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The Electric Charge and Magnetization Distribution of the Nucleon: Evidence of a Subatomic Turing Wave Pattern
01.05.2014 16:17
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The Electric Charge and Magnetization Distribution of the Nucleon:
Evidence of a Subatomic Turing Wave Pattern
Paul A. LaViolette, Ph.D.
© February 22, 2008
The Starburst Foundation, 1176 Hedgewood Lane, Niskayuna, NY 12309
Electronic address: gravitics1@aol.com
Published in: International Journal of General Systems, vol. 37 (6) (2008): 649-676.
Abstract
Subquantum kinetics, a physics methodology that applies general systems theoretic concepts
to the field of microphysics has gained the status of being a viable unified field theory.
Earlier publications of this theory had proposed that a subatomic particle should consist of an
electrostatic field that has the form of a radial Turing wave pattern whose form is maintained
through the ongoing activity of a nonlinear reaction-diffusion medium that fills all space.
This subatomic Turing wave prediction now finds confirmation in recent nucleon scattering
form factor data which show that the nucleon core has a Gaussian charge density distribution
with a peripheral periodicity whose wavelength approximates the particle's Compton
wavelength and which declines in amplitude with increasing radial distance. The subquantum
kinetics explanation for the origin of charge correctly anticipates the observation that the
proton's charge density wave pattern is positively biased while the neutron's is not. The
phenomenon of beta decay is interpreted as the onset of a secondary bifurcation leading from
the uncharged neutron solution to the charged proton solution. The Turing wave dissipative
structure prediction is able to account in a unitary fashion for nuclear binding, particle diffraction,
and electron orbital quantization. The wave packet model is shown to be fundamentally
flawed implying that quantum mechanics does not realistically represent the microphysical
world. This new conception points to the possible existence of orbital energy states below the
Balmer ground state whose transitions may be tapped as a new source of energy.
Keywords: self-organizing systems, Turing patterns, electromagnetic form factors, particle
diffraction and scattering, field theory, quantum mechanics, subquantum kinetics
1. Introduction
Subquantum kinetics is a unified field theory whose description of microphysical phenomena
has a general systems theoretic foundation (LaViolette 1985a, 1985b, 1985c, 1994, 2003). It
conceives subatomic particles to be Turing wave patterns that self-organize within a subquantum
medium that functions as an open reaction-diffusion system. In so doing, subquantum kinetics
presents a substantially different paradigm from that of standard physics which views particles
as closed systems. Whether these be subatomic particles bound together by force fields, or
quarks bound together by gluons, physics has traditionally conceived nature at its most basic
level to be composed of immutable structures. Unlike living systems which require a continuous
flux of energy and matter with their environment to sustain their forms, conventional physics has
viewed particles as self-sufficient entities, that require no interaction with their environment in
order to continue their existence.
This closed system paradigm has elevated the First and Second Laws of Thermodynamics
from laboratory rules to rigidly enforced universal laws of nature. The imposition of the First
Law, the law of energy conservation, accords well with a universe that behaves as a closed
system for in a closed system energy can be neither created nor destroyed, only converted from
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one form into another. Similarly, the Second Law enforces the idea of a closed system universe
whose entropy can only increase over time, never decrease. These rules, of course, are
conveniently put aside to accommodate the creation of the universe which otherwise should
never have been able to come into being.
Physics, though, is ripe for a conceptual revolution. String theory, the unified field theory that
for some time has been in fashion, has made no testable predictions in the 30 years of its
existence. Moreover its higher dimensional mathematics are so abstract as to be inaccessible to
most theoretical physicists, string theorists included. As a result, many physicists have become
disappointed with string theory and believe it is time for a change (Smolin, 2006; Woit, 2006).
By comparison, subquantum kinetics has made a number of testable predictions and twelve of
these were subsequently verified (LaViolette 1986, 1992, 1996, 2003, 2005); see Table 1.
The present paper focuses on the first prediction presented in table 1, namely that the electric
field in the core of a nucleon does not have an aperiodic cusp shape, as classical nuclear theory
had envisioned it, but rather is configured as a radially periodic stationary wave pattern which
may be termed a Turing wave. Not only does this Turing wave model solve long-standing
problems in modeling the phenomenon of particle diffraction, it also provides a new understanding
of hydrogen atom orbital energy levels, giving credence to emerging technologies that purport
to be tapping a source of clean energy from electron transitions to energy levels below the Balmer
ground state.
2. The Subquantum Kinetics Approach
Let us summarize the subquantum kinetics approach and examine how it predicts the existence
of these unique wave patterns in the core of the nucleon. Subquantum kinetics was inspired from
work done on chemical wave phenomena such as that observed in the Belousov-Zhabotinskii
reaction (Zaikin and Zhabotinskii 1970, Winfree 1974) as well as modeling work done on open
chemical reaction systems such as the Brusselator studied by several investigators (Lefever 1968,
Glansdorff and Prigogine 1971, and Prigogine, Nicolis, and Babloyantz 1972, Nicolis and
Prigogine 1977); see figures 1 and 2. Under the right conditions, the concentrations of the
variable reactants of these reaction systems spontaneously self-organize into stationary reactiondiffusion
wave patterns called Turing patterns, so named in recognition of Alan Turing who in
1952 was the first to point out their importance for biological morphogenesis. Alternatively,
they have been referred to as dissipative structures because the initial growth and subsequent
maintenance of these patterns is due to the activity of the underlying energy-dissipating reaction
processes.
The Brusselator is defined by the following four kinetic equations:
A ——❿
k1 X, a)
B + X ——❿
k2 Y + Z, b)
2X + Y ——❿
k3 3X, c)
(1)
X ——❿
k4 Ω. d)
The capital letters specify the concentrations of the various reaction species, and the ki denote
the kinetic constants for each reaction. Each reaction produces its products on the right at a rate
equal to the product of the reactant concentrations on the left times its kinetic constant. Reaction
species X and Y are allowed to vary in space and time, while A, B, Z and Ω are held constant.
This system defines two global reaction pathways which cross-couple to produce an X-Y
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Table 1
Twelve Apriori Predictions of Subquantum Kinetics that were Subsequently Verified
1) The prediction (discussed in this paper) that the electric field in the core of a nucleon
should be configured as a radially periodic Turing wave pattern of progressively declining
amplitude, and that a charged nucleon should have a Turing wave pattern whose core
electric potential is biased relative to the background electric potential.
2) The prediction that higher energy photons should travel slightly faster than lower energy
photons, in other words, that the velocity of photons traveling over astronomical
distances should be seen to increase inversely with their wavelength.
3) The prediction that the universe is cosmologically stationary and that photons passing
through intergalactic regions of space should progressively decrease their energy, that is,
that photons should continually undergo a tired-light redshift effect.
4) The prediction that photons travelling within galaxies should progressively increase their
energy, that is, blueshift their wavelengths, and consequently that the luminosity of
planets and red dwarf stars should be due to energy being spontaneously generated in
their interiors.
5) The prediction that the luminosity of brown dwarf stars should be due to the photon
blueshifting effect described in (4).
6) The anticipation of the Pioneer effect; the prediction that a spacecraft maser signal
transponded through interplanetary space should be observed to blueshift its wavelength
at a rate of about one part in 1018 per second.
7) The prediction that blue supergiant stars rather than red giant stars should be the
precursors of supernova explosions.
8) The prediction that galactic core emissions should come from uncollapsed matter-creating
stellar masses (Mother stars), rather than from matter-accreting black holes.
9) The prediction that stars in the vicinity of the Galactic center should be massive blue
supergiant stars as opposed to low mass red dwarf stars.
10) The prediction that galaxies should progressively grow in size with the passage of time
proceeding from compact types such as dwarf ellipticals and compact spirals to mature
spirals and giant ellipticals.
11) The prediction that a monopolar electron discharge should produce a longitudinal electric
potential wave accompanied by a matter repelling gravity potential component.
12) The prediction that the speed of the superluminal gravity wave component of a
monopolar electron discharge should depend on the potential gradient of the discharge.
reaction loop; see figure 2-a. One of the cross-coupling reactions, (1-c), is autocatalytic and
prone to produce a nonlinear increase of X, which is kept in check by its complementary
coupling reaction (1-b). Computer simulations of this system have shown that when the reaction
system is supercritical, an initially homogeneous distribution of X and Y can self-organize into a
wave pattern of well-defined wavelength in which X and Y vary reciprocally with respect to one
another. The concentration pattern produced by the computer simulation of the Brusselator in a
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Figure 1. Chemical waves formed by the Belousov-Zhabotinskii
reaction (photo courtesy of A. Winfree).
Figure 2. a) The Brusselator reaction system. b) One-dimensional
computer simulation of the concentrations of the Brusselator's X
and Y variables (after R. Lefever 1968).
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one-dimensional reaction volume is shown in figure 2-b. In a three-dimensional volume we would
expect that a supercritical Brusselator reaction-diffusion system would give rise to a periodic
structure having a Gaussian central core surrounded by a pattern of concentric spherical shells of
declining amplitude.
Subquantum kinetics postulates that similar reaction-diffusion processes take place among
subquantum units termed etherons which exist in various types, A, B, X, and so on, and which
together form a space filling ether (or aether) medium. It postulates that these etherons diffuse
through space and react with one another in a manner specified by the following set of reactions:
A ——❿
k1 G, a)
G ——❿
k2 X, b)
B + X ——❿
k3 Y + Z, c) (2)
2X + Y ——❿
k4 3X, d)
X ——❿
k5 Ω. e)
These five equations form a nonlinear reaction system called Model G which is mapped out in
figure 3. It is similar to reaction system (1) with the exception that step (1-a) is here replaced by
steps (2-a) and (2-b) which introduce a third intermediary variable G. We may write the
following set of partial differential equations to depict how all three reaction intermediates G, X
and Y vary as a function of space and time in three dimensions:
∂G(x, y, z, t)
∂t = k1A - k2G + Dg∇2G
∂X(x, y, z, t)
∂t = k2G + k4X2Y - k3BX - k5X + Dx ∇2X
∂Y(x, y, z, t)
∂t = k3BX - k4X2Y + Dy ∇2Y
, (3)
where the Dg , Dx and Dy values represent the diffusion coefficients of the respective variables.
Etherons in this reaction system play a morphogenetic role similar to Turing's morphogens.
Variations in the concentrations of the three reaction intermediates form observable electric and
gravitational potential fields which, in turn, form material particles and energy waves.
Subquantum kinetics identifies G concentration with gravitational potential, lower G
concentrations being correlated with more negative gravity potentials, a G etheron concentration
Figure 3. The Model G ether reaction system
investigated by subquantum kinetics.
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well corresponding to a matter attracting gravity potential field. The X and Y concentrations,
which are mutually interrelated in reciprocal fashion, are together identified with electric potential
fields, a positive electric potential being correlated with a higher Y and lower X concentration,
and a negative electric potential being correlated with a lower Y and higher X concentration.
Relative motion of an electric potential field, of an X-Y concentration gradient, generates a
magnetic (or electrodynamic) force (LaViolette 1994, 2003). As Feynman, Leighton, and Sands
(1964) have shown, in standard physics magnetic force can be mathematically expressed solely in
terms of the effect that a moving electric potential field produces on a charged particle obviating
the need for magnetic potential field terms. Also relative motion of a gravity potential field, of a
G concentration gradient, generates a gravitodynamic force, the gravitational equivalent of a
magnetic force.
The subquantum kinetics ether functions as an open system, etherons transforming irreversibly
through a series of "upstream" states, including states A and B, eventually occupying
states G, X, and Y, and subsequently transforming into the D and Ω states and from there
through a sequence of "downstream" states. This irreversible sequential transformation is
conceived as defining a vectorial dimension line termed the transformation dimension. Our
observable physical universe would be entirely encompassed by the G, X, and Y ether states,
which would reside at a nexus along this transformation dimension, the continual etheron
transformation process serving as the Prime Mover of our universe. According to subquantum
kinetics, the arrow of time, as physically observed in all temporal events, may be attributed to
the continuation of this subquantum transformative process. Since etherons both enter and leave
the etheron states that compose physical forms, the observable universe is open to the
throughput of etherons. Consequently, the universe's state of order is able to spontaneously
increase provided that its ether reaction system (Model G) operates close to or above a critical
threshold. Thus spontaneous matter/energy creation is allowed in subquantum kinetics.
Since etherons react and transform in a stochastic fashion, changing their individual etheron
states through a Markov process, the etheron concentrations characterizing any given etheron
state will vary stochastically above and below their steady-state value, the magnitudes of the
fluctuations conforming to a Poisson distribution. It is known that such fluctuations are present
in the chemical species of reaction-diffusion systems such as the B-Z reaction and the theoretical
Brusselator system and such would be true as well in the Model G reactive ether. Hence
subquantum kinetics predicts that stochastic electric and gravity potential fluctuations should
spontaneously arise throughout all of space, in regions both where field gradients are present and
where they are absent. This stochastic ether concept is similar to the conventional idea of a zeropoint
energy (or dark energy) background, with the exception that these fluctuations for the most
part are not large enough to nucleate the creation of material particles, fluctuations of such a large
magnitude being extremely rare. As described above, the zero-point energy background arises as
a direct result of the ether's regenerative flux and hence is conceived to be an indication of the
ether's open system character. At the same time, these emerging zero-point energy fluctuations
constitute the ether's incipient ability to create order, each fluctuation being a potential seed for
nucleating physical order.
We may assume that a large fraction of the zero-point fluctuations are of sufficient magnitude
to qualify them as the causal basis for quantum indeterminacy. Bohm and Vigier (1954) have
shown that random fluctuations in the motions of a subquantum fluid are able to generate a field
probability density |ψ|2 that provides an adequate causal interpretation of quantum theory.
Similar reasoning could be applied to subquantum kinetics, except for fluctuations that arise as
random concentration pulses (energy potential fluctuations) rather than as random mechanical
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impulses.
When the kinetic constants and diffusion coefficients of the ether reactions are properly
specified to render the system in the supercritical mode, the system is able to undergo a Turing
bifurcation. That is, a sufficiently large spontaneously arising zero-point electric potential
fluctuation (i.e., a critical fluctuation in the concentration of X or Y etherons), with further
growth would cause the initially uniform electric potential background field to break its
symmetry and nucleate a periodic structure. That is, it would cause the X and Y etheron
concentrations to spontaneously depart from their initial uniform distributions to form a steadystate
wave pattern. In subquantum kinetics this pattern would form the central field structure of
a subatomic particle. The particle's electric field would consist of a Gaussian central core, of
either a high-Y/low-X polarity or low-Y/high-X polarity, surrounded by a pattern of concentric
spherical shells where X and Y alternate between high and low extrema of progressively declining
amplitude. Being a reaction-diffusion wave pattern, we may appropriately name this the
particle's Turing wave.
Figure 4 depicts how the electric potential Turing wave might look in radial cross section
inside a proton (left) and antiproton (right).* This diagram is taken from the 1994 book
publication of the theory (LaViolette 1994 Fig. 8), a similar diagram earlier having appeared in the
original 1985 paper on subquantum kinetics published in the International Journal of General
Systems "Special Issue on Systems Thinking in Physics" (LaViolette 1985b Fig. 4). No ad hoc
assumptions need be introduced to produce such dissipative structures; they follow naturally
from the interplay of the reactions specified above in system (2). Subquantum kinetics predicts
that the electric fields of all subatomic particles, including nucleons, electrons, and their
antiparticles, should consist of such radial Turing wave patterns.
Furthermore subquantum kinetics has proposed that this Turing wave should have a
wavelength equal to the particle's Compton wavelength. The Compton wavelength of a particle,
λ0, is related to its rest mass energy Eo, or to its rest mass mo, by the formula:
λ0 = h c/Eo = h /moc, (4)
Figure 4. Radial electrostatic potential profiles for a charged subatomic
particle, positive matter state (left) and negative antimatter state (right). The
characteristic wavelength would equal the particle's Compton wavelength.
* This diagram was not generated from a computer simulation of Model G; it is a reasonable depiction based on
computer simulations and mathematical analyses that other researchers have performed on the Brusselator.
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where h is Planck's constant and c is the velocity of light. The Compton wavelength for the
nucleon calculates to be 1.32 fermis (λ0 = 1.32 × 10-13 cm). To generate a physically realistic
representation of the subatomic particle, the parameters of Model G (kinetic constants, diffusion
coefficients, and reaction concentrations) must be so chosen that this result is obtained.
The Compton wavelength is twice the wavelength of a hypothetical precursor gamma ray
photon capable of generating the particle. That is, in the course of pair production, a gamma
photon energy of approximately 2hc/λ0 would transform into a particle and antiparticle each
having an energy hc/λ0. In this way, the transition from the energy wave state to the matter state
becomes essentially a change of wave propagation geometry, the initially linear wave propagation
mode of the precursor photon changing into a radial wave propagation mode as the electric
potential field of the newly created particle expands radially outward from the particle's core.
Collision with a heavy nucleus provides the needed boundary condition to absorb the photon's
forward momentum and effect the resulting change of wave geometry with energy being
conserved.
Subquantum kinetics identifies positive charge density with an excess production rate of Y per
unit volume coupled with an excess consumption rate of X per unit volume. A negative charge
density would involve an excess consumption rate of Y and excess production rate of X per unit
volume. These production rate balances produce corresponding electric field potentials, either a
positive potential (high Y/low X) or a negative potential (low Y/high X). These terms, positive
and negative charge density, may be used to describe the X and Y concentration minima and
maxima that form a nucleon's electric potential Turing wave pattern. For example, the positive
electric potential (high-Y/low-X concentration) at the center of the Turing wave of a neutron or
proton would be produced by a local excess Y production rate per unit volume correlated with an
excess X consumption rate per unit volume, which subquantum kinetics identifies with positive
charge density. Similarly, the negative potential well prevailing in the spherical shell that
immediately surrounds this positive core would be produced by a local excess consumption rate
of Y and excess production rate of X which constitute a negative charge density.
The appearance of these charge densities necessitates the simultaneous appearance of the
particle's inertial rest mass. The shorter the wavelength of the Turing wave, and greater its
amplitude (greater its etheron concentration wave amplitude), the greater will be the inertial mass
of the associated particle (LaViolette 1985b). Since acceleration requires a structural shift and
recreation of the particle's Turing-wave dissipative space structure, the particle's resistance to
acceleration, its inertia, should be proportional to the magnitude of its Turing-wave charge
densities; that is, proportional to the amount of negentropy that must be restructured (LaViolette
2003).
The proton's electric potential pattern is positively biased relative to the ambient potential, as
indicated by the hatched region shown in figure 4. Correspondingly, the electric potential of the
antiproton's Turing wave is negatively biased. Subquantum kinetics identifies this biasing with
the origin of the particle's long-range electric field and predicts that such biasing should be absent
in the neutron's electric potential Turing wave pattern. The unbiased electric potential profile for
the neutron is portrayed in Figure 5 which shows two neutrons in a hypothetical state of nuclear
bonding (LaViolette 1985b Fig. 10, 1994 Fig. 15). It should be kept in mind that the charge
densities that generate the proton's Turing wave pattern, and that are associated with its inertial
mass, are distinct from and additional to the charge density that centrally biases its Turing
pattern and produces the proton's long-range electric field. The former periodic densities emerge
as a result of the particle's primary bifurcation from the homogeneous steady-state solution,
while the latter aperiodic bias emerges as a result of its secondary bifurcation from an existing
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Figure 5. Illustration of the electric field profile of a neutron (Particle No.
1). In close proximity, two such nucleon electric fields could interlock to
form a nuclear bond, producing an unstable di-neutron nucleus, in the case
of neutron-neutron binding, or a stable deuteron nucleus in the case of
proton-neutron binding.
steady state Turing solution. The origin of charge as a secondary bifurcation is described further
in section 5.
The subatomic particle, then, may be conceived to be an organized entity, or system, whose
form is created through the active interplay of a plurality of particulate structures existing at a
lower hierarchic level. Whereas quark theory postulates that a nucleon is composed of just three
quarks, subquantum kinetics proposes that a nucleon should be composed of a myriad of
constituent etherons, e.g. over 1025 per cubic fermi. Moreover whereas quark theory proposes
that quarks exist only within the nucleon, subquantum kinetics presumes that these are far more
ubiquitous, filling all of space and forming the substrate for all fields.
In open chemical reaction systems like the Brusselator, entropy can spontaneously decrease
without constituting a violation of the Second Law of Thermodynamics. For example, according
to the Prigogine equation, dS = diS + deS, where dS is the total change of entropy in the system,
diS is the entropy change due to irreversible processes (i.e., reactions) occurring within the
system and deS is the entropy transport across the system boundaries to the environment. Thus
if the entropy decrease due to irreversible processes is greater than the system processes that
increase the entropy of the system's environment, |–diS| > deS, it is possible for the system's
entropy to decrease. By analogy, the same would be true of a physical universe that is conceived
to function as an open, reaction-diffusion system. Although, methods of defining entropy change
at the subquantum, etheric level would need to be revised since measurable quantities such as
heat, energy, and temperature would exist only at the quantum (supersystem) level, not at the
etheric (subsystem) level. At the etheric level, concepts such as ether reaction affinity, action,
change, or flux might be applicable, but it is unclear how these would correspond to physically
measurable quantities such as potential energy or temperature.
Subquantum kinetics is incompatible with the idea of a big bang since a zero-point energy
fluctuation large enough to create all the matter and energy in the universe in a single event would
be a virtual impossibility. Rather, subquantum kinetics proposes the more likely scenario in
which subatomic particles (e.g., neutrons) spontaneously materialize throughout supercritical
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regions of space through a process of continuous creation (LaViolette 1985, 2003).* Also to be
conservative in our assumptions, we must assume that the ether is cosmologically stationary and
that galaxies, excepting their peculiar motions, are at rest relative to their local ether frame. The
cosmological redshift effect, which big bang theorists cite as evidence for cosmological expansion,
has been shown to make a better fit to cosmological data if it is interpreted as a tired-light energy
loss effect (LaViolette 1985, 1986, 1995, 2003). This tired-light phenomenon has been shown to
emerge naturally from the Model G ether which predicts that photons propagating through
subcritical intergalactic space should continuously lose energy. Thus unlike the big bang theory,
which predicts that the entropy of the physical universe as a whole should be progressively
increasing, subquantum kinetics predicts that the entropy of the physical universe should be
progressively decreasing as matter is continuously created.
For the same reason that the entropy of the universe is permitted to decrease through a
process of continual matter and energy creation, so too subquantum kinetics allows violations of
the First Law of Thermodynamics as well. But the photon redshifting or blueshifting rates that
subquantum kinetics models for subcritical and supercritical regions of space are so small as to be
undetectable in the laboratory, more than ten orders of magnitude smaller than what can
reasonably be measured in an Earth-based laboratory. Nevertheless subquantum kinetics had
correctly predicted the approximate rate of photon blueshifting that one should observe in maser
signals transponded through interplanetary space; see prediction 6 of Table 1 (LaViolette 1985),
because this prediction was subsequently verified by analyzing the blueshift in signals returning
from the Pioneer 10 spacecraft (Anderson 2002, LaViolette 2005).
The notion of an ether or of an absolute reference frame in space necessarily conflicts with the
postulate of special relativity that all frames should be relative and that the velocity of light
should be a constant for all frames. However, experiments by Sagnac (1913), Graneau (1983),
Silvertooth (1987, 1989), Pappas and Vaughan (1990), Lafforgue (1991), and Cornille (1998), to
name just a few, have established that the idea of relative frames is untenable and should be
replaced with the notion of an absolute ether frame. Furthermore the recent finding that the
velocity of light is frequency dependent, not only confirms one of the predictions of subquantum
kinetics (Prediction 2 of Table 1), it also refutes the relativistic notion that the velocity of light is
an absolute constant. Supporting evidence that higher frequency photons travel slightly faster
than lower frequency photons was found by studying the arrival times of gamma rays emitted by
flares in the core of Markarian 501, a galaxy that lies about 460 million light-years away. Gamma
rays in the energy range of 0.25 to 0.6 Tev were found to arrive 4 minutes after their higher
frequency counterparts whose energy was an order of magnitude higher in the range 1.2 to 10 Tev
(Albert, et al., 2007). Even so, subquantum kinetics does not negate the existence of "special
relativistic effects" such as velocity dependent clock retardation and rod contraction. These
emerge as expected results of its reaction-diffusion ether model (LaViolette 1985b, 1994, 2003,
2004).
3. Early Models of the Nucleon's Core Field
According to standard field theory, the electric and gravitational field of a subatomic particle
arises from a point source at the particle's center, the field potential ideally rising to an infinite
value as this center is approached. Even in the modern era of physics and astrophysics, the
assumption that a gravitational field singularity exists at the center of a particle is a fundamental
* In earlier writings it had been suggested that these primordially nucleating particles would be protons. It is more
likely, however, that they are neutrons that decay to protons and electrons through the process of beta decay.
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prerequisite to allow the formation of a black hole singularity. Einstein (1950), however, was
against the idea of point field sources, believing that the idea of a continuous field continuum
coexisting with mass or charge points led to a fundamental inconsistency of physical field theory
as a whole. He envisioned particles as "bunched fields," regions in which the field density was
particularly high. Thus Einstein's model is consistent with the predictions of subquantum
kinetics.
Physicists have sought to probe the interior of the nucleon by means of particle scattering
experiments to determine how its central electric and magnetic fields vary with radial distance.
Measurements of the resulting particle scattering angles and the transferred momenta Q yield a
function expressed in terms of the square of the momenta, F(Q2), which is called the electric form
factor and which equals the Fourier transform of the charge distribution of the target nucleon.
Thus it is possible to take electric form factor data and perform a reverse Fourier transform to
estimate the distribution of electric charge within the probed nucleon. Recent model fits to the
form factor data indicate that the nucleon's charge density is spatially distributed, rather than
point-like, and that it reaches a finite value at the particle's center, rather than rising to infinity.
Thus they show that early notions of the electric charge being point-like and having a potential
that rises to infinity at the nucleon's center is essentially a fiction and that the bunched field
particle model is more correct. The idea of a singularity-like charge could be retained if one were
willing to theorize that the charge stochastically danced around in such a fashion that allowed its
average, or probability density, to form the observed distributed charge density pattern. But, as
is seen below, the new measurements require that this dance be so complex as to undermine the
credibility of the singularity model.
Initially, particle scattering experiments analyzed momentum transfers at relatively low
particle momenta, for example, in the range of Q2 < 1 (E/c)2, where energy, E, is measured in
billions of electron volts. Thus this low range implies that the scattered particles have energies of
less than 300 million electron volts. At these energies, relativistic Lorentz contraction effects of
the scattered particle along the direction of particle momentum transfer may be neglected without
posing a problem. Consequently, early model fits were able to make reasonably good fits to
electric form factor data by predicting a simple exponential decline of charge density ρ with
increasing radius r, ρ ~ e-αr, where α is a model constant. Examples include Galster's model
published in 1971, Platchkov's (1990) representation of the 1980 Paris potential model, and
Schmieden's (1999) fit to the Mainz Microtron data. All predicted that charge density should
form a sharp central cusp, rising to a finite central value; see figure 6 (Kelly 2002).
As early as 1938, Gregory Breit (1938) had proposed that the charge distribution of a nucleon
rounds off at its center somewhat like a Gaussian curve. Licht and Pagnamenta came to a similar
conclusion in 1970. They noted that by taking relativistic effects into consideration, this central
cusp would become smoothed into a Gaussian shape, although they did not explicitly graph this
spatial variation. They noted that such a charge distribution would be consistent with
expectation if the nucleon was theorized to consist of a cluster of quarks trapped in a potential
well (Licht and Pagnamenta 1970).
Subquantum kinetics, which began its development in 1973, predicted a very different electric
field distribution within the neutron and proton. It predicted that a particle's electric field should
be Gaussian shaped at the particle's core, as in Breit's model, but in addition, that this field
should be periodic, that further out from the core it should form a spherically symmetric shell
pattern having a wavelength equal to the particle's Compton wavelength. The subquantum
kinetics prediction about the subatomic particle's core field was advanced at a time when fits to
nucleon form factor data were instead modeling the nucleon's electric field as having an aperiodic
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Figure 6. The relativistic neutron charge density model of J. Kelly made to
fit Jefferson Lab data (shaded profile) compared to three nonrelativistic
exponential model fits to earlier data sets (after Kelly 2002, Fig. 12).
cusp shape. Although Licht and Pagnamenta had earlier proposed that a nucleon with a Gaussian
central charge distribution would make a good fit to form factor data, their quark cluster model,
like the Gaussian chiral quark soliton model later published by Christov et al. in 1995 did not
anticipate that the nucleon's charge or magnetization distribution should have a surrounding
periodic component.*
It should be emphasized that these quark models were developed in an attempt to explain the
results of nucleon scattering data, whereas subquantum kinetics was devised from a systems
theoretic standpoint, namely with the belief that some version of the open reaction-diffusion
system model useful in describing the emergence of chemical wave patterns should provide a
useful description for the emergence and formation of a subatomic particle.
* While the model of Christov et al. did predict a reasonably good fit to proton form factor data available at the
time, it made a poor fit to form factor data then available for the neutron.
13
4. Confirmation of the Subquantum Kinetics Nucleon Model
As particle scattering experiments began studying collisions at higher particle collision
momenta (Q2 > 1), it became necessary to use polarized particle beams and increasingly
important to take account of relativistic effects. These more recent experiments made it possible
to more accurately assess the spatial variation of the nucleon's charge and magnetization density,
and to distinguish amongst the various competing models. Kelly (2002), for example, used a
method developed by Mitra and Kumari to perform a relativistic inversion of Sachs form factor
data from recent scattering experiments that employed the recoil-polarization technique (Jones
2000, Gayou 2001, Gayou 2002). His model fits to scattering data cast doubt on the cusp
models and instead pointed to a Gaussian charge density distribution modulated with a radial
stationary wave pattern, hence confirming the subquantum kinetics Turing wave model. His
charge density profile for the neutron is compared in figure 6 to the nonrelativistic cusp models
of Galster, Platchkov, and Schmieden. Kelly found that his relativistic model made a better fit to
data than the nonrelativistic models of Platchkov and Schmieden, although Galster's model
fortuitously also made a good fit to the data.
Kelly obtained a good data fit by representing the radial variation of charge and magnetization
density with a Laguerre-Gaussian expansion and obtained similar results with a Fourier-Bessel
expansion. His charge density profiles for the proton and neutron are shown in figures 7-a and
8-a. With his relativistic Laguerre-Gaussian expansion fit to electric and magnetic form factor
data, Kelly (2002) showed that the charge and magnetization distributions for the nucleon are
best characterized by a peripheral periodicity. The periodic character of Kelly's model fit is more
apparent when surface charge density (r2ρ) is plotted as a function of radial distance; see figures
7-b and 8-b. The surface charge density plot integrates the amount of charge contained within an
incremental spherical shell located a distance r from the particle's center and has the effect of
enhancing the magnitude of the charge density ordinate values at large radii along with their
associated periodicities.
Kelly pointed out that unless this surrounding periodicity is included, his nucleon charge and
magnetization density models will not make as good a fit to form factor data. In other words,
Kelly is saying that for a model to make a good fit to the most recent form factor data derived by
probing nucleons with high energy electron beams, such models must include this peripheral
periodic component, or what we here term the Turing wave. Based on his findings we may
conclude that the subquantum kinetics model offers a more detailed and accurate representation
of the nucleon field than the aperiodic predictions of the quark models of Licht and Pagnamenta
and Christov et al. All of the main features of the subquantum kinetics model are verified in
Kelly's model, the Gaussian shaped central charge distribution, the surrounding periodicity, and
the central bias characterizing the proton's positively charged field pattern. This peripheral
periodic feature was not anticipated by earlier quark models.
As in the subquantum kinetics model, Kelly's charge density model predicts that the proton and
neutron should both have a positive core potential. Compare figures 7 and 8 to figures 4 and 5.
In addition, Kelly's model, like subquantum kinetics, depicts the proton's electric potential wave
pattern as being positively biased compared to that of the neutron, with the bias increasing as the
center of the particle is approached. Furthermore, as in the subquantum kinetics model, Kelly's
model shows the amplitude of the proton's peripheral periodicity declining with increasing radial
distance.
Kelly models the nucleon's periodicity to have a wavelength of ~0.7 fermis for the proton and
~0.9 fermis for the neutron. Expressed in terms of the Compton wavelength, λ0,, this models the
14
Figure 7. a) Charge density profile for the proton predicted by Kelly's preferred
Laguerre-Gaussian expansion models and b) the corresponding surface charge density
profile (after Kelly 2002, Fig. 5 - 7, 18).
Figure 8. a) Charge density profile for the neutron predicted by Kelly's preferred
Laguerre-Gaussian expansion models and b) the corresponding surface charge density
profile (after Kelly 2002, Fig. 5 - 7, 18).
15
proton as having a periodicity of ~0.5 λ0 and the neutron as having a periodicity of ~0.7 λ0. This
is smaller than the 1.0 λ0 Turing pattern wavelength predicted in the subquantum kinetics model.
However, the form factor data that Kelly used to fit his models was obtained by bombarding
nucleons with relativistic electrons having Lorentz factors approximating the nucleon rest mass
energy. So at the moment of collision the nucleon's potential energy would momentarily double
causing the form factor data to predict a particle wavelength about half as long as would be
expected on the basis of the nucleon's rest mass. So Kelly's nucleon models do not portray the
particle's inherent wavelength, but one that arises as a result of the act of measurement. Doubling
Kelly's model wavelengths to correct for this collision energy effect yields values close to the
Compton wavelength values predicted by subquantum kinetics.
Kelly's findings that a periodic spatial pattern makes a good fit to nucleon charge form factor
data may be itself taken as evidence that the electric field pattern of a nucleon is generated from
reaction and diffusion processes continually taking place among constituents of a space-filling
subquantum medium. Hence Kelly's findings provide strong support for the subquantum
kinetics reaction-diffusion system approach. Unlike the linear wave packets of quantum
mechanics, the subatomic dissipative structures modeled by subquantum kinetics do not spread
out over time. Just as fast as X and Y etherons diffuse out from their respective shells, the
nonlinear reactions (3) rebuild the structure's form keeping entropy at bay. Thus subquantum
kinetics provides a simple solution to a problem that has plagued wave mechanics since its
inception.
Accounting for Kelly's findings in terms of the quark model instead proves to be quite
problematic. To represent the nucleon's wave-like electric charge distribution, one would have to
postulate a corresponding wave-like patterning in the probability distribution of up and down
quarks. Thus in the proton, two up quarks with an electric charge of +2/3 would be theorized to
occupy predominantly the Gaussian core region, and one down quark with a charge of -1/3 to
occupy predominantly the surrounding negative well, thus giving a net charge of +1. But
proceeding outward, it becomes difficult to account for the surrounding succession of hills and
wells. One now has to postulate that the three quarks dance about in such a manner that the two
up quarks spend their times predominantly in the vicinity of these successive hills and the down
quark spends its time predominantly in the vicinity of these successive wells. But it is difficult
to imagine what would cause quarks to move about in such a fashion to generate the observed
spherically symmetric wave patterns. Quarks themselves, or the "gluons" theorized to bind them
together, have no script to tell them they should behave in this manner. With the subquantum
kinetics approach, on the other hand, such subatomic particle periodicities emerge as a natural
consequence of the behavior of the subquantum reaction-diffusion processes that are postulated
to take place throughout all space, reaction processes that were proposed decades before there
was observational evidence of a subatomic Turing wave. The quark model not only failed to
anticipate the wave-like character of the nucleon's charge and magnetization distribution, but to
survive and accommodate Kelly's findings it must make a posteriori ad hoc assumptions dictating
that quarks behave kinetically in a very strange manner.
5. The Creation of Charge
Subquantum kinetics differs from the quark theory in several respects, one being the manner in
which it handles the origin of mass, charge and spin. Quark theory does not attempt to explain
how inertial mass, electric charge or spin arise. It merely assumes them to be physical attributes
present in quarks in fractional form and which in triplicate summation appear as corresponding
properties detectable in the nucleon. By comparison, etheron reactants of subquantum kinetics
16
have no mass, charge, or spin. Subquantum kinetics proposes that such properties are present
only at the quantum level, mass and spin emerging at the time the subatomic particle is created,
and charge emerging as a secondary bifurcation of the primary Turing bifurcation as described
below.
The practice of ascribing charge to quarks has met with some objection in that fractional charge
units (±1/3 or ±2/3) characterizing an unbound quark have never been experimentally observed.
Also the assumption that quarks have spin has been refuted on experimental grounds. For
example, quark theory predicts that protons whose quark spins are magnetically aligned should
interact 25% more frequently than those whose quark spins are unaligned. But particle scattering
experiments find an interaction ratio that is 20 fold greater, the spin-aligned protons being found
to interact five times more frequently (Jaffe 1995). A spin magnetization that is a property of
the particle as a whole and not present in any hypothesized subquantum constituents would be
more consistent with these experimental results. Since subquantum kinetics requires that spin
should emerge as a property of the particle as a whole, its concept of spin is more in line with
observation than that of the quark theory.
According to subquantum kinetics, the property of spin should emerge as a direct consequence
of the existence of the particle's Turing wave. Hence like inertial mass, it should appear at the
time the particle is formed through a Turing bifurcation. Following the emergence of the Turing
wave, radial etheron fluxes would extend between the particle's core (e.g., high-Y/low-X) and its
adjacent spherical shell (e.g., low-Y/high-X). That is, X would continually flow into the core and
Y would continually flow out of the core because of the core's X production rate deficit and Y
production rate surplus. Judging from a similar phenomenon occurring in macroscopic systems,
the inward flows would develop into a free vortex which could stimulate a rotational wave
pattern to propagate circumferentially. These would appear as rotating modulations of the X-Y
concentration pattern, or rotating electric fields, which would give rise to magnetic effects that
may be identified with particle spin (LaViolette 1985b, 1994, 2003). Since the particle's Turing
wave is periodic, its spin magnetization would also be expected to be periodic which is in
agreement with Kelly's findings.
This charge creation process may be better understood with the help of the bifurcation diagram
shown in figure 9. The horizontal axis plots the bifurcation parameter, β, which represents the
ether reaction system's degree of criticality, higher values of β indicating that the system is
operating increasingly far from equilibrium. Suppose that the system operates such that its
bifurcation parameter β surpasses critical thresholds βc and β', the first of these thresholds being
the Turing bifurcation. The ϕy ordinate in figure 9 charts electric potential which in etheric terms
is represented as the concentration magnitude of the Y etheron component relative to the ambient
Y concentration. Let us also assume that space is initially in its field-free vacuum state where the
X and Y reactants are initially uniformly distributed in space at their ambient levels; i.e., ϕy = 0.
Since the bifurcation parameter lies above, βc, the uniform state will be unstable so that an
emerging zero-point energy fluctuation (X-Y fluctuation) will tend to grow in size and
spontaneously break the prevailing symmetry. Past this Turing bifurcation, the reaction system
moves to a new steady state in which the X and Y reactants are inhomogeneously distributed as a
spherically symmetric reaction-diffusion wave pattern. The Y concentration would be at a
maximum and the X concentration at a minimum at the particle's center, the X-Y concentration
values, reciprocally alternating and declining in amplitude with increasing radial distance. This
solution, represented by the positive polarity upper branch in figure 9, is identified with the
neutrally charged neutron (n°). The neutron's Turing wave would have a maximally positive
electric potential at its center which would become negative at increasing radial distance and then
17
Figure 9. A hypothetical bifurcation diagram for nuclear particles.
Beyond the primary bifurcation threshold βc a fluctuation emerging
from the uniform steady state leads to the creation of a subatomic
particle, and beyond the secondary bifurcation threshold β ' leads
to the emergence of its electrostatic charge.
positive still further out, alternating with a characteristic wavelength of λo. This wave pattern
would not be biased relative to the background potential, but would appear externally to have a
zero charge. Kelly's charge density model for the neutron confirms this unbiased field
configuration.
At this intermediary stage, where the reaction system has just passed the Turing bifurcation
threshold, βc, the particle's field pattern will be unbiased; no charge will be present. But, as the
amplitude of the core wave pattern grows and eventually surpasses the secondary threshold β',
the primary branch neutron solution becomes unstable, as indicated by the dashed portion of the
line. This is consistent with the observation that the neutron, as a free particle, is unstable and
decays with a half-life of about 15 minutes. When the neutron at some point spontaneously
decays, the solution jumps to the left in the bifurcation diagram, to adopt a new stable state, the
electrically charged proton state (p+). Since the proton has a rest mass slightly lower than that of
the neutron, this neutron-to-proton transformation will result in the additional emission of an
electron and an electron antineutrino whose summed energy will equal the rest mass energy
difference. This transition from the primary bifurcation, or neutron state, to the secondary
bifurcation, or proton state, is commonly termed beta decay.*
It is significant that model fits that Kelly has made to electric form factor data show that the
peak core charge density for the proton is about four times that of the neutron; compare figures
7-a and 8-a. This is consistent with the bifurcation analysis shown in figure 9 which shows that,
with the emergence of the positively charged proton state, the ϕy core electric field amplitude
ultimately increases above that characteristic of the neutron solution.
Mathematical analysis of the Brusselator, which is applicable also to Model G, indicates that
at this secondary bifurcation the Turing wave pattern becomes biased relative to the ambient zero
* The proposal that the proton solution emerges as a secondary bifurcation of the neutron primary branch, as
depicted in figure 9, is new. In the original 1985 paper and in subsequent publications, the proton had instead been
theorized to emerge from a primary branch different from that for the neutron, but lying close to the neutron's
branch. The formulation presented here is felt to offer a better representation and leads naturally to an explanation
for beta decay.
18
potential state. In other words, in transforming into the newly adopted proton state, the
neutron's electric potential Turing wave pattern becomes positively biased. In this new state, the
proton's mean Y concentration will be increased and mean X concentration decreased from its pre
bifurcation value, the Y bias being indicated in figure 4 by the hatched region. To use the
terminology employed in mathematical treatments of the Brusselator reaction system
(Auchmuty 1975), we may say that X and Y are "nonconserved" in the transition to the biased
periodic steady state. However, since the neutron emits an electron in this process, we see that
charge is ultimately conserved when all components involved in the transition are taken into
account, that is the sum of all charges remains zero. If we were instead describing antineutron
decay ( n° —❿ p-) , the lower branch in figure 9, the negative polarity Turing wave pattern of the
antineutron would become negatively biased to yield the negatively charged antiproton state.
The proton's electric field bias arises because, Y is being produced in its core, at an excess rate
and X is being consumed there at an excess rate. Subquantum kinetics identifies these excess
production and consumption rates with the particle's electric charge, for they are ultimately
responsible for the central biasing of the particle's electric field pattern. As mentioned these
excess productions and consumptions occur when the particle's field amplitude surpasses
threshold β'. Thus in the subquantum kinetics methodology, there is no need to introduce
additional ad hoc assumptions to secure the existence of charge. Charge emerges as a direct
consequence of the behavior of equation system 2. It appears as a secondary bifurcation of the
primary bifurcation that brings the neutrally charged precursor particle into existence. The
proton's positively biased electric potential field consists of an upward biased positive Y core
potential and a downward biased negative X core potential and, as described in earlier
publications (LaViolette 1985b, 1994, 2003), the magnitudes of these biasings would decline
inversely with increasing radial distance to form a long-range 1/r electric potential field. Hence
subquantum kinetics produces a field result that conforms to the laws of classical electrostatics.
Subquantum kinetics qualifies as a unified field theory. First, its subatomic particles produce
long-range fields that conform not only to the classical laws of electrostatics, but to the classical
laws of gravitation as well (LaViolette 1985b, 1994, 2003). Furthermore it explicitly describes
what subatomic particle charge and gravitational mass are, how they arise, and how they produce
their fields. This is something that the quark theory does not attempt to do. Also magnetic and
gravitodynamic forces follow as corollaries of moving charges or masses. Second, subquantum
kinetics provides a reasonable model for spin magnetization. Third, it accounts for beta decay.
Beta decay is often connected with what is called the "weak force," although, in subquantum
kinetics there is no need to speak of a "force" being involved since the transition is simply a jump
from one steady-state solution to another. Fourth, subquantum kinetics also accounts for the
strong force since the Turing patterns of two nucleons, e.g., a proton and neutron, would
interlock with one another when in close proximity, the proton's positive core becoming trapped
in the negative potential well surrounding the neutron's core and the neutron's positive core
becoming trapped in the potential well surrounding the proton's positive core; see Figure 3. Thus
nuclear binding is attributed to the operation of very strong attractive and repulsive electrostatic
forces which come into play when nucleons are in close proximity to one another.
In discussing the charge distribution found for the neutron, theorists in the past have suggested
that the neutron has a positively charged core surrounded by a negatively charged shell and that
these neutralize one another to yield a net zero charge for the neutron. But this explanation
misses the point. Regardless of whether one deals with a proton or neutron, in either case, the
nucleon's electric potential distribution is periodic — a positive core potential hill being
surrounded by a negative potential well, which in turn is surrounded by a positive potential hill,
19
and so on. The reason that the proton projects a long-range positive electric field and the neutron
does not is that the proton's Turing pattern is biased positively, whereas the neutron's remains
unbiased, its potential wells exactly canceling its potential hills.
Subquantum kinetics predicts that the peak-to-peak magnitude of a particle's Turing wave
should decline with increasing radial distance r and that the nuclear force should drop off in
proportion to this wave amplitude decline. Kelly's neutron charge density model supports this
prediction in that it depicts the amplitude of the Turing pattern declining with radial distance. In
the context of subquantum kinetics, where nuclear force is due to the interlocking of Turing wave
patterns, we would expect nuclear force to follow a power law decline similar to that of the
particle's Turing wave pattern. Based on the surface charge density profiles shown in figures 7b
and 8b, we may conclude that Kelly models an approximately inverse fourth power decline at the
center of the Turing pattern. Standard nuclear theories, on the other hand, model a steeper
decline for the nuclear force, Fn ∝ 1/r7. In performing form factor model fits to nucleon scattering
data, it would be interesting to use a wave function with a 1/r-7 decline to investigate the
goodness of fit. To account for particle diffraction, it is necessary to presume that this initial
steep decline transitions further out to a more gradual decline approximating the 1/r2 decline of
the electric field. This subject is examined further in the next section.
Although we can never hope to measure the shape of the particle's gravity field by means of
particle scattering data, experimental evidence that points to the existence of electrogravitic
coupling would lead us to conclude that the particle's gravity field, like its electric field, is also
Gaussian in shape at the particle's center and hence does not form a central singularity. This is
consistent with subquantum kinetics which predicts that there should be a close coupling
between charge and gravity (between X-Y concentration polarity and G concentration polarity).
If the particle's gravity field is Gaussian, then we may rule out the possibility that black hole
singularities might form. Einstein would undoubtedly agree.
6. Particle Diffraction and the Wave-Particle Dualism
The existence of the particle's Turing wave eliminates the need to speak in terms of a "waveparticle
dualism." The subatomic particle simultaneously has both particle and wave
characteristics, its nuclear electric field having both a well-defined core and a surrounding
periodicity. This λ0 core wave by itself is able to account for the phenomenon of particle
diffraction (LaViolette 1985b, 1994, 2003).* The Turing wave forming a nucleon or an electron
would diffract from a diffraction grating in a manner identical to deBroglie's postulated phase
wave. In other words, the particle's advancing Turing wave pattern would establish an electric
potential interference pattern at the surface of the diffraction grating that would be identical to
that produced by de Broglie's hypothesized phase wave. For example, suppose that a subatomic
particle with its extended Turing wave periodicity approaches the diffraction grating at velocity
v. In the particle rest frame the Turing wave will have a wavelength equal the particle's Compton
wavelength, λ0 , whereas in the grating's rest frame it will appear to have a slightly shorter
wavelength λ0 ' = λ0 (1 - ß2)½ due to the Lorentz length contraction effect, where ß = v/c. As the
approaching Turing wave field continuously impinges on the diffraction grating with a velocity v
relative to the grating rest frame, it excites electric potential oscillations at the grating's surface at
a frequency, fe:
* The discovery that the Turing pattern adequately accounts for the phenomenon of particle diffraction was made
after this dissipative structure representation of subatomic matter had already been completed, hence indicating the
predictive potential of this approach.
20
fe = v/λ0 ' = v/ λ0 (1 - ß2)½,(5)
The faster the particle travels toward the grating, the higher will be the excited frequency. These
oscillations will occur coherently at each of the grating's grooves, and will radiate out longitudinal
electric potential waves traveling at the velocity of light. These together will create an electric
potential wave interference pattern in the direction of the approaching particle. Since the radiated
waves travel away from the grating at the speed of light, their wavelength, λv, will be longer than
the particle's intrinsic Compton wavelength, their wavelength, λv, being given as:
λv = c /fe = c—v λ0 (1 - ß2)½
. (6)
These radiated waves are termed "velocity waves" because of their dependence on the
particle's velocity relative to the grating rest frame, their wavelength varying inversely with
respect to the particle velocity (LaViolette 1985b, 1994, 2003). When equation (4) is used to
eliminate the Compton wavelength, λ0 , from equation (6), the velocity wave wavelength becomes
expressed as:
λv = h /mv. (7)
The wavelength of the velocity waves creating the electric potential interference pattern turns out
to be numerically equal to the deBroglie wavelength, λp, which is similarly related to particle
mass m and velocity v (or to momentum p = mv) as:
λp = h /mv = h/p. (8)
The electric potential gradients that make up the grating velocity-wave interference pattern
exert forces on the electric potentials that make up the dissipative structure of the approaching
particle/Turing wave, thereby perturbing the particle's trajectory, scattering it toward regions of
constructive interference. These scattering forces, which are electrostatic in nature, would
operate just as readily on charge densities making up a neutral particle, such as a neutron, as on
those making up charged particles, such as protons and electrons. Put another way, particle
diffraction is a phenomenon in which a subatomic particle interacts with the interference pattern
induced by its own periodic electric potential field. Since the velocity wave interference pattern
requires some time to be established and must already be in place to influence the particle's
trajectory as the particle nears the grating, we may conclude that the Turing pattern periodicity
extends a considerable distance outwards from the particle's core, perhaps on the order of tens of
millions of Compton wavelengths. In the case of the electron, this would amount to a distance of
several tens of microns.
This Turing-wave/velocity-wave explanation of particle diffraction, predicted by subquantum
kinetics, is indirectly validated by the electric form factor data discussed above, which reveals the
existence of the nucleon's Turing wave pattern, at least in the vicinity of the nucleon's core. As
we shall see below, the Turing wave description of the wave properties of matter avoids the
shortcomings of the quantum mechanical wave packet model which developed from deBroglie's
phase-wave model.
De Broglie's particle diffraction model conceived the subatomic particle, in the classical sense,
to be a localized bit of matter undergoing a periodic vibration. He suggested that this vibration
generated a phase-wave of wavelength λp = h /mv, which he represented as a continuous plane
monochromatic wave, having a constant amplitude and no spatial localization, that traveled
linearly at superluminal velocity in the same direction as the particle. Hence he did not take the
phase-wave as a description of the particle itself. Neither could it constitute a reasonable
description of a real electromagnetic wave since its velocity w computed to be greater than the
21
speed of light, i.e., w = E/p = c2/v, where E = mc2 is the particle's total energy and p = mv is its
momentum. Hence according to deBroglie, the phase wave's velocity would be greater than c by
the factor c/v. Although De Broglie (1962) associated this imaginary wave with the particle, he
did not assign to it any real physical significance; he merely meant it as a metaphor.
However, this changed when Erwin Schroedinger reinterpreted deBroglie's phase wave, viewing
it not as a metaphor, but as a concrete reality. Inspired by de Broglie's thesis, Schröedinger in
1926 conceived his famous wave equation which was to become the foundation of modern
quantum mechanics. He had the idea of representing a moving particle as a propagating wave
packet made up of several linear monochromatic wave functions, Ψ(x,t), traveling in the same
direction as the particle at a superluminal speed. They were chosen to have wavelengths that
differed from one another by a small amount, dλ, about an average value equal to the de Broglie
wavelength λp = h /p; hence they were essentially conceived as a group of deBroglie phase-waves.
Also the waves were assumed to have frequencies that differed from one another by a small
amount, dν, about an average value equal to the Compton frequency f0 = E/h. Hence like
deBroglie's phase waves, Schroedinger's wave packet waves were conceived to have a
superluminal velocity: w = λpf0 = E/p = c 2/v. In linear superposition the wave amplitudes
amplitudes would constructively interfere at the center of the wave group where their phases
would coincide, and destructively interfere further out where the wave amplitudes would
desynchronize and mutually cancel out due to their differing frequencies and phases. As a result,
these superimposed waves were conceived to produce a localized wave packet having a non-zero
amplitude over a finite region of space; see figure 10.
Figure 10. A wave packet representation of a material particle.
The breadth of the wave packet, Δx, was determined by a suitably choosing the wavelength
range dλ of the wave group (i.e., Δx = dλ). The velocity of the wave group vg = dν • dλ then
became identical with the particle velocity v (since dν • dλ = dE/dp = v). The ranges, dν and dλ,
were was justified as representing the uncertainty in knowing the frequency and wavelength of
the group's waves at any given instant.
But in proposing this representation, Schröedinger abandoned de Broglie's idea of a virtual
"associated" phase-wave and instead considered the wave packet waves as having physical
significance in the classical sense, the particle being conceived to be synonymous with this
traveling wave packet. Later, Born, Bohr, and Heisenberg adopted Schröedinger's model for their
own probabilistic interpretation. They proceeded to construct a wave-particle dualism in which
the wave packet, or more precisely the function Ψ2, was viewed merely as a description of the
probability of finding the point-like particle in a particular region of space. Whereas de Broglie
initially intended his phase-wave as an imaginary manifestation of a concrete particle singularity,
22
his insight became lost in the shuffle. The emphasis now became reversed. The Copenhagen
school of thought, as it came to be known, gave credence only to the wave packet. The breadth
of the packet Δx was identified with the observational uncertainty in locating the particle's
position, given that the particle's momentum was subjectively evaluated to an accuracy Δp. The
reality of the point-like particle itself was downplayed since the particle could not be accurately
resolved in terms of position and momentum.
Thus, in the space of a few years de Broglie's associated phase-wave had become reified into a
"concrete picture of reality." This presents an excellent example of what Whitehead has referred
to as the fallacy of misplaced concreteness. At a very early stage in its development,
Shroedinger's wave mechanics had become sidetracked by the probabilist school and to this day
has failed to extract itself from the Procrustean bed it helped to create.
However, this quantum mechanical reformulation of de Broglie's theory had several problems.
As in de Broglie's phase-wave, the individual waves forming the packet calculated to have a
velocity greater than the speed of light, thereby contradicting a basic tenet of special relativity
that an electromagnetic wave's velocity should always equal the speed of light, c. Furthermore,
the Schröedinger's wave packet was prone to spatial dissipation. In the words of de Broglie
(1960, p. 25, 1962, p. 121):
At the end of a sufficiently long period of time, the wave-group can no longer be
considered as moving without deformation, and, eventually, it spreads out further and
further into space with a diminution proportionate to its amplitude. In the usual (linear)
theory of wave propagation, wave trains spread into space with a consequent drop of
amplitude. Linear analysis shows that this process is associated with the fact that linear
theory considers wave trains as superpositions of plane monochromatic waves. These
plane waves are propagated through space independently, whence they are progressively
thrown out of phase and consequently dissipated.
A third difficulty lay in the fact that the extended hill-like character of the wave train provided an
inadequate description of the physical shape of the particle which according to classical theory
should be discrete and well localized. A fourth problem lay in the description of the particle at
low or zero velocity. Since the dimensions of the wave packet, Δx = dλ, must necessarily be
greater than the wave packet wavelength, λp, and since the value of λp in turn depends inversely
on the particle's momentum, see equation (8), then as v goes to 0, λp must approach infinity. As
a result, the location of the particle becomes completely indeterminate.
Moreover since the size of the wave packet and its de Broglie wavelength depended entirely
on the relative velocity of the observer, hence on the particular frame of reference chosen for
observation, the wave packet representation of a particle became highly subjective. This leads to
a fifth problem, which frames a serious paradox. Consider several observers each having an equal
opportunity for observing a remote wave packet (particle) and each bearing a diffraction grating
capable of scattering the wave packet (particle). Also suppose that each observer resides in a
different frame of reference such that the wave packet moves at a different velocity relative to
each. Depending on which observer ultimately ends up being the one to observe the particle, the
particle in advance would be characterized by different particle momenta and hence by different
wave packet wavelengths. In that case, which wave packet are we to say actually represents the
particle? Or conversely, how does the particle know in advance which wave packet it must be
formed as? In the case of Schröedinger's wave mechanical model if a particle were simultaneously
observed from many reference frames, there should exist a multiplicity of mutually contradictory
wave packets simultaneously defining the particle's position (Lande, 1971). In an effort to avoid
this problem, quantum theory has gravitated toward a monist view in which it regards the particle
23
as having no independent existence, its physical state being imminent until the moment of
measurement wherein its properties suddenly become defined in a real sense, the so called
"collapse of the wave function."* However, Dewdney et al. (1985) report the results of
experiments showing that the position of the particle is defined in a real sense prior to its
deBroglie scattering event. They conclude that this experimentally disproves the Copenhagen
interpretation and its wave-packet-collapse concept and favors a causal deterministic model such
as that proposed by deBroglie which theorizes that a subatomic particle and its wave aspect
coexist simultaneously.
These results also support the subquantum kinetics Turing wave model which proposes that
the subatomic particle's Turing wave is uniquely defined by its wavelength λ0 independent of any
act of measurement. By being uniquely characterized in this fashion, the particle is freed from
being defined by any future act of observation. The positivist paradigm, which has maintained an
active enclave in physics, may now come to an end. If a "tree" in the microphysics forest "falls",
and no one is there to observe it, the "tree" nonetheless does "fall". The Copenhagen wave
function has now collapsed, but for good. We can say now with some certainty that the Turing
wave model of subquantum kinetics effectively replaces the indeterminate quantum mechanics
paradigm, being now armed with particle scattering evidence which shows that the subatomic
particle's Turing wave does in fact exist.
This is not to say that there is not an indeterminate aspect to the Turing wave model. By its
nature, the underlying ether reactions are stochastic and the ether concentrations that form the
observable energy potential fields are in a continual state of fluctuation forming a zero-point
energy background. Subquantum kinetics proposes that these background fluctuations are crucial
since, under prevailing supercritical conditions, a critical fluctuation can lead to the creation of a
subatomic particle. This would occur through the morphogenetic process that Prigogine, et al.
(1972) term order-through-fluctuation.
The Turing wave model also avoids the localization problem that plagues Schröedinger's wave
packet. Whether in motion, or at rest relative to an observer, the particle's core would have a
fixed size, its diameter equaling the particle's Compton wavelength. The particle's center would
be defined by the central Gaussian maximum of its Turing electric potential wave pattern.
Moreover the troublesome concept of an entropically dispersing wave packet, may now be
dispensed with and replaced by the idea of the particle's Turing wave, or etheric concentration
pattern, continually maintaining its structural coherence as a result of the nonlinear reactiondiffusion
processes continually taking place in the underlying ether and continuously regenerating
the particle's form (LaViolette 1985b).
It is interesting that de Broglie became dissatisfied with the indeterminate quantum mechanical
description and in later years proposed a revision of his earlier ideas. Like subquantum kinetics,
he proposed the idea of a nonlinear, nondispersing core region as uniquely defining the position
of the particle which he conceived of as a "bunched field singularity", u0, having a radius
approximating the particle's Compton wavelength. But, unlike subquantum kinetics, he did not
elucidate what these nonlinear processes might be. Furthermore he theorized that the core field
potential was oscillating in clock-like fashion, having a characteristic frequency ν equal to the
Compton frequency and giving the subjective appearance to an outside observer of being a
translating wave of wavelength λp which he called the ψ-wave. The character of the ψ-wave
entirely depended on the observer's frame of reference and state of knowledge and had for him no
* Those interested in a general overview of the concept of the wave function collapse and its various conundrums are
referred to the site: https://en.wikipedia.org/wiki/Wavefunction_collapse.
24
physical significance except for its phase. He further theorized that a linear "v-wave"
immediately surrounded and was somehow "fused" to the particle's nonlinear core and shared its
characteristic frequency of oscillation. Further he proposed that this v wave was somehow
guided or piloted by the ψ-wave so that the phases of the v-wave and ψ-wave always matched
with one another. Mediated by the v-wave, the ψ-wave would guide the particle's core, steering
it towards regions of constructive interference in the ψ-wave interference pattern.
Although de Broglie's model had the particle core, u0, of Compton wavelength dimensions,
surrounded by a periodicity, the v-wave, unlike the steady-state Turing wave of subquantum
kinetics, this v-wave was theorized to have a far larger wavelength approximating that of the ψ-
wave, a wavelength that could change from one moment to the next depending on which remote
observer would choose to ultimately diffract the particle. Thus de Broglie's rather complex and
contrived formulation was substantially different from that of subquantum kinetics, and suffered
from many of the same externally determined multiple-reference-frame difficulties that plagued
Schroedinger's wave packet. Moreover, like the wave packet, de Broglie's linear v-wave was
prone to entropic dispersion.
7. Hydrogen Atom Orbital Quantization
Earlier publications also noted that the Turing wave formulation anticipates the atom's
quantized orbital electron states (LaViolette 1985b, 1994). That is, when an electron is in orbit
about a proton, its Turing wave will generate an orbital velocity wave in the proton's reference
frame having a wavelength λv = h/mv, where m is the electron's mass and v is its orbital velocity.
For this orbital oscillation to become self-reinforcing, this velocity wave wavelength must fit a
whole number of times within the circumference, 2\r, of the electron's orbit, leading to the
equivalence, 2\r = nh /mv, where n = 1, 2, 3,... is the principal quantum number. Without a need
for additional assumptions, this directly yields Bohr's orbital quantization formula:
mvr = nh/2\. (9)
Alternatively, this circling may be thought of as an electric potential oscillation that occurs at
every point in the electron's orbit. If we suppose, as before, that the velocity wave travels at the
velocity of light, the frequency, f, it excites in the atom's rest frame will be given as, f = c/λv =
mvc/h. Applying the restriction specified by equation (9), to eliminate mv/h, this becomes:
f = cn/2\r. (10)
When this equality holds, the velocity wave's phase will properly match with that of previous
orbital excursions so as to reinforce this oscillation.
To specify more exactly what orbital radii are allowed, we must take account of the
requirement that the outward centrifugal force due to the electron's orbital motion must exactly
balance the centripetal force of electrostatic attraction. This equality yields the following
velocity-radius relation: v = (ke2/mr)½, where k is the electrostatic constant, e is the electron's
charge, m is the electron's mass, and r is its orbital radius. For the Bohr ground state, n = 1, v
calculates to have the nonrelativistic value of 0.73% c. Using this relation to eliminate v in
relation (9), we get the following allowable orbital radii for the electron:
r = n2h2/4\2ke2m. (11)
In the case of the Bohr ground state (n = 1), the electron's orbit will have a radius of ao =
0.5292 Å, the so called Bohr radius, and its orbital circumference will equal 3.325 Å. This
circumference is such that a single velocity wave wavelength λv makes an exact fit to the orbit. In
the n = 2 orbit, the electron's velocity will be half as much and its velocity wave wavelength will
25
be twice as long. But since r increases according to n2, the n = 2 orbital circumference will be four
times as large, allowing two λv waves to fit within the orbital circumference a whole number of
times. Three velocity wave wavelengths will fit within the n = 3 orbital circumference, and so on.
Electron transitions between these orbital energy states may be treated by means of catastrophe
theory wherein each orbit is viewed as a stable attractor domain to which the system gravitates
unless sufficiently perturbed. For example, through the reception or emission of the proper
amount of energy the electron is able to transition to a new orbital attractor domain.
The Compton wavelength of the electron's Turing wave is far smaller than the orbital
circumference. For the n = 1 Bohr ground state orbit 137.06 electron Compton wavelengths are
able to fit around the orbital circumference. Coincidentally, this comes very close to the fine
structure constant which is estimated to equal 137.036. So we see that with the Turing wave
model, the compactness of the electron relative to its orbit allows a rather classical description of
the electron. Its orbital motion may be understood relatively concretely in terms of a charge
orbiting a proton of opposite charge, viewed in some respects like a planet orbiting a central star.
Such was not possible for Schrödinger's wave packet model. Figuring that the wave packet
would be approximately ten fold larger than its de Broglie phase wave, i.e., ~10λp, its size would
be so large as to envelope the entire atom, the electron wave packet for the n = 1 orbit being
roughly 63 Bohr radii in size. So a classical description was not possible. Indeed, in discussing
the application of his wave mechanics to the Bohr orbits, Schrödinger (1926, p. 1056)
acknowledged that his electron wave packet will "spread out in all directions far over the range of
the orbit". This led him to state (p. 1055):
"...ordinary mechanics will be no longer applicable to such an orbit than geometrical optics
is to the diffraction of light by a disk of diameter equal to the wavelength. ...the conception
of orbits of material points seems to be inapplicable to orbits of atomic dimensions."
Schrödinger is correct in saying that we cannot treat such a situation naively in terms of
ordinary mechanics, for ordinary mechanics would not anticipate the electron's restriction to
certain quantized orbital radii. Nevertheless his stated need of dispensing altogether with
ordinary mechanics is now seen to be unnecessary. Following the Turing wave conception, we
find that the electron core structure is over 8000 times smaller than Schrödinger had supposed.
Hence the classical picture of an orbiting charge is still feasible. It is merely the inherent periodic
character of this charge that requires it be treated specially at atomic dimensions.
We have seen in equation (9) above that due to the electron's production of velocity waves, its
atomic orbit angular momentum, mvr, is restricted to discrete whole number multiples, nh/2\,
where n adopts quantum values of 1, 2, 3,... But, what about quantum states n = 1/2, 1/3, 1/4.
Might fractions of the Bohr orbit angular momentum be allowed as well? If so, this would
correspond to Bohr orbital radii r = n2ao, leading to orbital radii of ao/4, ao/9, ao/16. Mills et al.
(2000) and Eccles (2000, 2005) have suggested that such sub ground energy states do exist and
that transitions to them are triggered through collisional excitations with nearby catalytic ions.
Mills uses the term "hydrino" state to characterize these more tightly bound orbital states in the
hydrogen atom. Catalysts for triggering such transitions in hydrogen atoms include elements
such as potassium, rubidium, and strontium which are capable of collisionally absorbing 27.2 ev
energy quanta from the hydrogen atom to produce an electronic transition in the catalyst atom
which would reradiate this energy in the form of heat or ultraviolet photons.
In one experiment, using a potential of only 2 volts, Mills et al. (2000) were able to generate an
ultraviolet emitting plasma in a hydrogen gas containing strontium as a catalyst. Without the
catalyst, they had to increase their threshold voltage to 250 volts and supply 4000 times more
power to generate the same amount of light output. Also Eccles (2005) reports the emission of
26
308 Å (40.3 ev) ultraviolet photons coming from rubidium catalyst electrolytes, a wavelength
that approximates the 40.8 ev transition between the n =1 and n = 1/2 sub ground state. Both
Mills and Eccles have developed technologies for releasing latent energy present in the hydrogen
atom through this collisional excitation process. Conrads et al. (2003) and Eccles (2000, 2005)
both suggest that this process could offer a new exploitable source of energy. In fact, some
companies state that they will soon have water heaters on the market that will power themselves
using Eccles' patented process.
Quantum mechanics and Schrödinger's wave mechanics both preclude the existence of such sub
ground orbital states, for they characterize the electron by its motion-dependent phase-wave, λp,
the equivalent of the subquantum kinetics velocity wave. Since the phase-wave is unable to fit
into an orbit smaller than the n = 1 Bohr ground state, quantum mechanics views any lower
energy states as being forbidden. However, the Turing wave model is not similarly restricted.
The electron, which in this case is characterized by the much smaller Compton wavelength, can
easily fit orbital circumferences smaller than the Bohr ground state. A fit to the n = 1/2 state is
possible if the electron's velocity wave circulates at c/2, and a fit to the n = 1/3 state is possible if
the velocity wave circulates even slower at c/3. Since these stimulated orbital velocity waves
would circulate at sub light speeds, their energy states would not be accessible via normal photon
emission and absorption processes. But, one might suppose that transitions to them could be
triggered through collisional excitations as research has experimentally demonstrated. Thus in
seeking an explanation for the energy source that will be powering water heaters that will soon be
entering public households, the answer will not be forthcoming from quantum mechanics, but
from subquantum kinetics.
It should be noted that no more than eleven such subground orbits should be allowable on the
basis of this model. That is, orbits having a circumference smaller than the electron's Compton
wavelength, the wavelength of the electron's Turing wave, would be prohibited. The smallest
allowable orbit, the n = 1/11 orbit, would have a circumference equal to 2.75 X 10-12 meters
whereas the Turing wave that would characterize the core electric field of the circulating electron
would have a wavelength slightly smaller, equal to 2.43 X 10-12 meters. Hence there is a limit as
to how much energy might potentially be extracted from electron transitions to such subground
orbits.
8. Conclusion
The subquantum kinetics dissipative structure model has successfully anticipated an
astounding array of details about the distribution of charge and spin in the core of a subatomic
particle, two decades before these features became known through the results of high-energy
particle scattering experiments; see Table 2 for a summary. While the comparisons made here
between the Model G Turing wave predictions for the nucleon and Kelly's charge distribution
models are qualitative, it is difficult to escape the conclusion that the Turing wave model offers
an excellent interpretation of the particle scattering results. Certainly, it offers a more accurate
description of the nucleon than the quark model. This experimental confirmation, together with,
to date, eleven other confirmations of the theory's a priori predictions, gives reason to believe
that quantum physics may greatly benefit from efforts to apply general systems concepts to the
microphysical domain.
As a next step, a more quantitative comparison should be made to form factor data. The
Brusselator-like Model G reaction-diffusion system should be simulated in three dimensions in
the marginally critical supercritical state to produce electric potential dissipative structures
having wavelengths equal to the nucleon Compton wavelength. The simulation for the proton
27
Table 2
Subquantum Kinetics Particle Field Predictions Versus Observation
Characteristic
Subquantum Kinetics
Prediction
Best Model Fit to
Observational Data
Gaussian core Yes Yes
Spherical symmetry Yes Yes
Surrounding electric field
periodicity, amplitude
declines with distance
Yes,
the Turing wave
Yes
Wavelength equals the
Compton wavelength Yes
Somewhat less, due to
probe particle effect
Positive electric field bias
for the proton Yes Yes
Positive potential for the
neutron core Yes Yes
should be such that the resulting structure resides past its secondary bifurcation where the
potential biased charged state emerges. The resulting simulated Turing wave profiles should then
be compared against form factor data. Based on the analysis presented here, it is expected that
the electric potential field simulated for the nucleon should fit form factor data at least as well as
Kelly's Laguerre-Gaussian expansion.
The confirmation of the Turing wave periodicity is particularly significant since this intrinsic
wave can account for the phenomena of particle diffraction, electron orbital quantization, and
nuclear binding, as subquantum kinetics had earlier proposed (LaViolette 1985b, 1994, 2003).
Furthermore we may now reasonably assume that the core electric field potential of all subatomic
particles is similarly characterized by a Compton wavelength Turing wave, not just nucleons, but
leptons and all baryons as well.
It has been shown above that, by effectively describing the phenomenon of particle diffraction
in a paradox-free manner, the Turing wave formulation offers a superior model to the troubleridden
wave packet formulation. Considering that recent particle scattering evidence compels us
to regard the Turing wave as a reality, we may accordingly regard the wave packet alternative as a
fiction that ultimately should be dispensed with. The Schrödinger equation, which has become
the fundamental equation of physics for describing non-relativistic quantum mechanical behavior,
plays a role for microscopic particles analogous to Newton's second law in classical mechanics
for macroscopic particles. Yet in view of the current insecure footing of the wave packet
concept, which is fundamental to the theory, quantum mechanics should no longer be considered
to offer a realistic description of subatomic particle structure but rather be viewed as merely a
heuristic model of subatomic particle interaction. The subquantum kinetics approach, which
provides the basis for a viable replacement, ultimately leads physics back towards a more
concrete, deterministic view of the microphysical world.
Furthermore the Turing wave model has been shown to provide a more classical view of an
electron's orbit in the hydrogen atom and allows for the possible existence of quantized
subground orbital energy states. Technologies being developed to excite electron transitions to
such sub ground states and thereby release the electron's store of orbital potential energy could
provide a revolutionary approach to clean energy production.
28
Finally, subquantum kinetics is able to explain the process of beta decay as being a secondary
bifurcation of the unstable primary branch solution forming the neutron. At this bifurcation, the
neutron abruptly transitions to the stable, charged nucleon state, to form the proton with the
additional creation of electron and anti-neutrino by-products. Thus subquantum kinetics is able
to account for the so called weak force, although subquantum kinetics views it as being due to a
transition of an unstable to a stable nonlinear solution, rather than as being due to any kind of
force. Since subquantum kinetics also accounts for the strong force, the electric field, the gravity
field, and spin, it forms the basis for a true unified field theory.
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30
Paul A. LaViolette received his BA in physics from the Johns Hopkins University, his MBA
from the University of Chicago, and his Ph.D. in systems science (general system theory) from
Portland State University. He is currently president of the Starburst Foundation, where in the past
has conducted interdisciplinary research in physics, astronomy, geology, climatology, systems
theory, solar desalination, aerospace propulsion, psychology, and ancient lore. He has also
authored six books, as well as edited a book of essays by Ludwig von Bertalanffy entitled A
Systems View of Man.
On the Notation of MAXWELL’s Field Equations
01.05.2014 16:06
copyright © (2000) by AW-Verlag, www.aw-verlag.ch Page 1
On the Notation of MAXWELL’s
Field Equations
André Waser* Issued: 28.06.2000
Last revison: -
Maxwell’s equations are the cornerstone in electrodynamics. Despite the fact
that this equations are more than hundred years old, they still are subject to
changes in content or notation. To get an impression over the historical development
of Maxwell’s equations, the equation systems in different notations are
summarized.
Introduction
The complete set of the equations of James Clerk MAXWELL[15] are known in electrodynamics
since 1865. These have been defined for 20 field variables. Later Oliver HEAVISIDE[11] and
William GIBBS[23] have transformed this equations into the today’s most used notation with
vectors. This has not been happened without ‚background noise‘[3], then at that time many
scientists – one of them has been MAXWELL himself – was convinced, that the correct notation
for electrodynamics must be possible with quaternions[5] and not with vectors. A century later
EINSTEIN introduced Special Relativity and since then it was common to summarize MAXWELL’s
equations with four-vectors.
The search at magnetic monopoles has not been coming to an end, since DIRAC[4] introduced
a symmetric formulation of MAXWELL’s equations without using imaginary fields. But
in this case the conclusion from the Special Theory of Relativity, that the magnetic field
originates from relative motion only, can not be hold anymore.
The non-symmetry in MAXWELL’s equations of the today’s vector notation may have disturbed
many scientists intuitively, what could be the reason, that they published an extended
set of equations, which they sometime introduced for different applications. This essay summarizes
the main different notation forms of MAXWELL’s equations.
* André Waser, Birchli 35, CH-8840 Einsiedeln; andre.waser@aw-verlag.ch
Page 2 copyright © (2000) by AW-Verlag; www.aw-verlag.ch
Maxwell’s Equations
The Original Equations
With the knowledge of fluid mechanics MAXWELL[15] has introduced the following eight
equations to the electromagnetic fields (the right equations correspond with the original text,
the left equations correspond with today’s vector notation):
1
1 1
2
2 2
3
3 3
d D
p p J j
dt t
d D
q q J j
dt t t
d D
r r J j
dt t
f
'
g
'
h
'
ü ¶ ü = + ï = + ï ï ¶ ï
ï ¶ ï ¶ = + ý ® = + ý Þ = + ï ¶ ï ¶
ï ¶ ï
= + ï = + ï þ ¶ þ
D
J j (1.1)
3 2
1
1 3
2
2 1
3
dH dG A A
H
dy dz y z
dF dH A A
H
dz dx z x
dG dF A A
H
dx dy x y
ü ¶ ¶ ü ma = - ï m = - ï ï ¶ ¶ ï
ï ¶ ¶ ï
mb = - ý ® m = - ý Þ m =Ñ´
ï ¶ ¶ ï
ï ¶ ¶ ï
mg = - ï m = - ï þ ¶ ¶ þ
H A (1.2)
3 2
1
1 3
2
2 1
3
d d H H
4 p 4 J
dy dz y z
d d H H
4 q 4 J
dz dx z x
d d H H
4 r 4 J
dx dy x y
'
'
'
g b ü ¶ ¶ ü - = p ï - = p ï ï ¶ ¶ ï
a g ï ¶ ¶ ï
- = p ý ® - = p ý Þ Ñ´ =
ï ¶ ¶ ï
b a ï ¶ ¶ ï
- = p ï - = p ï þ ¶ ¶ þ
H J (1.3)
( )
( )
( )
( )
1
1 3 2 2 3
2
2 1 3 3 1
3
3 2 1 1 2
dy dz dF d dA d P E H v H v dt dt dt dx dt dx
dz dx dG d dA d
Q E Hv Hv
dt dt dt dy dt dy
dx dy dH d dA d
R E H v H v
dt dt dt dz dt dz
t
æ ö Y ü jü = mç g -b ÷ - - ï = m - - - ï è ø ï ï
æ ö Y ïï jï = mça - g ÷ - - ý ® = m - - - ý
è ø ï ï
æ ö Yï jï = mçb -a ÷ - - ï = m - - - ïè ø ïþ þ
¶
Þ =m ´ - -Ñj
¶
A
E v H
(1.4)
1 1
2 2
3 3
P k E D
Q k E D
R k E D
f
g
h
= ü e = ü
= ï ® e = ï Þ e = ý ý
= ï e = ï þ þ
E D (1.5)
1 1
2 2
3 3
P p E j
Q q E j
R r E j
= -z ü s = ü
= -z ï ® s = ï Þ s = ý ý
= -z ï s = ï þ þ
E j (1.6)
copyright © (2000) by AW-Verlag, www.aw-verlag.ch Page 3
1 2 3 d d d D D D
e 0 0
dx dy dz x y z
f g h ¶ ¶ ¶
+ + + = ® r+ + + = Þ - r =Ñ
¶ ¶ ¶
gD (1.7)
1 2 3de dp dq dr j j j
0 0
dt dx dy dz t x y z t
¶r ¶ ¶ ¶ ¶r
+ + + = ® + + + = Þ - =Ñ
¶ ¶ ¶ ¶ ¶
gj (1.8)
This original equations do not strictly correspond to today’s vector equations. The original
equations, for example, contains the vector potential A, which today usually is eliminated.
Three Maxwell equations can be found quickly in the original set, together with OHM’s
law (1.6), the FARADAY-force (1.4) and the continuity equation (1.8) for a region containing
charges.
The Original Quaternion Form of Maxwell‘s Equations
In his Treatise[16] of 1873 MAXWELL has already modified his original equations of 1865. In
addition Maxwell tried to introduce the quaternion notation by writing down his results also
in a quaternion form. However, he has never really calculated with quaternions but only uses
either the scalar or the vector part of a quaternion in his equations.
A general quaternion has a scalar (real) and a vector (imaginary) part. In the example below
‚a‘ is the scalar part and ‘ib + jc + kd’ is the vector part.
Q = a + ib + jc + kd
Here a, b, c and d are real numbers and i, j, k are the so-called HAMILTON‘ian[7] unit vectors
with the magnitude of Ö-1. They fulfill the equations
i2 = j2 = k2 = ijk = -1
and
ij = k jk = i ki = j
ij = - ji jk = - kj ki = - ik
A nice presentation about the rotation capabilities of the HAMILTON’ian unit vectors in a
three-dimensional ARGAND diagram was published by GOUGH[6].
Now MAXWELL has defined the field vectors (for example B = B1i + B2j + B3k) as quaternions
without scalar part and scalars as quaternions without vector part. In addition he
defined a quaternion operator without scalar part
1 2 3
d d d
dx dx dx
Ñ = i+ j+ k ,
which he used in his equations. Maxwell devided a single quaternion with two prefixes into a
scalar and vector. This prefixes he defined according to
S.Q = S.(a + ib + jc + kd) = a
V.Q = V.(a + ib + jc + kd) = ib + jc + kd
The original Maxwell quaternion equations are now for isotrope media (no changes except
fonts, normal letter = scalar, capital letter = quaternion without scalar):
Page 4 copyright © (2000) by AW-Verlag; www.aw-verlag.ch
B = V.ÑA (1.9)
E = V.vB- A& -ÑY (1.10)
F = V.vB+ eE -mÑW (1.11)
B = H + 4pM (1.12)
tot 4pJ = V.ÑH (1.13)
J = CE (1.14)
1
D KE
4
=
p
(1.15)
tot J= J + D& (1.16)
B = mH (1.17)
e = S.ÑD (1.18)
m = S.ÑM (1.19)
H= -ÑW (1.20)
Beneath the new notation, the magnetic potential field W and the magnetic mass m was
mentioned here the first time. By calculating the gradient of this magnetic potential field it is
possible to get the magnetic field (or in analogy the magnetostatic field. Maxwell has introduced
this two new field variables into the force equation (1.11).
The reader may check that the equations above identical to the previous published equations
(1.1) bis (1.7), except the continuity equation (1.8) has this time be dropped. From the
above notation it is clearly visible why the quaternion despite the deep engagement for example
of Professor Peter Guthrie TAIT[19] did not succeed, then the new introduced vector notation
of Oliver HEAVISIDE[11] and Josiah Willard GIBBS[23] was much easier to read and to use
for most applications.
It is very interesting that Maxwell‘s first formulation of a magnetic charge density and the
related discussion about the possible existence of magnetic monopoles became forgotten for
more than half a century until in 1931 Paul André Maurice DIRAC[4] again speculated about
magnetic monopoles.
copyright © (2000) by AW-Verlag, www.aw-verlag.ch Page 5
Today‘s Vector Notation of MAXWELL‘s Equations
The nowadays most often used notation can be easily derived from the original equations of
1865. By inserting (1.1) in (1.3) it follows the known equation
t
¶
Ñ´ = +
¶
D
H j (1.21)
Equation (1.4) contains the FARADAY equation
E = v´mH ¾m¾=ko¾nsta¾nt® E = m( v´H) (1.22)
and the potential equation for the electric field
t
¶
=- -Ñj
¶
A
E . (1.23)
Together with the potential equation for the magnetic field (1.2) follows with applying the
rotation on both sides of (1.23)
( ) ( ) konstant
t t t
m= ¶ ¶ ¶
Ñ´ =- Ñ´ = m ¾¾¾¾® Ñ´ =m
¶ ¶ ¶
H
E A H E (1.24)
From (1.2) follows further with the divergence:
ÑgmH = 0 ¾m¾=ko¾nsta¾nt® mÑgH = 0 (1.25)
The six MAXWELL equations in today‘s notation are:
FARADAY‘s law
t
¶
Ñ´ = +
¶
D
H j (1.26)
AMPÈRE‘s law
t
¶
-Ñ´ =
¶
B
E (1.27)
COULOMB‘s law ÑgD = -r (1.28)
ÑgB = 0 (1.29)
0 0 r D = e E+ P = e e E = eE (1.30)
0 0 r B = m H +M = m m H = mH (1.31)
with
E: electrical field strength [V / m]
H: magnetic field strength [A / m]
D: electric displacement [As / m2]
B: magnetic Induction [Vs / m2]
j: electric current density [A / m2]
e: electric permeability [As / Vm]
m: magnetic permeability [Vs / Am]
Please note that the MAXWELL equation of today have became subset of the original equations
which in turn have got an expansion with the introduction of the magnetic induction (1.31).
Page 6 copyright © (2000) by AW-Verlag; www.aw-verlag.ch
Today traditionally not included in MAXWELL‘s equations are FARADAY‘s law and sometime
also OHM‘s law. Seldom the continuity equation (1.8) is even mentioned. But this
equation defines the conservation of charge:
( ) ( ) 0 0
t t t t
¶ ¶r ¶r ¶r
Ñ Ñ´ = Ñ +Ñ =- - = Þ =
¶ ¶ ¶ ¶
g H gD gj (1.32)
The electric and magnetic field strengths are interpreted as a physically existent force fields,
which are able to describe forces between electric and magnetic poles. Maxwell has – analogue
to fluid mechanics – this force fields associated with two underlying potential fields,
which are not shown anymore in the today‘s traditional vector notation. The force fields can
be derived from the potential fields as:
t
¶
- = Ñj+
¶
A
E (1.33)
B =Ñ´A (1.34)
with
j: electric potential field [V]
A: vector potential [Vs / m]
For a very long time scientists are convinced that the potentials do not have any physical
existence but merely are a mathematical construct. But an experiment sugested by Yakir
AHARONOV and David BOHM[1] has shown, that this is not true. There arises the question
about the causality of the fields. Many reasons point out that the potentials j and A really are
the cause of the force fields E and H.
Including the material equations (1.30) and (1.31) and with consideration of Ohm‘s law
j = sE (1.35)
with
s: specific electric conductivity [1 / Wm] = [A / Vm]
the Maxwell equations become for homogenous and isotrope conditions (e = constant,
m = constant):
t
¶
Ñ´ =e +s
¶
E
H E (1.36)
t
¶
-Ñ´ =m
¶
H
E (1.37)
eÑgE = r (1.38)
mÑgH = 0 (1.39)
copyright © (2000) by AW-Verlag, www.aw-verlag.ch Page 7
Real Expansions of Maxwell‘s Equations
The HERTZ-Ansatz
Recently Thomas PHIPPS[20] has shown that Heinrich Rudolf HERTZ has suggested another
possibility to adapt Maxwell‘s equations. During Hertz life this was hardly criticized and his
proposal was vastly forgotten after his death. Usually the differentials are partial derivative
and not total derivatives as shown in the comparison (1.1) to (1.8) between MAXWELL‘s
original equations and the today‘s vector notation. Now in the equations (1.26) and (1.27)
HERTZ has substituted the partial derivatives ¶ with the total derivatives d. With this the
Maxwell equations become invariant to the GALILEI-transformation:
d
dt
Ñ´ = +
D
H j (1.40)
d
dt
-Ñ´ =
B
E (1.41)
what wit the entity
d
dt t
¶
= + Ñ
¶
vg becomes
t
¶
Ñ´ = + Ñ +
¶
D
H vg D j (1.42)
t
¶
-Ñ´ = + Ñ
¶
B
E vg B (1.43)
Now the question arises about the meaning of the newly introduced velocity v. HERTZ has
interpreted this velocity as the (absolute) motion of aether elements. But if v is interpreted as
relative velocity between charges, then Maxwell‘s equations are defined for the case v = 0, hat
can be interpreted that the test charge does not move in the observer‘s reference frame.
Therefore Thomas PHIPPS explains this velocity as the velocity of a test charge relative to an
observer.
Consequently in equation (1.33) the partial derivatives has to be replaced wit the total derivatives,
too.
d
dt t
¶
- = Ñj+ = Ñj+ + Ñ
¶
A A
E vg A (1.44)
The invariance of (1.40) and (1.41) against a GALILEI-transformation for the case that no
current densities j and no charges are present can easily be seen. For v = 0 (a relative to the
observer stationary charge) always MAXWELL‘s equations will be the result:
t
¶
Ñ´ =
¶
D
H (1.45)
t
¶
-Ñ´ =
¶
B
E (1.46)
For a GALILEI-transformation is r‘ = r – vt and t‘ = t; thus for v > 0 is:
x x y y z z
'
' '
¶ ¶ ¶ ¶ ¶ ¶
= = = ® Ñ =Ñ
¶ ¢ ¶ ¶ ¶ ¶ ¶
and
t' t
¶ ¶
= + Ñ
¶ ¶
vg
Page 8 copyright © (2000) by AW-Verlag; www.aw-verlag.ch
from which for all v the equations
d
dt
Ñ´ =
D
H (1.47)
d
dt
-Ñ´ =
B
E
are valid. If the observer moves together with a test charge, this reduces again to the equations
(1.45) and (1.46). The first EINSTEIN postulate[5], that in an uniform moving system all
physical laws take its simplest form independent of the velocity, is in the example above
fulfilled. In each uniform moving reference frame the observer always measures for example
the undamped wave equation.
The DIRAC-Ansatz
The non-symmetry in MAXWELL‘s equation system always has motivated to extend this set of
equations. The most famous extension has originated form DIRAC [3], who suggested the
following extension:
e t
¶
Ñ´ =e +
¶
E
H j (1.48)
m t
¶
-Ñ´ =m +
¶
H
E j (1.49)
eÑgE = re (1.50)
m mÑgH = r (1.51)
Together with (1.51) this ansatz must lead to the postulation of magnetic monopoles, which
until today never has been (absolutely certain) detected. As a consequence of this ansatz the
force fields E and B are derived from potentials according to:
t
¶
= Ñj- -Ñ´
¶
A
E C (1.52)
t
¶
= Ñf- -Ñ´
¶
C
E A (1.53)
where f and C represent the complementary magnetic potentials. Therefore as another
consequence there must exist two different kinds of photons, which interact in different ways
with matter[14]. Also this has until today never been observed.
The HARMUTH-Ansatz
Henning HARMUTH[5] and Konstantin MEYL[17] have gone a step further and suggested new
equations, which differ to the DIRAC ansatz only in that point, that no source fields exists
anymore. Harmuth has used this proposition to solve the problem of propagation of electromagnetic
impulses in lossy media (impulses in media with low OHM dissipation) for the
boundary conditions E = 0 and H = 0 for t £ 0:
copyright © (2000) by AW-Verlag, www.aw-verlag.ch Page 9
t
¶
Ñ´ =e +s
¶
E
H E (1.54)
s
t
¶
-Ñ´ =m +
¶
H
E H (1.55)
eÑgE = 0 (1.56)
mÑgH = 0 (1.57)
with
s: specific magnetic conductivity [V / Am]
In the interpretation of this ansatz MEYL has gone again a step further and declares the
equations (1.54) to (1.57) to be valid in all cases, what says, that there exist no kind of monopoles,
whether electric nor magnetic. The alleged electric monopoles (charges) are then only
secondary effects of electric and magnetic fields.
From (1.54) to (1.57) HARMUTH[5]-GL.21 has derived the electric field equation to
( )
2
2 s s 0
t t
¶ ¶
D -me - ms+e - s =
¶ ¶
E E
E E (1.58)
and has shown, that this equation can be solved for a certain set of boundary conditions. The
same equation (1.58) is designated by Meyl as the fundamental field equation.
The MÚNERA-GUZMÁN-Ansatz
Héctor MÚNERA and Octavio GUZMÁN[19] have proposed the following equations (w º ct):
4
c
¶ p
Ñ´ = +
¶w
P
N J (1.59)
4
c
¶ p
Ñ´ =- +
¶w
N
P J (1.60)
ÑgN = 4pr (1.61)
ÑgP = -4pr (1.62)
with
N º B-E (1.63)
P º B + E (1.64)
From this follows MAXWELL‘s equations (1.26)-(1.29) as shown below:
FARADAY‘s law (1.26): (1.60) - (1.59) (1.65)
AMPÈRE‘s law (1.27): (1.60) + (1.59) (1.66)
COULOMB‘s law (1.28): (1.62) - (1.61) (1.67)
(1.62) + (1.61) (1.68)
In this notation the current density J and the charge density r are not understood as electric
only but merely as electromagnetic entities. With an analysis of MÚNERA and GUZMÁN it can
be shown, that beneath the electric scalar field also a non-trivial magnetic scalar field should
exist.
Page 10 copyright © (2000) by AW-Verlag; www.aw-verlag.ch
Imaginary Expansions of Maxwell‘s Equations
The Notation in Minkowski-Space
In electrodynamics the relativistic notation is fully established. Because of the second
EINSTEIN‘ian postulate[5] about the absolute constancy of the speed of light (therefore its
independency of the speed of the light source or light detector) the four-dimensional notation
has been developed. But the force field vectors E and H can not be used for four-vectors. But
the potentials and the charge densities have been regarded as very optimal to formulate the
electrodynamics in a compact form. If we first have an event vector
X = ict + x
then it follows in MINKOWSKI-Space the invariance of the four-dimensional length ds
2 2
1 1 2 2 3 3 ds dx dx dx dx dx dx dx dx c dt m m = = + + -
and
2 ( )
2 1 1 2 2 3 3
i 1
d dx dx dt dx dx dx dx dx dx
c c m m t = = - + +
From this follows the four-dimensional velocity vector to
( )
2
2
d 1
ic
d u
1
c
= = +
t
-
u
X
U
which gives the four-dimensional current density
0 ( )
0 2
2
ic
u
1
c
r
=r = +
-
J U u
With the four-dimensional gradient operator
4
1 2 3 ic t x x x
¶ ¶ ¶ ¶
Ñ= + + +
¶ ¶ ¶ ¶
i j k
follows with
4ÑgJ = 0
the continuity equation (1.8). With the four-dimensional vector potential
A = ij+ A
and with the D‘ALEMBERT operator
( ) 2
4 2 2 2
2 2 2 2
1 1
c t x x c t n n
¶ ¶ ¶
Ñ = =Ñ- = -
¶ ¶ ¶ ¶ W
follows the relation
2 1
c
W A = J
copyright © (2000) by AW-Verlag, www.aw-verlag.ch Page 11
But then the possibility for a compact and easy calculation within the MINKOWSKI-Space
comes to an end. To include the electric and magnetic fields, the following definition
A A
F
x x
n m
mn
m n
¶ ¶
º -
¶ ¶
is used to determine the electromagnetic field tensor:
3 2 1
3 1 2
2 1 3
1 2 3
0 B B iE
B 0 B iE
B B 0 iE
iE iE iE 0
ì - - ü
ï- - ï ï ï =í
- - ý ï ï
îï ïþ
F
With two equations with the components of the field tensor the four MAXWELL equations can
be derived. With the first equation
F F F
0
x x x
lm mn nl
n l m
¶ ¶ ¶
+ + =
¶ ¶ ¶
follows for an arbitrary combination of l, m, n to 1, 2, 3 the MAXWELL equation (1.29)
12 23 31 3 1 2
3 1 2 3 1 2
F F F B B B
0 0
x x x x x x
¶ ¶ ¶ ¶ ¶ ¶
+ + = ® + + =Ñ =
¶ ¶ ¶ ¶ ¶ ¶
gB
and if one of the indices l, m, n is equal 4 it follows the MAXWELL equation (1.27). With the
second equation
F 1
x c
mn
m
n
¶
=
¶
J
follow the non-homogenous MAXWELL equations (1.26) and (1.28).
Simple Complex Notation
One possibility to enhance the symmetry of MAXWELL‘s equations offers the inclusion of
imaginary numbers. INOMATA[13] uses the imaginary axis only for the „missing“ terms in
MAXWELL‘s equations. Thus they become:
m m
t
i i
t
¶
= e Ñ = r Ñ´ = +
¶
¶
= m Ñ =r -Ñ´ = +
¶
D
D E D H j
B
B H B E j
g
g
(1.69)
From this result an imaginary magnetic charge and an imaginary magnetic current density. In
this notation the imaginary unit i is used for variables, which are not physically existent (i.e.
are not measurable until now). Thus by using „i“ in the equations above the missing variables
are placed into an imaginary (non existent) space i(x1, x2, x3).
Page 12 copyright © (2000) by AW-Verlag; www.aw-verlag.ch
Eight-dimensional, Complex Notation
Elizabeth RAUSCHER[19] proposes a consequent expansion of the complex notation, so that for
each field and for each charge density a real and an imaginary part is introduced.
i
i
Re Im
Re Im
Re Im e m
Re Im e m
= +
= +
= + = +
r =r +r =r +r
E E E
B B B
j j j j j
(1.70)
Then, when using a correct splitting of the terms, two complementary sets of MAXWELL
equations can be formulated. The real equations are:
t
0
t
Re
Re Re Re Re Re Re
Re
Re Re Re Re
¶
= e Ñ = r Ñ´ = +
¶
¶
= m Ñ = -Ñ´ =
¶
D
D E D H j
B
B H B E
g
g
(1.71)
With an elimination of i on both sides we get for the imaginary parts:
0
t
Im t
Im
Im Im Im Im
Im
Im Im Im Im Im
¶
= e Ñ = Ñ´ =
¶
¶
= m Ñ =r -Ñ´ = +
¶
D
D E D H
B
B H B E j
g
g
(1.72)
As used by INOMATA also RAUSCHER uses the imaginary unit „i“ to sort the physical existent
variables from the physical non existent ones.
The Imaginary Quaternion Notation
An other possibility is the mixture of quaternions and imaginary numbers, what has for
example be done by HONIG[12]. With the vector potential and the current density
4 x y z
4 x y z
i A A A
i v v v
i j k
i j k
= j+ + +
= r+r +r +r
A
J
(1.73)
follows with the operator
q i
t x x x
i j k
¶ ¶ ¶ ¶
= + + +
¶ ¶ ¶ ¶ W (1.74)
and with the LORENTZ condition q 4 W gA = 0 the MAXWELL equations with
q
2
4 3 4 i i i i
t t
æ ¶ ¶ ö = Ñ +Ñ +Ñ´ +Ñ´ + ç- + ÷ = r + = è ¶ ¶ ø
E B
W A g E gB E B J J (1.75)
Actually this notation is very efficient. It is, for example, easily possible to formulate the
LORENTZ force or equations of the quantum electrodynamics with this notation. But now the
imaginary unit „i“ is not used to separate the observable variables from the non existent ones.
Interestingly there does not exists one single real number at all. Each real number is associated
either with the imaginary unit „i“ or with the HAMILTON units i, j and k. With some
additional rules also this notation can be expanded to eight dimensions. This should be
presented in another paper.
copyright © (2000) by AW-Verlag, www.aw-verlag.ch Page 13
Closing Remarks
Different notations to the MAXWELL equations are presented. Depending on the application
one or another notation can be very useful, but at the end the presented variety is not satisfactory.
This variety can be a hint, that the correct final form has not been found until now.
Many discussions have been presented about the existence of magnetic monopoles. But
either the electric field is only a subjective measuring caused by the relative motion between
charges -–as it is said by the Special Theory of Relativity – or the magnetic force field can be
derived from a scalar potential field. In the first case magnetic monopoles can not exist, in the
second case they can exist. Despite of extensive experiments no magnetic monopoles have
been found until now. So we can conclude, that no magnetic potential fields must be postulated
and that the non symmetry in MAXWELL‘s equations still are correct.
Proposals to enhance the symmetry with imaginary numbers are interesting but covers the
danger, that with the simple mathematical tool „i“ a symmetric formulation can be reached
vastly, but that the physical models do become nebulous.
References
[1] AHARONOV Yakir & David BOHM, „Significance of Electromagnetic Potentials in the Quantum
Theory”, Physical Review 115 /3 (01 August 1959)
[2] BARRETT Terence W., “Comments on the HARMUTH ansatz: Use of a magnetic current density in
the calculation of the propagation velocity of signals by amended Maxwell theory”, IEEE Trans.
Electromagn. Compatibility EMC–30 (1988) 419–420
[3] BORK Alfred M., “Vectors Versus Quaternions – The Letters of Nature“, American Journal of
Physics 34 (1966) 202-211
[4] DIRAC Paul André Morice, „Quantised Singularities in the Electromagnetic Field“, Proceedings
of the London Royal Society A 133 (1931) 60-72
[5] EINSTEIN Albert, „Zur Elektrodynamik bewegter Körper“, Annalen der Physik und Chemie 17
(30. Juni 1905) 891-921
[6] GOUGH W., „Quaternions and spherical harmonics”, European Journal of Physics 5 (1984) 163-
171
[7] HAMILTON William Rowan, “On a new Species of Imaginary Quantities connected with a theory
of Quaternions“, Proceedings of the Royal Irish Academy 2 (13 November 1843) 424-434
[8] HARMUTH Henning F. “Corrections of Maxwell’s equations for signals I,”, IEEE Transactions of
Electromagnetic Compatibility EMC-28 (1986) 250-258
[9] HARMUTH Henning F. “Corrections of Maxwell’s equations for signals II”, IEEE Transactions of
Electromagnetic Compatibility EMC-28 (1986) 259-266
[10] HARMUTH Henning F. “Reply to T.W. Barrett’s ‘Comments on the Harmuth ansatz: Use of a
magnetic current density in the calculation of the propagation velocity of signals by amended
Maxwell theory’“,IEEE Transactions of Electromagnetic Compatibility EMC-30 (1988) 420-
421
[11] HEAVISIDE Oliver, „On the Forces, Stresses and Fluxes of Energy in the Electromagnetic Field“,
Philosophical Transactions of the Royal Society 183A (1892) 423
[12] HONIG William M., “Quaternionic Electromagnetic Wave Equation and a Dual Charge-Filled
Space“, Lettere al Nuovo Cimento, Ser. 2 19 /4 (28 Maggio 1977) 137-140
[13] INOMATA Shiuji, „Paradigm of New Science – Principa for the 21st Century”, Gijutsu Shuppan
Pub. Co. Ltd. Tokyo (1987)
Page 14 copyright © (2000) by AW-Verlag; www.aw-verlag.ch
[14] KÜHNE Rainer W., „A Model of Magnetic Monopoles“, Modern Physics Letters A 12 /40 (1997)
3153-3159
[15] MAXWELL James Clerk, „A Dynamical Theory of the Electromagnetic Field”, Royal Society
Transactions 155 (1865) 459–512
[16] MAXWELL James Clerk, „A Treatise on Electricity & Magnetism“, (1873) Dover Publications,
New York ISBN 0-486-60636-8 (Vol. 1) & 0-486-60637-6 (Vol. 2)
[17] MEYL Konstantin, „Potentialwirbel“, Indel Verlag, Villingen-Schwenningen Band 1 ISBN 3-
9802542-1-6 (1990)
[18] MEYL Konstantin, „Potentialwirbel“, Indel Verlag, Villingen-Schwenningen Band 2 ISBN 3-
9802542-2-4 (1992)
[19] MÚNERA Héctor A. and Octavio GUZMÁ, „A Symmetric Formulation of MAXWELL’s Equations”,
Modern Physics Letters A 12 No.28 (1997) 2089-2101
[20] PHIPPS Thomas E. Jr, “On Hertz’s Invariant Form of Maxwell’s Equations”, Physics Essays 6 /2
(1993) 249-256
[21] RAUSCHER Elizabeth A., „Electromagnetic Phenomena in Complex Geometries and Nonlinear
Phenomena, Non-HERTZian Waves and Magnetic Monopoles”, Tesla Book Company, Chula
Vista CA-91912
[22] TAIT Peter Guthrie, “An elementary Treatise on Quaternions”, Oxford University Press 1st
Edition (1875)
[23] WILSON E. B., “Vector Analysis of Josiah Willard Gibbs – The History of a Great Mind”,
Charles Scribner’s Sons New York (1901)
The Heim Theory: Key to the Information Fields
01.05.2014 14:23
Exploring the Information Fields
Tools of Awareness for Successful Leadership and Management
Willy De Maeyer, PhD – Research Scientist
Gabriele Breyer – Life Coach
www.matrix-informational.com
Basel, Switzerland – May 2011
PAPER 1
The Heim Theory:
Key to the Information Fields
Leadership and innovation are the most powerful forces for driving success in any business. The main issue is: How to leverage management through these powerful forces and how to create sustainable magic in your business? We will share with you our experience in how to explore the information fields, the importance of fractals, some “time” considerations as well as the important role of global scaling.
In 14 different papers we will convey to you the knowledge and techniques of how to leverage management skills and how to transform your business into a company with continual drive and sustainable growth. The tools of awareness presented in these papers will probably be new to most of the readers; therefore we suggest NOT skipping a part before passing on to another one.
The 14 papers will guide you through a portfolio of new concepts combined with leadership and innovation skills. Take this opportunity to learn about and to
| 2
familiarize with information fields and the scientific background of how to communicate with them. We explain very clearly how and where on each management’s or entrepreneur’s level these new tools can be successfully applied.
In order to avoid questions such as “how is this possible?” or any other uncertainties please read carefully all papers one after the other.
Unknown to most people is the Heim theory, a collection of breathtaking ideas about the fundamental laws of physics proposed by the German scientist Burkhard Heim in the 1980’s. This valuable and remarkable theory was further refined and developed by Walter Dröscher and Jochem Häuser. The theory is a perfect unification of quantum theory and general relativity.
Heim became disabled after an explosive handling accident during the Second World War, which left him without hands and mostly deaf and blind. Apart from the geniality, the theory has two important drawbacks: its complexity and the fact that hardly anything was ever published in another language than German.
As Heim could not write and see, he dictated his work to his father (and later to his wife), who during many years worked day and night with him in order to publish this valuable theory. His severe handicaps led him to prefer isolation to a confined environment in order to avoid the stress of communication with the media and the outside world.
Burkhard Heim
The mathematics behind Heim’s theory requires the extension of space-time with extra dimensions. Various formulations involve six, eight or twelve dimensions. Elementary particles are represented as “hermetry forms” or multidimensional structures of space. In the Heim theory particle masses are yielded directly from fundamental physical constants. Heim’s pioneering work of the 12 dimensional structures of the universe is a first glance at how information fields can be defined and situated.
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Burkhard Heim was born in 1925 in Potsdam, Germany. As a young boy he was very interested in biology and explosive chemistry. We will use this as a first introduction to the background of information fields!
During one of the Potsdam winters a friend of Burkhard Heim studied the ice flowers that were formed on the (non-isolated, single) window glasses. Ice flowers are the structures of ice which form on single-layer glass windows during frost. Warm humid air on one side of the window pane will cool and sublimate ice crystals, which grow into flower-like structures. Since the surface of windows is usually flat and clean, the sublimation process goes slow, and the resulting ice structures can be quite beautiful.
He showed Burkhard that these ice flowers had different forms, very similar to the shape of the plants present in the room. They both studied these shapes, and came to the conclusion that indeed, they corresponded to those of the plants located in the room! They started an investigation in the neighborhood, and always came to the same conclusion: the shapes and forms of the plants present in a room were projected in the ice flowers of its windows! In other words: for one or another “strange” reason, the forms and shapes of plants were projected on window glass, and were materialized by means of the ice.
Although this was not the reason for Heim’s remarkable contributions, it is a valuable start for us to our exploration of information fields, especially as the Heim theory will be the key in understanding and working with these fields. But first, let’s introduce you further to this remarkable person and his ways of thinking that brought him to his theory of the 12 dimensions.
As we will mention it several times more in these papers, the dominance of modern materialism is due in large part to its association with the remarkable theoretical and practical power of classical physics as developed by Newton and his successors. According to this model, reality consists of a fixed and passive space containing localized material particles whose movement in time is deterministically governed by mathematical laws. Consequently, mental phenomena, in this picture, are nothing more than the complex functions of the material brain governed by physical law.
Luckily this narrow point of view was not the only one. Alternatives however had a bad time against materialism! Thanks to the developments leading to quantum physics the pure Newtonian, materialistic points of view needed to be revised in favor of the alternatives.
After the devastating blast in 1944, Heim tried to understand the connection and interactions between mental and material processes in our world. Heim realized that before doing this, he first needed to develop a complete theory of the surrounding material world. Therefore, he decided to become a physicist. Despite his severe handicaps, Burkhard Heim started to study chemistry in 1945 and 4 years later theoretical physics in Göttingen.
In 1950, at the age of 25 years, he married his guardian angel and wife Gerda who assisted him in his scientific work until his death. In 1954, Heim finishes his study in physics and worked during 2 years together with Carl Friedrich von Weizsäcker at the well-known Max Planck Institute in Göttingen. In 1958, he founded his own institute in Northeim where he tested his predictions regarding his gravitational theory.
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Burkhard and Gerda Heim in 1988
During his studies and research work, he aimed at describing all existing physical fields and their origins as dynamic properties of geometrical structures present in space. Between 1949 and 1959, he worked very hard on his unified field theory and published in 1959 his first article “Principle of Dynamic Contrabarie” in the German magazine “Flugkörper” where he presented the results of his research. In 1953, he was already able to represent his unified Hermitian field theory in a six-dimensional space (R6)!
Heim developed a mathematical approach based on quantizing space-time itself, and proposed the "metron" as a (two-dimensional) quantum of (multidimensional) space. Part of the theory is formulated in terms of difference operators; Heim called the mathematical formalism "selector calculus”. In developing his study of complex structures in the world that develop their own regularities, he visualized this during transition of geometry of metrons into geometry of syntrix. Then he developed his own mathematics in order to be able to make reliable calculations to higher dimensions. Stunning is the fact that Heim “invents” words to describe situations and theories never heard of before.
Heim found out that structure-giving processes in the material world are obviously controlled by non-material highly complex structures in a coordinate space (X5 and X6). They seem to exist as dynamically interchangeable “structural forms”. In other words dynamic structural signs, forms and shapes that exist both non-polar and non-local but that can always function at any moment anywhere in the Universe.
These structural forms are dynamic and in evolution. He understood the core of a human being as having a structure in X5 and X6 possessing a rather high complexity. Heim wondered what would happen with this structure by death or accident, when the material body disintegrates! Would in this case the core survive? How would it be able to contact a new developing biological body? Heim examined this question in
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great detail and studied which characteristics material structures must possess so that such unifications could take place. It is at this level of development that he discusses in great detail carbon chemistry as well as DNA.
In his publication “Postmortale Zustände” (Post-mortal Conditions), in 1980, he tackles these questions in 122 pages. However, this publication (just as all the others) is extremely difficult to understand, because he constantly refers to his syntrometric method which one need to understand (but anyone hardly knows it!) in order to even acknowledge his conclusions.
During his research on life processes, Heim noticed that new “forms” occurred at the same time within different animal species (for example the appearance of wings). These modifications as natural mutations were not explainable in a stochastic way. It seems that the mutation probabilities are somehow controlled in such a way that development takes priority into account. A mathematical estimation of natural mutation rates could be calculated by Heim!
How could such a controlling of the material world take place without violating the existing physical properties? Heim also wanted to find out where those structural forms in structured space came from.
And now it’s getting really interesting! If we consider X1 – X6 to be the material coordinates, than we also need to take into account the non-material dimensions X7 –X12. These non-material coordinates can be structured into two subspaces. If we assume X7 and X8 describing a space of ideas I2, there must be an imaging process from this space of ideas to the structure space S2. Heim could mathematically describe these imaging processes. The remaining coordinates X9 – X12 were probably the resident sources of what happened in X7 and X8. As there was not that much to say about the internal structure of X9 – X12 and its effects on the material world, Heim called this space G4 which stood for “Gott alleine bekannt” (God only knows).
A simple logical analysis shows us that:
Organization = materialized information
However, the description of the imaging processes from G4 was not possible. Heim had the idea of describing this by means of multidimensional Fourier transformations, but it was not clear how to handle these tools. Dröscher solved the problem and the description became possible.
In the 1980’s, Heim could prove the existence of a hyperspace with non-material dimensions (X7 – X12). This space is not a constituent of physical dimension, but it can affect the physical world! Further he gave an interpretation of the Cartesian coordinates:
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Space R3 (X1-X2-X3)
Time T (X4)
Structure S2 (X5-X6) Structure = manifested information
------------------------------------------------
Information I2 (X7-X8)
??? G4 (X9-X10-X11-X12) Origin of information
Along with a qualitative understanding of a possible control from G4:
G4
I2
Information
S2
Structure
T
Time
R3
Space
And finally, the deviation of the existence of the laws of quantum mechanics.
In 1996, Dröscher and Heim published the third part of what they called “Elementary Structures”. This part handles essentially about structures of the physical world and its non-material part.
It is a well-known fact that by the end of WW 2 (and just after), East and West both tried to get hold of German know-how. For more details we refer to the book “Reich of the Black Sun”. However, this “hunt” for German scientists was not just limited to that period! In the 1950’s, there have been many attempts to kidnap German researchers who would be operating from abroad. As Helmut Goeckel wrote:” Heim could write an evening filling detective story about his experiences regarding fraud, theft and even attempts of kidnapping!
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To distract the growing attention from the military industry, Heim turned towards metaphysical research especially in the field of instrumental communication with professor Bender. He defined paranormal phenomena as being transitions between the physical and psychological worlds.
According to Heim, the four dimensional (visible) world represents only the lower level of the Universe. X5 and X6 are the energetic stimulation fields and X7 and X8 can be designated as being the information fields. The communication between the three dimensions we live in with the organizational fields X5 and X6 as well as the information fields X7 and X8 happens via X4 (Time). Therefore time is an active force through which exchanges with higher fields (dimensions) happen.
The information levels as explained by Burkhard Heim can be considered as huge storing spaces, full of possibilities amongst those ones can be chosen that are necessary. The materialization of possibilities depends on the degree of necessity. X5 and X6 are considered as being the organizational level or also called the transition levels.
They are absolutely necessary, because in the information levels X7 and X8, there is no time and space or in other words there is non-locality and non-polarity. So, to be able to understand and work with the information in our three dimensions, they must be transformed into a local and polar structure. This is the reason that X5 and X6 are called structural levels.
The Heim theory shows us a world of organizational structures that are in permanent actualization. The complete information pattern of any creature or process can be found back in the higher information fields X7 and X8.
Through time (X4), the higher fields act on our material world we live in. In further papers we will discuss the findings of Nikolai Kozyrev, but we can already at this stage mention that Kozyrev defined time as a force that turns cause into effect, in other words time is a force whose patterns contain information about every process in this world. His research attributes to time a key role in the eventual communication between information – space – matter. The influence of time turns information into a material process and visa-versa. Every material process generates a pattern in time and Kozyrev claims that this pattern is accessible everywhere in the universe, independent of space and time.
Kozyrev also mentions processes of own vibration in space and vacuum that contain all levels (including time). Burkhard Heim considers these interactions as an exchange of photons. In other words, photons are to be considered as information carriers. It is photons that bring all necessary information into our three dimensions (via time of course). Photons are everywhere and can be focused through the use of crystals or any kind of psychotronic instrument.
In reality, photons are light particles that are nothing more than interactions of time (X4) with X5 and X6. The fact that light is a carrier of information can be found back in older cultures where light was considered to be synonymous of divinity and truth.
The great Burkhard Heim, who deceased on 14 January 2001 in Northeim, was the German equivalent to Stephen Hawking and one of the greatest German physicists.
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When he published his theory in two books (600 pages), nobody could believe that Heim discovered the unified mass formula. In addition to that, he presented in 1959 already a new revolutionary propulsion system for spaceflight.
So far so good! Using the research work of Burkhard Heim, we have been able to explain in a quit simple way the existence of information fields, as well as the important role they play in today’s Universe.
As mentioned earlier, Heim created some new terminology which is not so easy to understand. One of these concepts was metron mathematics, based on what he called metrons. Without going into too much detail we would like to present you how T. Auerbach and I. von Ludwiger explain this term in “Heim’s Theory of Elementary Particle Structures”. The reasoning behind this shows the genius in Burkhard Heim:
The existence of a field mass leads to a modification of Newton’s law of gravitation. Newton’s law is simple and specifies the force between two masses in terms of the distance separating them. As is well-known, the force is inversely proportional to the square of the distance. Due to the existence of field mass, the gravitational force in Heim’s theory is the solution of a so-called “transcendental” equation, i.e. an algebraic equation having no simple solution. Nevertheless, approximate analytical solutions, i.e. formulas, can be found for various ranges of the distance between two masses. Purely numerical answers on a computer can of course be obtained for all distances.
As is to be expected, Heim’s law is virtually indistinguishable from Newton’s law out to distances of many light years (1 light year = 5.91 trillion (1012) miles). Thereafter, the force begins to weaken more rapidly than Newton’s law and goes to zero at an approximate distance of 150 million light years. At still greater distances it becomes weakly repulsive. Finally, at a very great maximum distance it goes to zero and stays zero. This distance is significant for the size of the Universe, because at distances exceeding it the force becomes unphysical. Hence, greater distances cannot exist. The greatest possible distance in 3 dimensions is the diameter of the Universe, which will be denoted by the letter D.
A similar deviation from Newton’s law also occurs at very small distances, and there exists a very small minimum distance beyond which the force again becomes unphysical. This distance turns out to be just about 4 times smaller than the so-called Schwarzschild radius of general relativity, which is closely related to the formation of black holes. Even more significant than the maximum and minimum distances is a third distance relation derived from Heim’s law.
In the limit of vanishing mass, i.e. in empty space, a non-vanishing relation can be derived, involving the product of the minimum distance and another small length, known in quantum theory as Compton wavelength of a given mass (T. Auerbach, I. von Ludwiger: Heim´s Theory of Elementary Particle Structures, page 3 to 10). This product of two lengths clearly is an area, measured in square meters (m²). The product exists even when the mass goes to zero and turns out to be composed of natural constants only. Therefore it is itself a constant of nature. Heim called it a “metron” designated by the symbol t (tau). Its present magnitude is
t = 6.15´10-70 m²
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The significance of a metron is the fact that it exists in empty, 6-dimensional space. The conclusion is that space apparently is sub-divided into a 6-dimensional lattice of metron-sized areas. This is a radical departure from the generally held view that space is divisible into infinitely small cells. Independently of Heim, other authors in an attempt to quantize gravitation have found elementary areas of dimensions similar to that of a metron (Ashtekar et al. 1989).
References and suggested literature:
https://static.twoday.net/initiativevernunft/files/iv030-Vortrag-Das-Neue-Weltbild-des-Physikers-Burkhard-Heim.pdf gives you an idea of Heim’s theory of the twelve dimensions. It is a presentation including question time and answers given by Burkhard Heim.
https://www.engon.de/protosimplex/px_heimd.htm Olaf Posdzech gives a nice chronological overview with many pictures.
https://www.mufon-ces.org/text/deutsch/heim.htm Illobrand von Ludwiger, a former colleague of Burkhard Heim, who we met in Berlin, gives a lot of details on Heim and his research.
https://www.heim-theory.com/ is the website of the research group around Burkhard Heim, and offers a detailed insight in Heim’s theory in German or English language.
https://thescienceclassroom.wikispaces.com/Burkhard+Heim gives some more interesting information and a good reference list.
https://www.americanantigravity.com/documents/AuerbachJSE.pdf publishes an excellent 10 pages PDF in English, by T. Auerbach and I. von Ludwiger on Heim’s Theory of Elementary Particle Structures
Heim, B. (1989 (revised). Elementarstrukturen der Materie, Vol. 1. Resch Verlag, Innsbruck, Austria.
Heim, B. (1984). Elementarstrukturen der Materie, Vol. 2. Resch Verlag, Innsbruck,Austria.
Illobrand von Ludwiger, I. (1981). in: Heim’sche einheitliche Quantenfeldtheorie, Resch Verlag, Innsbruck, Austria.
Ehlersverlag - Raum und Zeit, Themaheft Energetisches Heilen; April 2011; Willy De Maeyer und Gabriele Breyer; “Heilimpulse aus dem Informationsfeld“
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ARCI trainings and seminars in Instrumental Information Field Communication in Europe, USA, South America and Australia by Willy De Maeyer and Gabriele Breyer, 2006 – 2011.
Much additional information we received through a meeting with Illobrand von Ludwiger in Schloss Kränzlin, Berlin, September 2010.
Most information on Burkhard Heim we learned during our year’s long collaboration with Burkhard Heim’s specialist and physicist, inventor and teacher Marcus Schmieke, who we are very grateful for sharing this knowledge with us.
Thank You…
To all our customers, partners, students and colleagues, all over the world, for sharing their enriching experiences with us.
This brought us to the knowledge standard we have today enabling us to write these papers and introduce those remarkable new concepts, for free, to all boardroom members, managers and leaders of either small or large companies.
These new concepts will give raise to unique innovations and install a new kind of leadership, this to the best of all.
Willy De Maeyer willy@matrix-informational.com
Gabriele Breyer gabriele@matrix-informational.com
Homepage: www.matrix-informational.com Facebook: https://www.facebook.com/pages/Matrix-Informational/147733098627701
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Links to the other “Exploring the Information Field” papers:
Paper 1 : The Heim Theory
Key to the Information Fields
Paper 2 : The Nature of Information fields
Properties and Communication
Paper 3 : The Logarithmic Space
Unusual Transitions
Paper 4 : The Global Time Wave
About some Properties of Time
Paper 5 : Information and DNA
Remote Transfer of Genetic Information
Paper 6 : Where Heim meets Kozyrev
About Time and Photons
Paper 7 : Sacred Geometry, Fractals and Holograms
The Unification of Everything
Paper 8 : A Healthy Company is a Fractal Company
Information is Managing Us
Paper 9 : The Disaster of Symptomatic Approaches in Business Management
Exploring Information Fields versus Helicopter Management
Paper 10: Reading the Information Fields
What They Don’t Teach You at School!
Paper 11: Decision Making Made Easy by Exploring the Information Fields
Making the Right Choices
Paper 12: Information Fields Make Creative Managers Think Differently!
Getting off the Hamster Wheel
Paper 13: Business Planning, a Unique Process
An Innovative Approach
Paper 14: Informational Leadership
Old Mac Donald had a Farm…
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