A structured wave model of electron
The electron, the lightest particle in nature, was discovered independently by several people, but J. J. Thompson was the first to show that it possessed a well-defined mass and charge. After more than 90 years, neither classical nor quantum physics have ever been able to create a satisfactory model of the mass and charge of the electron.
As early as 1921, Wolfgang Pauli stated: Electrodynamics as described by Maxwell and Lorentz is not compatible in general with the existence of charges unless it is integrated with elements that are alien to it.
Quantum mechanics was unable to integrate these elements.
In fact, although it constructed a quantum theory of electrodynamics that seeks to describe the field and dynamics of the electron, it was never able to express a model of either its mass or its charge.
Even today, we lack even the vaguest rational hypothesis based on any physical model of the electron. Such hypotheses have gradually either unconsciously or perhaps deliberately been replaced by a physics of “as if “.
Incomprehensible and contradictory properties have come to be attributed to the electron.
- As if it were a corpuscle
- As if it were a wave.
- As if it rotated on its own axis.
- As if its charge were distributed over the surface of its electric field.
- As if it possessed punctiform properties.
- As if the punctiform compression of is field increased its mass to infinity. And more of the like.
Among the many properties attributed to the electron, discussions of its spin can most easily be conceived of in a kind of model that could at least be visualized, but quantum mechanics is very careful not to admit that there can be a physical model of the electron.
In an attempt in 1925 to explain spectral anomalies in the x-rays emitted by Bohr’s atom, S. A. Goudsmit and G. E. Uhlembek formulated the hypothesis that the electron rotates on its axis.
Paul A. M. Dirac took it upon himself to use quantum mechanics to prove mathematically the value of the energy produced by what later came to be known as the electron’s “spin”.
In the course of his research, Dirac encountered a quadratic equation that appeared to provide the desired solution, but it was not compatible with the mathematical requirements of quantum mechanics, which demands that all energy equations be linear.
He therefore modified the first equation and obtained another one, which satisfied quantum mechanics’ requirements, but which still produced two different solutions for the energy of the spin, one negative and the other positive.
Dirac found it impossible to interpret these two solutions of the energy other than by positing the existence of two different particles possessing opposite charges. The first of which had to be the electron and possess the positive charge, while the other had to be a body that he, deeply conditioned by quantum mechanics, was unable to call a particle, but which had to possess a spin that was opposite to that of the electron.
Those who continued to explore the physics of “as if “ and ran into the interpretation of a hypothetical negative charge of the other particle, although they attributed to it a charge that was the opposite of that of the electron, understood it as a “hole” in quantum mechanics’ universe of “states”.
A “hole” in an anti-world into which the electron can hurl itself and, completing it, make both the electron and the “hole” disappear while both energies are set free and take the form of normal photons of radiation in the real world.
Once again we witness how difficult it is to interpret nature on the basis of mathematical intuition alone without the support of an adequate physical model.
In the bubble chamber, C. D. Anderson discovered traces of a particle coming from above under the influence of a magnetic field that accentuated the curvature of its trajectory as it passed through a lead plate. By comparing the curvature of the traces, he was able to establish that this particle had a positive charge, had lost velocity as it passed through the plate and that its mass was equal to that of the electron.
Picture 29. Discovery of a positon’s tracks
Dirac’ s “hole” later died a natural death as an unsatisfactory concept and was replace by a true particle with a positive charge called a positron, the existence of which was subsequently proven experimentally.
Interaction between the two particles with opposite charges would result in annihilation as Dirac had predicted and produce photons with the same energy that the electron and positron had possessed.
Dirac also predicted that the opposite reaction would be possible and the energy of radiation interacting with other particles and with a nucleus could take material form as pairs of particles and anti-particles. This produced what were later called “Dalitz pairs” consisting of an electron and a positron.
Millions of photographs made in the bubble chambers of every laboratory in the world bear witness to Dirac’ s successful prediction of the creation of pairs, but there is not one model of how this occurs.
For quantum mechanics, this too is a black box into which a photon enters with the appropriate energy level and from which an electron and a positron happen to exit.
Picture 30. Dalitz pair in a bubble chamber.
The sketchiness characteristic of Feynman’s famous diagrams express with extreme precision quantum mechanics’ inability to create a model of the phenomenon of the creation of Dalitz pairs.
Picture 31. Feyman diagrams of the creation and annihilation of electrons and positons.
No one has ever been able to say just what happens inside the box, but we will find a plausible model that is consistent with the dictates of new Wave Theory of the Field by continuing with the wave description of the Compton effect along a natural path that no one had suspected before now.
Countless dissertations have been written about whether physics can express itself in models, but no one has ever had any doubt that models can reflect the physical reality of real phenomena. Everyone has agreed in considering models to be a convenient ideal and fictional construction of some aspect of our knowledge of the phenomenon under investigation, while firmly denying that the model created in this way possessed any physical reality.
And yet, without models, all that would remain of our knowledge would be bare mathematical formulae that fail to create images in our minds, which believe they understand only when they succeed in “seeing” a mechanism made up of and out of all the models we had used to describe the phenomenon as we broke it down into separate parts that are as elementary as possible.
Our need for a model to take us from what is known to what we would still like to know leads our minds naturally through models of increasing complexity and no positivist or operationalist conditioning can change the way our brains work. Even the most obstinate of the advocates of the probabilistic description of the world of elementary quantum phenomena are forced to use models in their reasoning.
They claim to do this despite themselves and to be aware that they need to free themselves of this bad habit.
We on the other hand boldly state that the model is the pillar on which we will base the new wave physics we are about to establish.
Although we recognize that models are very frequently incomplete constructs subject to modifications and developments, we must establish a relationship that is different from what has existed up to now between physics and the models it uses.
The new theory maintains a different relationship with its models of the geometry of space and time and considers its primary objective to be the description of these very geometric forms into which the perturbations of Schild’s discrete geometry of space-time organize themselves.
When these representations were called “ideal physical models”, they referred symbolically to relationships between the physical phenomena under investigation.
The model is no longer merely a symbolic representation now. It still seeks to describe physical phenomena, but it uses a geometric representation of space-time to describe the very essence of reality — its geometric nature.
When the new paradigm has been developed, it will be supported by new geometric laws of discrete space-time that will completely replace the old laws of physics and its models will claim to be much more than convenient mnemonic devices.
The wave model of the Compton effect, that we will now show you at the next stage in its development, is a perfectly geometrical model. To demonstrate its validity, we can now produce a variant to this model that will lead us to a purely geometrical model of elementary particles and is the natural expression of the properties we have attributed to discrete space-time.
As we have already seen, the value of the term added to General Relativity’s formula for the deviation of light by the sun’s mass is the determining factor only when the ratio between the wavelength of the incoming radiation and the radius of the wave field of the particle that diffracts it l i/r is close to the unit.
To prove this, examine the possibility that the photon is deviated at an angle greater than the angles normally tested in the Compton effect and that the angle of diffraction equals:
λ i = 2 π.
As this is not impossible in principle, use our usual virtual experiment to observe as an electron is struck by a photon with a wavelength equal to half the wavelength of the electron at rest:
λ i = λe /2 and that it is deviated by a full 360 degrees.
Observe the phenomenon in four stages:
- As in the case of the Compton effect, the Principle of Relative Symmetry pushes the electron to velocity v1 as the photon transfers to it “half” of its momentum. The wavelength of the waves that pursue the electron is thus doubled:
λ i = λe .
- The electron’s field diffracts the decayed photon at a 180° angle and thus completes a half turn around the incoming obstacle.
- The wave train continues to be diffracted which rotates its direction another 180° and thus forces the electron to describe a complete circle.
Picture 32. Wave guide with a curvature that varies while it remains in a condition of resonance as long as its length is a multiple of the wave length.
At this point, the wave train has rotated along a circular orbit in a closed circuit around the source of the electron’s field of spherical waves. A wave-like condition now comes into effect that will direct the photon’s subsequent behavior.
If the length of the closed circuit traversed by the photon is equal to its final wavelength, the wave train will find itself subject to the “law of resonance” for waves.
No assimilations exist in the natural environment of discontinuous space-time. Thus the circular trajectory traversed by the photon in a space that is closed on itself can comply with the law of wave resonance that allows it to possess the condition of being stationary if its final wavelength is equal to the length of the close orbit traversed by the wave.
The circular photon, which now has a wavelength equal to the wavelength of the electron at rest, closes in on itself and perpetuates the circular motion of its wave “on the resonance orbit” with a length of:
2 π ro = n λo.
For n =1, we find the simplest case in which a single wave surface is predicted to circulate on the resonance orbit.
- Where n =1, the wave train that has completed one orbit can carry out several and superimpose all of its wave surfaces on the first one, which was already arranged in a condition of resonance on the radius ro.
This allows it to absorb what remains of the photon that created it, which disappears absorbed into the system of waves in resonance and superimposes all of its wave fronts on the wave front in resonance.
This raises a suggestive question for us. How specifically does this wave front circulating along a closed orbit in the condition of resonance?
“What part or section of the wave front is in the condition of resonance?”
Of the entire wave front, only the part closest to the center of the spherical wave field that diffracts the photon comes to be in the condition of resonance. In practice, only those portions of the wave that have the same trajectory of propagation as the orbit of resonance itself can achieve the condition of resonance.
The rest of the wave front consisting of elementary wave surfaces L2 propagates along trajectories that depend on the specific conditions of the propagation imposed on the elementary waves. And, as we have already seen, each discrete portion of the front of elementary wave’s propagation moves in a direction perpendicular to the tangent of the wave surface.
As it must comply with “both” conditions simultaneously.
The diffracted wave front is deformed during its propagation and takes on essentially a characteristic wave surface that brings to mind the spiral shell of nautilus consisting of the spherical projection of a particular curve known in geometry and mechanics as: Involvent.
Picture 33. An Ivolvent curve develops on the resonance plane of the wave that has been diffracted by 360° produced by projecting the wave surfaces that circulate on the resonance orbit.
As the resulting three-dimensional geometrical construction is rather complex, we must construct it in parts to understand it fully.
First work on two of its dimensions and describe first the curved Involvent plane that develops on the plane of the resonance orbit.
It turns out that the evolving plane produces wave fronts that increasingly approximate a circular shape at distances from the center in comparison with which ro, the radius of the orbit traversed by the wave front, becomes negligible.
A significant characteristic of the involvent plane lies in its ability to maintain unchanged the distance between two succeeding wave surfaces. This makes it possible to produce wave surfaces with constant wavelengths that are always identical to the length of the resonance orbit. 
Picture 34. Plane Involvent at varying stages of evolution.
On the basis of a bidimensional representation alone, the wavelengths illustrated by the involvent plane form a constant wave field, and, when described along with the energies of the elementary waves, they express the energy of a field of waves of mass according to the function of the natural constants “h” and “c”.
It now seems banal that we can get here in this way, but we can really identify this geometrical construction with the “perpetual wave mechanism” they were seeking that can produce waves with constant wavelengths continuously.
To construct the model in three dimensions, describe the rotation along the photon’s real resonance orbit that is made up of real two-dimensional wave surfaces.
Project ideally along the z-axis of the resonance orbit of this same orbit to obtain an infinite ideal cylinder, half of which is positive and half negative.
To make the evolution of the evolving plane’s surface three-dimensional as in reality, extend all vectors A which originate on the resonance orbit.
Increase their lengths with velocity “c” and are located within the solid angle “± m”, according to the temporal properties of the Schild reticular net, to obtain an almost spherical wave front like the one described in the figure.
Picture 35. An orthogonal spherical projection of an Involvent curve produces the structure of the Spherical Involvent.
This is the Spherical Ivolvent that is identified with an elementary particle and source of waves at rest with wavelength λ o and mass mo.
The wavelength produced by the Spherical Ivolvent is equal to the length of the orbit traversed by the wave front in resonance:
λ o = 2 π ro.
By observing the extremes of vectors A, we can describe two helixes with loops at constant intervals that, beginning at the resonance orbit twist around the ideal cylinder formed by projecting the resonance orbit.
One helix develops at + μ = π / 4 from the resonance orbit toward a positive z and is the mirror image of the first that develops at – μ = – π / 4 toward a negative z.
But the spherical image seems to shows were + μ = π / 2 and – μ = – π / 2 .
The spherical connection between the extremes of the vectors A in all their quantized intermediate positions down to the orbital plane constructs the surface of the spheroidal perturbation that we identify with the wave surface of a particle in expansion.
It looks like a snake biting its own tail, one wave after the other, without end to infinity.
They create a wave train that circulates permanently on the closed circuit of the resonance orbit that is made up of a single surface, whose wave surface constructs a spherical wave that evolves at the speed of light and expands into space at the same velocity.
As incredible as it may seem, given its extreme simplicity, there is really more than a slight possibility that a wave mechanism of this kind is produced by the evolution of the wave conditions we have just verified in the Compton effect.
These possibilities are realized when they are linked to the implementation of General Relativity, together with the resonance conditions and laws governing the propagation of perturbations in Schild’s space-time.
No known law prohibits this model — not even the Principle of Energy Conservation, which refers to secondary energy, i.e. the energy of the variations in primary energy, which in turn involve subquantum waves solely and exclusively.
It may seem incredible, but we now have a model of the electron in which there is much more than “something that spins”.
We have constructed a model that can react to the presence of photons, that can diffract them, in which mass varies in a consistent fashion according to the dictates of Relativity, and which possesses a precise spin value.
This model of the electron is based solely on waves.
In it, mass remains constant as the wavelength of the waves of its field is constant, which justifies its nature as the source of sherical waves.
Here too, there is “nothing new under the sun”.
Both de Broglie and Heisenberg had considered the existence of a probable wave mechanism at the subquantum level that possessed a frequency sufficient to justify the electron’s mass.
The Japanese Hideki Yukawa, who had predicted the existence and mass of the pion, also thought that it might be an expression of an elementary vibration in the field of space-time.
Now it is not necessary to Renormalize
Now, in the wave model of the electron we have created, it is no longer possible to reduce the field of mass to a mathematical point.
To resolve this last question, it is worth digressing into the history of research in microphysics, which for years sought to eliminate infinite quantities from the calculation of the energy of particle fields.
As far as was known, electrons had to be considered to be puntiform entities that lacked dimensions. But an electron is also its electrical field, and eighteenth-century physicists had already found an easy way to calculate the energy of the field in a single point in space by dividing the electrical charge “e” by the fourth power of the radius of its spherical field.
When one added up the energy of each infinitesimal section of the spherical surface of the field under examination using the integral method, one got the total energy of the field, which proved to be proportionate to the square of the electrical charge and inversely proportionate to the radius of the field.
Although this procedure proved adequate for the calculation of the real field normally described at macroscopic distances with respect to the dimensions of the electron.
It hid a trap when one sought to hypothesize a “puntiform” model for the electron for which the radius of the field had to decline and even be considered “zero” as the energy of a field calculated in this way became infinite.
This constituted a crisis in electromagnetic theory generally and involved physics as a whole in the uncertainty over quantum mechanics’ most elementary postulates. It was hoped that quantum electrodynamics would be able to eliminate infinite quantities in “autoenergy” (as the source of the quantum incongruity was called).
But it turned out that even quantum electrodynamics predicted that the interaction between the electron and its field would have to have an infinite intensity. This failure to reduce autoenergy also applied equally to particles’ gravitational fields.
If an elementary mass is considered punctual, the energy of its gravitational field in a mathematical point with radius zero must necessarily be infinite. The riddle appeared to defy resolution.
The question tormented generations of physicists for decades, but it was so important to quantum mechanics’ survival that a purely formal solution, based solely on mathematics and lacking any physical verification, had finally to be accepted simply to make it possible to continue to believe the theory.
They called it “renormalization”.
It was a mathematical sleight of hand by which a clever conjuror could make various infinite quantities disappear into the top hat.
It satisfied few people but was the only practical solution; a mathematical subterfuge that could never be proven physically, but the only formal safety net that allowed quantum mechanics to survive.
They found negative infinite quantities from which to subtract the positive ones and thus brushed the infinite quantities of autoenergy and all the other infinite quantities associated with it under the carpet.
Despite the absurdity of this expedient, they sought to use it to safeguard quantum mechanics “at least formally”.
It now seems clear that it is no longer necessary or even possible to reduce electrons to a mathematical point as the minimum dimensions of an electron are imposed by the radius of its resonance orbit ro, which has precise dimensions that can be derived from the wave interpretation of mass:
ro = h / 2 π mo c .
The infinite quantities of the autoenergy, and electromagnetic and gravitational fields of the electron disappear in the light of the wave model’s interpretation.
In the new model of the particle, the energy of the field is determined perfectly by the dimensions of the wave front’s radius of resonance that creates the Spherical Ivolvent.
Someone might raise a simple objection based on experimental results from experiments carried out in large accelerators that hurl electrons and positrons against each other at velocities very close to the speed of light, where calculations of their relativistic masses and cross sections seem to indicate that the electrons come to distances of less than
10-18 meters.
This would contradict with the dimensions the theory assigns to the electron’s resonance orbit of
2.42.10-12 meters,
but this objection can be explained in terms of the relativistic behavior of the model of the evolving plane, which we will illustrate later in the theory, when we verify the behaviors of the structure of the evolving spherical plane in motion at relativistic velocities in the context of our explanation of Lorentz forces in terms of waves.
At that time, we will see how relativistic conditions imposed on the particle can force the symmetrical center of the resonance orbit to shift within an area circumscribed by the orbit at velocities close to the speed of light, which would provide a theoretical justification of the dimensions which appear smaller than the particle that can be deduced from experiments with the large accelerators.
The intellectual and geometrical construction of the model of the particle as wave source has provided the author with considerable personal and aesthetic satisfaction, but the following mathematical description alone justified all the efforts involved.
Imaginary numbers (what an unfortunate name!), in addition to their numerous applications in analytic geometry, can also provide convenient symbols for the description of the rotation of the geometrical projection of vectors on Cartesian axes.
Their use provides a very precise physical proof in the Wave Theory of the Field.
They can be used to describe the rotation of the vectors perpendicular to each discrete portion of the wave surface that identify the points on the surface of the wave front diffracted by the resonance orbit and construct the evolving plane as a spiral that expands at a constant rate around an axis.
In the past, more than one person has sensed that the mathematical basis for this method of representing rotations had to have some fundamental significance in nature. Eulero used the five most significant numbers in the history of calculus to define the limits to infinite succession (e ¹ + 1 = 0).
If we use another formula to express the same thing
(e i π = -1), it can be applied to an infinite series of vectors that rotate the object by 90° and thus construct a spiral of segments of constantly diminishing dimensions around the point -1 on the Cartesian plane.
Referring to this formula, the mathematician Benjamin Peirce told his students when he had written it:
We cannot understand it and we do not know what it means, but we have demonstrated it so it is true.
Picture 36. The final formula of the Spherical Involvent which represents the particle mathematically in which the contour portion of the treatment expresses the possibility of varying the particle’s wave field as a function of the relativistic variation in its status of motion.
Now we can understand its fundamental importance for the wave description of the primary elements of matter, however. It can in fact be used to interpret elementary particles’ geometric structure mathematically and allows us to describe and understand the wave-like properties that determine their dynamics as mass.
We cannot fail to be amazed at the mysterious way in which the perceptiveness of geniuses anticipates events. There could have been no obvious connection in the past between these numbers and the idea that they could be associated with some fundamental reality of nature, although they appeared to be so special. They do not themselves contain any indication of their possible use.
The wave model of the evolving sphere contains much more than we have introduced here, however. It describes much more than the capacity to produce spherical waves with a constant wavelength, and thus much more than the proof that waves have mass.