5. Derivation and interpretation
Vortices cause big problems to every measuring technician. They have the unpleasant
property to whirl around the sensor even if it is as small as possible. Vortices avoid the
smallest disturbance and then can hardly be detected reproducibly.
From the well-known eddy current we know of this problematic. Instead of the vortex, we
are forced to measure and analyse any effects that arise from the vortex. These can be
measurements of the eddy losses or effects back on the stimulating field. But only
provided that the effect actually occurs.
The prerequisite for an increase in temperature by eddy losses is that the vortex falls apart.
In an ideal medium it unfortunately will not do us this pleasure.
As vrtex of the dielectric the potential vortex will find fairly ideal conditions in air or in
water. How should a vortex be detected, if it does not produce any effect? The classical
measuring technique is here at its wits' end.
From the duality to the well-known eddy current and by means of observation in the previous
chapters numerous properties of the potential vortex have been derived. But these
are not all the properties. The mathematical calculation of the electric vortex field, that we
want to turn to now, will reveal still further meaningful and highly interesting properties.
The observation is important, but it can't replace an exact calculation. A strictly mathematical
derived result has occasionally more expressiveness than a whole book full of
explanations. It will be a big help to derive and to discuss the field equation that all
considerations are based on.
We facilitate the mathematical work by vector analysis. Therefore it is useful that we
choose the differential form (equation 5.1 and 5.4) instead of the integral form (equations
3.1 and 3.2 resp. 3.8).
5.1 Fundamental field equation
We'll start from Ampere's law which provides a value for the current density at any point
pace and this value corresponds to the vortex density of the magnetic field strength
The new electric field vortices demand the introduction of a corresponding time constant
tau2 tha t should describe the decay of the potential vortices, as an extension. The extended
Faraday law of induction now provides a potential density, that at any point of space
corresponds to the vortex density of the electric field strength:
which according to the rules of vector analysis can still be further simplified:
= rot rot E = - grad div E , where we should remember that the divergence has to
vanish (div E = O. fig. 3.2, equation 3.7 ), should the corresponding field vortex be
Furthermore the following well-known abbreviation can be inserted: = 1/c2 (5.6)
With that the relation with the speed of light c simplifies to the sought-for field equation:
This equation describes the spatial (a) and temporal (b, c, d) distribution of a field vector.
It describes the electromagnetic wave (a, b) with the influences that act damping. As
dumping terms the well-known eddy current (c) and in addition the newly introduced
potential vortex (d) appear.
5.2 Mathematical interpretation of the fundamental field equation
Every specialist will be surprised to find the Poisson equation (a, e) again as a term in the
wave equation. This circumstance forces a completely new interpretation of stationary
fields upon us. The new damping term, that is formed by the potential vortices (d), is
standing in between.
Let us start with a mathematical analysis. We have applied the curl to equation 5.4*, then
inserted equation 5.1* and obtained a determining equation for the electric field strength
E. Of course we could as well have applied the curl to equation 5.1* and inserted equation
5.4*. This would have resulted in the determining equation for the magnetic field strength
If we insert Ohm's law (5.2) and cancel down the specific conductivity, or we put in the
relations of material (3.5) or (3.6) and cancel down by u respectively then the field
equation can likewise be written down for the current density j, for the induction B or for
the dielectric displacement D.
It just is phenomenal that at all events equation 5.7 doesn't change its form at all. The field
vector is thus arbitrarily interchangeable! This circumstance is the foundation for the claim
of this field equation to be called fundamental.
It does make sense to introduce a neutral descriptive vector as a substitute for the
possible field factors E, H, j, B or D.
The fundamental field equation 5.7 consists of three different types of partial differential
equations: a hyperbolic (b), a parabolic (c and d) and an elliptic (e) type. On the left-hand
side each time the Laplace operator (a) is found which describes the spatial distribution of
the field factor.
The potential equation of the elliptic type (e) is known as Poisson equation. It describes
the stationary borderline case of a worn off temporal process ( resp. = O).
With this equation potentials and voltages can be calculated exactly like stationary electric
The hyperbolic equation (b). known as wave equation, shows a second derivative to time.
which expresses an invariance with regard to time reversal; or stated otherwise: the direction
of the time axis can be reversed by a change of sign of t, without this having an influence
on the course of frequency. Wave processes hence are reversible. Equation 5.7 makes
clear that a wave without damping by no means can exist in nature. For that both time
constants would have to have an infinite value, which is not realizable in
practice. Seen purely theoretical, undamped waves could withdraw themselves from our
measuring technique (5.9).
Both vortex equations of the parabolic type (c and d) only show a first derivative to time.
With that they are no longer invariant with regard to time reversal. The processes of the
formation and the decay of vortices, the so-called diffusion, are as a consequence irreversible.
Seen this way it is understandable that the process of falling apart of the vortex,
where the vortex releases its stored energy as heat e.g. in form of eddy losses, can not take
place in reverse. This irreversible process of diffusion in the strict fhermodynamic sense
increases the entropy of the system (5.10).
Because it poses an useful simplification for mathematical calculations, often the different
types of equations are treated isolated from each other. But the physical reality looks
5.3 Physical interpretation of the fundamental field equation
In nature the different types of equations always occur in a combined manner.
1. Let's take the concrete case of the particle-free vacuum. Here the specific conductivity
is zero. The relaxation time constant responsible for the decay of vortices tends
towards infinity according to equation 5.3 and the terms (c) and (e) are cancelled from the
field equation 5.7. What remains is the by potential vortices (d) damped wave equation (b)
2. The reversed case (with ) will consequently occur in materials without
resistance, super conducting materials. We now are dealing with the well-known case of
the wave damped by eddy currents (equation 5.12*).
Vi r t u a l l y all in nature existing materials however don't fulfil these boundary conditions,
from which it follows that both damping terms always occur together and in addition the
stationary term (e) becomes active.
It is true that every antenna demonstrates that the electromagnetic wave is convertible in
high-frequency alternating currents and voltages, which then are amplified in the receiver.
But until this fundamental equation was written down it however was not understood that
this transition takes place by means of a vortex. Used are either antennas from well conducting
material, or wave guides and horn radiators, which only have a minimal conductivity,
because they are filled with air. Actually the wave can be converted in two dual
ways; by means of the rolling up to current eddies or to potential vortices (fig. 1.4).
Now we finally are capable to explain, why wave guides make possible a better degree of
effectiveness: Owing to the concentration effect of the potential vortex the HF-power is
bound in the inside and not emitted until the antenna is reached as happens for a wire for
reason of the skin effect.
Therefore, physically, one has to imagine this relation, which describes the transition of an
electromagnetic wave into a vortex, in the way that the wave spontaneously can roll up to
a vortex in case it is disturbed from the outside. The more vortices are generated, the
larger consequently is the damping of the wave (equations 5.12 and 5.12*).
3. The life span of the vortices is limited and is determined by the electric conductivity.
The at first stored vortices decay with their respective time constant This process is
described by the diffusion equation 5.12**. The final stage of the decaying vortices
finally is described by the Poisson equation (a, e: equation 5.8).
If the vortex falls apart, it converts the in the vortex stored energy in heat. These processes
are known from the eddy current. We speak of heating losses, that the stationary currents
cause in the conductor material.
But new is the concept that such vortex phenomena can occur as dielectric losses in
capacitors or in the air. The microwave oven or induction welding are good examples of
5.4 Phenomenological interpretation of the fundamental field equation
How does a damping by vortices take effect in practice? First of all we notice that the
reception of broadcastings gets worse. "The information signal is neglectable regarding
the noise" explains the radio engineer and means, the number of vortices increases at the
expense of the wave intensity.
Why, does the pupil ask, is it so cold in space? There the sun shines day and night and in
addition much more intensely than on earth! The correct answer would have to read that
because of the extremely small conductivity no diffusion process can take place. We owe
the warmth on our earth solely the here taking place decay of vortices. Responsible is the
conductivity of the atmosphere.
In 60 km to 500 km height over the earth's surface, the region which is called the
ionosphere, the gases predominantly exist in ionized form. There a very good conductivity
prevails and eddy current losses are the result. Correspondingly high are the measurable
temperatures. Besides the diffusion process the eddy currents carry out a damping of the
cosmic radiation. We say the sunlight is filtered and reduced to a for nature bearable
But not all frequencies are damped in the same way (fig. 2.8). We observe a blue shift, if
we look into the actually black sky. The blue sky doesn't show any spots or clouds. The
reason is to be sought in the skin effect of the eddy currents, which strive outwards. Since
no edge of a conductor is present here, no skin can form. The vortices spread evenly over
The potential vortex however is able to structure. It merely needs a bad conductivity and
this it finds in lower heights between 1 km and 10 km. It damps the wave and with that
also the light, for which reason we say it becomes darker, the sun disappears behind
The clouds well visibly form the discussed vortex balls and vortex strings. Clouds can
form virtually from the nowhere during intense solar irradiation, i.e. the waves can roll up
to vortices. But as a rule this takes place above the oceans. Here also the phenomenon of
transport has an effect. Because of the high dielectricity the water surface favours the
formation of potential vortices. So the vortices bind individual water molecules and carry
them away. If a diffusion process takes place, in which the vortex decays, then it rains.
This can happen in two different ways:
1. Either the conductivity increases. If for instance during intense solar irradiation air ions
form, the sun is able to break up clouds and fog. Or when the air is raised in higher
layers with better conductivity, because a mountain forces this, then it rains at the
2. For potcntial vortices the electric field is standing perpendicular to them. If at one point
an exceptionally lot of vortices join together, which let the cloud appear particularly
dark to black, then the danger exists that the ionization field strength for air is reached,
in which case a conductive air channel forms along which the stored up charges
discharge. Also lightning is a diffusion process, in which potential vortices decay and
rain can form.
In connection with the electromagnetic environmental compatibility great importance is
attributed in particular to the storage and the decay of electric vortices. There not only is
an academic-scientific interest in the question, how many potential vortices are generated,
how many are stored and how many decay, if we make a telephone call with a handy, if
we are staying under a high-tension line or if we are eating food, which has been heated
up in a microwave oven. The necessary mathematical description is provided by the
fundamental field equation 5.7.
5.5 Atomistic interpretation of the fundamental field equation
Let's again turn to the smaller, the atomistic dimensions. Here positively charged protons
and negatively charged electrons are found. Both are matter particles and that means that
seen from the outside both have the identical swirl direction. For reason of the unequal
charge conditions they attract each other mutually and according to fig. 4.9 rotate around a
common centre of mass as differently heavy pair. Chemists say: "the light electron orbits
the heavy atomic nucleus". With small balls they try to explain the atomic structure.
But the model is no good: it contradicts causality in the most elementary manner. We are
dealing with the problem that according to the laws of electrodynamics a centripetally
accelerated electron should emit electromagnetic waves and continuously lose energy, to
eventually plunge into the nucleus.
Our experience teaches that this fortunately is not true - and Niels Bohr in order to save
his model of the atom was forced to annul the laws of physics with a postulate founded in
Actually this state only exists for a very short time and then something unbelievable
happens: the electron can't be distinguished as an individual particle anymore. "It is
smeared over the electron orbit" do certain people say; "it possesses a dual nature" says
Heisenberg. Besides the corpuscular nature the electron should in case of its "second
nature" form a matter wave, "the position of the electron is to be looked at as a resonance
which is the maximum of a probability density", do explain us de Broglie and
These explanations can hardly convince. If the electron loses its particle nature in its
second nature, then it also will lose its typical properties, like for instance its mass and its
charge. but this is not the case.
THE vortex theory provides clear and causal answers: if the electron were a ball it continuosly
would lose energy, therefore another configuration forms that does not know
this problem. Here the phenomenon of transport takes an effect. The electron opens its
vortex centre and takes the tiny protons and neutrons as atomic nucleus up into itself. The
Bohr electron orbit with that is not a path anymore, but is occupied by the whole particle
as spherical shell. This is confirmed by the not understood measurements exactly like the
photos of individual atoms with the scanning electron microscope.
But now an electron does in its inside have the opposite swirl direction as the proton seen
from the outside. As a consequence a force of repulsion will occur, which can be
interpreted as the to the outside directed current eddy, the force of attraction for reason of
the opposite charge works in the opposite direction and can be interpreted as the potential
If both vortices are equally powerful: (5.13)
or if both forces are balanced, as one usually would say, then the object which we call an
atom is in a stable state.
It probably will be a result of the incompatible swirl direction, why a very big distance
results, if the electron becomes an enveloping electron. On such a shell not too many
electrons have room. Because of the rotation of their own, the electron spin, they form a
magnetic dipole moment, which leads to a magnetic attraction of two electrons if they put
their spin axis antiparallelly.
As a "frictionless" against one another rotating pair they form two half-shells of a sphere
and with that occupy the innermost shell in the hull of an atom. If the positive charge of
the nucleus is still not balanced with that, then other electrons is left only the possibility to
form another shell. Now this next electron takes the whole object up into itself. The new
shell lies further on the outside and naturally offers room to more as only two vortices.
5.6 Derivation of the Klein-Gordon equation
The valid model of the atom today still raises problems of causality, as has been
explained. An improvement was provided by an equation, which was proposed by the
mathematician Schrodinger 1926 as a model description. This equation in this way missed
the physical root, but it nevertheless got international acknowledgment, because it could
be confirmed experimentally. Looking backwards from the result then the physical
interpretation of the probability density of the resonance of the waves could be pushed
The Schrodinger equation is valid for matter fields (of mass m), while the interaction
with a outside force field the energy U indicates. It can be won from a wave equation by
conversion, which possibly is the reason why it usually is called a wave equation,
although in reality it is a diffusion equation, so a vortex equation!
For the derivation Schrodinger gives the approach of a harmonic oscillation for the
complex wave function
if the entire time dependency can be described by one frequency f = W/h
(de-Broglie relation): (5.16)
The high-put goal is: if the structure of the atom is determined by the fundamental field
equation 5.7 then one should be able to derive the experimentally secured Schrodinger
equation and to mathematically describe the discussed special case. Also we select at first
an approach periodic in time:
with . (5.18)
We insert the approach 5.17 and its derivations into the field equation 5.7 and divide by
the damping term e-wt:
If as the next step the angular frequency according to equation 5.18 is inserted, then
summarized the provisional intermediate result results:
The derived equation 5.20 represents formally seen the Klein-Gordon equation, which is
used for the description of matter waves in quantum mechanics and which particularly in
the quantum field theory (e.g. mesons) plays an important role. Even if it often is regarded
as the relativistic invariant generalization of the Schrodinger equation, it at a closer look is
incompatible with this equation and as "genuine" wave equation it is not capable of
treating vortex problems correctly, like e.g. the with the Schrodinger equation calculable
quantization of our microcosm.
5.7 Derivation of the time dependent Schrodinger equation
With the Schrodinger approach 5.15 and its derivations the derivation is continued:
The for a harmonic oscillation won relations according to equation 5.21 and 5.22 are now
inserted into equation 5.20:
This is already the sought-for Schrodinger equation, as we will see in a moment, when
we have analysed the coefficients. Because, besides equation 5.16 for the total energy W,
also the Einstein relation is valid (with the speed of light c):
we can replace the coefficients of the imaginary part by:
To achieve that equation 5.23, as required, follows from the Schrodinger equation 5.14, a
comparison of coefficients is carried out for the real part:
If thc kinetic energy of a particle moving with the speed v is:
then acccording to De Broglie this particle has the wavelength h/mv. The consideration of
the particle as matter wave demands an agreement with the wave length c/f of an electromagnetic
wave (with the phase velocity c). The particle hence has the speed v, which
corresponds with the group velocity of the matter wave:
According to equation 5.24 on the one hand the total energy is W = w • h and on the
other hand the relation 5.28 gives resp.:
Inserted into equation 5.27* the sought-for coefficient reads (according to eq. 5.26):
5.8 Derivation of the time independent Schrodinger equation
The goal is reached if we are capable to fulfil the comparison of coefficients 5.26:
The angular frequency w is given by equation 5.18. Therefore has to be valid:
As is well-known the arithmetic and the geometric average only correspond in case the
variables are identical. In this case, as already required in equation 5.13:
has to hold.
From this we can draw the conclusion that the Schrodinger equation is just applicable to
the described special case (according to eq. 5.13), in which the eddy current, which tries
to inc rea se the particle or its circular path and the potential vortex, which keeps the atoms
together and also is responsible for the stability of the elementary particles, are of
identical order of magnitude.
As a check equation 5.23 is divided by c2 and equations 5.30 and 5.25 are inserted:
This is the time dependent Schrodinger equation 5.14 resolved for
Next we replace according to equation 5.21 with acc. to equation 5.24:
If we separate the space variables from time by the Schrodinger approach 5.15 we
This quation 5.34 for the function of space coordinates is the time independent
The solutions of this equation which fulfil all the conditions that can be asked of them (of
finiteness, steadiness, uniqueness etc.), are called eigenfunctions. The existence of
corresponding discrete values of the energy W, also called eigenvalues of the Schrodinger
equation, are the mathematical reason for the different quantum postulates.
5.9 Interpretation of the Schrodinger equation
The interpretation of the Schrodinger equation is still disputed among physicists, because
the concept of wave packets contradicts the corpuscular nature of the elementary particles.
Further the difficulty is added that wave packets at a closer look never are connected, run
apart more or less fast, and really nothing can hinder them doing that. But for a particle the
connection represents a physical fact. Then there can be no talk of causality anymore.
The monocausal division into two different levels of reality, in a space-timely localization
and in an energetic description, does not represent a solution but rather the opposite, the
abolition of the so-called dual nature. As has been shown, the potential vortex is able to
achieve this with the help of its concentration effect.
But from the introduction of this new field phenomenon arises the necessity to interpret
the causes for the calculable and with measuring techniques testable solutions of the
Schrodinger equation in a new way. Laws of nature do not know a possibility to choose! If
they have been accepted as correct, they necessarily have to be applied.
Three hundred years ago the scholars had an argument, whether a division of physical
pheomena, like Newton had proposed it, would be allowed to afterwards investigate
them in the laboratory individually and isolated from other influences or if one better
should proceed in an integrated manner, like for instance Descartes with his cartesian
vortex theory. He imagined the celestial bodies floating in ethereal vortices.
One absolutely was aware that the whole had to be more than the sum of every single
realizato n, but the since Demokrit discussed vortex idea had to make room for the
overwhelming successes of the method of Newton. And this idea after 2100 years was
stamped, to in the meantime almost have fallen into oblivion.
Today, where this recipe for success in many areas already hits the limits of the physical
possibilities, we should remember the teachings of the ancients and take up again the
vortex idea It of course is true that only details are calculable mathematically and that
nature, the big whole, stays incalculable, wherein problems can be seen.
If we consider the fundamental field equation 5.7, we find confirmed that actually no
mathematician is capable to give a generally valid solution for this four-dimensional
partial differential equation. Only restrictive special cases for a harmonic excitation or for
certain spatial boundary conditions are calculable. The derived Schrodinger equation is
such a case and for us particularly interesting, because it is an eigenvalue equation. The
eigenvalues describe in a mathematical manner the with measuring techniques testable
structures of the potential vortex .
Other eigenvalue equations are also derivable, like the Klein-Gordon equation or the
Lionville equation, which is applied successfully in chaos theories. So our view opens, if
chaotic systems like turbulences can be calculated as special cases of the same field
equation and should be derivable from this equation.
The in pictures recorded and published structures, which at night should have come into
being in corn fields, often look like the eigenvalues of a corresponding equation. The ripe
ears thereby lie in clean vortex structures flat on the soil. Possibly potential vortices have
charged the ears to such high field strength values that they have been pulled to the soil by
the Coulomb forces.
Consequences resulting from the derivation of the Schrodinger
equation from the fundamental field equation 5.7:
The relation between the energy of oscillation and the mass is
described by the relation named after Albert Einstein