Pekka Virtanen

1
Pekka Virtanen Studies of physics and
mathematics in University of Helsinki,
Finland D-Theory - Model of cell-structured
space Part ? ? ? Electromagnetism Hypothesis
The cell-structured physical space is a
four-dimensional hyperoctahedron. It has a three
dimensional surface and it is quadratic and
absolute in comparison to the observed space. The
three spatial dimensions of its surface are
closed and isotropical. The fourth spatial
dimension is open and edged and it is observed
indirectly as the quadratic speeds of
bodies. (The observed space is an emergent
property of absolute space and it appears from
the absolute space through coarsening and
observing light and bodies and it is the
three-dimensional surface of Riemann's
hypersphere.) Abstract The cellular structure
of absolute and quadratic space were defined.
Appearing of observers space from the absolute
space as its emergent property were described
.The Lorentz's transformations were derived from
the space model . A solution to the measurement
problem of quantum mechanics were proposed. The
uncertainty principle and the gauge invariance of
the wave function are derived from the space
model. A new interpretation of wave function
collapse and of violation of Bell's inequality
were proposed. The structure and motion of the
space grid outside the 3D-surface were defined in
direction of the fourth dimension. The charge and
the spin of an elementary particle were defined
in cell-structured space. The four-dimensional
atom model and its all quantum numbers and
projections on the 3-dimensional surface of the
hyperoctahedron were defined. The accurate values
for proton diameter, Rydbergs constant and
hydrogen atom radius were derived. The geometric
structure of quarks, of three families of
particles and of photon were defined. Three
invariance equations and the asymmetric wave
function of a particle were defined. The locality
of mass, length and time were introducted in
absolute space with help of the asymmetric wave
function. It was shown that the electromagnetic
fields are caused by the effects of the moving
space grid and that the model is compatible with
the Maxwell's equations and Lorentz's
transformations. This is the version v1.30
published 29.11.2008.email pekka.virtanen_at_webinf
o.fi
2
Contents, part 3 The space grid or the
ether 3The electric charge in the
grid 4The virtual particles 5The
Einstein's claims against the ether 6More
symmetries 7 The structure of elementary
particles 8The
colours of the quarks 11Mesons and weak
interaction 12Parity 13The components of
atom in reciprocal space 14The
four-dimensional atom model 16The orbital
angular momentum of the electrons in an
atom 18The spin of an electron 20The
projections of particles on the 3D-surface
21Annihilation 29 The potentials of the
space grid 30Amperè's law 32The
direction of the magnetic force 33The dynamic
properties of the grid 34Electromotive
force 35The mechanics of the Farady's law in
the ether 36Biot-Savart's law 38Entropy
and the ether 38Summary 39Sources 41

3
The space grid or the ether In the first part of
D-theory we already considered the grid lines
moving outside the 3D-surface. They penetrates
the 3D-surface and form there space grid or the
famous ether. The particles of the grid are
electrons e- and positrons e. These elementary
particles organize a regular grid.
4.D
Two grid lines




The grid is made of positive and negative grid
lines perpendicular to each other in space and
antispace. There exists no forcies or
interactions between the perpendicular grid
lines. The forcies appear only, when the grid is
not homogenous. The grid tries to stay homogenous
and resists all forcies, which will change the
homogeneity of the grid. The grid allways tries
to reach a state, in which it has the minimum
number of energy. The pull and push forcies
determine the structure and the properties of the
grid. The particles of the grid with the same
signs, the positive ones or the negative ones,
pull each other, and the particles with different
signs push each other. The regularity and the aim
to become homogenous gets the particles to behave
in the grid in the opposite way of their nature
the particles with different signs seem now to
pull each other and the particles with the same
signs seem to push each other. The particles have
in the grid both the opposite properties at the
same time! The internal forcies of the grid
affect only on the particles parallel to the grid
(or 4.D), which are electrons and positrons, and
more heavy particles, which include the electron
or positron. According to the disobservation law
this kind of grid is not possible to observe
directly. The grid, however, is observed
undirectly in many different ways, of which the
light and electromagnetic fields are not the
slightest. The grid stands in cell-structured
space so that it is possible to move in grid
without to collide with the particles of the
grid. The structure of space includes the layers
of two unit cells. The grid lines are on these
layers at continuous orbital motion around the
space, which creates the continuity of events or
the time. According to the D-theory the space
is the only substance in physics. Thus the
particles of the grid must be understood as a
wave in 1-dimensional space. The waves has not so
far any detailed description.
4
The internal forces of the grid are either push
or pull forces. The forces get weaker along the
distance or F(x) 1 / x. In such grid the
particles move in relation to each other and
search for their balance. When the observer
travels through the grid, he does not observe the
grid as complete homogenous.
The electric charge in the grid
4.D
When the electrically charged body (in the
picture), which consists more negative than
positive particles parallel to the grid, moves
perpendicular to the grid, appears in the grid a
force parallel to the grid, which pulls the
positive particles of the grid and pushes the
neagtive particles. The force causes the grid
current parallel to 4.D. The reason is the aim of
the grid to become homogenous. The grid current
consists of the grid lines, which rotate into the
opposite directions. (D-theory, parts 1. and 2.)




3D-surface
e-
e-
?
?
The grid current has sgot the grid lines to
rotate to angle ?. A grid charge appears as a
result grid current.
The motion of the gharged body is even or in its
motion has not any component parallel to 4.D. The
change of homogeneity caused by the grid current
is reversible or when the body has passed the
place, the grid returns homogenous again. The
charge, however, creates into the grid a
potential, which appears in all directions of
4-dimensonal space. The grid current and the
potential together creates the electric and
magnetic fields of the 3-dimensional space. The
electrical polarisation is considered more
carefully in connection with an atom model, as
the mechanics of the fields as well.
5
4.D
Id
The negative grid current Id moves in the picture
as a straight line away from negative charge at
the 3D-surface. Correspondingly the positive grid
current moves to the opposite direction. Thus the
grid currents of the opposite signs polarize the
grid and the grid charge Qi is created into the
grid. The grid charge Qi creates into the grid a
potential V, which is described with a flux
vector, as later is told. The potential is
observed the bigger the more far the observer is
from the 3D-surface in direction of 4.D. The
potential V does not penetrate the 3D-surface,
then for any closed 2D-surface S ? V dS 0 .
V
Qi
3D-surface
The aim of the grid to be homogenous in direction
of the grid lines is observed as a grid current,
or the grid itself moves and will not transfer
the charges. It means that in direction of the
grid lines two particles, which have different
signs and which are not a part of the grid,
behave as there would not be any grid around
them, or they push each other. The aim of the
grid to become homogenous is observed in
different way in all other directios of space,
because in those directions the grid current can
not appear. The grid instead aims to transfer the
charges of the different signs together. The grid
becomes thus more homogenous. Correspondingly the
grid aims to transfer the charges of the same
signs far away from each other. This aim is
observed as Coulomb's force of electric
field. The virtual particles The modern physics
explains, for example, the push force between two
electrons with help of a virtual photon, which
the electrons will change all the time. The
distance between the electrons may be large as
well (for example, 150 km). The virtual (
assumed) photons can not be observed. Also the
space grid in direction of 4.D is not possible to
observe or it is also vitual in a way. D-theory
proves later that the reason for the push force
between the electrons is the aim of the grid to
become homogenous. The grid causes also other
effects, which in modern physics are explained
with help of the virtual photons.
6
The Einstein's claims against the ether Albert
Einstein has in his book "The evalution of
Physics" written two claims against the existence
of the ether. In place of the ether he proposes
the field, which would be a substance. (The
substance means here a thing or idea, which does
not need to explain with any other thing.)
Claim 1. The ether can not exist, because the
medium would cause a friction and it is not
onserved. The space grid or the ether does not
consist of the atoms, but the electrons and
positrons. The friction exists between the atoms.
The space grid does not cause any friction. The
direction of the grid is 4.D and a body moves in
that direction only in acceleration, when a
charged body emits an energy to the grid. The
speed of the grid is allways measured to be the
speed of light in all directions.
(Michelson-Morleyn experiment).
Claim 2. If the ether does not influence to the
motion of matter, there can not exist any
interaction between the ether and the particles
of matter. Only the particles and the bodies,
which includes a component in direction of 4.D,
like electrons and positrons, can have an
interaction with the ether.The interaction causes
an electric and magnetic field and an
electromagnetic wave. This all supposes that the
particles (quarks) are shared parallel to the
dimensions or the main axes as the forces caused
by them. The fourth dimension differs from the
other ones. It and the grid causes all electric
effects.
At the end Einstein writes " Would there
possibly exist interaction between the ether and
matter only in optical but not in mechanical
events. It would anyhow be quite paradoxical
conclution." D-theory shows that the interaction
is optical or electric and in addition that
electromagnetic effects are mechanical in their
nature. The electromagnetic effects have in space
their own direction, which separates them from
other mechanical effects referred by Einstein.
The arguments like these have seriously
prevented to develope idea of the ether. As a
result, for example, "Dirac's field" created by
Paul Dirac is not taken as a physical fact.
7
More symmetries According to the disobservation
law the radius 4.D of space is not possible to
observe. We have considered the issue
theoretically and considered that there exists
the directions up and down parallel to 4.D. We
can not, however, know in what direction are the
upside or downside. When we look at the issue
from point of view of the positive or negative
particle, the issue will change. The sign of a
particle determinines, which side of the
3D-surface is the upside and downside. When we in
addition notice that the coordinates of the grid
lines with the opposite signs, which travel in
the opposite directions around the space, are
rotated in relation to each other 180º, we get a
theoretical idea of the symmetry of an atom. The
quadratic space can be "opened" mathematically by
taking a square root of it. Then we get a
positive and negative space, as in the next
chapters is told.
In the quadratic space the polarization of the
basic particles to positive or negative particles
can be observed indirectly only on grounds of
their spin. The space is divided into the normal
space and the reciprocal space like in picture.
In the reciprocal space the absolute speeds w are
replaced with the speeds 1/w. The observer can
theoretical consider the space grid, which
consists of the lines of positive and negative
particles. The straight lines are overlaped and
stand physically side by side. The space can be
"opened" mathematically so that the signs of the
particles differs and are found out. The
particles are positive or negative depending on
if they stand at a layer on antilayer. The
3D-surface forms a symmetry axis for the opened
space. The axis divides the space to the normal
space and the reciprocal space. The reciprocal
space (in the picture) is considered next. In the
reciprocal space appears an atom model, in which
a part of the particles of an atom travels at a
layer, and a part at an antilayer. This symmetric
model explains the Pauli's rule for the electrons
in an atom.
Reciprocal space
4.D
c
2
3D-space
Normal space
0
4.D
c
3D-space
4.D
The reciprocal spaces are opened to positive and
negative space
8
The structure of elementary particles We have
mentioned, that on the 3D-surface the unit cells
of the space are 3- dimensional and on the both
side of the surface they are 1-dimensional. The
free electrons and the nucleus move on the
3D-surface. A common feature for them is that
they are called fermions and they have the spin
½. Let's consider next the structure and the spin
of these particles.
A proton p consists of three quarks parallel to
3D-space and they form the 3-dimensional positive
or negative base of a proton. In addition a
proton includes one positive or negative basic
particle e parallel to 4.D, which moves on both
sides of the 3D-surface. The basic particles of
3D-surface are quarks made of one cell of the
3D-space, but the particle parallel to 4.D is
spin ½-particles and its length is a ½-layer or
half a cell. The proton can move only at the
3D-surface. A proton, which is standing in
antispace, has the opposite sign and is a proton
as well. The magnetic moment of a proton is
positive ( 2.79 ?i ), because the positron e
inside the proton has the same sign as the
three-dimensional base of a proton. In the
picture we can see the relation of 4D-component
inside a proton to the space grid around them.
The motion is similar to the motion of electron.
The charge of a proton is positive, because of
its positron e. A proton polarizes the grid and
causes a grid current. The motion of
4D-components of the protons from left to right
and back means even parity.
positron
3D-base of proton
p
The protons have the opposite spins
1.
2.
e
3.
4.
In picture the positron e in a proton moves up
and down through the 3D-surface.
9
electron
When the positron e in a proton is changed to an
electron, it becomes a neutron. The mass of a
neutron and a proton are about the same. The
neutron n has no electric charge, because in the
space in direction of 4.D above it or below it
does not exist any grid line. The reason is in
its path through the grid. The magnetic moment of
a neutron ( -1.91 ?i) has the opposite sign
compared with a proton, because the particle
parallel to 4.D has the opposite sign. The
3D-particle of a neutron has different sign than
the sign of an electron inside it. In neutron the
phase of an electron has turned 90 degrees and
therefore the path of an electron is shifted 90
degrees in comparison with the path of the
positron in a proton. The 4D-particles of proton
and neutron must travel along different paths in
relation to the grid lines. The neutron travels
in the grid like in the picture. Along the path
of a neutron stands no grid lines, so the neutron
can not polarize the grid. The electron in
neutron moves up and down through the 3D-surface
and it has four states because of its way of
motion.
n
The neutrons have the opposite spins
A free neutron decays to a proton so that its
electron becomes free to the 3D-surface and from
an upper grid line transfers a grid particle
(green in picture) to its position. A hole
appears into the grid. The hole is then filled
with a particle from a grid line below (red),
where appears now a hole. The grid particle (red)
is transfered over the 3D-surface and it forms
now with the hole a neutrino.
e-
10
In first part of D-theory the time was defined as
an effect, which appears from the motion of the
grid lines. The model means at the same the
realizing of Lorentz's transformations in
cell-structured space. The grid lines would not
have any interaction, for example, with a proton,
if the proton would not include a component
parallel to 4.D. The motion of the grid lines
past the proton on both sides of the 3D-surface
creates the interaction between the grid and a
proton. As concequence of this interaction the
proton has a certain wave length, electric charge
and spin. Because of this interaction the proton
can be described with help of a wave function
like in second part of D-theory. The same is
valid for all other particles, which have a
component parallel to 4.D. Without this component
the particle would be only parallel to 3D-surface
and the observing would be possible only with
help of the mass. It strongly looks that the
fourth dimension is at the point of view of
observing the most important direction and
without the mass the only important direction.
The grid lines travel past the particle into the
opposite directions at speed c. When the particle
itself moves at speed v to the right, the
asymmetric speed of particle is in relation to
the grid w² (c - v)( c v) c² - v². The
asymmetry of a particle is described by the ratio
v/c and the speed of time passing is determined
by the number of passings made by the grid
lines t' t / 1 - v²/c² .
v
When the interaction between a particle and the
grid lines is based on the positron of the
proton, the interaction to the positron at speed
v is asymmetric. The positron moves on the other
side of the surface a longer way depending on the
speed of a particle. This asymmetry explains all
prognosis for the relativity of basic quantities
in theory of Special Relativity. (D-theory, part
2.)
11
The colours of the quarks The 3-dimensional
surface made of the regular octahedra includes
the main axes parallel to diagonals of the
octahedra. The main axes are not possible to
observe, so their meaning is abstract in quantum
physics. An abstract colour code is given for
each main axis. Let the colours be blue, red and
green. The sum of these colours is zero or
colourless. Because the main axes are fully
isotropic, also the colours are isotropic. When
the cell-structured space has its anti-space, we
can consider the anti-main-axes to the side of
anti-space. A single quark is allways
parallel to its dimension or to its main axis. So
we can give a colour code for each quark as well
according to their main axis. The colours of the
quarks are isotropic as well. A single quark can
never exists alone in 3D-space. Three quarks are
needed for a proton. They all are parallel to the
three main axes. Then a proton is a colourless
particle. The colours of the antiparticles are
correspondingly anti-colours. The sum of colour
and anti-colour is colourless. The anti-colour
corresponds then to the anti-main-axes of
anti-space. Also the grid lines can have a colour
according to their main axes.
Neutrino
The neutrino ? is a grid particle in a grid line.
Its sign is opposite compared with the rest of
the grid line. The neutrino is its own
antiparticle. The neutrino moves as a part of the
grid at the speed of light and it has not any
mass. Its spin is ½ or -½. The neutrino is
called also the neutrino of an electron. The hole
of the grid is a part of the neutrino. The hole
is created, when the grid line moves over the
3D-surface to fill there the original hole.
Because of the hole, the weak interaction is
connected to the neutrinos. It may be possible
that the grid can combine several neutrinos
together to a myon's neutrino. The neutrino is
right-handed or left-handed depending on which
grid line it happens to stand. The similar
"handed" is already defined for the grid lines.
?
?
12
Mesons and the weak interaction A grid line
appears on the 3D-surface as a particle like a
point and it is called ? -meson or a pion.
According to modern physics the nucleons, proton
and neutron, change between themselves a particle
called ?-meson, of which mass is about 273.3 x
me, where me the mass of an elecron. The mass
is approximately the same as the mass of one pair
of the grid lines 2 x 137 x me 274 x me. The
difference is 0.25. There exists an electric
neutral ?º-meson and a positive and negative ?
and ?- meson. The ?º-mesons are lighter (264.3 x
me ). A meson is not observed until it comes off
the grid and leaves there a hole. Both a meson
and a hole move in relation to the grid.
The ?º-mesons are like the neutral grid lines.
They can not have antiparticles, because they
would collide the grid immediately. When two
?º-mesons change between themselves either the
upper half or the lower half, they become charged
? and ?- -mesons. In high energy collisions the
grid lines may start moving as ?-meson, and a
hole is leaved in place of them. Also the hole
can travel in the grid. The grid aims to put
together the meson and the hole. This aim is
named weak interaction. The weak interaction
means for a charged mesons ?- and ? a certain
life time 2.55 x 10 (exp -8) s. When the hole
then catch the charged meson, it will change
itself with the meson to a ?-meson (or the myon)
and a neutrino of a myon, which fills the hole in
the grid. The meson ?º decays in electromagnetic
interaction, because it is polarized like any
another grid line by any charged particle. So its
life time is only 1.8 x 10 (exp -16) s. The
charged ?-meson can not be polarized, because of
their structure and they will decay only during
the weak interaction of the grid.
?º ?º ?- ?
The meson ?- moves like in the picture. We can
see that the right and left side of the layer are
not symmetrical for the meson. The meson thus
makes the difference between the right side and
the left side! Up-down-motion does not happen
here. The way of motion means an odd parity. For
the charged mesons it means a charge e or e-,
when they polarize the grid. When we consider the
way of motion of the protons and the neutrons,
the observer in 3D-space can think that the
protons and neutrons change between themselves
the mesons or the grid lines. He thinks the meson
to be built of quark and antiquark or the
positive and negative grid line, which are
particles like a point in 3D-space.
13
Parity
Antilayer Layer
In connection with the meson we noticed that
left-right-symmetry does not realize. There is a
symmetry violation, which is called parity
violation.
grid lines in reciprocal space
grid lines in normal space
The weak interaction appears as parity violation
during the decay of some particles. A
conservation of CPT-symmetry is known in physics,
which is fundamental like the conservation of
energy. ( C is a symmetry of charge changing, P
is a symmetry of parity changing and T is a
symmetry of changing the direction of time
passing.) CPT-symmetry expresses that if the
world has a mirror image, where all particles are
changed to their antiparticles and the direction
of time passing is changed as well, the original
world and its mirror image are completely equal.
In the previous model (and in picture) it will be
realized and means only that the 3D-surface of
hyperoctahedron and the grid lines outside it are
turned inside out. When this is done at a layer
and an antilayer, the result is an image, where
everything will work in the same way. We have
before presented the way of the 4D-components of
the proton, the neutron and the free electron to
move diagonally fro-and-to on the 3D-surface and
through it. The mesons do not stand at the
3D-surface and they travel on a layer without
penetrating the 3D-surface and the grid lines are
standing on the right or left side of them
depending on the orbital direction. Let's define
the parity of a particle on grounds of the way of
movement. The parity gets then the values even or
odd. We can define the parity of a proton, a
neutron and an electron to even parity. Then the
mesons have odd parity.
The way to move defines at the same the form of a
wave function of a particle. An even particle
moves in two perpendicular direction. An odd
particle moves only in one direction. The
left-right-symmetry does not appear for an odd
particle because of way of its movement. The
symmetry violation is opposite for an
antiparticle. The odd particles can thus be
defined to left- or right-handed particles.
?
?
even odd
14
In modern physics the weak interaction and the
electromagnetic interaction have been connected
in "Great unity theory" to one force, which have
these two ways to appear. The same unification is
made also in the model of D-theory. The weak
interaction is based on the aim of the ether to
combine a particle from a grid and the hole in
the grid leaved by the particle. The grid aims to
homogeneity with help of combining. Thus a life
time, which depends on the properties of the grid
and the particle, can be defined for the
particle. The electromagnetic interaction is also
based on to the aim of the grid to homogenous.
Then the grid will be polarized by a charged
particle on the 3D-surface. The result is an
electric and magnetic field. Both ways of
interactions can be understood as properties of
the grid. The grid is there a unifying factor.
The grid charge
In the picture the protons p on a layer and
antilayer are standing side by side. The spins of
the protons are opposite. The protons create on
the layers the grid charges with the opposite
signs. So the grid does not resist the appearing
of the grid charges, altough it aims to be
homogenous. If the other one of the protons is
changed to antiproton, which has the charge e-,
would the grid charges on layers side by side
have the same signs and now the grid would resist
the appearing of the grid charges. So the charges
e and e- side by side would eliminate each other
and no grid current or the grid charge would be
created.
p
p
The components of an atom in reciprocal space The
previous components, protons, neutrons and
electrons build in cell-structured space a whole
called an atom. The simplest atom includes one
proton and one electron and is called a hydrogen
atom. In the atom a proton at 3D-surface binds
with help of the grid lines an electron to
itself. The electron moves mostly at the distance
of 1 layer from the proton or the nucleus in
direction of 4.D. It does not travel around the
nucleus, but moves with it in direction of the
3D-surface. The electron lies at the hollow of
potential and the projection of the electron is
observed around the nucleus. Let's consider next
the structure of an atom on grounds of the space
model and take a closer look to the spin of an
electron.
15
The next picture shows an opened reciprocal space
so that only the positive and negative spaces are
visible. There is a diagram about the positive
and negative half of an atom. The halfs are
symmetric in 3D-space. The nucleus of the atom
includes 4 protons, 4 neutrons, (the neutrons are
not shown in the picture,) and in the grid
outside the 3D-space travels 4 electrons. In the
picture the positive particles are green and the
negative are red. The sign of a particle is
determined by the layer. On a layer the sign is
positive and on an antilayer it is negative. The
sets of 3D-coordinates of the halfs are rotated
180º in relation to each other.
The grid lines travel past the nucleus to the
right and left. The electrons e- (red) in the
picture move with the nucleus at their own
layers. The 4.D-components of the nucleus and the
electrons e- move in the grid in four phases. In
the picture the negative electron above the
positive proton feels the aim of grid to combine
opposite particles and a pull force to the
3D-surface. On the other hand in direction of the
3D-surface the electric field of the nucleus
holds the electron e- beside the nucleus p.
Reciprocal space
e-
e
Layer
Reciprocal space
e-
e
Antilayer
The halfs of an atom at a layer and antilayer are
symmetric. The spins of the particles are
opposite. Also the negative electrons e- polarize
the grid and the atom is electrically neutral.
We have defined a proton of the nucleus to
electric positive and an electron at a layer to
electric negative. An atom is thus electrically
polarized. The polarization appears from the
symmetry violation mentioned before and is found
out, when in a positive half of the atom the
positive particles stand in nucleus and the
negative particles stand far away as electrons.
The corresponding polarization is found in a
negative half of the atom, where the signs of the
spins are opposite. This structure of an atom
leads to the electric polarization. The positive
charge of the protons and the negative charge of
the electrons are created in the atom. The three
quarks of the proton and the component with the
same sign parallel to 4.D makes the proton
"everlasting". When the number of dimensions
sometimes increases, the proton already has a
base made of four quarks with the same signs.
16
The four-dimensional atom model Let's consider in
four-dimensional space an atom, which has its
layers full of electrons. Let's consider first
the positive half of the atom in opened
reciprocal space. The picture presents the full
electron layers of a half of an atom, which is
much more heavy than the hydrogen atom. All the
spins of the electrons in the upper half of the
atom have the same signs.
4.D
The profile of an atom in opened 4-dimensional
reciprocal space
fdps dp s ps s
N
4.D
3D-surface
M
L
n 1
K
4.D
When the stack of the electrons of a half of an
atom is observed from down in direction of 4.D
and the layers are thought to be planes parallel
to 3D-surface, we can share the electrons at the
different planes as circles around the the
vertical 4D-axis like in picture. Every layer
corresponds to one plane or the surface parallel
to 3D-surface. The planes lies outside of
3D-space. The nucleus stands on the 3D-surface
and for every electron there exists one proton in
the nucleus. In the half of the nucleus the
protons stand in the same way at different layers
of 3D-space. The layers are in 3D-space spheres
inside each other. On the layers the numbers of
protons increases with the radius of sphere
1,4,9,16,25... The negative half of the atom
stands in four-dimensional space on the surfaces
of the spheres, which are turned inside out and
have the same centre. It can not be distinguished
from the positive half in any way but from the
signs of the spins of the particles. In this
model the mechanical structure of the auxiliary
quantum number is found out. The auxiliary
quantum number (or orbital angular-momentum
quantum number) describes the distance of an
electron from the vertical 4D-axis of the atom.
The distance is parallel to 3D-surface. The
auxiliary quantum number 0,1,2 are named with
letters s,p,d,f. The main quantum number ( or
principal quantum number) describes the distance
in direction of 4.D.
17
We have already defined the three quantum numbers
of the electrons in an atom. They are spin, main
quantum number and orbital quantum number. They
all define the location of the electron in a
four-dimensional atom. The sign of the spin
determines the half of an atom, the main quantum
number defines the distance of the electron from
the 3D-surface and the orbital quantum number
defines the distance of the electron from
vertical axis of an atom and also the radius of
the circle, on which the electrons stand around
the vertical axis. The circle is cell-structured
and consists of 1-dimensional cells. When all
these are determined, there is still left the
location of the electron at the circle, which
surrounds the vertical axis of an atom. The
locations are equal to each other around the
vertical axis, but they all have their own
direction in 3D-space in relation to the centre
of an atom. When one direction z of 3D-space is
selected around the nucleus of an atom, the
locations are not any more equal in relation to
that direction. In addition the directions are
quantized. The location of an electron in
relation to the direction z is described with the
magnetic quantum number m. The number of the
electrons at the circle is 2l 1, where l is the
orbital quantum number, which defines the circle.
The magnetic quantum number m gets as many values
as there stands electrons at the circle. It gets
the values m 0, 1, 2, 3, ... until to the
value l. According to the Pauli's rule the
quantum numbers of two electron in an atom can
not be equal. When the quantum numbers determine
the location of the electron in a
four-dimensional atom, the rule actually
expresses that two electrons can not be in the
same place in the same time in absolute space.
The location of an electron means here the real
place and not the place of the projection of the
electron at 3D-surface. Also the concept "state"
of quantum physics can be understood as the
location of a particle in four-dimensional
cell-structured space. A particle transfers from
one state to another or from one cell of the
cell-structured space to another cell.
Vertical axis of an atom
When N 3 and l 2, there exists 2l 1 5
electrons at the circle around the vertical axis
in an atom. Their magnetic quantum numbers get
the values m 0, 1, 2 or together 5 different
values. The quantum number m defines the
locations at the circle, when the direction z is
selected at the 3D-surface.
m 1
m 2
m 0
m -1
m -2
z
3D-surface
We can thus mention that all quantum numbers of
an atom can be described geometrically in a
four-dimensional atom model. Each quantum number
is uniquely connected with the location of an
electron.
18
The orbital angular momentum of the electrons in
an atom The atom has a vertical axis parallel to
4.D. The axis is two unit cells thick. The
vertical axis stands in 3D-space in centre of an
spherical atom. The electrons of an atom have
gathered symmetric around the vertical axis. The
angular momentum of an atom depends on gathering
of its electrons. Each electron has its own
orbital angular momentum and the angular momentum
of an atom is their sum. Not any angular
momentum exists at the vertical axis in the
centre of an atom. All locations of the
electrons, where an auxiliary quantum number has
the value l 0 , have the value L 0 of orbital
angular momentum.
rds
L
Vertical axis
L
l 0 l 1 l 2
s p d
r
n 3 n 2 n 1
The direction of orbital angular momentum L is
the direction of grid lines and the dimension is
Js
nucleus
The main and auxiliary quantum numbers at the
vertical axis of an atom.
In cell-structured quadratic space the length r
is written (D-theory, part l ) r ds or
r ds , where d and s are integers, which
expresses the length and d s 1. The orbital
angular momentum around the vertical axis is
calculated by multiplying the distance r and the
quantum interaction h. The distance of an
electron from the vertical axis is expressed as
number of the layers. When in an atom the
auxiliary quantum number l means the distance
from the vertical axis, we get for the orbital
angular momentum
2
L r h ds h l (l 1) h , where
l is an auxiliary quantum number l
0,1,2,3...(n-1). The value comes near to the
orbital angular momentum of de Broglie's model L
mvr n h , when l is big. Here h h / 2?.
19
The magnetic quantum number m determines the
quantization of the direction z of orbital
angular momentum. We get for the component Lz of
the orbital angular momentum Lz m h, where m
l, l -1, l -2, ... 0,... - l
Lz 2 h
Lz h
Lz 0
Lz -2 h
Lz - h
The vector, which describes the orbital angular
momentum of the electron in an atom, is quantized
in relation to the direction z. In picture N 3
and l 2.
z
The electrons in an atom move towards the
vertical axis and back the faster the more far
from the axis their position is. The motion makes
the electrons in the direction of motion
asymmetric and their projections at the
3D-surface are asymmetric too.
The calculated projection of an electron on the
3D-surface, when l 4 and m 1.
20
The spin of an electron The spin of an electron
is in modern physics an abstract quantum
interaction, which has no physical equivalent in
macroscopic world. We have already defined for
the spin the opposite abstract directions "up"
and "down" or the signs positive and negative.
What issue will determine the magnitude of
spin? When we change the frequency f in the
formula E hf to the angular frequency ? 2?f,
we get the energy of quantum interaction in
form E h ? , where h h / 2? , as before is
already told.
The electron is also parallel to the grid lines
and it has its length in this direction. The
length of an electron is a half of the height of
one layer or h /2. It is the internal property of
an electron, which can be observed only
undirectly at its projection.
h
electron
h /2
The length of an electron is a half of the height
of one regular layer.
The electron is said to be a spin ½-particle,
because of its length in direction of 4.D. The
quantum interaction of an electron can be
positive or negative according to the sign of a
particle. The electrons can not have any
individual properties. The sign of a particle or
its position at a layer or antilayer shares the
electrons, however, into two groups and differs
one electron from an other. The Pauli's rule for
the electrons of an atom shares the electrons
into two groups according to their quantum
interaction or spin. Thus we at last get the
mechanical explanation for this rule. The same
explanation is valid for the proton and neutron
as well. They have also the component parallel
to 4.D and they stand in space or antispace. "If
you will learn the idea of the real electron, you
need to cut all connections to common sense".
The D-theory proves that it does not need to be
that way.
John Wheeler, at year 1984 "The most
revolutionary invention of the sience is not done
yet! And it is not done by questioning the
quantum, but revealing the extremely simple
thing, of which consequence the quantum is."
21
The projections of the particles on the
3D-surface The dualism of the particles or Bohr's
principle of complementarity has been difficult
to understand with the common sense. The model of
D-theory aims to explain this idea in simple way
by using projective geometry in cell-structured
four-dimensional space. The wave nature of the
particles concerns all particles from a photon to
neutron and still more heavy particles like the
nucleus of the atoms. The common thing for all of
them is that they include one or more component
parallel to 4.D. This component allways has a
projection at 3D-surface. The projection is
described by de Broglie's formula for the matter
waves ? h / mv . The extent of the projection
or the wave length ? is inversely proportional to
the absolute hollow ? caused by the mass m of the
particle and to the relative hollow ?' caused by
the relative speed v. The hollows get the
particle to grow away from the observer in
direction of 4.D, and get thus its projection ?
to seem shorter. We get ? k / ??', where k is
a constant. Both the hollows are already
described in the second part of D-theory.
e
c
Let's consider with help of a hydrogen atom the
projection of an electron on the 3D-surface. The
auxiliary quantum number of an electron is zero,
so the projection of an electron is symmetic.
That means that not any direction of its
3D-projection has a privileged status. The
projection describes the vector e on the
3D-surface in limits of the uncertainty and is a
sphere. The magnetic moment of an electron is
zero in the atom. When the electron is described
by a vector e parallel to 4.D, the vector is
always imaginary in 3D-space.
3D-surface
c
The radius rn of the sphere determines the size
of a projection. We can define a cycle time for
the projection of the electron. We get for the
cycle time at orbital speed vn Tn rn / vn n
h n h n h kme k e
mk e The cycle time increases as cube
with the distance n, when n is the number of the
layers. Let's consider first the projection of a
particle, when n 1. The distance of a particle
from 3D-surface is then 137 layers. The
projection or the 3D-sphere follows the motion of
the grid lines past the particle. The size of the
sphere corresponds now to one wave ?1. The wave
appears, when the grid lines travels in relation
to the electron at speed c. The sphere or the
projection thus describes the particle in its
entirety and is in the same phase with it in
relation to the grid lines.
2 2 3 3
2 2 2 4
22
When n 2, the projection of a particle is still
a sphere. The projection of a particle is
possible to share in limits of uncertainty into
room limited by two spheres. The room is limited
in picture by the spheres B1 and B2. The centres
of these spheres are on the surface of an other
r2-radius sphere. The wave length of a projection
is now a half of the r2-radius circle or ?2. When
the grid lines move one cell ahead in absolute
motion, the projection or the two spheres are
rotated in 3D-space allways a 1/8-cycle.
radius r2
B1
?2
?2
B2
When Tn n³ T1 and n2, we need 2³ 8 times
more time that the projection of a particle could
travel at the r2-radius orbit a complete circuit
in comparison with the previous projection with
the radius r1. It means that the grid line must
travel a way 8 layers long in their orbital
motion before one circuit in a projection is
done. Every transition in direction of the
3D-surface corresponds to the time T1. The two
spheres B1-B2 form a symmetric surface of 2³ 8
pairs of spheres around the projection of a
particle and corresponds to the cycle times
t1...t8 T1.
t1... t8
The pairs of spheres in picture should lies
partly inside each other. The pairs of spheres
are rotated in eight phases around the r2-radius
sphere in centre.
t1
B1
B2
t8
When the pairs of spheres corresponding the times
t1...t8 are rotated a whole circuit, the result
is a pair of symmetric waves or the waves ?2 in
length. The places of the projection seem to
travel on the surface of the r2-radius sphere one
circiut in timeT2 8T1. The sphere limits the
size or the wave length of both projections to
?2.
23
The projection of a particle has no exact place
on 3D-surface and the size of a projection
depends on the momentum of a particle. We can
thus mention that the projection of a particle is
a wave, which has a certain wave length and cycle
time. The amplitude is its spin or the magnitude
of quantum interaction. In part 2. of D-theory
we considered the double slit experiment, in
which the electron travels simultaneously through
two slits in 3D-space. When the projection of the
electron is now proved to be a wave, can the
interference-effect in two slits be explained
with wave nature of the projection. The details
of the interference is not considered in
D-theory, but the effect is possible to explain
with means of the wave theory. When n3, the time
to travel around the projection is now T3 3³ T1
27T1. The r3 -radius sphere is shared into
three parts. The wave length is ?3. The times of
transitions from a cell to another are t1...t27
T1. The 27 spheres B1...B27 are now needed for
each cycle
B1
B3
de Broglie's model of matter wave is possible to
explain with help of projective geometry. The
particle has an exact place outside of the
3D-space in cell-structured space. The projection
of the particle at 3D-surface is the matter wave
of the model. The dependence of a wave length on
the mass and speed of the particle can be
explained with help of the absolute and relative
hollows in four-dimensional space. At the same
the Bohr's principle of complementarity or the
nature of a particle as a wave and as a particle
gets an explanation.
radius r3
?3
?3
?3
B2
The projection of a particle on the 3D-surface
expresses the location of a particle and also the
place, where the particle may be realized to the
measureable quantity in 3D-space. The sharing of
the projection to the parts ?1... ?n, is caused
by the four-dimensional cell-structuted space.
The probability for a particle to be found in one
of all the spheres B1...Bn is the same, when the
projection is a symmetric sphere. The particle
has, however, in four-dimensional space a
location, which is desrcribed exactly with four
coordinates. The disobservation law limits the
observation and only the projection is possible
to observe. When n increases, begins the
projection of a particle to resemble a surface of
a sphere, which is limited by relatively still
smaller spheres B1...Bn.
In part 1. of D-theory the projection is
described with the quantum interaction h. The
quantum interaction h forms a one layer high
surface, of which side parallel to 4.D is the
momentum p mc or the total energy E mc² of
an electron and the other side is the length or
the time parallel to 3D-surface. Next we consider
the projection of an electron in aan atom and the
potential energy of an electron with help of the
space model and quantum interaction.
N 1/2 (Spin 1/2)
h h / 2?
h h / 2?
h
r 137d
Eh mc²
The projection of an electron
24
h
The height p of the area is in direction of 4.D
proportional to the orbital speed at a layer or
to momentum p mv, where m is a constant. The
length of the area at 3D-surface is the circuit
?c 2?r of the r-radius projection. The area h
of the circuit is h ?c p 2? r mv
2?137dmc or h/ 2? 137dmc h. The length
of the circuit ?c 2?137d 0.0243 Å h / p h
/ mc is so called "Compton's wave length". This
wave length ?c defines the quantum interaction h
at all layers in normal and reciprocal space and
is the wave length of an electron in normal space
at layer n1. For all layers in reciprocal space
is valid the known formula n h n²137²dm x c
/137n m x n² r1 x v1/ n (D-theory,
part1.) , where n 1,2,3...,137 and r1 137²d
and v1 c /137. The areas nh form at
3D-surface the circuit Ln in the projection at
layer n. The projection includes n pieces of
areas h.
pmv
?c
n2
h
n3
3h
Ln
?n
1 layer
e-
e
The length of a circuit at different layers n is
Ln 2? rn 2? n² x 137²d. The area of the
projection is divided into n part so that the
size of each area is h and the length of the area
is ?n Ln / n 2? x 137²dn and a speed
corresponding the height of an area vn c / 137n
, which again gives for the area h m vn ?n
2?137dmc (and h 137dmc). The momentum ph and
total energy Eh of the quantum interaction h h
/ 2? are calculated with help of the space model.
The momentum ph is the height of the area h in
direction of 4.D and it is different for each
layer ph mv h / ?n. The energy Eh is for a
particle the same at all layers in normal and
reciprocal spaces and comes from the formula h
137dmc by multiplying both sides with the speed c
or h c 137d mc² 137d Eh , where mc²
Eh. For the total energy comes Eh h c / 137d
hc / 2?137d. When 2?137d / c T, where T is
the time, which the light uses for the length ?c
2? r 2?137d, the pevious is written Eh h
/ T .We know that on the 3D-surface the width of
one layer is the same as the diameter of a
proton, which is d 2.82 fm. We calculate from
this the total energy Eh of an electron.
25
When T 137d / c, we get T 2?137d / c 80.97
x 10 s. For the total energy Eh Eh
mc² h / T 81.8 x 10 J 0.51 MeV. The
total energies of an electron and positron are
Eh, when their mass is on grounds of the space
model m Eh / c² 9.1 x 10 kg. We can
now mention that the direction of an electron in
space is 4.D and its total energy and mass can be
expressed on grounds of the space model with help
of Planck's constant h and the speed of light.
The formula h 137dmc is valid at all layers in
normal and reciprocal space. From the formula Eh
T h we get by setting the time T to cycle time
T 1 / f Eh hf. The formula expresses, how
is the energy of a particle parallel to 4.D
projected with help of the area h to energy on
the 3D-surface. The energy may be the total
energy of a particle or it may be also the
potential energy in direction of 4.D, which is
considered next. When an electron of an atom
changes its layer, a photon is describes the
potential difference between the two layers. The
energy of a photon is thus a potential energy. An
electron of a hydrogen atom stands normally in
reciprocal space at the layer n 1 and a proton
stands in a nucleus at the 3D-surface. An
electron has a negative potential energy -Ep,
which is described with a vector parallel to 4.D,
and which is determined by the charge number of a
nucleus Z1. Let the height of a vector H be the
same as the distance of an electron to a proton
or H1 layer. Let's use the length of a place
vector to describe the magnitude of potential
energy Ep of an electron. The electron at the
layer n1 has a projection, of which radius R at
3D-surface is as before R 137² layers. Let's
calculate the area A, which corresponds to the
energy Ep. The area A is defined by the radius R
and the height H. A HR 137 x 137² 137³. In
such space the ratio between the energy of a
particle and the areas of the projections is
inversely proportional. We got before for a
½-layer high vector the area A corresponding its
total energy Eh (D-theory, part 1., page 34).
A½ n x N 137 x 1. We can cange the area
inversely to area of 1-layer high particle A1
137 /2. (Such particle does not exist!)
-22
-15
-31
Now we get for the ratio of energies Eh and Ep
inversely with help of the areas Eh / Ep A /
A1 137³ / (137/2) 2 x 137². Now we can solve
the pontential energy Ep Ep Eh / (2 x 137²)
81.8 x 10 J / (2 x 137²) Ep ( 0.00218 x
10 J) / (1.6 x 10 J/eV) 13.6 eV
-15
-15
-19
26
Ep
The potential energy Ep of an electron at the
layer n1 changes inversely with the quadratic
radius R of the projection of an electron. The
radius R is determined by the charge number Z of
a nucleus. When Z 2, R 137²d / 2 and Ep' 4
Ep. The radius R is then not a multiple of d and
it is not needed to be. We consider here the case
of a hydrogen atom or Z1. Let's consider the
potential energy Ep of an electron with help of
the area A at different layers n 1,2,3...137.
A
H137
R137²
R
The area A HR of the potential energy Ep.
We get for the potential energy of an electron at
the layer n 1 as before Ep ?² mc² / 2
,where ? 1/137 and m is the mass of an
electron. We have got for the ratio of energies
Eh / Ep A / A1 2 x 137². We get the radius Rp
of the projection of a photon corresponding the
potential energy Ep 13.6 eV by multiplying the
radius r 137d of the projection of an electron
with this ratio. We get Rp r Eh / Ep 2 x
137³d. We use now the length of a circle of a
projection as the wave length ? of the
projection. When d 2.82 fm, we get for the
wave length of a photon corresponding the energy
Ep 13.6 eV
-10
? 2? Rp 2? x 2 x 137³d 914 x 10
m. The frequency of the wave ? is f c / ?.
With the formula E hf we get inversely Ep
hf hc / ? 13.6 eV. This gives 1 / ? Ep /
hc R , where R is so called Rydberg's
constant. We substitute now to the formula, which
we already got, Ep ?² mc² / 2 in place of
total energy Eh mc² h c / 137d and we get
Ep ?² h c / (2 x 137d) , which gives for the
Rydberg's constant R Ep / hc ?² / (2? x 2 x
137d) 1 / (4 ? x 137³d) 1.097 x 10 Å We
can now make the conclusion that in space exists
a structure like a grid parallel to 4.D, where
its internal forces spreads as the surfaces in
four-dimensional space.
-3 -1
27
(Note! The energy Eh is compatible with the
energy operator E of Schrödinger's wave function,
and the momentum ph with the differential
operator p, when 4.D is imaginary and E -h /
(2? i ?/?t), p -h / (2? i ?/?x), because the
wave function describes the projection of an
electron at the 3D-surface.) We can write for
the radius R of the projection of an electron in
reciprocal space at layer n 1 R D
, where D 137²d. Let's broaden the previous
equation with numbers n 1,2,3, ...137, when d
h /137dmc n²R n²D n²137² h / 137mc h c /
(137mc²/137²n²). The orbital speed at different
layers n is v c /137n or v² c² /137²n² ,
when we get n²R h c / 137mv². We got before in
connection with the fine structure constant ?
ke² / h c 1 / 137 or ke² h c / 137. Thus
we get n²R ke² / mv² and
mv² ke² n²R mv² ke² .
n²R (n²R)² When n²R rn, the previous
result is the same equation as we got before with
help of the forces Fs me v² / rn and Fe
ke² / rn² for the orbital motion of an
electron in an atom Fs Fe. In normal space,
however, it is valid for the radius R of the
projection R n x 137d and for these radius the
equivalent ofthe forces Fs and Fe is not valid in
the same way (, exept at the layer n 137). An
electron can thus not stand in an atom at the
side of the normal space. Let's at next consider
the potential energy of the electric fields of an
electron anda positron on grounds of the space
model, and the annihilation of these particles.
28
According to the Coulomb's law the potential
energy Ep of an electron in the electric field of
a charge q e or of a proton is Ep -ke² /
r . Let's calculate Ep, when the distance r
between the electron and nucleus is r d, where
d 2.82 fm. We get Ep -ke² / r -ke² /
d. We can substitute to the previous formula
ke² h c / 137 , which is already known, and
we get Ep -ke² / d - h c / 137d. When we
got the total energy of an electron Eh mc² h
c / 137d, we can now write Ep - h c / 137d
-mc² -Eh . We can now mention that at the
distance d from the charge e standing electron
has the same potential energy as the total
energy of an electron or Ep -Eh . According to
the space model the electron can not stand any
closer to the nucleus. Thus we get to the maximum
value of potential energy of the charged
particles e- or e in electric field of the other
similar charge allways Ep -Eh . When the two
particles with the opposite charges are united
and the potential is disappeared, must thus an
energy E be released according to this maximum
value or E 2Eh 2mc². Let's presume that an
electron has at the beginning been in place or in
rest, when its mechanical energy is E 0. The
electron then rush free towards the charge e and
its mechanical energy E is finally the sum of the
kinetic and potential energy or E Ek Ep
Ek - mc² 0. The kinetic energy is now Ek mc².
Let's presume that the electron e- and the
positron e get closer each other from the rest
and then collides. In the collision E Ek Ep
0. When the particles are unified, their charges
disappear and at the same the potential energies
Ep of electric fields disappear as well. The
kinetic energies Ek mc² of both the particles
are left. When the antiparticles collides, two
photons are created. Both the photons have the
energy E mc² hf. When f c / ? or ? c /
f, we get
? hc / mc² h / mc 2.43 x 10
m. This wave length of a photon is the same as we
got before ?c 2?137d 0.0243 Å h / mc, which
is so called Compton's wave length. The masses
and charges of the particles e- and e have
disappeared and changed to two photons. Both the
photons stand on the side of the normal space at
the layer n1 and their length of the circle of
the projections are ?c. When the momentums of the
particles were the same but in the opposite
directions, also the directions of motion of the
photons are opposite.
-12
29
It is important to understand that when the
antiparticles e- and e are united, the number of
the released energy is explained by the
disappearing of the charges. The masses of the
particles only switch off each other or the sum
of masses of the opposite sign is
zero. Annihilation The annihilation is understood
that a particle and its antiparticle are unified
and changed to energy. This idea is, however, not
the whole thruth. When the electron-positron-pair
disappears, the masses of the particles switches
each other off. The uncharged matter particles
pull each other because of the gravitational
force, for example. Correspondingly the uncharged
antimatter particles pull other antimatter
particles. The matter and antimatter
correspondingly push each other. The pushing is
observed as an expanding system. The closed
system, which includes uncharged particles of
3D-space and their antiparticles, is labil. The
relative speeds of the particles and
antiparticles start to increase in collisions. It
seems, as energy would be increased to the
system, but the the labil mass only increases.
The kinetic energy of the particles and
antiparticles increases and the masses grow also
according to the invariance equations. The
phenomenon is inverse to the another one, when
two particles with the same signs are connected
together and their rest mass will reduce and
change to binding energy. Now instead appears the
"fly energy". The kinetic energy Ek must be equal
to the rest energy Eo of particle, when the total
energy E of particle is double to the rest energy
Eo E Ek Eo 2Eo .
moc² 2 moc² 1 - v² / c² 1 -
v² / c² 1 / 2 v 0.87c . When the relative
speed between a particle and an antiparticle is
over 0.87c, the particles are destroyed in their
collision and the kinetic energy is released.
The number of total energy is a constant during
the whole process or E 2Eo. So that the
particles like a neutron and antineutron could
meet each other, they must have a kinetic energy
to win their mutual push force. They get their
kinetic energy from the collisions before the
annihilation. The kinetic energy is released in
annihilation. It is as big or bigger than the
rest energy E moc ² of their rest masses. The
masses of the particles switches each other off
and do not produce any energy.
30
The potentials of the space grid The internal
force Fi of the grid aims to keep the grid
homogenous in all directions of space. In the
homogenous grid Fi is constant and the internal
potential of the grid is zero. When, for example,
an additional number of electrically positive
particles or a positive charge is brought into
the grid, a potential