The Paradoxes Of The Electron Point Source

The Paradoxes of the Electron Point Source Robert D. Morrison, Revised Sept 9, 2009 Revised Aug 22, 2010, added field description Abstract: The Standard Model quantum representation of the electron as a point source has been mathematically successful with the fullest verification possible with experimental collision results, but has logical paradoxes that have not been resolved. These paradoxes have been relegated to the “not yet understood” class by researchers, but the author investigates a unitary vector field model that gives logical solutions to the paradoxes.

The Electron Paradoxes The electron, and its anti-matter counterpart, the positron, are elementary particles with a constant charge and magnetic moment which enables versions that are either spin up or spin down when detected. Extensive experimentation, in particular, scattering and photon excitation experiments, have shown that the electron has dimensional characteristics of a singularity—the region of homogeneity and internal structure of a collision cross section has been shown to be less than an epsilon radius that is smaller than measurable with the best available colliders of our time. Standard Model physicists believe that the best logical conclusion, based on theoretical analysis of these results, is that the electron physically is a point singularity, that is, it is a zero dimensional object (or possibly very near zero dimensional with internal structure at the Planck length, i.e, 10-37 cm, some 24 orders of magnitude smaller than an atom). However, such a conclusion gives rise to several paradoxes. Some paradoxes are less severe than others, but there are a sufficient number of these that the conclusion of the electron as a point particle has been often questioned over the last century. There is no question that experimental results show a knife edge character, but from a geometrical standpoint, it is easily shown that it is not possible to determine whether the knife edge is a point, a line, a ring, or other shape—we only can know, and do know, that the crosssection of a collision is zero dimensional. There is another class of experiment that might be called excitation experiments, where a high energy photon is absorbed by a particle. The resulting behavior of the particle, where some portion of the photon is absorbed such that not all of the energy is immediately applied to the particle’s momentum, can show internal structure. For example, if the particle obtains additional angular momentum, or enters an excited energy state, this generally implies internal structure, but the electron clearly shows neither. The first and most obvious paradox comes from the Heisenberg Uncertainty relation, which in one of its many corollaries to the generalized form says that a discrete object cannot have a pair of orthogonal physical parameters such as momentum and width multiply to less than Planck’s constant. Experimental collision experiments show that the measurable point representation of an electron violates this to an extreme degree. The Standard Model represents the electron as a zero dimensional object, although currently

physicists expect to find internal structure at the Planck length. The problem with this is that if this is true, the uncertainty relation then specifies a minimum momentum of absurdly large values that only approach validity as the electron approaches the speed of light. The second paradox, equally serious, is the problem of momentum conservation. Electrons are repelled by sources of negative charge (such as another electron) and attracted by sources of positive charge, such as a proton or positron. The Standard Model (in particular, Quantum Field Theory) concludes that the force on an electron is mediated by photons traveling from the source of the charge to the electron. The problem is that when attraction is considered, momentum of the system is not conserved—the momentum of all entities prior to the interaction is the inverse of the momentum of the system after the interaction. The third paradox, also quite serious, is the question of electron-photon absorption. A photon of any energy will interact with an electron if the experimental circumstances permit it. The quantum nature of both particles is such that if a photon is absorbed, it is completely absorbed by only one electron, regardless of nearby particles that are also capable of absorbing the photon. If an electron is a point source, the geometrical ability to uniquely capture a complete photon whose wavelength may be billions of times larger than that of the target electron, instead of billions of other electrons within the photon wavelength, strains the imagination—and raises a serious question of the accuracy of the point source model of the electron. An important permutation of this is the quantum interference property that causes an electron to have wave properties when heading through two closely spaced slits—a clear violation of the believed point dimensionality of electrons. Even if the electron had structure at the Planck length, the wave effect would be far too small to cause the experimentally observed wave properties of the electron. The fourth significant paradox is that there are four permutations of particles that have the characteristics of the electron—the spin-up electron, the spin-down electron, the spinup positron, and the spin-down positron. All of these particles are currently concluded to have the zero-dimensional point size. The problem is that there are two degrees of freedom here that under the Standard Model are not allowed to have a physical dimensional element. Quantum field theory resolves this by assigning the particle properties that do not have physical attributes. Magnetic moment (responsible for spin) is problematic (and is the paradox) here, because in every other case it results from motion, yet if there is no dimension, it is not possible in theory to give rise to a magnetic moment. It is possible to say that the electron is not a true point but is very close to it (structure at the Planck length) to get around this, but this is pure speculation at this point in time. The fifth very important paradox is the charge renormalization problem that occurs with a point source. If the electron were truly a point source, the 1/r2 nature of electrostatic forces introduces infinities as the interactions near the singularity, causing severe inconsistencies with theoretical levels of energy that do not appear in reality. Many techniques to renormalize the infinities, that is, make them go away, are used to make the

point source model match reality, but this should be a red flag that there is a problem with the point source model. There has been considerable investigation into “clothing” the bare electron singularity with a shell of virtual electron-positron particle pairs. Similar to the methods shown in quantum field theory, this becomes a recursive problem that can compensate or hide the infinite 1/r2 nature of electrostatic attraction at r=0. While this clothing model could explain this fifth hypothesis, scattering experiments do not reveal evidence of such a two component structure to the electron. In summary, there are clear problems and questions about representing the electron as a point singularity or close to it. Consequently, researchers point to the success of the Standard Model’s ability to predict results and take the view that these are questions that are not yet known and don’t really have to be answered to generate predictions. This approach, sometimes called the “shut-up-and-calculate” approach, shuts down the question-and-answer method of logical analysis, with the result that short-term, we are able to get results, but long term, this will result in a stalling of theoretical progress. This is the case for elementary particles—the last significant breakthrough really was the discovery of quark substructure more than 30 years ago. It is time to look at these paradoxes and see if logical analysis will move our understanding of elementary particle physics forward.

Why the Point Source Model Is Accepted In Spite of Paradoxes There is no question that scattering experiments show a knife edge characteristic for electron internal structure, and that photon excitation experiments show no evidence of internal modes for the electron. Scattering experiments can show if there is a homogeneous region for a particle by analyzing the distribution of scattering angles, and it can show evidence of internal composition, such as multiple point sources, by analyzing the distribution of scattering angles combined with the momentum tracks of post collision particles (this is how evidence for two and three quark particle composition was found). However, it is not possible for collision experiments to control specifically where, in a given cross section, a particle will collide—thus it is not geometrically possible to distinguish a point from a line or ring of zero width. To do that, it would be necessary to hold a particle still and then exert control of source particles to hit distinguishably different points on the target particle—neither of which is remotely possible due to the coarseness of control of either the source or the target particle position in experimental setups. The other very effective method for detecting internal structure is done by imparting energy to a particle to see if excited states can be reached—particles with no internal structure cannot generate detectable angular momentum or other excitations of a ground state. These experiments show no exceptions, even for brief periods of time, to the electron case—all the absorbed photon energy is immediately converted to the electron’s momentum. While the collision results for electrons have always shown that its size is at or very near zero, the paradoxes, particularly the Heisenberg uncertainty relation problem mentioned previously, have stimulated interest in a variety of alternate electron models. In 1927, a model for the electron was proposed that was a shell, a sphere, with a radius (Compton

radius) that corresponded to the measured energy wavelength of the electron. It was postulated that there was a charge distribution on the shell that kept the shell from collapsing, and further refinement of this model suggested that the electron mass momentum of the shell gave it the inertial component necessary to sustain the shell radius. This model was repudiated in the 1950s when the magnetic moment of the electron was discovered and when scattering experiments showed that a surface or homogenous distribution of the charge was not possible. It turns out that a dipole will also give the Compton radius, but once again, scattering and excitation experiments showed that no evidence for internal structure with two nodes was present. About 15 years ago, I proposed something called a charge-loop, a zero width rotating ring variation of the dipole that also has the Compton radius, and uses the momentum of the rotation to counteract the inward attraction of opposite sides of the ring. This looked promising because it shows the knife edge characteristics demonstrated by experiment, but did not explain the specific mass of the electron, the two degrees of freedom for spin and matter/anti-matter, or the attraction of oppositely charged particles. It also was problematic because of the dependence on inertial mass in the ring traveling at relativistic speeds. It depended on the electrostatic attraction of E field components, which is not consistent with Maxwell’s equations. These efforts to form particles from Maxwell’s field equations fail because they are linear. Linear systems of equations cannot produce stable, physically localized states (a soliton), nor can they produce integral (quantum) energies at a given frequency required by E=hv. The only way to form a delta function of any type in a Maxwell field is by a Fourier or equivalent composition of an infinite number of waves, and these are not stable over time and are not quantized. The latter constraint (E=hv) implies that a degree of freedom needs to be removed from Maxwell’s field equations—since there is no constraint on photon frequency, the only other possibility was to constrain EM field magnitude. A reasonable candidate for a field solution with frequency as an independent variable is to constrain field amplitude, which suggests a unitary vector field obeying Maxwell’s field equations. This unitary vector field which I will call SU3 (unit magnitude complex value over R3) turns out to be very satisfactory because it not only generates quantized energy as a function of frequency, it also permits localized field twists that are topologically stable. Only complete 2 Pi twists are possible because this field SU3 has a (localized) default background direction, so partial twists must either dissipate or complete. I then discovered the linear and twist ring solutions of this field, which turns out to overcome all of the listed paradoxes for the point source model of the electron, and fixes the problems with the original charge loop model. And most interesting of all, the twist ring is a true stable soliton independent of the twist quantization—a single twisting, selfcorrecting field wave solution that happens to have the Compton radius and energy of the electron and does not need mass to explain its structure, nor does it need attraction of E field components. This SU3 unitary field twist solution forms the basis for the twist ring model of the electron and the linear twist solution for photons.

Why the Twist Ring Resolves the Paradoxes

After describing the field solution of the twist in a unitary vector field in detail, I will examine how the twist ring model forms the four related electron particles and photons, and addresses why there are antiparticles for electrons but not for photons. I then will examine each of the paradox cases to show how the twist ring is a more effective model for electron behavior. Further, I will then show how the twist ring has only one possible frequency in this subset of a Maxwell EM field that is directly tied to the coupling force ratio of electrostatic forces to magnetic forces, and additionally is stable to perturbative forces. Finally I show how the twist ring model of particles geometrically gives rise to the Lorentz Transforms of special relativity for both photons and electrons. First, note that the twist ring for the electron, and the corresponding photon straight line quantum twist, is a unitary complex valued vector field (called SU3 in this document) such that the twists move only at speed c (structures of twists such as circulating rings can go slower or stop, but the twist itself propagates at c). As mentioned, it should be clear that the general Maxwell field (without the particle equations) cannot yield either solitons or quantized photons obeying E=hv, since this has a linear solution space that will dissipate and allow a continuum of energies at a given frequency. Thus, the SU3 field is a vector field in R3 that is a continuum of unit length vectors. A physical field with no particles such as photons or electrons has a default state, and a SU3 field with full twists will form particles within the default state. This field is analytic and the gradient of the field is always finite (no field discontinuities are possible). It is easy to show that these constraints cause topological states to be stable—if there is a twist in the field, it is not possible for that twist to disappear without a discontinuity, nor is it possible for a single twist to spontaneously appear (but pairs of twists can spontaneously form if they are complementary). There is no constraint on the length of the twist, nor on the assembly of multiple twists in sequence—but because there is a default field direction, topologically any partial twist must dissipate (be equivalent to the default field), only full 2 Pi twists can remain stable in this field. There are many stable twist possibilities in this field model, two of which are a linear twist and a twist ring. It should be clear that this field solution describes both quantized twists and energy states that are only a function of frequency, and thus should be an ideal model for photons and electrons. Models of other particles are possible and further research is underway for these. It is necessary and sufficient to have each point in R3 described by two angles that are constrained to +/- Pi. For convenience, this will map to a vector field U3 of z on R3. This vector field is a phase field transformation of the actual physical field, and exhibits a mapping that provides the physical field property that every location has a unit magnitude vector. A photon is the simplest twist structure, and therefore quantization of the photon must occur since only one twist of 2 Pi radians will yield the required full rotation of the complex valued field. If there were more than one twist to a photon, then fractional energies should be possible in Einstein’s photomultiplier experiment (note that photons are often drawn showing multiple twists but such a system of n complete twists should

have energy multiples that are harmonics of n twists and thus is not correct). Note that the twist rotation circle of the complex valued field entity can be oriented with two degrees of freedom about the axis—thus permitting the twist model of the photon to correctly represent photon polarization. Different photon energies are represented by faster or slower twist spins. There are no other degrees of freedom for the photon that might give rise to an antiphoton. Obviously, the electron is the more complex case, but will be represented by a field entity that has a ring shape. The ring consists of exactly one complete twist. It is tempting to think that twist quantization (the requirement to return to the starting twist vector direction) is sufficient to establish the electron mass and radius, but it is not, since a larger radius can fit a slower twist rate. Finding the characteristics of the Twist Ring model that define the specific energy of the electron was critical to establishing the viability of the Twist Ring model, and I will show that momentarily. First, though let’s make sure that the Twist Ring has the right number of degrees of freedom to represent the four electron variations, and explore why photons have no antiparticles, but electrons do. This case is tricky since there is one degree of freedom for twist rotate direction relative to the motion of the ring—and it might be tempting to say that the second degree of freedom is the direction of spin of the ring, clockwise and counterclockwise. Unfortunately this can’t be the case, since the clockwise motion ring is identical to the counterclockwise motion from the other side of the ring—you can’t form two unique particles just from ring rotation direction. However, since the ring twists are complex valued, there is another degree of freedom that comes from whether the positive real part or the imaginary part is first in progression around the twist (see Fig. 1). Thus there are two true degrees of freedom and no more, thus showing that the twist ring is the only possible model that could give the spin-up electron, the spin-down electron, the spin-up positron, and the spin-down positron. Clearly, a point source electron model can give us no guidance as to why there are four permutations, but the twist ring shows how we get exactly four.

Fig. 1: The second degree of freedom for twists

We still need to deal with the question of why the electron only comes with one particular rest energy (mass). The twist ring, as mentioned, has every possible radius as a solution as long as the twist rate about the ring circumference can vary. The twist quantization requirement is upheld for every possible case. What is it about the twist ring that would point to why every electron at every location in the universe has exactly one particular rest energy? To answer this, and to show why the twist ring won’t dissipate or otherwise break up, let’s look at the forces within the ring. We cannot assume a mass, like Compton did with his Compton electron model, because of two serious problems: first, the elements within his shell are moving at speed c, causing relativistic inconsistencies, and any perturbation of shell components is not stable (equipotential central force solutions are not stable under perturbation, for example, planetary motions show that perturbations will cause elliptic variations in the orbits). In addition, even if somehow we force a circular path, this is a 1/r2 – 1/r2 differential equation which has a LaGrange minimum that is not unique to a specific radius. Finally, the distributed charge was assumed to be repulsive to prevent the shell from collapsing, but E field levels do not repel (Maxwell’s equations show a repulsive force only for particles). But the addition of vector twist motion to the twist ring means that an internal repulsive magnetic field with a field strength that varies as 1/r3 will be present on the opposite side of the ring, thus the ring will follow the Lagrangian minimum of a 1/r3 – 1/r2 differential equation. This is not a central force differential equation—it is far better,

because it is a restoring force equation that not only guarantees stability, but it only has one solution. It is important to note that the electrostatic attraction term is not caused by opposite sides having an electrostatic attraction, which would cause inconsistencies with Maxwell’s equations, which show attraction only for particles, not field components. The electrostatic term is due to the force caused by a loop or twist moving through an E field. And far better still, this solution, this one valid circular solution to the twist ring, happens to have the same value as the experimentally measured rest mass energy of the electron. Here is a simple 2D derivation that assumes field elements lie in a common plane: The Lorentz force equation is F = q E + q/c (v x B) When traversing the minimum energy path, the net radial force is zero but the velocity in the direction of motion cannot change--in the twist mode, twists always propagate at speed c. Therefore, the solution to the Lorentz force equation for a twist ring can be modeled to a first order by assuming that the field components of the twist are rotating such that opposite sides of the ring have complex vectors that are always pointing in opposite directions, and thus have a constant electrostatic attraction. In addition, each point of the twist ring has a complex vector that is rotating at a constant rate, thus generating a magnetic field that varies as Me/r3, where Me is the experimentally determined magnetic moment of the electron, that is, q h / (4 π me c r3). We then have: Fr = q2/rr2 – q c Me / (c r3) = q2/rr2 – q c q h / (4 π me c r3) = q2/rr2 – q2 h/ (4 π me rr3) In the 2D case, this will solve to a stable solution at the Compton radius, given the correct initial conditions: rr = h / (4 π mec) This radius forms a circumference that will be traversed in time tr, the reciprocal of which gives the twist ring frequency: c tr = 2 π rr = 2 π h / (4 π me c) = h / (2 me c) tr = h / (2 me c2) = 1 / fr f r = 2 m e c2 / h

Since Ee = h v = h v / 2 π, and v = 2 π fr, E e = 2 me c 2 for each of the two field elements, giving the expected rest energy of the electron as represented by the twist ring. Because the twist loop is self restoring, even incorrect initial conditions will asymptotically approach the stable solution. This is easily shown by realizing that if the actual r is greater than the stable r, there is a restoring force that is dominated by the 1/r2 term (the electrostatic term) over the 1/r3 magnetic repulsion term, causing a net attractive force that will push the field parts toward the stable r. If the actual r is less than the stable r, the 1/r3 repulsion force will dominate and push the field parts back toward the stable r. The solutions for these cases are somewhat more complex, but are easily demonstrated with iterative simulations. Figures 2 and 3 show simulation results that demonstrate how opposing field elements will eventually settle into the twist ring due to the self-restoring nature of the twist ring. It is especially important to note that any electromagnetic system of components that obeys Maxwell’s equations, not just electromagnetic field vectors, will observe a singularity of behavior at the Compton radius. This radius forms a soliton, a ring, in any field that obeys Maxwell’s equations that is solely a function of the ratio of the electrostatic attractive force to the magnetic normal force of an element in motion. This soliton can only happen at the Compton radius and is independent of twist frequency, mass, or even initial conditions.

Fig. 2: Simulation of EM field twist components approaching quickly to the Compton radius

Fig. 3: Simulation of EM field twist components, magnified, with different initial conditions that were somewhat unstable before settling into the Compton radius.

Now that we’ve established that the twist ring has the right parameters to represent the electron, unlike the point source model, let’s look at what the twist ring does for our paradoxes. The first paradox is the point source model of the electron violates at least one corollary of the Heisenberg uncertainty relation, especially for electron speeds much less than the speed of light. The twist ring does not; in fact it upholds it exactly regardless of the speed of the electron. This can be shown in the case of width versus momentum by computing Δx·Δp as the ring moves—any object with a ring of radius h/(4 Pi mec) will observe the uncertainty relation from standstill to just below the speed of light. The Δx·Δp relation becomes the resolvable size of the ring (Δx = 2 * h/(4 Pi mec))

times the smallest resolvable component momentum (mec) of the twist going around the ring gives h/2 Pi. As mentioned, the point source (or even the commonly believed Planck length electron structure) gives momentum values in the uncertainty relation that are at least 20 orders of magnitude too large. There have been efforts to get around this (see the Feynman checkerboard theory or other quantum jittering solutions) but these introduce internal motions and accelerations of such high energy that abandoning any semblance of known physics is necessary and thus is pure speculation, a problem the twist ring solution does not have. The second paradox for the point source is very clearly resolved in the case of the twist ring. Instead of trying to show motion by the exchange of photons, which will always yield a conservation of momentum paradox when managing the particle attraction case, the twist ring twists generate Lorentz force law motion in an EM farfield (Quantum field theory specifies that this also is done by the complex exchange of photons, but now there is no violation of overall conservation of momentum as is the case for the electron point source). As shown in Fig. 4, the twists that are parallel to the change in the EM field will have a net force in the direction either toward or away from the EM field source. Both sides of the ring will have the same force because the twist rotations (red arrows) are opposite, causing opposite direction magnetic fields for the opposite side, but so is the charge current flow, thus causing both sides to experience a Lorentz magnetic field force in the same direction. Note that current rings, of which the twist ring is a quantum version of, will align magnetic moments (coming out of the page in the figure) and thus the figure would correctly represent the stable case.

Fig. 4: Twists on opposite side of the twist ring generate magnetic fields that induce net force (orange arrows) in an electromagnetic field The third paradox where the electron must absorb a giant photon is a very interesting one. The basic case for the point source electron isn’t really a paradox, researchers just don’t know how it could work or just handwave it away—for example by saying that there has to be one electron more likely than all the others in the region to absorb a giant (very low energy) photon. But geometrically this is very problematic, and the usual resort is to say to quit thinking classically, this is a quantum mechanical issue. That shuts down any attempt to logically think through a solution. In addition, the corollary to this paradox is even more important—a point source electron going through a two point slit apparatus simply cannot yield a viable solution, geometric or otherwise, and leads to the great philosophical discussion of non-local action at a distance. The twist ring does not have this problem—the two slit apparatus should be a textbook proof of why the point source model cannot be the correct representation of the electron. The twist ring model clearly can retain its particle characteristics while interacting with (and getting interference from)

both slits, unless a detector causes a chaotic disruption of the phase characteristics of the ring. The twist ring does provide an attractive solution to the giant photon absorption problem which the point source model simply cannot address. A giant photon headed toward a large group of electrons is still a zero dimensional twist (it is tempting to think of the radial complex field values of the twist occupies a non zero radius because it is often drawn as arrows normal to the direction of propagation, but remember this is just a direction indicator, not a measure of physical distance). This giant photon twist thus will be able to penetrate through only one electron twist ring before being absorbed, providing a clean geometrical answer to why only one electron will absorb the photon. The fourth paradox where two degrees of freedom are needed to represent the electron particle variations has already been discussed. The fifth paradox is resolved with the twist ring because the twist ring has no internal subparticles or states that must adhere to Maxwell’s equations for particles. Unlike the point source particle model, there are no components of the ring that have electrostatic attraction, which in the point source model causes energy/force infinities due to E field components increasing as one approaches the singularity. It is the twist alone that interacts with the electrostatic field, and this field magnitude is finite by definition of the twist ring (twist ring of unitary EM field components). All of the renormalization efforts necessary for point source electron model interactions close to the singularity are not needed for the twist ring, no infinities arise. An important question to ask is why experiments do not show evidence of twist ring structure—if the twist ring were the correct model for the electron, it would seem probable that experiments would show differences than what a point source would do. Both collision and excitation experiments are commonly used to find internal structure of a particle. As mentioned previously, an excitation experiment involves shooting a high energy photon at a particle to cause an excited state or to add detectable rotation momentum. If the photon is absorbed in such a way that the energy does not show up instantly and entirely as outgoing particle momentum, some of the energy must have gone into particle rotation or some other internal excitation. It would seem that the twist ring would have to have a different momentum profile than a point source electron, but some examination of the twist ring shows that it cannot be distinguishable—the twist ring is a soliton, there is no solution with alternate energy levels, rotations, or other internal variations. Even if the twist ring had a momentary absorption of a photon before retransmitting it (implying an internal excited state) this would be detectable by experiment —but the soliton solution only has one valid geometry and radius, there is no other stable solution. Since the ring rotates at speed c, it is not possible to add to or subtract ring circumference momentum—and trying to spin the twist ring about the ring diameter is also not a stable solution. As a result, the traditional method of discerning internal structure in a particle has no distinguishing trait from the point source solution, and both the twist ring and the point source have the same zero dimensional cross section in

collisions. As a consequence, photon excitations or collision experiments cannot distinguish between a point source and the twist ring. One additional note: the photon is modeled as a quantized EM unitary field twist. If the twist ring responds to the E field presented by the opposite side of the ring, why doesn’t the twist of a photon respond to an E field source? This can be seen by realizing that the E field component of both sides of the ring are always pointing in opposite directions, whereas an E field from a distant source points in one direction. A photon twist will see an E field vector alternating parallel and anti-parallel to the twist E field vectors, and never see a net attractive or repulsive force. Finally, the twist ring shows a geometrical basis for the equations of special relativity. Since the twist within the ring must propagate at speed c, even if the ring as a whole is moving relative to a given frame of reference, it can be shown that the ring will obey the Lorentz transforms. The clearest way to see this is if the ring moment is parallel to the direction of motion of the ring. If a marker were placed on the ring where the real vector is maximum, this marker will trace a spiral on a cylinder. If one revolution of the spiral is unrolled, geometrically you will get a right triangle. This right triangle proportionally defines the rotation velocity as slowing down (because the twist rotation now must travel the spiral distance at speed c rather than just the ring circumference) according to the composite ring velocity v0 (see figure 5). When the ratio of ring rotation time in the observer’s frame of reference to the rotation time in the ring’s frame of reference is computed, the following results: Sqrt[c2 – v02] * tr’ = c * tr Thus, the apparent time to complete a revolution of the ring from the observer’s frame of reference will be tr’ / tr = c / Sqrt[c2 – v02] = 1 / Sqrt[1 – ß2] which is the special relativity time dilation factor. Similar computations can be computed to get the other Lorentz transform equations. Notice that the photon in the twist model is a straight line twist, there is no radial component such as in the electron’s twist ring, so the only effect of a changing frame of reference is to change the apparent twist rate. This will geometrically explain why particles see relativistic contractions due to the Lorentz transforms but photons do not. Photons appear to the observer to be moving at speed c regardless of the motion of the frame of reference in this model, just like in real life.

Fig. 5: The unrolled spiral of a twist ring moving radially at speed v0

SUMMARY: There are many paradoxes and unexplained consequences of assuming an electron point source model. While scattering experiments clearly show a knife edge zero dimensional attribute to the electron, this defines a set of a solutions only one of which is the point source. Another potential solution is the twist ring, and I show how this solution yields results that are a far better representation of the electron substructure. I have found that a unitary vector field that obeys Maxwell’s field equations has both quantized energy states and the capability of forming localized and topologically stable twists. These twists can be assembled in many ways, two of which are the linear twist and the twist ring. I have shown how the twist ring is a viable single, stable field solution that forms a stable soliton at the electron Compton radius. I have shown how a twist ring derives both electrostatic repulsion and attraction without resorting to photon exchanges that induce a momentum conservation violation. I have shown that twist rings provide an effective and simple answer to quantized absorption of giant photons. I have shown how the twist ring has exactly the right number of degrees of freedom to represent the four electron permutations, and shows why there is no photon antiparticle. I have shown how the questionable tactics of renormalization are not needed for the twist ring. I have shown why the twist ring would be indistinguishable from a point source model of the electron in experiments such as scattering or excited energy level experiments. Finally, I have shown how the linear twist and twist ring model yields the special relativity relations for electrons and for photons. I have made a clear case for why the twist field solutions in SU3, rather than the point source, are indicated as the correct representation of electrons, positrons, and photons.

CURRENT RESEARCH: A computer simulation model of the twist ring has been created that has been extensively used to verify (or deny) analysis of the SU3 field behavior and is currently undergoing testing for pair production and other twist combinations. If successful, this model will test various interactions and provide a path for creating an analytic solution to the twist ring field. I expect to find additional possible geometrical twist combinations for other elementary particles such as the various quarks, muons, and neutrinos.