The Physics of Negative Mass Tachyons

Ernst L. Wall

Institute for Basic Research

Palm Harbor, FL

To see the broad spectrum of the ongoing work at the Institute for Basic Research, visit their web site by clicking here.

This particular web site was originally created circa 1995 by Ernst L. Wall

Last Updated on April 30, 2011

1. Introduction

This web page provides a derivation of a very simple tachyon based, unified particle model that produces general agreement with experiment for most of the known subatomic particles, namely the electron, the proton, the neutron, and the mesons. It also produces the binding energy of the deuteron. Further, it suggests an electrodynamic origin of the electron’s de Broglie waves, and from that, an attendant longitudinal electrodynamic impulse neutrino associated with the and transitions.

Strangely enough, it turns out that this basic model is very easy follow, for the most part, for a reader with a relatively modest background in physics. This is because it is a Bohr-like model of the electron, and its complexity is even less than that of the Bohr model of the Hydrogen atom.

To begin with, we assume that the reader already knows what a tachyon is, so we dispense with the usual definition of a tachyon at this point even thought the results of this page would not be possible without a tachyon. We will present the derivation later.

The reason for this is that, if we present the derivation of the tachyon itself model up front, it will obfuscate the simplicity of the results for the reader. Therefore, we present a few of the results up front in order that the reader can see where we are heading with this model. Then we will provide the derivation.

But if the reader so desires, he may jump directly to the tachyon derivation in Section 8.

First, using the cutoff energy of the conversion curve and a tachyonic model, we provide a unique derivation of the magnetic moment of the electron, which is

namely, the Bohr magneton.

In order to produce this result and the following results, we show that the electron is actually spinning. (To say its tiny charge is revolving internally would be a more accurate term, but we will also, for the moment, use the term “spinning”.)

On top of that, we show that the internal structure’s dimensional change with velocity inherently provides a unique derivation of the mass-energy of the electron, namely the well known

Further, this model suggests an electrodynamic origin of the electron’s de Broglie waves. Finally, it suggests an attendant longitudinal electrodynamic impulse neutrino associated with the πμ and transitions.

From the tachyonic proton and neutron model, we show that there exists pion energy states whose excited energies behave in a manner similar to the Bohr Atom’s excited energy states. The energy levels are given by

· These energy levels are an analog to the energy levels of the Bohr Hydrogen atom and correspond to several mesons ranging in mas from 156 MeV to 4415 MeV.

· The first order transitions (an analog of the Bohr Atom’s Lyman series) account for the Psi mesons to within 5 %.

· The second order transitions (an analog of the Bohr Atom’s Balmer series) account for those mesons having masses ranges from the h through the a₀(980) to within 3%, except for the W(783), which is in error by 9.5 %.

· There are many others that fall out of this, namely, a binary pion series, and they too are discussed below.

The data that is used to verify this model is obtained from the standard physics literature, especially the Review of Particle Physics, published by the Particle Data Group of the American Physical Society. (See Tables 17-1 and 17-2 below.)

Here, we will simply state that these models of an electron, a muon, a proton, a neutron, and the basic nuclei through the alpha particle all stem from one graph and its cutoff energies. This graph is shown in Figure 1, below.

Figure 1. This graph shows the experimental transition of rates (relative) as a function of energy for the muon to electron and the rare, direct pion to electron conversions. This graph heart of this model in that the magnetic moments of the electron and muon are obtained directly from the cutoff energies of the two curves. (Yes, we know the V-A Model describes the μ e curve quite well.)

Finally, when it comes to the description of subatomic particles, the Standard Model includes such terms as flavor, strangeness, color, and charm

Here, we use none of these terms, but yet we get results that agree with experiment. In fact, the Standard Model, in spite of its successes elsewhere, has nothing at all like the meson model we will show below. In saying this, we do not disparage the Standard Model. We merely say this is a different approach.

In any case, because we do not use flavor, if one wishes, he could refer to this model as a “truly tasteless” particle model.

Also, because we do not discriminate among particles based on strangeness and color, this model could also be referred to as a “politically correct” particle model.

The meson model is produces mesons, but no charmed mesons. (No witches, warlocks, shamans, demons, etc., are required to conjure up results.) It just works. (Arthur C. Clark stated that “Any sufficiently advanced technology is indistinguishable from magic”. Perhaps some of the early theorists that looked at these particles thought that there was some magic in them. However, there is nothing magic, or charmed about these particles. They are simple pion resonances.)

Also, there are no quarks. They don’t seem to be needed here.

But, none of the comments above should be construed in any way to imply that there is anything wrong with the Standard Model. It has been a spectacular success. What we have here is a different method for obtaining information about subatomic particles. It is nothing more, nothing less. However, we do state that the Standard Model produces nothing like the meson model described here.

Finally, all of this has been published in The Bulletin of the American Physical Sociey, The Hadronic Journal and in the book, The Physics of Tachyons, Ernst L. Wall, 234 pp., ( Hadronic Press, 1995). See the list of publication

This model is described in more detail in a book, The Physics of Tachyons, by Ernst L. Wall, and in the published papers provided at the end of this web page. The book itself is available directly from the publisher, the Hadronic Press, (which is part of The Institute for Basic Research) or Amazon.com.

Terminology: tachyon, magnetic moment, spin, electron, muon, pion, meson, proton, sigma hyperon, psi meson, neutrino, de Broglie wave, Bohr magneton, quantum mechanics, orbit, mass energy, relativity.

Contents

1. Introduction

2. A Summary of the Electron’s Mass Energy and Bohr Magneton as Derived From the Tachyonic Electron Model Using the μe Curve’s Cutoff Energies for Energy Levels

3. The Proton Model

4. The Neutron Model

5. The Light Nuclei

6. The Basic, Single Particle Meson Model

7. The Binary Mesons

8. The Derivation of the Magnetic Moments of the Electron and the Muon by Means of the Tachyonic Model

9. The Semi Classical Revolving Charge Model

10. An Electrodynamic Model of Electron de Broglie Waves

11. Electron-Electron Interactions and the Derivation of the Electrodynamic de Broglie Wavelength

12. Interactions of the Wavelets with Apertures and Edges

13. Electron-Lattice Site Scattering of Compton Wavelets - The Davison Germer Effect.

14. A Longitudinal Electric Field Model of the Neutrino

15. The Imaginary Mass Tachyon Model: Not Even Wrong

16. A Brief Comment on Constants and Units

17. Final Comments

18. Publications by Ernst Wall

Appendix 1. A Digital State Machine Simulation of the Universe and the Difficulties of Time Travel

_____________________________________________________________________________

2. A Summary of the Electron’s Mass Energy and Bohr Magneton as Derived from the Tachyonic Electron Model Using the μe Curve’s Cutoff Energies for Energy Levels

Before we begin, we would like to note that the mass-energy relationships and the Bohr magneton, as given here for the electron, might appear to be ad hoc relationships that were brought forth from thin air. However, they are most assuredly not ad hoc. They were derived from the conversion curve’s cutoff energy using a negative mass tachyon model. These results were as much a surprise to the author as they may be to the reader. That derivation is presented later.

But to begin, consider that high energy scattering experiments have shown that the electron, a charged particle, has a cross section of some 10^-7 to 10^-11 barns, depending on the collision energy. Simplistically, we will say that the charged particle’s diameter is as small as some 10^-18 cm.

This would seem to be an extremely small entity to generate the relatively large magnetic field due to the electron spin. Further, it is extremely small compared with another fundamental length associated with the electron, namely, the Compton wavelength which is given by

= 2.42631023767219E-10 cm. (2-1)

Based on the these considerations, we could simply posit here that the very tiny charged particle within the internal structure of the electron behaves as if it were trapped in a photon-like circular orbit and that it revolves at the speed of light. ( If it revolved any slower than the speed of light, it would radiate its energy away. If it were any faster, it would be a tachyon. )

However, it goes a step beyond that. Based on the cutoff energy of the conversion curve (above) and the tachyon model to be described below, the circumference of the orbit is required to be precisely the Compton wavelength. Hence, the frequency of the revolving particle is given by

. (2-2)

Henceforth, we will refer to this as the Compton frequency.

If we continue treating this structure as if it were a photon and apply the Einstein relation to the Compton frequency, we have for the energy of the electron’s charged particle

c². (2-3)

Hence, we have derived the rest mass energy of the electron based on the cutoff energy of the

curve. But to take it a step further, we observe that if we simply multiply the Compton wavelength in Eq. 2 by the Lorenz-Fitzgerald contraction, we have that

. (2-4)

That is, we treat it as if the Lorenz-Fitzgerald contraction for a closed particle structure affects both the dimension transverse to the dimension longitudinal to the velocity dimension as well as the longitudinal dimension itself. Normally, we would expect that the contraction of a circular orbit such as this would only be in the direction of motion so that the orbit would become elliptical. However, if we take these results seriously, the elliptical contraction would not appear to be the case for the internals of closed particle system. That is, it appears that the Lorenz-Fitzgerald contraction may apply to the whole composite shape of a subatomic particle as opposed to contraction along the direction of motion. Eq. 2-4 is as observed to be the case experimentally for a particle system as viewed from the outside. We make no comment other than this at this time.

Next, if an observer were standing near the orbit of the revolving particle, the current that he would see passing him would be given by the number of charges per second passing by him, or

(2-5)

where we use the cgs units of charge, q = e/c. The magnetic moment of an orbiting charge, μ, would be the product of the current passing a point on the orbit multiplied by the area of that orbit. I.e., we would have

, (2-6)

where we have used and where the revolving charge’s orbital radius is . This expression gives, of course, the well known magnetic moment of the electron, namely, the Bohr Magneton.

Hence, it is not unfair to say that the electron actually spins. (Perhaps it would be more accurate to say its tiny charge revolves internally, but nevertheless, it still “spins”!)

r_c is also referred to as the reduced Compton wavelength and is, in effect, the orbital radius of the electron’s revolving point charge.

But the critical issue here is why the electron’s charge was assumed to revolve at all, let alone revolve in an orbit having a Compton wavelength as the circumference. The reason was, quite simply, that the tachyonic model and the cutoff energy of the μe curve required it, as will be shown below. But note that the tiny charge, in high energy collisions, will still appear to be tiny. The fact that it is revolving will not cause it to appear larger in a scattering experiment than if it were not revolving.

It should be noted that this value is the same as the orbital magnetic moment of the ground state orbit of the Bohr Hydrogen atom. That was first derived by Neils Bohr around 1913. Why these two different states of the electron produce the same magnetic moment is not clear at this time.

To this author’s knowledge there is no other derivation of the Bohr Magneton for the electron spin itself based on its internal structure. That is, there is no other derivation that states that the electron has an internal structure as opposed to being a simple point particle. There are some angular momentum (quantum mechanical) requirements that state that the magnetic moment is given by the Bohr magneton, but no derivation of a revolving particle with a finite orbital radius whose value is obtained from some basic, measured energy level.

However, the fact that the electron’s magnetic moment had this numerical value was well known since the early 1920s. This was based on the observations of the splitting of Hydrogen’s spectral lines (the fine structure) and the on the Stern-Gerlach experiment.

What was not known was why it had this value.

Also, it should be noted, the magnetic moment as calculated here is somewhat smaller than observed experimentally. To have the correct value, it must be multiplied by the gyromagnetic ratio, g_e/2, where g_e= 2.002319394367. This value has been measured out to some 12 decimal places. That is, the Bohr Magneton as calculated here is too small by about 1 part per thousand.

We have no interest in pursuing a one ppt error at this time insofar as it would apply to this model because there are far more interesting things to pursue. Some approaches were taken to earlier to add this correction are described in this author’s book, The Physics of Tachyons that is listed below.

We have no further comment on the electron at this time.

______________________________________________________________________________

3. The Proton Model

The nearest analog to the conversion for the proton is the Σp curve. However, that curve has no clearly defined cutoff energy such as there is for the electron and the muon. Therefore, an inverse approach must be used for the proton. That is, using the electron configuration but the magnetic moment of the proton, μ_p= 1.4106076 x 10^-23 ergs/gauss, we find that the charged particle's orbital radius, r_c , is 0.58736077 fm.

The masses of the proton and sigma hyperon are 938.27231 MeV and 1189.37 MeV, respectively. The result is that their mass ratio is R_P =1.2676, and the mass of the tachyon is -251.10 MeV. Using these values in the tachyon model, we find that the radius of the tachyon's orbit is 2.782 fm. (More will be said about calculating the tachyon radius later.) High energy and low energy scattering experiments indicate that these two radii agree with experiment to within 3 %.

See the Figure 2, the composite proton/neutron diagram, below.

4. The Neutron Model

Adding a similarly orbiting, but smaller, negatively charged pion with its tachyon to the center of the proton, and we have a neutron. That is to say, it is a coaxial model with the orbits sharing the same orbital plane and revolving in the same direction.

While quantum mechanical considerations indicate that the pion is a spin 0 particle (zero magnetic moment) we treat it here as if it has a tiny magnetic moment, too small to interact with other particles except as below.

Subtracting the magnetic moment of the neutron ( μ _N = 9.6623707 x 10^-24 ergs/gauss ) from that of the proton, we find that the orbiting pion's magnetic moment is μ _π = 0.4443705 x 10^-23 ergs/gauss. (Note, incidentally, that this value is within 2.5 % of the magnetic moment of the deuteron. More will be said about that shortly.)

Using this value to calculate the radius of the orbiting pion's charged particle, we find it to be 0.18503077 fm. High energy scattering experiments have verified this value. Equating the pion's de Broglie wavelength to the circumference of its orbit, its energy level is found to be 4076 MeV. Its excited levels are found to be

(4-1)

with values of the index, n, ranging from 1 through 9. This accounts for energy levels of the meson model previously shown. The first of these resonances to be discovered was a neutron resonance and was called the J particle by S. Ting. Then, the same resonance was found in e-p collisions by B. Richter. Hence, it appears that the meson family consists of various states of the pion, both within the neutron and in the electron.

Figure 2. This is a composite of the neutron and the proton.

It shows the orbits of the proton’s sigma hyperon and its tachyon.

When the revolving pion is added to the center of the sigma

hyperon’s orbit, we have a neutron. The pion’s tachyon is not

shown because its orbit is only fractionally larger than that

of the pion. The addition of the pion does not change the

dimensions of the proton’s components significantly. The

drawing is to scale.

(See publications 1, 17, and 20, below)

5. The Light Nuclei

Now consider combining a neutron and a proton to form a deuteron. In that case, the negatively charged pion from inside the neutron would be attracted to the proton so that the two positive charges would be mutually attracted to the pion. Although the positive charges would repel each other, they would reach a balance position wherein their mutual repulsion would balance their attraction to the pion. ( For the calculation of the potentials between the revolving charges, consider them to behave as if they were rings. For the treatment of potentials between rings, the reader is referenced to Kellog, Foundations of Potential Theory, Dover Books 1953. ) This is shown below.

A better approach would be to treat them as point charges and use a computer simulation to calculate the energies. That has not been done as of this time.

Figure 3. The deuteron consists of a proton and a neutron, the neutron being a combination of a proton and a pion, as shown in the prevous drawing. Therefore, we show two positive protons and a negative pion. The protons are counter revolving, leaving only the revolving pion to generate a magnetic moment. (While the pion is generally is believed to be a spin 0 particle with no magnetic moment, it is predicted here that if an accurate, direct measurement is made of its magnet moment, it will be found to be of the order the deuteron’s magnetic moment.)

If a proton approaches a neutron, its sigma hyperon will attract the neutron's pion, thus axially deforming the neutron and causing it to behave as a deformable dipole. While the sigma hyperons electrostatically repel one other, they are both attracted to the pion, thus causing this model to be somewhat similar to the Yukawa model.

This produces a highly nonlinear attractive force, so that an experimental evaluation of the force would cause it to appear to have no relationship to simple electrostatic forces with the result that it would not be recognized as an electrostatic force. In that case, you might name that force, say, a “nuclear force”.

Based on the quadrapole moment of the deuteron and the diameter of the proton’s charged particles, we find that the spacing of the sigma hyperons is 1.323 fm, and their magnetic energies are 0.2276 MeV each. Using these, the sum of the calculated electrostatic and magnetostatic binding energies is 2.381 MeV, as compared with the measured deuteron's binding energy of 2.2246 MeV, a 7.0 % difference.

Similarly, crude calculated values for the binding energy of tritium is 28.3 % less than the experimental value, and for the helium-three binding energy is 43 % less than the experimental value. This is discussed in detail in The Physics of Tachyons.

These values are not precise because they are based on crude estimates rather than carefully integrated algorithms. However, in spite of the lack of precision, an argument can be made that these light nuclei could be at least partially bound by electromagnetic forces, and not totally by a separate nuclear force. It is likely that with more careful calculations, better agreement will be obtained.

But if these large errors seem excessive, it should be noted that such errors are not uncommon in the particle physics literature where errors of 50 % or greater are not unheard of.

The calculated magnetic moment of deuterium is within 2.5 % of experiment, the calculated magnetic moment of helium-three is within 1.7 % of experiment, and the magnetic moment of tritium is within 3.5 % of experiment. These are shown graphically in Figure 7.

Figure 4. The rather trivial chart of measured and calculated magnetic moments of the lighter nuclei.

(See publications 1, 17, and 20, below)

6. The Basic, Single Particle Meson Model

This meson model consists of various resonances of a pion. It was originally derived from the tachyonic neutron’s pion as a set of excitation energy levels, or resonances, and it was only after the paper was published that it was realized that these resonances were, in fact, mesons. (The tachyonic neutron is discussed later in this web site.)

Later, it was realized that the pion, as a parent particle of the electron and muon (within the context of this model), should resonate in the case of colliding electrons. This is, of course, the case in that large numbers of mesons are produced. In addition, electron-electron collisions produce pions, muons, and gamma rays as would naturally be expected based on this model.

Because of its extreme simplicity and its excellent fit to the experimental meson data, we will present the meson portion of the model prior to the actual derivation of the basic tachyonic particle model.

The first theoretical prediction of a meson resulted from Hideki Yukawa’s proton model which was published in 1935. In 1937 a particle of mass close to that of Yukawa's prediction was discovered in cosmic rays by Anderson & Neddermeyer and in a cloud chamber by Street & Stevenson. These were independent experiments. This particle was, at first, thought to be the Yukawa particle but it was later concluded that it was not the Yukawa particle. 10 years later, Lattes, Muirhead, Occhialini and Powell discovered the pion in a photographic emulsion that was exposed at high altitudes. It was concluded that this was the sought after Yukawa particle.

Still later, other mesons were later observed in various high energy particle collisions as interaction energies, or even as free particles. These resulted typically from pion-proton collisions, K meson-proton collisions, or electron-positron collisions. Some of the earlier and more spectacular observations were made inside Hydrogen bubble chambers. Typically, these reactions produce pions as by products, although K mesons and other mesons are also produced.

In this model, quite unlike the standard model, all mesons arise from a resonating pion, whose internal binding energy is 4076 MeV. But what is so interesting is that the pion (again, unlike the standard model) is the mother particle of the muon and the electron, so that one would expect that electron-positron collisions must produce at least pions, muons, and gamma rays, as well as the other meson energies. The most obvious would be the psi resonances. In fact, some of them were originally published in a table of excitation energies in a tachyon-hadron paper (see references below), but their significance was overlooked at the time.

These internal pion excitation levels are given by

, (6-1)

where the index, n, ranges from 1 through 9. (This expression is similar to the expression for the energy levels of the Bohr model of the atom.) To obtain the various meson energies, you may use the above equation as follows:

(NOTE: The meson graphs shown below were put together in the 1990s, so many of the mesons detected since that time have not been added as of yet. They will be reworked as time permits.)

Figure 5. The transition diagram showing the basic mesons as they arise from transitions among the various resonant states (energy levels) of the pion.

The first 9 levels of the pion, along with their transition energies are shown in Figure 5, below.

· The first three levels of E _m correspond to the energy levels of the y(4415) , the f(1020) , and the K- mesons to within - 8 % to +8 %. The next two levels correspond to resonances that arise from a K p collision and a p p collision, these resonances having energies of 280 MeV and 156 MeV, with agreements of -9.3 % and -4.3 %, respectively.

· The first order transitions, corresponding to the so-called "charmed" psi mesons, are within -1.3 % to +4.7 % of the observed experimental values. (The Bohr Atom's analog is the Lyman series.) These transitions are shown in the transition diagram above, and the values are plotted in the graph below (Figure 6) along with the corresponding experimental values where the index is the value of the energy level, n, that is differenced with the value for n=1.

· The second order transitions ( corresponding to the Bohr Atom’s Balmer series ) produce the seven light mesons, i.e., the h through the ao(980). The agreement with experiment ranges from 0.5 % to -2.3 %, except for the omega(783), which is within +9.5 % of experiment. These are shown in Figure 7, below, along with their corresponding experimental values, where the index is the value n that is subtracted from the value n = 2.

THE READER IS INVITED TO TRY THIS WITH A SIMPLE HAND CALCULATOR.

1. FIRST, SIMPLY CALCULATE THE ENERGY LEVELS OF EQ. 6-1 FOR N = 1 THROUGH 9.

THAT WILL PRODUCE THE ENERGY LEVELS OF FIGURE 5, ABOVE.

2. THEN, SUBTRACT EACH OF THE ENERGY LEVELS FOR N = 2 THROUGH 9 FROM THE ENERGY LEVEL FOR N = 1.

THESE ENERGY DIFFERENCES WILL GIVE YOU THE MASSES OF THE PSI MESONS SHOWN IN FIGURE 6, BELOW.

3. SIMILARLY, DO THE SAME BUT SUBTRACT THE LEVELS FOR N = 3 THROUGH 9 FROM THE ENERGY LEVEL FOR N = 2 AND YOU WILL HAVE THE MASSES OF THE LIGHT MESONS SHOWN IN FIGURE 7, BELOW.

To reiterate, it is not necessary to understand quarks, gluons, etc, to achieve this systematic agreement with experiment.

(NOTE: The meson graphs shown below were put together in the 1990s, so many of the mesons detected since that time have not been added as of yet. They will be reworked as time permits.)

Figure 6. The first order transitions of the pion resonances, or the psi mesons ( the charmed mesons). The error ranges from -1.3 % to +4.7 %.

(NOTE: The meson graphs shown below were put together in the 1990s, so many of the mesons detected since that time have not been added as of yet. They will be reworked as time permits.)

Figure 7. The second order transition of the pion resonances. The agreement with experiment ranges from 0.5 % to -2.3 %, except for the omega(783), which is within +9.5 % of experiment.

(See publications 17 and 18 below.)

7. The Binary Mesons

Because many of the mesons studied here arise from relatively large energies that produce two or more pions, we must consider that at least some of these collision should produce energy levels that are the sum of two particles excited mass-energies.

Since the energies can be a combination of any two levels, we combine all possible energies of the lighter mesons (the second order transitions, above) and obtain binary energy levels of the electron-positron collisions.

These binary levels are graphically shown in Figures 8 and 9, below, along with their corresponding experimental values. Here, the index n arbitrarily picks up from the value n = 9 in the graph above. Both the experimental energy levels and the summed values of the light mesons are arranged in ascending numerical order and plotted. None of the experimental mesons are named here simply because there are too many of them. They are shown in detail in The Physics of Tachyons.

The agreement with experiment ranges from -16 % to + 12 %. While this might appear to be only crude agreement between experiment and theory, it should be noted that no attempt was made to compensate for any binding energies between the positive and negative excited pions.

Further, many of these mesons were not discovered at the time this model was originally developed, so that this model predicted more binary mesons than were know at the time.

There are, however, a number of mesons above the binary set that it does not explain. These are also shown here. No attempt has been made at to account for them at this time, although it is likely that they arise as excitations from an even more massive particle than the pion.

Note that the first 19 binary mesons have a scalloped shape that a reflection of the parabolic shape of the light mesons energy curve. The experimental values, while somewhat crude, seem to correspond to this scalloped shape. The first 20 levels are shown below in more detail to illustrate this shape.

(NOTE: The meson graphs shown below were put together in the 1990s, so many of the mesons detected since that time have not been added as of yet. They will be reworked as time permits.)

Figure 8. The binary pion resonances that arise from electron electron collisions. The agreement with experiment ranges from -16 % to + 12 %. It is to be noted that the error includes those of the two particles that are combined into one resonance. Further, no particle-particle interactions are considered. If those were included, it is likely that the agreement would be even better.

There is no other model that produces as many mesons and is so simple a manner as this model, especially when you add the binary mesons, as described below.

(NOTE: The meson graphs shown below were put together in the 1990s, so many of the mesons detected since that time have not been added as of yet. They will be reworked as time permits.)

Figure 9. Details of the binary resonance masses showing the scalloped shape of the calculated value curves and how they have a semblance of appearance similar to the curve of the experimental particles.

(See publications 1, 13, 15, and 17, below.)

8. The Derivation of the Magnetic Moments of the Electron and the Muon by Means of the Tachyonic Model

In order to produce the results shown above without resorting to ad hoc methodologies, it is necessary to take a contrarian approach to the πμ and μe transitions. Instead of saying that a pion “decays” into a muon and a muon “decays” into an electron, we take the approach that the pion captures a negative mass particle and becomes a lighter muon, and the muon in turn captures still another negative mass particle and becomes an electron. (Note that for those who worry about neutrinos, not only do we acknowledge that they have been observed, we have an appropriate model that we discuss later.)

Note that the dimensions of this model are precisely defined insofar as its spatial and velocity dimensions are concerned. This is an issue that is likely to severely try the patience of any self respecting quantum mechanic. He would assume that any particle such as this must be described by a wave function and that its dimensions could not be precisely determined. More will be said later about why this is not necessarily so within the context of this model.

Hypothesis: A muon converts to an electron by capturing a negative mass tachyon. A pion converts to a muon by capturing a negative mass tachyon. A pion converts directly to an electron by capturing two negative mass tachyons.

To begin the derivation of the characteristics of the electrons, the masses of the muon's and electron's tachyons are obtained by subtracting the heavier particle from the lighter particle, i.e.,

(8-1)

(8-2)

Next, we will need to utilize half of these masses as binding energies. I.e., we have

(8-3)
.

(8-4)

The sum of these energies is

(8-5)

The right most curve, the direct π e conversion curve, is less well known and describes the relatively rare, direct conversion of a pion into an electron. This event occurs about one in 10⁴ pion conversions.

Next, examine Fig. 1. It is a composite of two particle conversion curves. The μe curve on the left is well known and is contained in most particle physics books.

It should be noted that accurate fits to the μe curve have been produced by the V-A theory, so that we make no attempt to claim that the positive results of this model invalidate the Standard Model . This model is simply a different approach.

The generally accepted assumption is that two neutrinos are produced by the decay of an electron into a muon, and the shape of the curve is determined by the relative angles of emission of the two neutrinos. That is to say, the curves are normally considered to be decay spectra.

Furthermore, neutrinos have been observed, and the residual energies of this model are 20 eV for the electron model, and 123 MeV for the muon model, more than enough to account for the generally estimated masses of the neutrinos.

The interpretation used here is that the reaction during the capture of a tachyon by a muon has a residual energy whose distribution is described by the μe curve. However, if the reaction energy is greater than that of the binding energy of the electron's tachyon to the charged particle, there will be no capture and hence, no electrons will be produced. The point at which this happens, 52.6 MeV, is the cutoff energy of the μe curve. This compares favorably with the energy of Eq. 4.

But having said that, the possibility of a neutrino carrying away part the energy but leaving a tachyon is not precluded. (See the neutrino model, below.)

Figure 7. This shows an analog for the balance condition that is used to calculate the center of mass for the positive mass charged particle and the negative mass tachyon. Note the use of parallel strings to attached to the weight and the balloon to the shaft. Probably the only particle model in which strings have proven to be useful! (Well, at least we kept our word in the abstract wherein we stated that we used strings in this model, so we cannot accused of total misrepresentation. ) (W. Niblack is thanked for pointing out the above balance condition for a negative mass.)

The πμ capture, on the other hand, produces monoenergetic muons at an energy 4.119 MeV, so that there is no cutoff energy. Therefore, another approach must be taken. So compare Eq. 5 with the 69.5 MeV cutoff energy of the μ e curve. The double tachyon capture implies that the total binding energy of the muon and electron's tachyons is half of sum of their masses, and hence, the binding energy of the muon's tachyon is also half of its mass energy. Note, incidentally, that the difference in the two cutoff energies is 16.9 MeV, which is half the muon's tachyon's mass energy as given in Eq. 3.

Again, as in the case of the the πe conversion, a neutrino is emitted. But in any case, we have no state transition model as of this time that will give the energy balance between the neutrinos and the tachyons.

Because of its negative mass, a revolving tachyon will have an inwardly directed force, not an outwardly directed force. This inwardly directed force of the tachyon balances the outwardly directed force of the orbiting charged particle, thus maintaining the particle systems in tightly bound orbits. The balance conditions are similar to that of a helium balloon (a negative mass analog) on one end of a massless rod balanced by a less massive weight placed between the balloon and a pivot on the other end of the rod. Because of the negative mass, the center of mass of the system is at the pivot, and is thus external to the line connecting the charged particle's orbit and the tachyon. This is shown in Fig. 7.

Figure 8. This shows the rather bizarre behavior of the electron’s revolving charge around the center of mass that is external to the line joining the charge q and the tachyon. From the the tachyon’s perspective, it revolves around the charge with an orbital circumference equal to its de Broglie wavelength, λ _Te .

Based on the above, in general, the magnitude of the binding energy, which is the same as the ground state energy, is given by

(8-6)

Considering the above, the de Broglie wavelength for the tachyon is given simply by

(8-7)

where h is Planck's constant, M_T is the mass of the tachyon in grams, and E_T is the energy of the tachyon. Using Eq. 6 for the energy in Eq.7, we have

(8-8)

It could be argued that it is naive to apply this simple equation to tachyons and ignore relativity. But there is no experimental evidence one way or the other as to how they behave. Certainly it is no more naive than extending the Lorenz transformation to hyperluminal regions and concluding that tachyons have an imaginary mass as has been the accepted practice. Therefore, we will work with what we have and see how the model develops.

If we assume a single de Broglie wavelength, lambda, for the circumference of the tachyon's orbit around the charged particle, we may divide equation 8 by 2 p. This gives us the tachyon's orbital radius, r_l_T, as it orbits the charged particle in the charged particle's frame of reference. That is,

(8-9)

Here, the subscriptrefers to the de Broglie wavelength of the tachyon, and .

While the original model used this concept, another way of looking at it is to consider that both the tachyon and charged particle revolve around the common, external center of mass. The tachyon has some 207 de Broglie wavelengths in its orbit, which is, in this case, larger than that of the charged particles orbit.

We will now explore the balance conditions for a negative mass particle that is coupled to a positive mass. This is illustrated in Fig. 2. For the electron, we define

(8-10)

For the muon,

(8-11)

The equations describing the balance of this system for the electron model is

(8-12)

where we used the fact that . Using Eq. 2 ( for ) in Eq. 12, we have that

(8-13)

The terms cancel, so that Eq. 13 becomes, after a little rearrangement,

(8-14)

Dividing both sides of 14 by m _e , and then using Eq.10, we obtain

(8-15)

Also, rewrite Eq. 2 using Eq. 10 to obtain

(8-16)

Using Eq. 9 for , Eq. 15 becomes

(8-17)

Using M_Te as defined by Eq. 16, we eliminate (R_e - 1) and M_Te from Eq. 17 so that we have for the electron

(8-18)

Using an identical approach for the muon model, the orbital radius of the muon's pion is

(8-19)

The magnetic moment of a current loop is, in general,

(8-20)

where I is the current in the loop, and A is its area. (Note that using for the magnetic moment is not to be confused with the subscript representing the muon.)

Current is, in general, given by the number of charges passing a point multiplied by the charge per particle. Also, recall that in the Gaussian system of units, the charge in statcoulombs divided by the speed of light is the unit of charge used to calculate the magnetic field. Hence, the current at a point caused by a single charged particle revolving about a center point is

(8-21)

where f is the frequency of the particle's rotation, and for a light speed particle is given by

(8-22)

where c is the velocity of the charged particle and r_c is its orbital radius. Hence, the magnetic moment of a single, revolving charged particle is obtained from Eqs. 20, 21, and 22, as

(8-23)

where is used for the area, A, of the current loop of Eq. 20. Eq. 23 then becomes

(8-24)

Using equation 18 in Eq. 24, the magnetic moment of the electron is

(8-25)

Using Eq. 20 in Eq.24, the magnetic moment for the muon is

(8-26)

These are the Bohr magnetons for the electron and muon respectively. These values for the magnetic moments agree with experiment to within 0.17 % for the electron and 0.12 % for the muon. No particular significance is attached to the plus and minus versions of the magnetic moments at this time.

But to take it a step further, by requiring that the electron's charged particle have an integral number of wavelengths, the accuracy of the electron's magnetic moment is improved to within 39 parts per million. That is, the gyromagnetic ratio is g/2 = 1.0011208. (QED does better than this, but with hundreds or workers and almost 60 years, this should be the normal course of events.)

It should be noted, for contrast, that the self-energy calculation for the electron provides the well known classical electron radius of 2.8179 fm, which is far smaller than that of the electron as given above. However, it is less than twice that of the muon. No particular significance is attached to this, however. But it is interesting to note that if we divide the electron's charged particle's radius (the reduced Compton wavelength) by the classical electron radius, the result is the fine structure constant. Again, the significance of this with respect to this model, if any, is not clear at this time.

One objection that may be raised is that the electron is much larger than the high energy scattering data indicates it is rather small. The electron's charged particle's orbit has a radius of 386.15933 fm, and the muon's charged particle's orbital radius is 1.8675947 fm. In spite of these large orbital radii, the actual scattering cross section of muons and electrons would be expected to be much smaller at high energies because the actual charged particle itself is no larger than the pion. That is, the upper most limit of its radius is 0.185 fm (2.15 Mb). This does not contradict the much lower experimental value of 5 - 30 Nb. (No lower limit is available from the model.)

In the next section, we address the issues with synchrotron radiation in the case of a revolving charged particle.

(See publications 1, 18, 19, and 20, below.)

9. Comments on the Semi-Classical Revolving Charge Model

To retiterate: The muon and electron models are Bohr-like revolving particle models that utilize a negative mass tachyon in conjunction with a revolving, but very tiny (10 ^-18 cm diameter or less), charged point particle that revolves in a circular orbit exactly at the speed of light and behaves like a photon trapped in its orbit. The charged particle does not radiate because it revolves exactly at the speed of light, as will be discussed later. It generates a magnetic moment equal to the Bohr Magneton. Associated with the revolving charged particle is a negative mass tachyon whose orbital radius around the center of mass of the system is larger than that of the charged particle. It is not clear if this tachyon is a captured particle or if the transition from the muon to the electron creates a ‘hole’ in space or in the electromagnetic field surrounding the particle.

A free pion captures a negative mass tachyon and becomes a lighter muon. The muon, in turn, captures another negative mass tachyon and becomes an electron. This is, of course, very much in contradiction with the standard particle model. That is to say, the pion is the mother particle of the elctron and muon.

Note that a negative mass particle is inherently an antigravity particle.

Further, based on this model, it is mandatory that colliding electrons and positrons would produce at least muons, pionsm, both of which are different states of the same particle, at least in context with the present model. This agrees with observation.

All of this, of course, makes the pion the mother particle of the lepton family, and again, this is very much in contradiction to the standard model but is in agreement with experiment as interpreted by this model.

Also, if there is enough energy, e-p collisions should produce the same or similar psi resonances that are produced by neutron scattering experiments. This has been observed.

Similarly, a proton consists of a heavier sigma hyperon ( Σ ) that has combined with a negative mass tachyon, but one that has a different mass from that of the electron and the muon. Because the conversion curve has no precise cutoff energy, we must use a converse methodology to that of the electron and the muon models. In this case, the magnetic moment of the proton is used to determine its dimensions with the result that the dimensions of the proton agree to within 3% of the experimental dimensions that are determined from both high and low energy scattering experiments.

A proton captures a pion and becomes a neutron which has a smaller magnetic moment. It is the resonances of this pion that produce the mesons that were described earlier. Further, based on this model, the pion has a very small magnetic moment, very much in contradiction to the standard model that assumes it to be a spin zero. In fact, if a direct measurement of its magnetic moment is made, it is predicted that its magnetic moment will be very close to that of the deuteron.

This is not to say absolutely that it has to be case of capturing a negative mass tachyon. As one might have noticed in the discussions above, the mass-energy of the particle is contained in its electromagnetic field and the smaller the radius, the greater the energy of the particle.

However, this model was developed by taking this contrarian approach with results that continually surprise the author himself. Therefore, we use it here as if it were an absolute truth.

But to continue, if there were free standing negative mass particles, they should have been noticed by now. Hence, they either exist in conjunction with a positive mass particle or they are tachyons that interact with the subluminal universe only under special cases, or both. Within the context of this model, the negative mass particle is required to be a tachyon. It could be a case of either capturing an existing tachyon or creating one during the conversion process.

Because these transitions from pion to muon and from muon to an electron behave like monopole transitions as opposed to dipole transitions, no radiation would be expected of them. This is observed to be the case experimentally.

It will be noted in the development below that the electron’s charge is revolving at the speed of light, which would normally be thought to be forbidden by relativity. Why it is not the case for an internal charge is discussed below.

Therefore, from the perspective of the this model, the electron may be viewed as being a revolving point charge that is trapped in a Compton wavelength orbit, and this revolving point charge behaves like a bound photon. But what is most important is that this says is that the rest mass of the electron is contained in the photon-like revolving charge, which is of the order of 0.185 fm in diameter. As the electron, as a whole system, is accelerated to higher and higher speeds, its angular velocity increases and its energy increases to infinity as the velocity approaches the speed of light.

As a result, relativity does not preclude the particle’s internal, constituent charge from revolving at the speed of light. In contrast, however, the electron, as a revolving system, is precluded from being accelerated to the speed of light.

The balance of the revolving charge energy with that of the internal magnetic field has not been investigated at this time.

Finally, the question of synchrotron radiation must be mentioned insofar as why the revolving charge does not radiate its energy away. The classical synchrotron model is covered quite well in Jackson’s book, Classical Electrodynamics, 2^nd Ed. There, it is important to note that the classical electrodynamic model is developed for sublight speed particles, not light speed particles. In fact, the model becomes meaningless for light speed particles.

In addition, synchrotron radiation also supposes an emission of a field from a sublight speed particle in the direction of its instantaneous velocity. But a light speed particle would not emit a field ahead of itself because the particle is moving as fast as the field itself. So again, the synchrotron model is meaningless for a light speed particle.

Beyond that, we postulate that the energy of the light speed charge constitutes a ground state energy that simply does not radiate.

(See references 1, 3, 5 and 16.)

10. An Electrodynamic Model of Electron de Broglie Waves

As was stated above, there is the charge in the electron revolves at the speed of light. But, we must ask what the field around such a charge would look like in its near vicinity.

First of all, before considering the revolving electron, consider the field around a charge at three different velocities as shown in Figure 11. First is the case of the non moving charge at illustration A. The second, illustration B, is the field around the charge as it approaches the speed of light. Here, the field begins to bunch up perpendicular to the velocity vector. The third case, illustration C, is the extrapolation of the behavior at B to the speed of light. I.e., it would tend to be completely perpendicular to the velocity vector.

Figure 12. Three illustrations of the field around a charge at three different velocities.

Based on the above, and because the electron’s charge revolves at the speed of light, the electric field it emits would be expected to be perpendicular to its instantaneous orbital velocity. Because of this, a nearby “observer” would not experience an increasing/decreasing field as the particle revolved, but would experience an impulse from the charge only when it passes by his location. That is to say, the electric field near arising from an electron is not a steady, uniform field, but is instead dynamic field.

More specifically, it is a dynamic impulse field that arises from the revolving point charge and spirals outward at the speed of light after the manner shown in Figure 12 below. This spiraling field is not unlike the spiraling stream of water ejected from a spinning water sprinkler.

Figure 13. The spiraling, dynamic electric field of the revolving electron in

its orbital plane. The red circle is the charge’s orbit and the green dot is

the charge, and the cross is the center of the orbit. Note that the spacing

between succeeding wavelets in the spiral is equal to the Compton wavelength

for the electron. This could well be referred to as a “water sprinkler” model.

The important issue here is not just that the field is a spiraling field, but that the spiraling field forms wavelets that move outward at the speed of light with a spacing that is the same as the orbital circumference, λ_C, which is equal to the Compton wavelength for the electron. Because of this, we refer to them as Compton wavelets, or as, simply, wavelets. This also implies that the spiraling field or an electron fills all space out to infinity. Otherwise, the electrostatic extend of the electron’s field would be limited.

But it is to be emphasized that the above is a very simplified description of the dynamic field around the charge. For an indication of the complexity of the field, see Figure 13, below for a three dimensional view of the field.

Note further, that this would imply an almost circular polarization along the polar axes, to use an antenna analog. Thus, it is quite possible that the detailed behavior of the interaction with wavelets from other electrons is dependent on the polarization of the affected electron with respect to the normal of the wavelet.

More on this is so is discussed in publications 2 and 4 below.

Figure 14. This is the three dimensional field of a revolving point charge as it

revolves in an orbit of radius r around an axis, A, with a velocity v. The

Electric field, E, is emitted at the speed of light at the time the charge is at

point Q, and by the time it has reached the point Q’ in the orbit, the field has

propagated to some boundary, P. The blue circle is the H field, the red circle

is the charge’s orbit, and N is an instantaneous virtual Poynting vector. We

say “virtual” in that it is assumed that there is no net radiation from the

field.

11. Electron-Electron Interactions and the Derivation of the Electrodynamic de Broglie Wavelength

Having described the wavelets emitted by an electron, we now look at a model of how these wavelets might interact with another electron. These are illustrated in Figure 11 below. For the case of electron A, the wavelet is approaching the charge from within the orbit, for electron B the wavelet is approaching the charge head-on. In both cases the wavelets normals are parallel to the electric field of the charges. In these cases, we hypothesize that electrons A and B will receive an impulse from the wavelet.

In the case of electron C, however, the wavelet’s normal is not parallel to the charge’s field. In this latter case, we hypothesis that there is little or no impulse transmitted to the electron.

As a result of this, an electron in a field of other electrons would experience a cacophony of wavelets passing by with an occasional impulse being acquired from these other electrons as its phase happened to match their phases in a purely probabilistic manner.

Figure 15. Here we have three different electrons, A, B, and C

interacting with a wavelet, W, from another electron. The

wavelet and its normals are shown in blue, the electron’s

charge’s orbits are shown in red, and the charges themselves

are shown in green.

Now that we have described how electrons interact with wavelets from other electrons, we will no use its dynamic characteristics to derive the electron’s de Broglie wavelength.

The interactions between the wavelets of two electrons, A and B, are shown in Figure 15, below. If the electrons are stationary, then the wavelets from the two electrons will move outward at the velocity of light with a constant phase relationship with one another. But if B is stationary and A moves slowly with velocity v, then a nearby observer will see the relative phase of subsequent wavelets change, going in and out of phase. In one unit of time t, the electron will move a distance d, and the relative phases of the wavelets will change continuously.

Figure 16. The phase interactions of the de Broglie waves of two electrons.

When the relative phase is 0 (or 2p), then a double wavelet will occur. (This model will not work otherwise. It is at this point that the electron’s charge is perpendicular to the wavelet it is bumping into.) The number of these doublets, or phase crossings, that will occur in that time unit is

. (11-1)

Using the definition of the Compton wavelength,, we have that the frequency of crossing, f, is

. (11-2)

Because the wavelets are light speed entities, we may solve Eq. 32 for c/f. We have that the spacing of the phase crossings (the doublets) of the wavelet are propagated outward with spacings between them of

, (11-3)

where is the de Broglie wavelength.

Obviously the wavelets do not simultaneously move in and out of phase in the entire pulse train as the charge moves. Only those wavelets that are emitted after a given incremental movement will have a new phase relationship, so that an observer at some distance away in the direction of v would see the passing wavelets moving at the speed of light and going in and out of phase at the frequency f and with a resultant spacing between phase crossings of λ _D.

The significance of the crossing of these wavelets is that they are correlated with the revolving charge, so that it is not, in fact, the crossing of the wavelets from the two electrons that is important. It is the coincidental head-on, perpendicular collision of a wavelet from one electron with the revolving charge of the other electron that imparts an impulse to the charge that encountering the wavelet.

Figure 17. This illustrates the case of an electron’s interaction

with its own correlated, scattered wavelets from a scattering site, S.

The reflected wavelets are shown on the vectors x.

The red circle is the charged particle’s orbit and the green dot is

the charge. The same electron is shown at three different times,

t1, t2, and t3 where for simplicity we only show those wavelets

emitted toward the scattering site when the electron

is at phase angle 2π and not at phase angle π. bw refers to the

backward wave emitted by the electron. As previously stated,

the model we use here is based on the assumption that the

charge interacts with a wavelet when the charge is on the

same side of the orbit as the impinging wavelet. Other

approaches are also worth considering, such as the assumption

that the interaction occurs when the charge is on the opposite

side of the impinging wavelet.

But for the case of an electron’s wavelet that is reflected by a nearby scattering center, the autocorrelation of the electron with its own wavelets causes a probabilistic scattering in the direction away from the scattering center. The simplest case of this is shown at three different sequential times in Figure 16, above.

Note that in the above and the discussion below we have not, as yet, provided a more detailed analysis of the conditions whereby preferential scattering in one direction or another would occur due to this effect. That is to say, we provide a qualitative discussion only, not a quantitative discussion. It is hoped that a quantitative model can be provided in the near future.

12. Interactions of the Wavelets with Apertures and Edges

Having established a model wherein an electron interacts with its own wavelets as they are reflected off of nearby scattering sites, we consider the case of a charge traversing a double slit aperture.

This is shown in Figure 17 which illustrates the possible effects of the reflected wavelets on an electron that has just traversed slit B in a double slit diffraction experiment. Note that the relatively slow electron is soon overtaken by the wavelets reflected from the edges of the slits, and with null reflections from the open area of the slits and, in an idealized case, specular reflection elsewhere. (Specular reflection is used rather loosely here, because a surface would, in fact, consist of atoms with some reflection back to the electron.)

Figure 18. A very simplistic illustration of the spiraling

field of an electron that has just traversed a slit and the

correlated reflections of the field as they reflect off the

edges of the slits.

It should be noted that as the electron approaches the slits before traversing them it would also be influenced by those wavelets that are reflected off the slits as it approaches. But that is irrelevant here because it has only have two possible paths to take if it is to traverse the slits, either slit A or slit B. The probabilistic interference pattern at some planar screen to the right of this drawing would be determined only by the reflected wavelets that are emitted after the electron has traversed the slits.

While we have presented this as an alternative to the currently accepted model as is presented in the standard quantum mechanics book, it is important to note that it is also possible that the text book model is the correct model in that the spiraling wavelets penetrate the slit that the electron is not penetrating. It is also possible that the same is true in the case of the electrons reflected off the crystalline surface as described below. I.e., wavelets from electrons reflected off of atoms several layers below the surface exit the surface along with the electron but are scattered from nearby lattice sites according to the standard text book models.

To test this reflection model for the slit, we propose a double slit experiment where two complete slits are photo etched into a thin piece of metal. However, half of one of the slits, say the upper section, should covered by conductive layer on the electron source side of the slits, but displaced from the slit, i.e., a square channel parallel to the slit. That way, the lower half of the slit pair will permit electrons to penetrate both slots while the upper half permits electrons to penetrate only one slit. However the upper section of the slit opposite the source will appear to the exiting electron’s wavelets to be open so that the model shown in Fig 18 can be verified. If the reflection model is correct, then there will be little or no difference in the interference pattern on the covered section or the uncovered section of the slits.

13. Electron-Lattice Site Scattering of Compton Wavelets - The Davison Germer Effect.

In this section, we will make a qualitative description of the phenomena that arises from the scattering of wavelets by crystal lattice sites. It is important to note that we are concerned here only with the scattered wavelets because they can be correlated with the phase of the electron that produces them. This model is not concerned with the wavelets from the electrons contained in the lattice because they are random and uncorrelated with the electron of interest.

Figure 18 shows an electron moving directly towards a lattice scattering site at some velocity, v, while emitting direct wavelets towards it. Here, we hypothesize that the Compton wavelets of a given electron will be reflected from the lattice atoms, thus forming “interference” patterns. However, there is no constructive or destructive interference as in the case of electromagnetic waves; there are only phase differences between the Compton wavelets. Those regions wherein the wavelets are in phase we will call Compton ridges. Those regions wherein they are completely out of phase we will call Compton channels. When the ridges impinge on an electron, it will get a slightly greater impulse that in the channels so that it tends to be probabilistically scattered in the direction of the ridges. This is not simply because the wavelets in these directions are all in phase with each other but because they are all in phase with the electron itself and give it a slightly greater probability of being scattered in this preferred direction. See Figure 16, below.

Figure 19. Here, an electron approaches a crystal lattice from

location D. It is later reflected through an angle q in a probabilistic

direction based on the effects of the combined self correlated Compton

waveletts reflected from lattice sites A, B, and C. Note that the

electron moves relatively slowly while the wavelets move at the

the speed of light.

Figure 20. Here, we have the de Broglie waves reflected from

three lattice sites, A, B, and C. They are all in phase with

one another, and for the right electron energy, they are also

in phase with the electron. For the Davison-Germer

experiment, the angle from the vertical here would be about 50 degrees.

The Davison-Germer experiment was an experiment that succeeded in making the first direct measurements of the effect of de Broglie waves. Published in 1925, it was the first physical evidence for the existence of de Broglie waves. In that experiment, currents of some 10 microamps at 50 – 100 volts from a 1 mm cathode bombarded a nickel crystal and the intensity of their directional dependence followed that calculated by the de Broglie model.

Applying the parameters from this model to the Davison-Germer experiment, the electron revolves about 68.8 times each time it traversed a Compton wavelength. But if it revolved through an angle p an integral number of times n during this interval, then it collided with a reflected wavelet at such a phase angle as to receive a maximum impulse. In such case, the wavelet (either the forward wave or the backward wave) emitted by the electron in the direction away from the lattice will travel outward with the reflected wavelet. This produces correlated double wavelets with spacings between each succeeding doublet equal to a de Broglie wavelength every 68, 68.5, or 69 revolutions. However, those correlated wavelets that have passed the electron will have no further effect on it.

The spacing of the atoms in nickel is of the order of 2.5 angstroms, whereas the Compton wavelength of the electron is 0.024 angstroms, so that multiple orders (about 100) of the reflections of the Compton wavelengths would occur if the lattice were diffracting a plane wave so as to form ridges.

It is important to note that at the currents used here, the spacing between impinging electrons is extremely large, so that there no significant interaction between the electrons in the beam.

Finally, we note that it is likely that there is a preferred phase angle of the electron’s charge with respect to the impinging wavelet. We have examined the case where it contacts the wavelet just as it is emitting a new wavelet against the impinging wavelet. However, this alone produces, in the reflected wavelet case, a behavior that would produce an additional line between the first peak of the Davison-Germer experiment and the normal to the crystalline surface. This peak, of course, was not observed. This would imply either that the probability of an interaction was ½, or that the impinging wavelet blocks the emission of a wavelet from the electron, this blocking being the source of the repulsion of the electron. This latter effect is now under investigation.

In contrast to the case of correlated wavelets impinging on an electron, the case of the cacophony of wavelets from multiple nearby electrons colliding with a particular is purely statistical providing that we assume, as we do for the moment, that the collision must be head on with the charge, i.e., when the radial direction from the center of the orbit out to the charge is pointing to within a few degrees of head on to the colliding wavelet. (This is the assumption at the moment. More work is being carried out to investigate this aspect of the model.)

The Aharonov-Bohm effect has not, as yet, been explored insofar as how it may relate to this model as opposed to a simple point electron.

(See publications 3, 5, 6, 11, and 12, below)

14. A Longitudinal Electric Field Model of the Neutrino

It is likely that a neutrino is a speed particle within a few parts per billion. This is based on the fact that the optical observation of the Supernova 1987A occurred within hours of the detection of its neutrinos after a journey of some 163 thousand years. We make this statement without arguments about the time for a photon to travel from within interior the supernova versus the time for a neutrino to travel from the interior. We assume a few hours for both particles because a supernova is a violent event as opposed to a stable star. (It is to be noted that it may take a million or so years for a photon to make its way from the center of a stable star to the exterior because of the scattering. ) Based on this, we propose a light speed neutrino model that is consistent with this model.

When a pion converts into a muon, we hypothesize that part of the spiraling impulse field is separated from the revolving particle so as to from a longitudinal impulse field that is independent of the electron and that travels outwardly at the speed of light. This results in a neutrino model that is consistent with this particle model. A crude, qualitative illustration of this model is shown below in Figure 20.

The details of the E and H fields are shown in the figures below.

Figure 21. This is the fundamental longitudinal electric impulse neutrino model having a radius r. The E field is directed parallel to the velocity vector. At the front and back, where the field is rapidly changing, the cylindrical region is surrounded by magnetic fields

.Figure 22. This shows the relationship between the primary E field and

the magnetic fields, H, that result from the increasing E field at the front

of the neutrino and the decreasing E field at the rear. These changing H

fields produce counter emfs, e, that oppose the primary E field in the

front and reinforce it at the rear.

Figure 23. This is the graphical version of the fields shown in Fig. 13. Here, we

see the E field along the longitudinal cross section of a neutrino. The increasing

E field generates a changing circumferential H field. When the E field drops off,

it generates an H field in the opposite direction. Also shown is the counter

emf, e. We use here a Gaussian E field for convenience.

It should be noted that there is no spin associated with the neutrino model. However, there is a definite orientation with respect to its direction of propagation.

Also, we do not make any judgments as to the direction of emission from the electron with regards to its spin axis. Because experiment indicates that neutrinos have a preferential emission in the direction of the spin axis, we accept that is the most likely emissions direction.

(See publications 1, 2, 7, and 10, below.)

15. The Imaginary Mass Tachyon Model: Not Even Wrong

The bottom line for this model is that there are no other models like it.

In order to obtain this agreement with experiment, it was necessary to abandon the imaginary mass tachyon and to use, instead, a simple negative mass tachyon.

While a negative mass tachyon model contradicts the thinking of the physics community, it does not contradict any experiment. The initial imaginary mass tachyon model as suggestion by Bilaniuk, Deshpand, and Sudarshan back in 1962 was an excellent start in the search for superluminal particles. But in the nearly 50 years since that time, that model has produced no agreement whatsoever with experiment. While it could be argued that a few papers might have produced a vague suggestion of physical reality, for the most part there have been little or no specific models that could be compared with experiment. I.e., it would not be totally incorrect to say that the imaginary mass tachyon is “not even wrong”, to use Wolfgang Pauli’s phrase.

Furthermore, there is no a priori reason to assume that a derivation, based on the subluminal domain where photons are faster than all other particles, should be capable of describing phenomena in a superluminal domain where photons have a velocity slower than all other particles and they cannot catch tachyons.

If anyone knows of such a proof, this author would definitely like to hear about it.

It goes without saying that reasonable agreement with experiment is necessary for a physical model to be considered viable, and the imaginary mass tachyon is certainly no exception.

But before you can have agreement with experiment, you have to have a model that produces a measureable observable, which is what this model is.

But to continue, Wolfgang Pauli was noted for his frequently blunt and uncomplimentary assessment of other people’s work. “That is ridiculous!”, he would exclaim. However, in one particular instance someone asked him what he thought of a paper that he was reviewing. “Its not even wrong!”, he said in referencing its lack of a model that was testable through experiment. This same phrase was utilized as a title by Woit in his book that discusses the failure of string theory to produce anything that is testable via experiment.¹

The imaginary mass tachyon has never produced any significant theory or model that was testable by experiment. Therefore, it is appropriate to use Pauli’s phrase here.

The first attempt in relatively recent times to describe a tachyon, or hyperluminal particle, was carried out by simply using a velocity greater than that of light in the Lorentz transformation. This extension of the Lorentz transformation to hyperluminal velocities was first published by Bilaniuk, Deshpande, and Sudarshan in 1962, some 45 years ago. (We will call this the extended Lorentz transformation.) In the time since then, hundreds of papers based on the extended Lorentz have been produced by many very capable people. But in all of that time, and in spite of the obvious talent of those authors, no agreement with experiment whatsoever was achieved. This comment is meant to be a disagreement with and is not meant, by any stretch of the imagination, to demean the early pioneers or the subsequent workers in this field and their efforts in any way. They gave it a valiant try and should be applauded for doing so. In fact, this author himself spent many months attempting to apply that theory, but to no avail.

The difficulty is this: When the Lorentz transformation is extended above the speed of light, it gives rise to an imaginary mass tachyon, an entity for which there is no physical meaning. There is no empirical justification for this extension whatsoever. There are no experimental curves showing what the energies of tachyons would be as their velocities are varied. Further, it is also frequently stated that it is mass-squared that is negative, not mass itself. Again, that too is based the extend Lorentz transformation.

In the subluminal domain, interactions between atoms and atoms, particles and atoms, and particles and photons, etc., are generally of an electromagnetic nature. (We will avoid the mention of weak and nuclear forces for reasons that will become clear later.) But in the case of the hyperluminal domain, the particles are traveling faster than the photons so that an interaction cannot take place between a photon and a receding particle. While we cannot definitely make the same statement about a head-on photon-tachyon collision, we can certainly question that it will behave in the same manner as in the subluminal interactions.

In short, there is no a priori reason to assume that relativity will necessarily hold in the hyperluminal domain. ( If there is, please send the proof to the above email address. )

But having discussed the total failure of the imaginary mass tachyon, it should be noted that in 1974 Recami and Mignani published a paper, based on the extended Lorentz model, that stated that a tachyon would manifest itself to a subluminal observer as a negative mass particle ². That observation was used as an initial justification for using a negative mass in one of this author’s early papers. The utilization of that observation even during that time may appear to some to be somewhat disingenuous in view of the above negative statements. Nonetheless, this paper by those two very capable theorists was extremely helpful at that time. If it should ever turn out that the extended Lorentz transformation is valid, it is not inconceivable that their observation might be the only thing useful that ever came out of the imaginary mass model in that it would provide a validation of the negative mass tachyon model. It remains to be seen.

Regardless of the validity or invalidity of the extended Lorentz transformation, we simply state that if one simply posits a negative mass tachyon and uses it to develop a particle model, then that model can produce agreement with experiment. It is not necessary to extend relativity into a domain in which it has no empirical validation. We will demonstrate that below.

In addition, we also note that quantum mechanics has been extraordinarily accurate in its description of the atom. Like relativity, it was one of the great achievements of the 20^th century. However, the model we present here utilizes only simple quantization and no attempt, as yet, has been made to arrive at a wave function for the internal structure of this particle systems. But in spite of that there is little to place this model in direct conflict with quantum mechanics as it applies to atomic structures, although its very definite structure will undoubtedly be disputed by many quantum mechanics.

Ultimately, however, a logical consequence of developing the electron model’s detailed field produces an electrodynamic model of waves that behave similarly to de Broglie waves. This was presented above and should clarify why it is not necessary to develop wave functions to describe this very basic particle model at this stage of its development.

But if that is not enough and if the lack of a wave function is bothersome, it should be quite possible to devise a simple wave equation that will fit this model. However, the result might well be a wave mechanical description of the particle that had lost all information about the structure of the particle, but it still might provide some useful insight into the model. Such a model would be well worth investigating at some time in the future.

1. Not Even Wrong, Peter Woit, 2006. Basic Books, New York.

3. E. Recami and R. Mignani, Rivista Del Nuovo Cimento 2, 209 (1974).

16. A Brief Comment on Constants and Units

For those with minimal experience with subatomic particles, a few comments should be made on the mass terminology used here. For example, the mass of an electron is 9.1093896 x 10^-28 grams. But this is a little clumsy for human beings to deal with on a daily basis, especially verbally. It is easier to express the mass in terms of electron volts, which for the electron is 0.511 MeV, where MeV is an abbreviation for million electron volts. Further, the early particle accelerators, such as the Van der Graaf generator and the Cocroft-Walton machine used high voltages to accelerate the particles. From this an electron volt was defined as the amount of work done when a charged particle moves through a potential of one volt. Hence, it was natural to express the energy in terms of the voltage with which the particle was accelerated.

The equivalent mass energy relationship is obtained from the Einstein relationship, namely E = mc². To calculate E, we use the particle mass in grams along with the speed of light which is c= 2.99792458 x 10¹⁰ cm/sec. The resulting energy, E, is in ergs. However, from electrodynamics we know that one erg is equivalent to 6.24150636 x 10¹¹ eV, where eV is the abbreviation for electron volts. Hence, the calculation is quite simple, so the reader should have a try at it with his hand calculator.

Table 14-1. Particle Masses ¹

Particle	Mass (gms)	Mass-Energy (MeV)	Magnetic Moment (Ergs/gauss)
electron	9.1093896x10^-28	0.51099906	9.2847701x10^-21
proton	1.6726231x10^-24	938.27231	1.4106076x10^-24
neutron	1.6748286x10^-24	939.56563	9.6623707x10^-24
muon	1.8835327x10^-25	105.658387e-24	4.4904514x10^-23
pion	2.488018x10^-25	139.5675	4.3x10^-24 (No, it's not zero, quantum mechanical spin 0 or not.)
Deuteron	3.3435860x10^-24	1875.61339	4.3307375x10^-24

Table 14-2. Physical Constants ¹