The Physics of Negative Mass Tachyons

Referenced Terms and Names

Tachyons, antigravity, FTL, superluminal, faster than light,
faster-than-light, quarks, gluons, mesons, spin, charm, relativity, neutrinos,
photons, electrons, protons, neutrons, pions, muons, leptons, hadrons, Bohr
Magnetons, nucleus, chromodynamics, hyperluminal, quasar, Planck, Einstein, de
Broglie, Schroedinger, Heisenberg, Dirac, Kepler, Ptolemy, Ptolemaic

Abstract

o This web page provides a summary derivation of the first and only
theoretical tachyon model to agree with experiment. No other tachyon model, to
date, has produced agreement with experiment. It is a unified model that
produces agreement with experiment for the electron, the proton, the neutron,
the light nuclei, and the mesons.

o The data that is used to verify this model is well known from the standard
physics literature, especially the Review of Particle Physics,
published by the Particle Data Group of the American Physical Society. No source
that is controversial or possibly bogus is used. This model is, in short, a
reinterpretation of existing particle data that is accepted by and used by the
physics community.

o All attempts have been made to make the presentation as simple as possible
so that even someone with a relatively modest background in physics can carry
out the calculations with a simple hand calculator. Hence, it is shown just how
simple basic subatomic physics can be if presented properly.

o The model is a Bohr-like revolving particle model that utilizes a negative
mass tachyon in conjunction with a revolving, but very tiney,
charged point particle. The tachyon, being unable to drop below
the speed of light, causes the positive mass charged particle to revolve in a
relatively large orbit, thus generating a magnetic moment while maintaining
agreement with experiment in that the charged particle for the electron is know
to be extremely small.

o Specifically, the heavier positive mass muon captures a negative mass
tachyon to form an electron. The muon, in turn, consists of a positive mass pion
and another negative mass tachyon. The radii of the revolving electron and the
muon are determined from the cutoff energies of the well known mu -> e and
the rare, direct pi -> e conversions curves. These radii determine the
magnetic moments of the electron and the muon. The resulting magnetic moments
are found to be the well known Bohr magnetons, identically. That is, the
magnetic moments are

These values are well known to agree with experiment. By requiring that the
electron's charged particle have integral multiples of its de Broglie
wavelength, its magnetic moment is improved to within 39 ppm of experiment.
(Quantum electrodynamical corrections do much better in terms of accuracy, but
there is nothing obvious to preclude its use as a correction factor for this
model.)

o The residual energies of the conversions are 20 eV and 123 KeV for the
electron and muon conversions, respectively, somewhat larger in magnitude than
the experimentally estimated masses of their respective neutrinos, but more than
enough to account for the neutrinos' masses.

o Based on this, it would be expected that colliding electrons and positrons
would produce at least muons, pions, and gamma rays. This agrees with
observation. All of this, of course, makes the pion the mother particle of the
lepton family. This is quite at odds with the standard particle model. And while
indirect evidence has generally been interpreted to mean that the pion is a spin
0 particle, the implication here is that the pion has a magnetic moment, albeit
extremely small.

o Similarly, a proton consists of a heavier sigma hyperon that has combined
with another negative mass tachyon. Converse to the electron/muon model, the
magnetic moment of the proton is used to determine its dimensions. These
dimensions agree to within 3% of the with the experimental dimensions that are
determined from both high and low energy scattering data.

o By adding a revolving pion-tachyon combination to the center of the proton,
we form a neutron. The ground state energy of the pion is 4076 MeV, and its
excitation energies are found to be

E = 4075/n2 ,

where n ranges from 1 - 9. As shown below, the first order transitions of
this bound pion, an analog of the Bohr model's Lyman series, produce the
"charmed" psi mesons within 5 % of the experimental values. The second order
transitions, an analog of the Bohr model's Balmer series, produce the eta
through the ao(980) mesons generally to within 4 %. Theses energies are also
produced by colliding electrons and positrons, so that one must conclude that
the unbound pion itself resonates. Again, there is no relationship between this
model and the standard model.

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

o And while this model agrees quite well with experiment, there is no
relationship whatsoever between it and the generally accepted standard particle
model. This model is described in more detail in a book, The Physics of
Tachyons, and in the published papers provided at the end of this web
page.

o As can be seen, comparing this extremely simple particle model to the
extremely complex standard particle model is like comparing the extremely simple
Kepler planetary model to the complex Ptolemaic planetary model that was
prevalent in the middle ages. And while both particle models may agree with
experiment, there is no comparison beyond that. There are no quarks, gluons, or
strings here, just as there were no epicycles in the Kepler model. And here,
mesons are not classified according to strangeness, or flavor. This model is
even politically correct in that there is no reference to chromodynamics, or
color!

Ernst L. Wall

Istituto per la Ricercia de Base

Monteroduni, Italy

Email - [email-masked]

(Note: Comments are solicited.)

To see the broad spectrum of Monteroduni Research, see
preliminary web by clicking
here.

Last Updated on March 12, 1997 by Ernst L. Wall

The Definition of a Tachyon

Tachyons are particles whose velocity exceeds the velocity of light. While
many believe that the existence of particles with hyperluminal (superluminal)
velocities (FTL, or Faster-Than-Light velocities) is precluded by relativity,
this is not the case if the particle is created with a velocity already
exceeding the velocity of light. What relativity precludes, within the
boundaries of our present technologies, is the acceleration of a subluminal
particle to hyperluminal velocities. What is also not precluded is the
possibility that technology might someday be developed that will permit the
relativistic limitations to be overcome, and hyperluminal velocities to be
achieved by subluminal objects.

The most well known tachyon model uses an imaginary mass. As interesting as
that model is, mathematically speaking, an imaginary mass has no physical
meaning. As a result, that model has, as yet, produced no agreement with
experiment. That model and its mathematics is discussed extensively in an
excellent review by Recami and Mignani in Rivista Del Nuovo Cimento
4, 209,

(1974).

A Brief Comment on Units of Mass 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 abreviation 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, and an
electron volt is 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 abreviation for electron volts. Hence, the calculation is quite simple, so
the reader should have a try at it with his hand calculator.

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, spin 0 or
not.)

Deuteron

3.3435860x10-24

1875.61339

4.3307375x10-24

In addition, the following constants will be needed:

Constant's Name

Symbol

Value

Unit

Speed of light

c

2.99792458x1010

cm/sec

Elementary Charge

e

4.80320680x10-10

statcoulombs*

Elementary Charge

e

1.60217733x10-19

coulombs

Planck's Constant

h

6.6260755x10-27

erg-sec

Planck's Constant/2pi

h_bar

1.05457266x10-27

erg-sec

Nuclear and subatomic dimensions are usually expressed in fm, which can stand
for Fermis, which is10-13 centimeters, or femtometers, which is
10-15 meters. (Obviously the same length, but different units.)

The Meson 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 overall model.

The first meson to be physically observed was the muon, and some 10 years
later, the pion. Still later, other mesons were 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.

From the standpoint of this 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 from the standpoint of this 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

Em(n) = 4076/n2,

where the index, n, ranges from 1 through 9. To obtain the various meson
energies, you may use the above equation as follows:

1. First, calculate the 9 levels of Em using the values 1 - 9 for
n.

These are shown in the energy level and transition energy diagram, below. The
first three levels of Em correspond to the energy levels of the
psi(4415), the phi(1020), and the K- mesons to within - 8 % to +8 %. The next
two levels correspond to resonances that arise from a K- proton collision and a
pi- proton collison, these resonances having energies of 280 MeV and 156 MeV,
with agreements of -9.3 % and -4.3 %, respectively. These latter resonances two
will be discussed in more detail in an upcoming revision of The Physics
of Tachyons.

2. Next, subtract each resulting value of Em from the first value
(for n = 1), and you will have the first order transitions, or the "charmed" psi
mesons to within -1.3 % to +4.7 %. (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 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.

3. Finally, subtract each subsequent value from the second value (for n = 2),
and you will have the second order transitions, These produce the seven light
mesons, i.e., the eta 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. (The Bohr atom's analog is the Balmer series.) These are shown in the graph, below, along with
their corresponding experimental values, where the index is the value n that is
subtracted from the value n = 2.

These transitions, i.e., the transitions between the level for n = 1 and the
lower levels correspond to the "charmed" psi mesons. The second order
transitions, the transtions between the n= 2 level and the lower levels
correspond to the lighter mesons from the eta through the a0(980).

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

To reiterate, it is not necessary to understand quarks, gluons, etc, to
achieve this systematic agreement with experiment. Nor is reference to
strangeness flavor, or color needed.

The Binary Mesons

Because the many 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 energy levels.
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 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.

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

A Brief Comment on Magnetic Moment

The origin of the magnetic moment of particles, , is the heart of this model,
so that a brief discussion of it is warranted before proceeding to the actual
derivation of the magnetic moments of the electron and the muon.

In the simplest case, a wire loop with an electrical current flowing through
it causes a magnetic field to be generated along its axis. A measure of the
amount of torque that the loop would be subjected to when placed in a magnetic
field is called the magnetic moment. It is calculated by multiplying the area of
the loop by the current flowing through the wire. That is, = I x A ergs/gauss,
where A is the area and I is the current. The current I is normally given by I =
ne where n is the number of charges passing a point in the wire, and e is the
charge per particle. For current as we normally use it, e =
1.602`7733x10-19 coulombs. However, in order to have a magnetic
moment that is correct for the cgs system of units, we must use e =
4.8032068010-10 statcoulombs, and then that must be divided by the
speed of light. I.e., I = ne/c.

The amount of torque on the loop is the T = Bcos dyne-cm, where is the angle
between the axis of the magnetic moment and the direction of the magnetic field,
B.

Insofar as its application to particles is concerned, the proton, electron,
neutron, and the nuclei all have magnetic moments. The fact that the electron
has a half-integral value of spin prevents any two electrons in an atoms from
occupying the same energy/angular momentum. This fact causes the various atoms
to combine chemically in the way that they do, thus making our world the way it
is. That is, without tachyons, there would be no particle spin, and so we would
not exist. (Note that, here, I am not making a distinction between spin as a
quantum mechanical term and magnetic moment as a physical phenomena in this
simple discussion. This may enrage quantum mechanics, but is for them to worry
about their blood pressure.)

The Origin of the Magnetic Moments of the Electron and
the Muon

To summarize the revolving electron and muon models, a pion captures a
negative mass tachyon and becomes, overall, a less massive muon. The muon
captures still another negative mass tachyon, and becomes an even lighter
electron. The orbital velocities of the revolving charged particles are constant
at the speed of light, with only the orbital dimensions and the overall energy
of the system changing during the transition from one particle system to
another.

Because these transition from pion to muon and from muon to 3 electron are
monopole and not dipole transitions, no radiation would be expected of them.
This is observed to be the case experimentally.

Further, based on this model, it is mandatory that the byproducts of
high energy electron-positron collisions include muons and pions and gamma rays.
This is observed to be the case experimentally.

NOTE: (With apologies) At this time there is a some
difficulty getting some of the Greek characters to print on the html page, so we
are forced to use pi and mu for pion and muon for the moment, except in graphics
or those equations that derived from the equation editor and imported from
another document. This also holds for h bar, or Planck's constant divided by 2
pi.

The masses of the muon's and electron's tachyons are

MTmu = Mmu - Mpi = -33.9091
MeV,

(1)

MTe = me - Mmu = -105.147388
MeV.

(2)

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

ETmu = - 16.9546 MeV.

(3)

ETe = - 52.573694.

(4)

The sum of these energies is

ETmu + ETe = - 16.9546 - 52.573694 = -
69.5283 MeV.

(5)

Next, examine Fig. 2. It is a composite of two particle conversion curves.
The mu -> e curve on the left is well known and is contained in most particle
physics books.

The right most curve, the direct pi -> e conversion curve, is less well
known, the direct conversion of a pion into an electron is relatively rare,
about one in 104 pion conversions.

It should be noted that accurate fits to the mu -> e
curve have been produced by the V-A theory.

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.

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 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
mu-> e 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 mu -> e curve. This compares
nicely with the energy of Eq. 4.

The pi -> mu 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 -> 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.

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. 2.

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

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

where h is Planck's constant, MT is the mass of the tachyon in
grams, and ET is the energy of the tachyon. (NOTE, again that we will
be forced to sometimes use the word lambda for the wavelength.) Using Eq. 6 for
the energy in Eq.7, we have

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
pi. This gives us the tachyon's orbital radius, r lambda T, as it
orbits the charged particle in the charged particle's frame of reference. That
is,

Here, the subscript lambdaT refers to the de Broglie wavelength, and h_bar =
h/2 pi.

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

For the muon,

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

Mmurce + MTerTe = 0,

(12)

Mmurce + MTe(rce +
rTe) = 0,

where we used the fact that rTe = rce + rlambda
Te. Using Eq. 2 (for MTe) in Eq. 12, we have that

Mmurce + (me -
Mmu)(rce + rTe) = 0,

(13)

Mmurce + merce +
merlambda Te - Mmurce -
Mmurlambda Te = 0.

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

rceme = (Mmu -
me)rlambda Te .

(14)

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

rce = ( Re - 1 )rlambda Te .

(15)

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

MTe = me - Mmu = (1 -
Re)me .

(16)

Using Eq. 9 for rlambda Te, Eq. 15 becomes

Using MTe as defined by Eq. 16, we eliminate (Re - 1)
and MTe from Eq. 17 so that

for the electron.

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

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

mu = IA,

(20)

where I is the current in the loop, and A is its area. (Note that mu is not
to be confused with the subscript mu 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

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

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

where pi*rc2 is used for the area, A, of the current
loop of Eq. 20. Eq. 23 then becomes

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

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

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 many
years and hundreds of workers, it should!)

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.)

Insofar as the question of a revolving charge radiating its energy away, we
could, after the manner of Bohr with his hydrogen atom, merely hypothesize that
its orbit is a bound, ground state orbit, and leave it with no further
explanation. After all , this model uses a revolving charged particle to produce
extensive agreement with experiment in spite of the radiation problem, so that
obviously, conventional electrodynamics would appear not to hold here.

But to suggest two paths of investigation to address this radiation problem,
first consider the significance of Eq. 27. The possibility that an electron or
muon is a pseudo photon looping back on itself is one possible path of
investigation. The second is to provide a detailed analysis of a light speed
particle's radiation characteristics. There is no a priori reason to assume that
a light speed particle should radiate in the same manner as a subluminal
particle, nor is there a definite a prior reason to assume that it does not. But
to reiterate, this model does produce extensive agreement with experiment in
spite of this radiation question.

The Proton Model

The proton consists of a sigma hyperon and still another negative mass
tachyon. However, the sigma hyperon -> proton conversion has no curve with a
clearly defined cutoff energy such as there is for the electron and the muon.
Therefore, an inverse approach must be used for the proton. Using the above
configuration and the magnetic moment of the proton ( muP = 1.4106076
x 10-23 ergs/gauss ) in Eq. 24, we find that the charged particle's
orbital radius, rc , 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 RP =1.2676, and the mass of the tachyon is
-251.10 MeV. Using these values in Eq. 15, the radius of the tachyon's orbit is
found to be 2.782 fm. High energy and low energy scattering experiments indicate
that these radii agree with experiment to within 3 %.

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. Subtracting the magnetic moment of the neutron (
muN = 9.6623707 x 10-24 ergs/gauss ) from that of the
proton, we find that the orbiting pion's magnetic moment is mupi =
0.4443705 x 10-23 ergs/gauss. (Note, incidentally, that this value is
within 2.5 % of the magnetic moment of the deuteron.) Using this value in Eq. 24
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

Em = 4076/n2 MeV,

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 ep 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.

The Light Nuclei

Now consider the attraction of neutrons and protons to form a deuteron. 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.

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 are disturbing, it should be
noted that a perusal of the Physical Review will reveal many models with errors
of 50 %, or worse, in those cases where the authors have the nerve to make a
comparison with experiment. In many cases, they conveniently ignore that little
matter of comparison with experiment, but the mathematical elegance can be
impressive, even if it is useless.

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.

"Space Warps"

And for those who might ask, there is no "space warp" of any kind involved in
this model, nor any similarity to anything mentioned in Star Trek outside of the
word "tachyon". But of course, the demonstration of the existence of tachyons
might well suggest the possibility of someday breaking the light speed barrier.
Obviously, that would require some advancement beyond our present technology.

How to Obtain Detailed Information on the Tachyon
Model

The negative mass tachyon is described in various publications by Ernst L.
Wall in the standard physics literature. It is also described in the 241 page
book, The Physics of Tachyons, Ernst L. Wall, 1995, ISBN I-57485-001-6. It is
published by the Hadronic Press, 35246 US 19 North #115, Palm Harbor, FL 34684,
USA. Phone: (813) 934 - 9593. Special Price for Individuals, $30.

The book is also carried by:

o The International Tesla Society, PO Box 5686, Colorado Springs, CO 80931.
Phone (800) 397-0137;

o Quantum Books near MIT for Boston area residents, (617) 494-5042.

Additional Publications by Ernst Wall

o "The Role of Tachyons in Electron Spin and Muon Spin", Bulletin of the
American

Physical Society, 30, 729 (1985).

o "The Role of Tachyons in Proton Spin", Bulletin of the American Physical
Society, 30,

729 (1985).

o Indirect Evidence for the Existence of Tachyons: A Unified Approach to the
mu->e and

the pi->mu Conversion Problem", Hadronic Journal 8, 311 (1985).

o "On Tachyons and Hadrons", Hadronic Journal 9, 239 (1986).

o "Unresolved Problems of the Tachyonic Models of the Electrons and the
Muon",

Hadronic Journal 9, 263 (1986)

o "Time Cancellation Hypothesis", Bulletin of the American Physical Society,
33, 1076

(1988).

o "Charm, Other Resonances, and the Tachyonic Particle Model", Bulletin of
the American Physical Society, 33, 1076 (1988).

o "On Pion Resonances and Mesons, Time Cancellation, and Neutral Particles",
Hadronic

Journal 12, 309 (1989).

o "Hamming Code Error Correction for Microprocessors", Chapter 3,
Microprocessor Applications Handbook, edited by D. Stout. McGraw-Hill, 1981.

o "Applying the Hamming Code to Microprocessor- Based Systems", Electronics
(McGraw-Hill) 52, p. 103 (1980). (Note that this was the feature (cover) article
of this issue.)

o "Edge Injection Currents and Their Effects on 1/f Noise in Planar Schottky
Diodes", Solid State Electronics 19, p. 389 (1976).

o E. D. Adams, G. C. Straty, and E. Wall. "Thermal Expansion Coefficient and
Compressibility of Solid Helium-three", Physical Review Letters 15, p. 549
(1965)

o E. D. Adams and E. L. Wall. "Thermal Expansion Coefficient and
Compressibility of Solid Helium-three", Bulletin of the American Physical
Society 10, p. 519 (1965). (Note that these were the first measurements of the
PVT surface of solid helium-three.)

Other Interesting Technical Web Sites

The Institute for New Energy - Click
Here

Phil Gibb's Old Cyclotron Laboratory - Click
Here

(More to be added later.)

APPENDIX

Causality - While many physicists worry about the
causality associated with the reverse time direction of tachyons, it is
pointed out here that in this model, the tachyon is bound to a positive mass,
charged particle in a closed system, and so the effects of causality need not
be worried about, at least in the case of a first order calculation. Insofar
as time travel is concerned, the random nature of atomic processes would
likely preclude anyone getting younger if he achieved hyperluminal velocities.
This is not to say that reverse time has no effect whatever. It might well be
that there is a cancellation effect between the reverse time of the tachyon
and the forward time of the charged particle, but only at the microscopic
particle level. However, that possible effect has not been explored
extensively, as yet. More is said about this in The Physics of Tachyons. But,
consider that the Starship Enterprise, on a brief junket to the colony on
Neptune, goes to warp factor 1 until it gets near Neptune, and then it drops
below light speed. Have the colonists on Neptune gone backwards in time just
to accommodate the Enterprise? Not likely! Have the crew members become
younger? Not likely, because molecular processes will still be random
processes.

Precedence - This model began as an ad hoc model, and
many physicists object ad hoc models. However, when it was extended to other
areas than the lepton family, it produced surprisingly good results. But for
those who still object, they should recall that the Kepler model was pure ad
hoc, but it influenced Isaac Newton in his development of classical physics.
Planck's black body radiation model was pure add hoc. Einstein described his
photoelectric equation as "heuristic", i.e., one step above ad hoc. Bohr's
model was ad hoc. De Broglie's thesis committee considered his thesis to be ad
hoc. In fact, Einstein's recommendation was necessary in order for them to
accept the thesis. Without these developments, Schroedinger's and Heisenberg's
work would not have been developed, or if it had, it would have been pure ad
hoc because there would have been no rationale for the development. Further,
some would argue that Dirac's model was ad hoc because there was no a priori
reason for his utilizing matrices the way he did. He himself said of it that
"I was only looking for some pretty mathematics." So if anyone wishes to
retain his intellectual "purity of essence" and not use ad hoc models or
anything derived from them, then that person fobidden to utilize Newtonian
mechanics and quantum mechanics.

Light Speed Charged Particles - It is inherent in this
model that the magnetic moment of particles arises from very tiny charged
particles revolving in very much larger orbits at the speed of light. This
allows the revolving charge to have sufficient area to generate a magnetic
moment while remaining consistent with the experimental observations that the
charged particle is extremely small. As will be shown, the radius of the
electron's charged particle's orbital radius is 386.15933 fm, or the reduced
Compton wave length for the electron. Likewise, the muon's charged particle's
orbital radius is 1.8675947 fm, or its reduced Compton wavelength. The
absolute maxim diameter of the charged particle, based on this model, is 0.185
fm, with no lower limit. (High energy scattering measurements indicate that
its radius is actually of the order of 10-4 fm.)

This is a radical departure from conventional models. These charged particles
are not accelerated to these velocities, but when the particles are created,
they are created with these orbital velocities, much in the same sense that
tachyons are created with hyperluminal velocities.

Charged Particles as Stationary Photons - These revolving
systems, such as electrons, appear to behave as stationary photons, but they
are bound by the laws of relativity when they move as a system. But for the
extreme skeptic, recall that there is some indication that the jets ejected
from quasars appear to have light speed velocities, so that there may be some
prior precedent for luminal, or even superluminal, light emitting particles.
(Note also that Lucas and Bergman have a particle model that does not use a
point charged particle, but uses a ring charge that revolves at light speed.
It does not use tachyons to maintain the system balance. See their web site at
www.cormedia.com/css/info.html.)

Radiating, Revolving Charged Particles - A common
objection expressed about this kind of model is that revolving charged
particles would be expected to radiate their energy away. However, it is to be
emphasized that this is a light speed particle, and there is no a priori
reason to assume that it must radiate its energy away as would be expected for
sublight speed particles. But in the simplest case, consider its orbit to be a
ground state orbit. (Shades of Neils Bohr! It worked for him!!) But a more
specific explanation is to be found in the fact that the revolving light speed
particles appear to be stationary photons. (For more on this, see Eq. 27 and
the discussion following it.) But more important, the equation that describes
revolving charged particles does not hold for light speed particles. It goes
to infinity at these speeds.

Relativity has passed every test to which it has been
subjected, but it has been tested only in the subluminal domain. While it
might be possible that relativity is relevant to the luminal and
superluminal domain, there is no a priori reason to assume that it must
be particularly relevant to these domains. This model is concerned only
with luminal and hyperluminal velocities and is, therefore, not at variance
with any measured aspect of relativity.

Quantum mechanics has been extraordinarily accurate in
its description of the atom. This model utilizes only simple quantization, and
it does not attempt to arrive at a wave function for the internal structure of
the particle systems. However, there is little to place it in conflict with
quantum mechanics as it applies to atomic structures, although its definite
structure may bother some quantum theorists.

Neutrinos have been detected, so therefore, they exist.
However, estimates of the of the neutrino masses via Kurie measurements give
upper limits of about 10 - 20 eV, so it does seem rather strange that it could
be believed that the shape of the mu-> e curve (up to 52.6 MeV) can be
determined by such small, non-photonic particles. Further, it should be
pointed out the residual energy from the tachyonic mu-> e conversion is
19.83 eV, and the residual energy of the pi -> mu conversion is 122.97.
These numbers are within range of the respective experimental estimates of the
upper limits of neutrino masses. In any case, they are not needed to account
for the Bohr magneton in this model.