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 tachyons and takionics

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MessaggioOggetto: tachyons and takionics   Lun 12 Ott 2009 - 16:27

FONTE: http://www.skepdic.com/takionic.html

tachyons and takionics

A tachyon is a theoretical particle or wave which travels faster than the speed of light. Tachyons exist in a theoretical world where objects have negative mass and time goes backwards. Tachyon energy is used to scan "subspace", among other things, on the sci-fi fantasy program, Star Trek Voyager. So far, there is no empirical evidence for the existence of tachyons. "If they do exist, tachyons would be extremely difficult to utilize under our current understanding of physics," says NASA scientist Tom Bridgman. Despite being theoretical and, if real, difficult to utilize, and if utilized, of unknown value, tachyons are the main ingredient in a feature line of New Age products that range from beads, belts and shoe inserts, to sweatbands, power pillows, massage oils and vials of tachyon water. And, of course, there are books, such as Tachyon Energy: A New Paradigm in Holistic Healing by Gabriel Cousens and David Wagner (2000).
A few enterprising New Agers claim that they know tachyons exist and they have harnessed its power. For example, Fred Pulver who runs the Carbondale Center for Macrobiotic Studies (formerly known as Biotech Industries of Carbondale Colorado) claims to know that "The Tachyon Field supplies the energy needs of all living organisms until balance is achieved, then it eases until called upon again. As it is needed, and a depletion occurs, it rushes in until balance is achieved once again."1 But just in case Nature fails to keep you in tachyon balance, you can get all the tachyon power you need from one of Biotech’s swell takionic products. (The reason for the spelling difference has to do with the fact that common words such as 'tachyon' cannot be trademarked and 'tachyonized' was taken. 'Takion' and 'takionic' can be trademarked and thereby used to identify one's products.)
Pulver makes some incredible claims. For example,
Motors have been built which draw upon the Tachyon Field for energy. They exhibit strange behavior, such as increasing in speed the longer they run, even though they are connected to no visible power source.2
Where are these motors? No one has seen them but we are to take it on faith that they exist. Perhaps they are in the vehicles bringing all those UFOs to earth for the cattle mutilations and reproductive experiments.
Here's another claim:
Takionic products, with their aligned atomic polarities, enhance the body's natural ability to draw from the Tachyon Field for its energy needs. Athletes have discovered that Takionic products allow them to perform faster and longer, and shorten recovery time. As conduits for input from the Tachyon Field, Takionic products are proving themselves in the sports performance arena.
Who are these athletes? The same ones who wear magnetic shoes and wear crystals to ward off bad energy?
What is this "Tachyon Field"?
The Tachyon Field is extremely dense. This density cannot be measured because it is a negative state, mirroring the universe of positive density which we inhabit. The theory of negative density is supported by an observable phenomenon: a perpetually expanding physical universe which is brought into being through pressure exerted by expansion of the invisible one. Pressure exerted by the Tachyon Field upon our physical universe indicates the existence of an invisible, highly dense universe, the Tachyon Field.3
The tachyon field may be dense but it would have to go a long way to match the density of an "alternative physicist who finds this stuff compelling. There seems to be something missing here--such as a fundamental grasp of reality. We are asked to believe in an invisible universe causing pressure on ours, thereby forming energy which “cannot be proven by instrumentation currently available” but which we can use to explain just about anything we want.
But wait, there's more!
Tachyon theory is holistic because it accepts the notion of two interdependent universes which are actually indivisible: the visible, sub-light speed universe and an invisible, faster-than-light one. Tachyon theory also substantiates omnipresence, a purely metaphysical concept. God is omnipresent (simultaneously existing everywhere). Omnipresent existence can only occur at faster-than-light speeds, since slower-than-light travel takes time to cross space. Therefore, omnipresence can only be an attribute of a Tachyon Universe where time and space are uniform.4
Pulver is also an "alternative" metaphysician! The speculation seems infinite. "The nervous system and brain are nothing but an extremely sophisticated antenna and receiver to absorb, process and transform the resources of the Tachyon Field." "Healers have learned to access the Tachyon Field's resources for its healing powers more successfully than the average person has."
All the above claims are by way of introduction by their author to the line of takionic products he has for sale. New Age Energy is especially marketable in alternative health care, where the products to make us healthy and happy are as boundless as the imagination, as endless as New Age Energy itself. (Note: the following are 2002 prices and may not be accurate any more.) There are takionic beads, 10 for $118.95, which are said to have antennae which "focus the beneficial tachyon energy". There is a takionic belt for $268.95 which is said to help improve circulation and increase strength.. Takionic water comes in a small vial for $27.95 and is "pure and cluster-free".
Another New Age business, Advanced Tachyon Technologies (ATT) of Santa Rosa, California, offers a much broader array of products, including some for cats and dogs. ATT has products which can enhance your love-life (including one called panther juice) and your athletic skills, not that the two are mutually exclusive. They have products to ease your pain and improve your brain. They have chakra balancing kits for those who would like a little takionic boost to their meditation. You can purchase a personal tachionized cocoon for $396 or a Tachyonized Silk Meditation Wrap for $298 to use while meditating "with your lover before making love."
See also Ayurvedic medicine, chi, confirmation bias, energy, pragmatic fallacy, reiki, therapeutic touch, and Transcendental Meditation.
________________________________________
reader comments
further reading
books and articles
Baadhio, Randy A. Geometrical tachyon and grafted string world sheet (Berkeley, Calif.: Mathematical Sciences Research Institute, 1991).
Forward, Robert L. Future Magic: How Today's Science Fiction Will Become Tomorrow's Reality (Avon
Books, 1988).
Herbert, Nick. Faster Than Light: Superluminal Loopholes in Physics (Plume Books, 1988).
Pickover, Clifford A. Time: A Traveler's Guide (Oxford University Press, 1998).
Wall, Ernst L. The Physics of Tachyons (Hadronic Press, 1995).
websites
Reality Check: On the Internet by Milton Rothman
Reality Check: Pseudoscience on the Internet by Milton Rothman
Tom Bridgman of NASA on Tachyons
Alternative Medicine and the Laws of Physics by Robert L. Park
Particle Data Group
The Physics of Negative Mass Tachyons by Ernst L. Wall
Tachyon Energy (Advanced Tachyon Technologies)
Takionic.com

The Question
(Submitted June 12, 1997)
My question comes as a result from reading "About Time" by Paul Davies. In it he talks of time travel and other such things. How can one prove the existence of tachyons and once it is proven, can they be implemented for space travel like in Star Trek (i.e. faster than light space travel)?
The Answer
Probably two of my favorite popular level treatments of the possible technological implications of tachyons are:
"Future Magic: How Today's Science Fiction Will Become Tomorrow's Reality" by Dr. Robert L. Forward, Avon Book, 1988.
and
"Faster Than Light: Superluminal Loopholes in Physics" by Dr. Nick Herbert, Plume Books, 1988.
Personally, I think Bob Forward's is the better of the two.
As to proving the existence of tachyons, one basically has to discover a particle interaction which can *only* be explained by the presence of one or more tachyons. Some theoreticians argue that if tachyons exist, the universe could be filled with them but they interact so weakly with ordinary matter that we can't detect them. Physicists have searched through some experimental records and so far none of the high-energy accelerator labs have detected an interaction which can *only* be explained by tachyons. This means that tachyons must be far more weakly interacting than neutrinos. If they do exist, tachyons would be extremely difficult to utilize under our current understanding of physics.
You could travel faster than light if you could turn yourself (and your starship) into a tachyon. However, special relativity indicates that if you did this, you could travel back in time and violate causality - the idea that causes must precede their effects. You could wind up in the "Grandfather Paradox": What if you go back in time and kill your grandfather before your father is born? But if you're never born, how could you go back and kill your grandfather?
There seems to be a lot of bogus science on the Web surrounding the subject of tachyons. A number of companies seem to like the name in their product so be careful what you read.
Tom Bridgman
CGRO Science Support Center
for Ask an Astrophysicist


FONTE: http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970612b.html





FONTE: http://www.tachyonmodel.com/

The Physics of Negative Mass Tachyons and the Fundamental Electrodynamic Origin of Electron de Broglie Waves Plus a Longitudinal Electrodynamic Neutrino Model
________________________________________

Abstract


• This web page provides a summary 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.
• In order to get this agreement with experiment, it was necessary to abandon the imaginary mass tachyon and to use, instead, a simple negative mass tachyon. This should be a minimal problem because after some 45 years since it was introduced, the imaginary mass tachyon has produced no agreement whatsoever with experiment. 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.
• 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.)
• All possible attempts have been made to make this presentation as simple as possible so that even someone with a relatively modest background in physics can verify the model for himself by carrying out some of the calculations with a simple hand calculator. In fact, the reader is encouraged to try the calculations for himself. (Physical constants and particle data are provided in Section 17.)
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MessaggioOggetto: Re: tachyons and takionics   Lun 12 Ott 2009 - 16:28

FONTE: http://www.tachyonmodel.com/

The Physics of Negative Mass Tachyons and the Fundamental Electrodynamic Origin of Electron de Broglie Waves Plus a Longitudinal Electrodynamic Neutrino Model
________________________________________

Abstract


• This web page provides a summary 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.
• In order to get this agreement with experiment, it was necessary to abandon the imaginary mass tachyon and to use, instead, a simple negative mass tachyon. This should be a minimal problem because after some 45 years since it was introduced, the imaginary mass tachyon has produced no agreement whatsoever with experiment. 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.
• 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.)
• All possible attempts have been made to make this presentation as simple as possible so that even someone with a relatively modest background in physics can verify the model for himself by carrying out some of the calculations with a simple hand calculator. In fact, the reader is encouraged to try the calculations for himself. (Physical constants and particle data are provided in Section 17.)


• Figure 1. This single graph is at the heart of this model in that the experimental accuracies of the dimensions and the resulting magnetic moments of the electron and muon are obtained directly from the cutoff energies of the two curves. In addition, the geometric structure of these particles has their origin in these curves.

• Therefore, it is critically important that we state unambiguously that this model could not exist without these curves.

• To reiterate, the rather abrupt inclusion of Figure 1 in an abstract, i.e., the μ e curve and the rare, direct e conversion curve ( 1 event out of 10 4 events ), was done because it is important to stress that the electron and muon models would not be possible without it. In turn, the dimensions of the de Broglies waves also arise directly from these curves. The geometries of the proton and neutron models are derived from the electron and muon models, but their dimensions and energies are derived from their magnetic moments. The meson energies, in turn, are obtained directly from the neutron model’s dimensions.

• All of these results arise very naturally and effortlessly, one from another, with no ad hoc add-ons or adjustments.

• For those that may have wondered what really “waves” when you have a de Broglie wave, a possible answer is provide here in a model of an electrodynamical field around an electron’s revolving charge that spirals outward at the speed of light with a Compton wavelength. It is these spiraling waves that produce a de Broglie probabilistic scattering of an electron exiting from a double slit experiment. That is, a self correlating, spiraling, electrodynamic field arising from the electron reflects off of the edges of the slit and causes a probabilistic scattering of the electron in preferred directions after the electron has exited the slit. In addition, the reflection of an electron’s spiraling field from lattice points on the crystalline surface causes a probabilistic scattering as the electron interacts with these autocorrelating wavelets.

• Based on neutrinos associated with Supernova 1987A, it is likely that a neutrino’s velocity is within a few parts per billion of the speed of light. Using this, we propose a light speed, longitudinal impulse, electric field neutrino model that is consistent with this overall particle model. That is, in a conversion from a muon to an electron, a portion of the electric field detaches itself from the muon’s revolving charged particle and produces the longitudinal electric impulse.

• It is to be emphasized that while this extremely simple model agrees quite well with experiment, there is no relationship whatsoever between it and the generally accepted but much more complex standard particle model. In saying this, we do not question the validity of nor the accuracy of the results of the standard model. We merely use a different approach to describe fundamental particle behavior. Further, it is probably not inaccurate to say that, for the most part, this model is “orthogonal” to the standard model. That is to say, it merely looks at particles from a different perspective than that of the standard model. But at the same time, we do acknowledge that there are a few conflicting interpretations of phenomena but ultimately, this model agrees with experiment.

• The electric and magnetic binding energy of the components of the tachyonic deuteron is 2.381 MeV, about 7% more than the 2.2246 experimentally observed binding energy. The radial dimensions of the deuteron are determined from the magnetic moment of the proton, and the longitudinal dimension was determined from the quadrapole moment of deuteron. This was an unexpected result that could be frankly be regarded as a form of heresy by many. Nevertheless, it still holds, and the calculations are easy enough for any undergraduate physics major to perform using the data shown here.
• Not only does this very simple model appear to give a reasonably accurate picture of accounting of meson masses, but it should be acceptable in polite society because it is politically correct in that it does not discriminate among mesons based on strangeness or color. However, it is totally devoid of charm and it does not assign flavor to particles, and so for this reason some might considered it to be a truly tasteless particle model. It has tachyons but no gluons, which we sincerely hope is not a sticking point with everyone, although we realize that there are many that are already have strong bonds to the standard model. Quark is an ugly (by human standards) little Ferengi bar keeper and is already employed on Deep Space Nine, so he not employed here. This is not to contest the validity or value of those characteristics. It is only that they are not needed here for what we are trying to do.
• Reverse time, insofar as tachyons are concerned is, in reality, an irrelevant issue. See Appendix 1.
• 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.

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 by clicking here.
Web site originally created circa 1995 by Ernst L. Wall
Last Updated on May 3, 2007 by Ernst L. Wall

CopyrightErnst L Wall 2007, All Rights Reserved.
________________________________________
Table of Contents

1. The Definition of a Tachyon
2. Some Preliminary Comments on the Imaginary Mass Tachyon Model and other Issues
3. The Basic Meson Model
4. The Binary Mesons
5. A Brief Comment on Magnetic Moments
6. The Semi Classical Revolving Charge Model
7. The Derivation of the Magnetic Moments of the Electron and the Muon by Means of the Tachyonic Model
7. The Revolving Light Speed Charge as a Bound Photon.

9. The Proton Model
10. The Neutron Model
11. The Light Nuclei
12. An Electrodynamic Model of Electron de Broglie Waves

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

14. Interactions of the Wavelets with Apertures and Edges
15. Electron-Lattice Site Scattering of Compton Wavelets.
16. A Longitudinal Electric Field Model of the Neutrino

17. A Brief Comment on Constants and Units
18. Final Comments
19. How to Obtain Detailed Information on the Tachyon Model
20. Additional Publications by Ernst Wall
Appendix 1. A Digital State Machine Simulation of the Universe and the Difficulties of Time Travel

©Ernst L Wall 2007, All Rights Reserved
________________________________________

1. 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 object 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 so that hyperluminal velocities can be achieved by large, macroscopic subluminal objects.



©Ernst L Wall 2007, All Rights Reserved
________________________________________



2. Some Preliminary Comments on the Imaginary Mass Tachyon Model and Quantum Mechanics
Relativity is one of the greatest theoretical discoveries of the 20th century and is surely one of the greatest achievements of the human mind. Further, it has passed every experimental test to which it has been subjected. The fact that this web site is about tachyons in no way challenges relativity insofar as the subluminal domain is concerned because here we only deal with luminal speed entities (photons and neutrinos) and hyperluminal speed entities.
Furthermore, at the risk of pointing out the obvious, all of those tests that relativity has passed have all been carried out in the subluminal domain. Of course, this includes the Michelson-Morley experiment that gave rise to relativity in the first place as well as those experiments on which Maxwell based his work in electromagnetics.
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,
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MessaggioOggetto: Re: tachyons and takionics   Lun 12 Ott 2009 - 16:28

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 said that, 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 1. 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 20th 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 is presented below 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 E. Recami and R. Mignani, Rivista Del Nuovo Cimento 2, 209 (1974).
©Ernst L Wall 2007, All Rights Reserved
________________________________________
3. The Basic Meson Model


We will present the meson model before the derivation of the electron, muon, proton and neutron models. That is because of its simplicity and because its energy levels fell out of the neutron 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


,



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:


1. First, calculate the 9 levels of E m using the values 1 - 9 for n.
These are shown in the energy level and transition energy diagram, Figure 1, 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. These latter resonances two will be discussed in more detail in an upcoming revision of The Physics of Tachyons.


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



Figure 2. The transition diagram showing the basic mesons as they arise from
transitions among the various resonant states (energy levels) of the pion.
2. Next, subtract each resulting value of E m 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 (Figure 2) 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 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. (The Bohr atom's analog is the Balmer series.) These are shown in Figure 3, 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" y mesons. The second order transitions, the transitions between the n= 2 level and the lower levels correspond to the lighter mesons from the h through the a0(980).

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.

Figure 3. The first order transitions of the pion resonances, or the psi
mesons ( the charmed mesons). The error ranges from -1.3 % to +4.7 %.


Figure 4. 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.
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MessaggioOggetto: Re: tachyons and takionics   Lun 12 Ott 2009 - 16:29

(See publications 17 and 18 below.)

©Ernst L Wall 2007, All Rights Reserved
________________________________________
4. 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 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.


Figure 5. 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 as this, especially when you add the binary mesons, as described below.





Figure 6. 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.)
©Ernst L Wall 2007, All Rights Reserved
________________________________________



5. A Brief Comment on Magnetic Moments
The origin of the magnetic moment of particles, m, is at 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 factor used to determine 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 = 1.6027733x10-19 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 = n e / c.


The amount of torque on the loop is the T = B cos( q ) dyne-cm, where q 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 would not set well some quantum mechanics.)


©Ernst L Wall 2007, All Rights Reserved
________________________________________

6. The Semi Classical Revolving Charge Model
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 -15 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.
Further, based on this model 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, and again, this is very much in contradiction to the standard model but is in agreement with experiment.
Similarly, a proton consists of a heavier sigma hyperon ( ) that has combined with a negative mass tachyon, but one that has a different mass from that of the electron and the muon. Converse to the method of the electron/muon model, the magnetic moment of the proton is used to determine its dimensions. The dimensions of the proton agree to within 3% of the experimental dimensions that are determined from both high and low energy Note that a negative mass particle is inherently an antigravity particle.


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 are described later. 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.

7. The Derivation of the Magnetic Moments of the Electron and the Muon by Means of the Tachyonic Model
To summarize the revolving electron and muon models, a pion captures a negative mass tachyon and becomes, overall, a less massive muon. The muon then 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 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.


Further, based on this model, it is mandatory that the by-products of high energy electron-positron collisions include muons and pions, both of which are different states of the same particle, at least in context with the present model. Gamma rays would also be expected to arise from the collisions. 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 in Section 7, below.
Note that the dimensions of this model are precisely defined insofar as its spatial and velocity dimensions are concerned. This is an issue that will 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.
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.,

(1)
(2)
Next, we will need to utilize half of these masses as binding energies. I.e., we have


(3)
.
(4)
The sum of these energies is


(5)

The right most curve, the direct 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 curve on the left is well known and is contained in most particle physics books.
It should be noted that accurate fits to the 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.
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 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 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 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 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 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
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MessaggioOggetto: Re: tachyons and takionics   Lun 12 Ott 2009 - 16:30

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
(6)
Considering the above, the de Broglie wavelength for the tachyon is given simply by
(7)


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


(Cool
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 lT, as it orbits the charged particle in the charged particle's frame of reference. That is,
(9)
Here, the subscript lT refers 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

(10)


For the muon,
(11)


The equations describing the balance of this system for the electron model is
(12)



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



(13)

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


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


(15)


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


(16)


Using Eq. 9 for , Eq. 15 becomes


(17)


Using MTe as defined by Eq. 16, we eliminate (Re - 1) and MTe from Eq. 17 so that we have for the electron
(18)
Using an identical approach for the muon model, the orbital radius of the muon's pion is
(19)
The magnetic moment of a current loop is, in general,


(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
(21)


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


(22)
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
(23)


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


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

(25)

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

(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.)
© Ernst L. Wall 2007, all rights reserved.


8. The Revolving Light Speed Charge as a Bound Photon.

To explain how it is possible for a subatomic particle’s constituent charge to revolve at the speed of light, we will summarize some of the highlights of the revolving charge model. As previously stated, the charge in an electron revolves at the speed of light in a quantized orbit whose radius is equal to the reduced Compton wavelength of the electron. Using this, recall that the magnetic moment of a current loop is, in general, μ = IA, where I is the current and A is area of the loop. The current at any point in the orbit of a revolving charge is given by I = e f / c, where the frequency of rotation for a Compton wavelength orbit is

Hz . (27)

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

Next, if we treat the revolving charged electron as if it were a bound photon and use the Einstein photoelectric equation ( ) with Eq. 27, we find that this bound photon’s energy is

E = m c 2 MeV, (28)

which is the same as the rest mass of the electron.
But now, if we let the electron move with some velocity v and we replace r in Eq. 27 with , the mass of the electron in Eq. 28 becomes

MeV . (29)



One would expect the Lorentz contraction to hold only for the radius in the direction of the velocity. That is, one would expect the circular orbit to become oval shaped, whereas Eq. 29 implies that the radius actually contracts uniformly in all directions. There is no a priori reason to expect this uniform radial contraction, but we will accept it for the time being as a fact that produces agreement with experiment.

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 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, 2nd Ed. There, it is important to note that that 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.
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MessaggioOggetto: Re: tachyons and takionics   Lun 12 Ott 2009 - 16:31

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.)
©Ernst L Wall 2007, All Rights Reserved
________________________________________
9. The Proton Model


The proton consists of a sigma hyperon and still another negative mass tachyon. However, the sigma hyperon to 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 %.


Figure 9. 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)
©Ernst L Wall 2007, All Rights Reserved
________________________________________
10. 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 ( 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.) 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


(30)


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.

(See publications 1, 17, and 20, below)
©Ernst L Wall 2007, All Rights Reserved
________________________________________



11. 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 they 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.

Figure 10. 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”.
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 11. The measured and calculated magnetic moments
of the lighter nuclei.
(See publications 1, 17, and 20, below)
©Ernst L Wall 2007, All Rights Reserved
________________________________________

12. 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, λC, 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.



©Ernst L Wall 2007, All Rights Reserved
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MessaggioOggetto: Re: tachyons and takionics   Lun 12 Ott 2009 - 16:32

13. 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. The number of these doublets, or phase crossings, that will occur in that time unit is

. (31)

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

. (32)

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

, (33)

where l D 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 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.

©Ernst L Wall 2007, All Rights Reserved
________________________________________

14. 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. 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.

©Ernst L Wall 2007, All Rights Reserved
________________________________________

15. Electron-Lattice Site Scattering of Compton Wavelets.

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)
©Ernst L Wall 2007, All Rights Reserved
________________________________________

16. 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 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
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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.)

©Ernst L Wall 2007, All Rights Reserved
________________________________________

17. 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 17-1. 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 17-2. Physical Constants 1
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/2p h 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.)
1 J. J. Hernandez, et. al., Phys Letters 239B (Holland) ( This is the bi-annual Review of Particle Physics published by the American Physical Society’s Particle Group. )
©Ernst L Wall 2007, All Rights Reserved
________________________________________

18. Final comments

We do not present his model as a finished product. We state merely that we have explored the possibility of using negative masses to describe characteristics of known, quantifiable subatomic particles.

Unresolved is the issue of the extra peak predicted near the zero degree direction in the case of the Davison-Germer experiment. They saw no such peak. But yet, we feel that the different approach to de Broglie waves presented is compelling enough that it warrants further exploration.

There is also the case of the narrow electric field lines associated with the de Broglie waves. That would make the binding energy of the deuteron somewhat problematic if these narrow lines were a applied to the deuteron. This is an issue that warrants reconciliation.

________________________________________

19. 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.


©Ernst L Wall 2007, All Rights Reserved
________________________________________



20. Additional Publications by Ernst Wall


1. Ernst L. Wall. Book, The Physics of Tachyons, 234 pp., ( Hadronic Press, 1995)

2. Ernst L. Wall, “A Longitudinal Electrical Impulse Field Neutrino and its Origin in Virtual Quanta of the Tachyonic Electron, Muon, and Pion”, Hadronic Journal 24, p. 207 (2001).

3. Ernst L. Wall, “The Tachyonic Electron’s Revolving Light Speed Particle as a Non-Radiating, Bound Photon”, Hadronic Journal Supplement 15, p. 419 (2000).

4. Ernst L. Wall, “A Digital State Machine Simulation of the Universe and the Difficulties of Time Travel”, Hadronic Journal Supplement 15, p. 231 (2000).

5. Ernst L. Wall, “The Fundamental Electrodynamic Origin of Electron de Broglie Waves”, Hadronic Journal Supplement 15, p. 123 (2000).

6. Ernst L. Wall, “Electrodynamics of Revolving Light Speed Particles and A Fundamental Basis for de Broglie Waves”, Hadronic Journal Supplement 14, p. 79 (1999).

7. Ernst L. Wall. “Radial Stability in a Longitudinal Electrical Field Neutrino”, Bulletin of the American Physical Society 44, p. 34 (1999).

8. Ernst L. Wall. “Origin of de Broglie Waves in a Tachyonic Electron Model”, Bulletin of the American Physical Society 44, p. 35 (1999).

9. Ernst L. Wall. “A First Tangible Step in the Quest for Hyperluminal Space Travel”, Proceedings of NASA’s Breakthrough Propulsion Physics Workshop, NASA/CP - 1999-208694, p. 349 (Jan 1999).

10. Ernst L. Wall. “A Longitudinal Electromagnetic Impulse Neutrino Model”, Bulletin of the American Physical Society 43, p. 2163 (1998).

11. Ernst L. Wall. “A Possible Fundamental Origin of the de Broglie Equation”, Bulletin of the American Physical Society 43, p. 2163 (1998).

12. Ernst L. Wall. “Electrodynamics of Revolving Light Speed Particles”, Bulletin of the American Physical Society, 43, p. 1399 (1998) .

13. Ernst L. Wall. "On Pion Resonances and Mesons, Time Cancellation, and Neutral Particles", Hadronic Journal 12, p. 309 (1989).

14. Ernst L. Wall. "Time Cancellation Hypothesis", Bulletin of the American Physical Society 33, p. 1076 (1988).

15. Ernst L. Wall. "Charm, Other Resonances, and the Tachyonic Particle Model", Bulletin of the American Physical Society 33, p. 1076 (1988).

16. Ernst L. Wall. "Unresolved Problems of the Tachyonic Models of the Electron and the Muon", Hadronic Journal 9, p. 263, (1987).

17. Ernst L. Wall. "On Tachyons and Hadrons", Hadronic Journal 9, p. 239, 1986.

18. Ernst L. Wall. "Indirect Evidence for the Existence of Tachyons; A Unified Approach to the π ® μ ® e Conversion Problem", Hadronic Journal 8, p. 311 (1985).

19. Ernst L. Wall. "The Role of Tachyons in Electron Spin and Muon Spin", Bulletin of the American Physical Society 30, p. 729 (1985)

20. Ernst L. Wall. "The Role of Tachyons in Proton Spin", Bulletin of the American Physical Society 30, p. 729 (1985)

21. Ernst L. Wall. "Hamming Code Error Correction for Microprocessors", Chapter 3, Microprocessor Applications Handbook, edited by D. Stout. McGraw Hill, 1981.

22. Ernst L. Wall. "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.)

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

24. 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)

25. 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).

US PATENT:
U. S. Patent 3,800,412 awarded to Walter K. Niblack and Ernst L. Wall, "Process for Producing Surface Oriented Devices", April 2, 1974.

©Ernst L Wall 2007, All Rights Reserved
________________________________________

Referenced Terms and Names
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MessaggioOggetto: Re: tachyons and takionics   Lun 12 Ott 2009 - 16:33

Tachyon, antigravity, FTL, superluminal, hyperluminal, faster than light, faster-than-light, quark, gluon, meson, spin, charm, relativity, neutrinos, photon, electron, proton, neutron, pion, muon, lepton, hadron, Bohr Magneton, nucleus, chromodynamics, hyperluminal, quasar, Planck, Einstein, de Broglie, Schroedinger, Heisenberg, Dirac, Kepler, Ptolemy, Ptolemaic

©Ernst L Wall 2007, All Rights Reserved
Note that all of the material, concepts, and ideas expressed on this page and within the reference material by Ernst L. Wall is copyrighted. In addition to the general copyright of the material described: this copyright specifically includes the concept of the revolving charge, the magnetic field it generates, and its contribution the particle mass both in the stationary case and the relativistic case insofar as it applies to subatomic particles and nuclei. This copyright includes the term “longitudinal electrodynamic electric field neutrino” as well as the concept of the neutrino described here. This copyright includes the concept of the electrodynamic wave model of the de Broglie waves and the specific term “Compton wavelet” and the concept of auto correlated wavelets and their reflection off of nearby lattice sites and nearby slits, edges, and surfaces. This copyright includes the tachyonic meson model, the meson equation, the tachyonic electron and muon model, the tachyonic proton and neutron model, as well as the small nuclei models bound together by electric and magnetic fields. This copyright includes the concept of negative mass tachyons. This copyright includes the concept of and use of a state machine to investigate the flow of time.
However, this material may be freely quoted provided full credit is given to Ernst L. Wall.
Appendix 1

A Digital State Machine Simulation of the Universe and the Difficulties of Time Travel
Ernst L. Wall
The Institute for Basic Research
Palm Harbor, FL 34684
April 26, 2000

Published: Hadronic Journal Supplement 15, p. 231 (2000).

Abstract.
The flow of time, in previous scientific literature, has been discussed in terms of classical thermodynamics and statistical mechanics. Here, we propose a new approach to the study of time flow by taking advantage of concepts derived from modern computer science. We devise a thought experiment that uses a hypothetical, gigantic digital state machine to simulate the universe. This simulation will, at least in concept, process objects that include atoms, nuclei, particles, and photons. These objects change state on a regular basis at a rate determined by a clock whose period is based on the frequency of a gamma ray. This clock provides a high time resolution so that the total state count, as it progresses from one discrete state to the next most probable discrete state, provides a new definition of absolute time. Absolute time is a count of the all states of the universe from its beginning to any given count. Based on this state machine argument, time travel to some absolute past would require that copies of all past states of the universe be stored in some medium, somewhere, so that the time traveler could rewind the universe. This would seem unlikely with today’s technology as well as the technology of the foreseeable future, so that time travel would seem to be an unlikely possibility. Further, we demonstrate that time could not exist without the existence of matter.

1. Introduction.
In many publications in recent years, especially in the popular press, science fiction articles, and even the movies, much has been presented about human beings undertaking reverse time travel that ostensibly occurs as a consequence of such diverse phenomena as traversing wormholes and exceeding the speed of light.
But reverse time is a very real concern today for those who investigate tachyons, or particles whose velocity exceeds the velocity of light. This has been a consideration from the earliest days of the investigations of these particles because of the causality issues, or the assumption that these particles travel backwards in time and cause difficulties with the present( 1, 2, 3, 4 ).
The usual classical thermodynamic counter to the argument for the possibility of reverse time travel, at least for large macroscopic bodies, is to simply state that increasing entropy, the arrow of time, is always in the direction of increasing time, so that reverse time movement is impossible (5).
While a study of time flow using the concept of increasing entropy is not a difficult concept, we will develop a new method that is conceptually even simpler than the entropy argument, but at the same time, it provides a far greater conceptual extent. This methodology easily demonstrates that the phenomena of time travel for a macroscopic body is a highly questionable possibility, at least based on physics as we know it today. This new method is based on a more modern concept, namely, state machines as implemented by modern computer technology.

2. Scope of Investigation
In this work we will describe a method of simulating the universe by means of a hypothetical digital state machine. We will use this state machine model to arrive at a new definition of time, specifically, a definition of absolute universal time. This definition of time will show that matter is necessary for time to exist.
We will use this simulation to demonstrate that to go backwards in time, you would either have to rewind the entire past universe while the future universe continues its forward trajectory, or you would have to have a record of all states of the universe from the present to the point in the past that you wished to visit. We also demonstrate that merely exceeding the speed of light, or transiting a worm hole, does not rewind the universe, nor access hypothetical records of the past. We will use these to demonstrate that time travel is inherently impossible in the physical universe as we know it today.
In this work, we are only interested in introducing a new, basic concept. We are not interested in answering all possible questions that arise from this model. We are not interested advancing computer science, or even in providing an optimum methodology from computer science. We are only interested in a very simple, very basic state machine concept that will illustrate time flow from the standpoint of basic physics. And, it is not necessary to consider relativistic or quantum mechanical aspects of this model in order to introduce it. These would be interesting enhancements of the model, and including them in it would not be extraordinarily difficult. But neither of these are necessary in order to illustrate the basic state machine method of studying the flow of time, and so we will not consider them in this present work.


3. A Simulation of The Universe by Means of a Digital State Machine.

In order to arrive at an improved method of analyzing the difficulties associated with time travel, we describe a hypothetical model of the universe that is a gigantic digital state machine that will simulate the general behavior of the universe as time advances.
State machines are commonly used in the analysis of modern digital logic systems. Not only are they simple to understand, they also provide a more definite methodology for general simulation of statistical phenomena than generalizing from a statistical ensemble. And because a state machine implementation of physical phenomena is generally scaleable, a computer simulation can be implemented at various levels of complexity that range from huge simulations on complex multiprocessor systems to simple simulations in household computers.
This state machine can be sufficiently general as to process a covariant model when it is desired to do so for a large scale, relativistic model of the universe.
This state machine will process a set of objects. Specifically, these objects are particles, including atoms, nuclei, alpha particles, beta particles, electrons and photons, and even tachyons, if desired. Each of these objects has a state that is uniquely determined by parameters that include its mass, cross section, position, velocity, and spin.
We will define the state of the universe at some integral time, t, as

, (1)
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where s t , i (m, r, v, k) is the state of some particle i at time t. The state includes mass m, position r, velocity v, and spin k. N is the total number of particles in the universe. Because each particle is in motion, the state of the universe will change from instant to instant. The nature of this change will determine the new state of the particle as it progresses to the next time interval. The new state can be generalized as

. (2)

Here, I(s t, i , s t, j ) represents an interaction that relates a particular particle, i, at some time, t, to all other particles, j, in the universe. Conceptually, at least, it is inherently symmetric with respect to time reversal because time is merely the sequential progress of the universe from state to state, regardless of whether the state count goes backwards or forwards.
However, digital numbers are inherently limited in precision. As a result, the limited precision of the specification of the targets state could cause motion under time reversal to have a slightly different trajectory than the exact reverse of the trajectory of a preceding, forward state. This provides a built in randomness, of sorts, to I(s t, i , s t, j ).
But even so, it would still be necessary to provide a time independent random number generator in order to model a more probabilistic trajectory to the next state for each particle, i, as opposed to a definite path. This is because the randomness build into the real universe allows for many possible trajectories into the future. Without this inclusion of randomness, each time the simulation is started from the same point, the forward trajectory would be exactly the same. This randomness must be very small, however.
For a realistic simulation of the universe, the states of all objects, near and far, must all change before a universal state is complete. This is simultaneity of state change. Because of the simulation of simultaneity, the interaction, I(s t, i , s t, j ), of any two objects must be processed in such a manner as to account for the time of propagation of the interaction from one object to the other. I(s t, i , s t, j ) is, in fact, an object of simulation in itself.
However, if we wished to simulate a synchronization of distant clocks by means of light signals, then time delays at the macroscopic level would have to be considered as measured by the simulated clocks in the same manner as is used in a typical textbook introduction to special relativity.

4. The Nature of Forward Time Flow

In a digital simulation, the time, t, is an integer value, not a continuous value. Further, the division of time into intervals of seconds is meaningless for this state machine. It is too gross a quantity to calculate the effects of atomic and nuclear transitions because the state of the universe will change millions, or even billions of time in one second. Therefore, a rational calculation of one state based on the previous state is not possible for time divisions or one second or greater. That is, the end state based on such a gross sequence interval is a completely random state with respect to its starting state. What we must have is a time division that is smaller than that of the interval of the fastest changing object in the state set that composes the universe. Therefore, we will define:

The fundamental universal time sequence interval is the minimum time that is required to resolve the state change of the fastest changing object in the set of all objects that constitutes the universe.

In order to implement this definition, we propose that a hypothetical clock having the time sequence interval based on the frequency of a high energy gamma ray be used to separate one nuclear state from the next. In this, we have a mechanical definition of time that is a natural, fundamental state change integer through which the universe can unfold. This fine division of time does increase the difficulties of simultaneity insofar as the sheer size of the model we must process, but we are dealing with a generalized hypothetical model that will deal conceptually with the general passage of time, and this model will be very adequate for that purpose.
But first, we must relate this to the real physical universe. Here, we make simultaneous, hypothetical digital “samples” of the all of the parameters of an object, and store the data in a computer memory. This defines the state of the object. This hypothetical sampling would be done in the same manner as the analog-to-digital sampling that is used in modern day digital signal processing, where we would use the above clock to trigger the samples
Using this, we define a non-subjective, or non-anthropomorphic time as follows:

Absolute universal time is the total count of the state transitions that occur, starting at some initial time of t = 0 at the beginning of the universe and continuing forward to any specified time. These state counts occur when the universe makes regular transitions from one discrete state to the next discrete state.1

This definition is not dependent on an anthropomorphic definition of time as derived from earth based intervals. There are no years, days, seconds, etc. It is based only on the requirements that the simulation provide for the most probable trajectory of one state of the universe to the next state based on the behavior of the smallest, fastest objects in the universe.
It is to be noted that in the definition, we specified the “next discrete state” of the universe. But it is important to note that it is also the “next most probable state” of the universe. If our hypothetical computer were used to implement Eq. 2 with the intent of simulating the real universe, then the simulation would calculate each object’s new state based on its current state and I( s t, i , s t, j ), which provides for the most probable next state, not a predetermined, definite state. It is because of the slightly probabilistic nature of I(s t, i , s t, j ) that the future in the simulation is not absolutely ordained in advance.
Based on the definition of absolute universal time, it is obvious that without physical matter, time has no states to count. And with no state count, there is no passage of time. Therefore, we state that:

The timeless, eternal void hypothesis: In the absence of matter , there are no state transitions to count. Without a state count, there can be no time. Therefore, in the absence of matter, time is devoid of any meaning, and hence, is nonexistent.

5. Reverse Time Flow

Suppose we were to reverse the clock in the simulation and begin processing the state machine in reverse. Starting from the last state that occurred during positively advancing time, the objects would begin to retrace their previous trajectories. However, the randomness that is built into I(s t, i , s t, j ) would cause them to follow trajectories that are slightly different from their original trajectories. The reverse path would be random, and entropy would continue to increase, just as it did while time was moving forward. However, time reversal would also imply velocity reversal, which would have the effect of reversing the velocity of the objects.
But this velocity includes the not only the velocity of the individual objects, but the composite velocities of all objects composing a macroscopic body. As a result, this macroscopic body would also reverse its velocity, providing that the precision of the digital state specification is sufficient to include the large particle velocities and the slower velocities of the macroscopic objects that are composed of these particles.
While there might be a trajectory to an approximate near past point, there would be no trajectory to any previous, but distant, exact point in the past. As time advances in reverse, the effect on the universe would, in time, behave similarly to the forward movement of time in that the same random state changes and movement of events would be the same as if clock had been counting forward.
For example, suppose we simulate a billiard game. The balls are racked on the table into a triangle, the triangle is broken, and the balls scatter randomly on the table. Several shots latter, we reverse the simulation. Because of the time independent, very small randomness built into I, the balls will not go back to their exact original triangular, racked condition. Disorder, or entropy, has increased.
Similarly, we could simulate the process of adding a drop of milk to a container of water. After a few minutes, the milk will be dispersed. If we reverse the simulation, the randomness built into I will not permit the milk molecules to re-coalesce into the spatially bound drop of pure milk that they started out as.
What is more difficult to predict is the effect of simulated humans and their free will on the progress of reverse time. We will not cover this subject in this work.
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MessaggioOggetto: Re: tachyons and takionics   Lun 12 Ott 2009 - 16:36

6. Tachyons and Time Travel

As previously noted, a tachyon is a particle whose velocity exceeds the speed of light, and in the literature of the past, it it has generally been assumed to travel backwards in time(1). This is another object whose effects are suitable for a very simple simulation within the hypothetical universe.
We make no assumptions about the characteristics of a tachyon, only that it has a velocity greater than the speed of light, and that it has the ability to interact with a subliminal particle. (To date, there has been no direct detection of a tachyon, although indirect evidence for their existence has been proposed (2, 3). )
In a simulation involving a tachyon, two interacting particles, A and B, might have tachyons that serve to carry information back and forth between them. While it would be true that the tachyons would carry information faster than photons, particles A and B still exist in their environment in the present state, not the past or the future. If a tachyon and a photon were simultaneously emitted from particle A and both of them travel toward particle B, the tachyon would scatter B before the photon was able to reach it. This is not to say that there is a causality violation. The tachyon merely beat the photon to the target. Only if an observer at A were attempting to measure a characteristic of particle B by using a photon based signal would there be any reason for an uninformed observer at A to question whether or not causality was violated. This would be a measurement problem, not an actual case of time reversal.
Further, the trajectories of the particles A and B would still progress in a near random fashion before and after the collisions. The presence of the tachyon would merely serve as a different signaling mechanism. A more mundane analogy would be the use of optical observation of an object that was simultaneously being observed by a sonar scan. The light does not present a causality issue with regards to the sonar scan.
In a simulation, a tachyon, even though its velocity exceeds the velocity of light, will not go backwards in time. Neither will the two particles, A and B, above, backwards in time.

7. The Plight of a Would Be Time Traveler

Next, consider the spatial extent of the present day universe, and an individual who wishes to return to some point in the rather gigantic past. If we were travel to some time and location in the past, and if he has the means and the desire to move about the galaxy to any random point, then the entire galaxy must be available to him. That would constitute true time travel. Or, if his means of transport is to be limited, at least he should be able to use a high powered telescope and be able to view the entire galaxy as it existed back at that time. (But even that reduced capability in a real universe would still be a rather substantial achievement.)
There would be two hypothetical options available to the traveler. He could try to rewind the universe itself, or he could try to find a record of the past history and use that to recreate a point in the past.
The probable past would be different from the absolute recorded past because of randomness built into I(s t, i , s t, j ). In fact, as already stated, the mere attempt to run the universe in reverse would produce, after a short interval of counts, a different past that the actual past. In fact, after a time, the randomness of the rewind of the universe would make it difficult to say that time was really reversed. It is more likely that after a short time of disorientation, the residents of the reverse universe would begin to carry on as if nothing had happened. They would continue to age, have children, and do their jobs.
To simulate our traveler’s visit to an exact point in the past, he must stop the entire universe and then rewind a record it for some specified number of state counts. This requires that a copy of the entire universe for all the past times must be saved somewhere, somehow 2. That is, it requires that all previous absolute recorded states of the entire physical universe must be recorded. We specify the need to use absolute, recorded states to visit the real past because he does not wish to revisit a mere probable past.
Having reached some point in the past, if our traveler is to move forward from that past point to exactly where he came from in the present, not only must he not cause any influence on the past, he must travel forward in a recorded time sequence, or he will arrive at a substantially different point than that which he departed from because of the random nature of the state change. That is, if the recorded sequence is not allowed to replay, and the universe begins its progress forward in a random manner, then he will progress forward to a present that might be quite different than the one he departed from. This is especially true if he interferes with some critical event in the past.
Further, while the traveler is rewinding the past, the universe must continue to move forward from the point in the present time from which he departs on his journey, and the events of this unfolding reverse state sequence must also be recorded if further visits are to be made to correct any problems that a “previous” traveler may have caused. Further, a new recording of the universe must be made after the present point is reached in order to account for the changes that he caused going forwards from the past, as well as the future point from his departure point.
It could be argued that if a time traveler has only a limited part of space available in a simulation, then he might be able to regenerate a small spatial part of the universe at a particular past time, and then let it move forward in time. This would be a localized time journey. But what would happen if he moved to the edge of this localized spatial environment? What would happen to past residents of this region whose paths crossed over the borders of this region? Would they step into another universe, or vanish? What would that do the future of that local region? These might present severe difficulties for the traveler as well as the previous occupants of the time-spatial region near his trajectory. Obviously the future of this local region might be severely disturbed during the return trip to the traveler’s original point of departure, especially on both sides of its borders.
But to complicate matters further, suppose there were multiple time travelers who start out on their journeys at the same time but from different locations. We must ask, which time traveler gets to rewind the universe first? Or, which one gets to go to which copy of which part of the universe at what time?
This problem can become even more complicated if one time traveler has rewound the past universe and moved backwards in time, and is followed some time later by another time traveler who begins to unwind this past universe. We must ask what happens to the previous time traveler in his rewound past universe, and what happens as he returns to the time from which he started his journey.
It is to be noted that we have utilized the term “rewinding” the universal record as an analog to a rewinding a VCR tape or a binary tape from a computer. This is because it is a closer analog to running the universe in reverse. But in these times of random access computer storage, our simulated time traveler could pick a point in the past and return there immediately.
But the simulation of traversing a black hole and jumping back to some time in the past could be done by, essentially, accessing a random point in a mass storage system. This would be an example of near immediate access to a specific point in the past that involved no rewind.
As a brief aside, it is to be noted that as time progresses forward in the recorded universe, the residents have no free will. The traveler, assumedly would have free will, but this depends on the simulation. In any case, it is suggested that some interesting philosophical points could be raised from this issue of free will versus predestination.
These are some of the questions that are more clearly enunciated by the use of a digital state machine simulation than we could obtain from a continuous time, statistical ensemble model of the universe. A continuous time model (i.e., an analog model ) that is developed from a statistical mechanical ensemble has no definite transition from one particle state to another particle state (6). An analog recording of the state of a universe and the interaction of its components, or a recording of even a small ensemble of objects, is rather difficult to envision. Therefore, the classical analog model does not permit a hypothetical storage methodology that will permit the concept of storing and rewinding the universe that is as conceptually simple as that obtained from the digital model. The illustrative capability of the analog model is severely limited as compared to a digital state machine.

8. A Digression On Macroscopic Bodies at Hyperluminal Velocities

To depart somewhat from a pure state machine argument for a moment, we will consider a more general discussion of the argument that an object that moves faster than the speed of light would experience time reversal(1,4). For example, the space ship Enterprise, in moving away from Earth at hyperluminal velocities, would overtake the light that was emitted by events that occurred while it was still on the earth. It would then see the events unfold in reverse time order as it progressed on its path. This phenomena would be, in effect, a review of the record of a portion of the Earth=s history in the same manner that one views a sequence of events on a VCR as the tape is run backwards. But this does not mean that the hyperluminal spacecraft or the universe is actually going backwards in time anymore than a viewer watching the VCR running in reverse is moving backwards in time.
Further, it must be asked what would happen to the universe itself under these circumstances. To illustrate this, suppose a colony were established on Neptune. Knowing the distance to Neptune, it would be trivial, even with today’s technology, to synchronize the clocks on Earth and Neptune so that they kept the same absolute time to within microseconds or better. Next, suppose that the Enterprise left Earth at a hyperluminal velocity for a trip to Neptune. When the crew and passengers of the Enterprise arrive at Neptune, say 3 minutes later in Earth time, it is unlikely that the clocks on Neptune would be particularly awed or even impressed by the arrival of the travelers. When the Enterprise arrives at Neptune, it would get there 3 minutes later in terms of the time as measured on both Neptune and Earth, regardless of how long its internal clocks indicated that the trip was. Neither the Enterprise nor its passengers would have moved backwards in time as measured on earth or Neptune.
The hands of a clock inside the Enterprise, as simulated by a state machine, would not be compelled to reverse themselves just because it is moving at a hyperluminal velocity. This is because the universal state machine is still increasing its time count, not reversing it. Nor would any molecule that is
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MessaggioOggetto: Re: tachyons and takionics   Lun 12 Ott 2009 - 16:36

not in, or near the trajectory of the space ship, be affected insofar as time is concerned, provided it does not actually collide with the space ship.
In the scheme above, reverse time travel will not occur merely because an object is traveling at hyperluminal velocities. Depending on the details of the simulation, hyperluminal travel may cause the local time sequencing to slow down, but a simulated, aging movie queen who is traveling in a hyperluminal spacecraft will not regain her lost youth. Simulated infants will not reenter their mother’s wombs. Simulated dinosaurs will not be made to reappear. A simulated hyperluminal spacecraft cannot go back in time retrieve objects and bring them back to the present. Nor would any of the objects in the real universe go backward in time as a result of the passage of the hyperluminal spacecraft.
The mere hyperluminal transmission of information or signals from point to point, nor objects traveling at hyperluminal velocities from point to point, does not cause a change in the direction of the time count at the point of departure nor at the point of arrival of these hyperluminal entities, nor at any point in between.

9. Conclusion.

Based on concepts derived from modern computer science, we have developed a new method of studying the flow of time. It is different from the classical statistical mechanical method of viewing continuous time flow in that we have described a hypothetical simulation of the universe by means of a gigantic digital state machine implemented in a gigantic computer. This machine has the capability of mirroring the general non-deterministic, microscopic behavior of the real universe
Based on these concepts, we have developed a new definition of absolute time as a measure of the count of discrete states of the universe that occurred from the beginning of the universe to some later time that might be under consideration. In the real universe, we would use a high energy gamma ray as a clock to time the states, these states being determined by regular measurements of an object’s parameters by analog-to-digital samples taken at the clock frequency.
And based on this definition of time, it is clear that, without the physical universe to regularly change state, time has no meaning whatsoever. That is, matter in the physical universe is necessary for time to exist. In empty space, or an eternal void, time would have utterly no meaning
This definition of time and its use in the simulation has permitted us to explore the nature of time flow in a statistical, non-determinate universe. This exploration included a consideration of the possibility of reverse time travel. But by using the concept of a digital state machine as the basis of a thought experiment, we show clearly that to move backward in time, you would have to reverse the state count on the universal clock, which would have the effect of reversing the velocity of the objects. But this velocity includes the not only the velocity of the individual objects, but the composite velocities of all objects composing a macroscopic body. As a result, this macroscopic body would also reverse its velocity, providing the state was specified with sufficient precision.
But if you merely counted backward and obtained a reversal of motion, at best you could only move back to some probable past because of the indeterminate nature of the process. You could not go back to some exact point in the past that is exactly the way it was. In fact, after a short time, the process would be come so random that there would be no real visit to the past. A traveler would be unable to determine if he was going back in time, or forward in time. Entropy would continue to increase.
But doing even this in the real universe, of course, would present a problem because you would need naturally occurring, synchronized, discrete states (outside of quantized states, which are random and not universally synchronized). You would need to be able to control a universal clock that counts these transitions, and further, cause it to go back to previous states simultaneously over the entire universe. Modern physics has not found evidence of naturally occurring universal synchronized states, nor such an object as a naturally occurring clock that controls them. And even if the clock were found, causing the clock to reverse the state transition sequence would be rather difficult.
Without these capabilities, it would seem impossible to envision time reversal by means of rewinding the universe. This would not seem to be a possibility even in a microscopic portion of the universe, let alone time reversal over the entire universe.
But aside from those difficulties, if you wished to go back to an exact point in the past, the randomness of time travel by rewind requires need an alternative to rewinding the universe. This is true for the simulated universe, and a hypothetical rewind of the real universe. Therefore, the only way to visit an exact point in the past is to have a record of the entire past set of all states of the universe, from the point in the past that you wish to visit onward to the present. This record must be stored somewhere, and a means of accessing this record, visiting it, becoming assimilated in it, and then allowing time to move forward from there must be available. And, while all of this is happening in the past, the traveler’s departure point at the present state count, or time, must mover forward in time while the traveler takes his journey.
Even jumping back in time because of a wormhole transit would require that a record of the past be stored somewhere. And, of course, the wormhole would need the technology to access these records, to place the traveler into the record and then to allow him to be assimilated there. This would seem to be a rather difficult problem.
This then, is the problem with time travel to an exact point in the past in the real universe. Where would the records be stored? How would you access them in order just to read them? And even more difficult, how would you be able to enter this record of the universe, become assimilated into this time period, and then and have your body begin to move forward in time. At a very minimum our time traveler would have to have answers to these questions.
Still another conundrum is how the copy of the past universe would merge with the real universe at the traveler’s point of departure. And then, if he had caused any changes that affected his departure point, they would have to be incorporated into that part of the universal record that is the future from his point of departure, and these changes would then have to be propagated forward to the real universe itself and incorporated into it. This is assuming that the record is separate from the universe itself.
But if this hypothetical record of the universe were part of the universe itself, or even the universe itself, then that would imply that all states of the entire universe, past, present, and future, exist in that record. This would further imply that we, as macroscopic objects in the universe, have no free will and are merely stepped along from state to state, and are condemned to carry out actions that we have no control over whatsoever.
In such a universe, if our traveler had access to the record, he might be able to travel in time. But he were to be able to alter the record and affect the subsequent flow of time, he would have to have free will, which would seem to contradict the condition described above. We obviously would be presented with endless recursive sequences that defy rationality in all of the above.
This is all interesting philosophy, but it seems to be improbable physics.
Therefore, in a real universe, and based on our present knowledge of physics, it would seem that time travel is highly unlikely, if not downright impossible.
We do not deny the usefulness of time reversal as a mathematical artifact in the calculation of subatomic particle phenomena(7). However, it does not seem possible even for particles to actually go backwards in time and influence the past and cause consequential changes to the present.
Further, there is no reason to believe that exceeding the speed of light would cause time reversal in either an individual particle or in a macroscopic body. Therefore, any objections to tachyon models that are based merely on causality considerations have little merit.
For the sake of completeness, it should be commented that the construction of a computer that would accomplish the above feats exactly would require that the computer itself be part of the state machine. This could add some rather interesting problems in recursion that should be of interest to computer scientists. And, it is obvious that the construction of such a machine would be rather substantial boon to the semiconductor industry.
We already know from classical statistical mechanics that increasing entropy dictates that the arrow of time can only move in the forward direction (5). We have not only reaffirmed this principle here, but have gone considerably beyond it. These concepts would be extremely difficult, if not impossible, to develop with an analog, or continuous statistical mechanical model of the universe.
We have defined time on the basis of a state count based on the fastest changing object in the universe. But it is interesting to note that modern day time is based on photons from atomic transitions, and is no longer based on the motion of the earth. Conceptually, however, it is still an extension of earth based time.
But finally, history is filled with instances of individuals who have stated that various phenomena are impossible, only later to be proven wrong, and even ridiculous. Most of the technology that we take for granted today would have been thought to be impossible several hundred years ago, and some of it would have been thought impossible only decades ago. Therefore, it is emphasized here that we do not say that time travel is absolutely impossible. We will merely take a rather weak stance on the matter and simply say that, based on physics as we know it today, there are some substantial difficulties that must be overcome before time travel becomes a reality.

References:
1. G. Feinberg, Phys. Rev. 159, 1089 (1967).
2. Ernst L. Wall, Hadronic Journal 8, 311 (1985).
3. Ernst L. Wall, The Physics of Tachyons., (Hadronic Press, 1995).
4. P. Davies, About Time, p. 234 (Simon & Schustere, 1995)
5. P. Davies, op. cit. p. 196.
6. K. Huang, Statistical Mechanics, p. 156 (John Wiley & Sons, 1963).
7. E. Condon & H. Odishaw, Encyclopedia of Physics, p. 9-139 (McGraw-Hill, 1967).
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MessaggioOggetto: Re: tachyons and takionics   Oggi a 15:19

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