scalar voltage field.

The increase of the waves' period length is redshift, which occurs because the energy coming from the oscillator's focus cannot propagate with infinite speed. Thus, a tiny part of the forwarded energy is reflected exactly backwards. This causes an intrinsic energy-backflow. Exactly backwards, because its velocity is also c, it's a wave within the medium.

We can further associate energy with the scalar voltage field:

E = e∙U ,

where e is the conversion factor (elementary charge).

Furtheron, we can associate the oscillator's energy with the wave period:

E = h∙f ,

where h is the conversion factor (PLANCK constant).

The intrinsic energy-backflow is the basis of gravity.

Additionally, because there is a minimum wave period time, the described medium is non-linear!

This property allows waves to interact in superpositions, which is very different from usually considered linear waves. This non-linearity is very very tiny, thus not measurable at short distances.

But, the most important consequence is, that resonance can take place!

The very first such resonance of our tiny PLANCK-sphere-oscillator simulated above, occurs at HIGGS-level (VEV=246.2965(6)GeV), the basis of all particles. (VEV: Vacuum Expectation Value)

Further resonances, based on VEV, build all the known (plus yet unknown) particles.

This very first VEV-resonance can be seen in the following equation:

K = 2∙G∙mᵥₑᵥ²/(ħc) = 8.13434(6)∙10⁻³⁴

which is the ratio between the gravitational energy between two masses, related to VEV

E_G = G∙mᵥₑᵥ²/r

and a wave passing the medium

E_λ = hc/λ,

where

2πr = λ/2 .

The VEV-masses seem to appear in both wave's nodes (λ/2), independent of their wave period length. This ratio K is constant for any wave, passing the "vacuum".

The result is that per wave's period length this factor K of the forwarded energy is reflected.

[Actually, it is the wave's period length, which increases by (1+K) per its wave period, for any wave: intrinsic redshift, caused by the medium itself; the resulting energy reflection per wave period actually is

1 - 1/(1-K).]