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D-theory
New interpretation of quantum mechanics
based on quantized space and time
&
combining gravitation with quantum
mechanics
Pekka Virtanen
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email: [email protected]
1
Introduction
One important theoretical achievement of natural sciences is the idea of atom. The matter can
not be divided endlessly into still smaller parts. The idea of atom hints that in the world exists a
special spatial scale, the scale of atom. The physicists believe that all physical phenomena
appear from the effects of quantum level. The scale is always connected to the space.
What is empty room or space? What kind of structure and of properties does the empty space
have? Does the shortest indivisible length exist? Are the directions quantized in the smallest
scale? The existence of a special scale refers to a quantized space or cell-structured space. In
that case the space can be described with help of the background independent unit vectors,
which span the cells. This kind of space is absolute, but it is not the same as Newton’s absolute
space. It is not possible to observe the empty space directly, but it is possible to examine its
structure theoretically. When the space is depicted as cell-structured, several strange quantum
effects can be interpreted in a new way. Coarsened observations are needed to make the
observer’s space appear from the cell-structured space. The classical observer’s space
appears geometrically as an emergent property of the absolute space. There exist two different
images of one space, coarsened and noncoarsened, linear and nonlinear.
D-theory is a new interpretation of quantum mechanics. It is based on the hypothesis, which
defines the structure of space. The cell-structured space of D-theory will solve the
measurement problem of quantum mechanics. It will, for example, produce the Lorentz
transformations, which the Theory of Relativity is based on.
When mathematics is suitable to describe the effects of nature and it is an abstract part of the
world, must the exhaustive physical theory be able to describe also the basics of mathematics,
such as the origin of the sets of numbers. The space is also a mathematical concept and the
absolute space combines the physical world with the basics of mathematics.
The physicists have tried to interpret the quantum mechanics over 80 years and no satisfactory
interpretation is found. Observer’s consciousness seems to be a part of the measurement
process. The model of the cell-structured space gives a new point of view on the role of
consciousness in quantum mechanics. Also another issue in interpretation, the non-locality, is
cleared up with help of the space model and of the violation of Bell’s inequality. The non-locality
is a strong evidence for validity of the used model. The third issue in interpretation is the wave
function of a particle. It is a mathematical abstraction. It has in D-theory a direct connection to
the complex absolute space, which is not unique and observable for a macroscopic observer
because of its structure (Manhattan-metric). Thus for example the place of an undetected free
particle is not unique and the particle looks like a wave. A measurement however gives for the
particle its place in the linear and unique observer’s space or in other words the wave function
of the particle “collapses” simultaneously everywhere.
Rotations in symmetry spaces are fundamental in Standard model of QM, as well the so called
gauge principle. They have direct connections to the properties of quantized space and time.
When the physical space includes also the quantized complex space, the rotations of a
macroscopic stick in the cell-structured space are length-remaining.
Finally stays left a modest question “What is everything?". D-theory shows that it is impossible
to get answer to this question. One abstraction stays always left in the model. But only one.
2
Pekka Virtanen
Studies of physics and mathematics
in University of Helsinki, Finland
D-Theory - Model of cell-structured space
Part  : Space and time
Hypothesis of theory:
In large scale the physical space is a background independent, cell-structured, threedimensional surface of a four-dimensional hyperoctahedron. It is absolute and quadratic in
comparison to the Euclidean observer’s space. Inside and outside the closed surface exists a
cell-structured complex space extending to a limited distance from the surface. Manhattanmetric is valid in the space.
(The observer’s space is an emergent property of absolute space. It appears from the absolute
space by coarsened observations and it is different for every observer depending on the
observer’s motion. It is the three-dimensional surface of Riemann's hypersphere.)
Abstract :
The background independent cellular structure of the absolute space were defined. Appearing
of the observer’s space from the absolute space as its emergent property were described. The
Lorentz's transformations were derived from the space model. The rotations of a macroscopic
stick were proved to be length-remaining in a cell-structured space.
A solution to the measurement problem in quantum mechanics were proposed. A new
interpretation of wave function collapse and of violation of Bell's inequality were proposed. The
uncertainty principle and the phase invariance of a wave function are derived from the space
model.
The structure of the cell-structured complex space outside the 3D-surface were defined. The
charge, the spin and the rotations of an elementary particle and the symmetry groups in the
cell-structured complex space were defined. The geometric structure of the fine structure
constant were defined. Time and the momentum of a particle were quantized. Influence of
gravitation were described on appearing of the observer’s space.
The four-dimensional atom model and its all quantum numbers and projections on the 3dimensional surface of the hyperoctahedron were defined geometrically. The accurate values
for proton diameter, Rydberg’s constant and the radius of a hydrogen atom were derived.
The geometric structure of quarks and of the three families of particles were defined.
The locality of mass, length and time were introduced in absolute space with help of the
asymmetric wave function.
It was shown that the electromagnetic fields are caused by the effects of the complex space
and that the model is compatible with the Maxwell's equations.
3
The new D-theory 2.12 is published
D-theory presents a new way to deal with all physical effects. The theory is based on
geometry, algebra and logic.
The general theory of relativity is based on geometry, but in quantum mechanics a geometric
description is missing. The geometrisation of quantum mechanics based on abstract algebra
is necessary to combine these two theories.
At the beginning the geometry of absolute space based on Manhattan-metric is defined in
large and in small scale. The absolute space is described cell-structured and quadratic in
comparison to the observed space. It is shown that it is not possible to observe the absolute
space or time by any observer. The realization of Lorentz's transformations in the absolute
cell-structured space is a strong evidence of the validity of the used model. Also the violation
of Bell's inequality in experiments gives support to the space model.
According to the model of D-theory the background independent cell-structured space is the
only substance (base of reality) that is needed. Even time and elementary particles are a part
of the space. The absolute space, however, does not prove to be unique, which explains for
example so called wave function collapse in measurement.
D-theory shows mathematically that the world is reductionistic. All macroscopic phenomena
appear from the effects of quantum level.
D-theory explains the birth of the Universe with the increasing number of dimensions. The
known world did not appear directly as three-dimensional. The increasing number of spatial
dimensions from 0 to 4 is an idea, which has been missing from the story of the Universe.
For example, the exact value of Rydberg’s constant, the mass of electron and of proton are
derived with help of the space model, as well the diameter of proton and of hydrogen atom
are derived with help of the space model.
The four-dimensional atom model produces geometrically all quantum numbers of the
electron in an atom and for example the geometric description of Higgs’ doublet field.
The Euclidean 4-dimensional space defined by mathematicians, where the 3-dimensional
bodies can appear from emptiness, is not the space of the D-theory and does not match with
the observations. The four-dimensional space can be defined with several ways. The
Minkowski's space-age is only one example of all these. The space of the D-theory will
match better with the observations and gives answers to many open questions of physics.
The cell-structured space is defined background independent. It means that the space is
observed only from inside. The cells, which form the space, do not need any background. A
cell has its location and properties only in relation to the other cells, not to the background.
The observers themselves are made of the cells and are completely determined by the
properties of the cells. The observers belong then to the same set as the objects of the
observations.
The theory is divided into two parts or files: 1. Space and time. 2. Gravitation and
electromagnetism.
4
Contents, part 1:
D-theory versus the Standard model of quantum mechanics
The unreasonable efficiency of mathematics in physics
Background of D-theory
Cell-structured absolute space
Complex space
Calculating a distance in cell-structured quadratic space
Calculating a speed in cell-structured quadratic space
Calculating a time in cell-structured quadratic space
Dualism
Classes of phenomena
The smallest scale
Isotropic and quadratic space
Schrödinger’s equation for elementary particles
Neutral gravitational wave
Conservation of momentum in Manhattan-metric
Disappearance of interference
Compton-wave length
Particles
Quantization of momentum
Non-locality in quantum mechanics and ”spooky action-at-a-distance”
Chaos and determinism in cell-structured space
Rotations and gauge principle in cell-structured space
The uniqueness of space and " wave function collapse"
Localization of body or how does the observer’s space appear
Normal and reciprocal space
Uncertainty principle in cell-structured space
Geometric derivation of proton mass
The absolute orbital motion in a loop-space
Asymmetric particle
Time is not a substance
The principle of simultaneous
Lorentz's transformations in loop-space
Spin-rotations
Rotations in the lattice space
Charge symmetry
Electron in a lattice box
The families of particles
Virtual photon
Projection of electron on the 3D-surface
Geometric derivation of Rydberg’s constant
Quantum interaction
The lattice lines form the famous ether
The structure of photons
Properties of the lattice
Atom model
Sources
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D-theory versus Standard model of quantum mechanics
Understanding of D-theory does not insist on deep familiarity with Standard model. One
foundation of Standard model is the wave function and the global and local invariance of its
phase or so called gauge principle. According to the global gauge principle the phase of a wave
function can be changed at all points in space and in time only at once and only with the same
number. The Standard model does not give any physical meaning for the wave function. Dtheory explains geometrically with help of its space model what are the wave function and its
complex phase and how does the gauge principle occur. The developers of the gauge principle
considered it to be against the principle of relativity. D-theory proves that the gauge principle is
only apparently against the principle of relativity. The gauge principle is applied in D-theory to
electromagnetism and to gravitation.
An other foundation of Standard model are the rotations in symmetry groups U(1), SU(2) and
SU(3). D-theory explains so far the meaning of the groups U(1) and SU(2) in electromagnetic
interaction and as well the reason why the rotation groups are important. The features of the
rotation group SU(2) are applied to the geometric description of spin-½-particles together with
an abstract isospin-space. Symmetry group SU(3) is used in depiction of the color force.
Energy and fields are quantized according to the Standard model. According to D-theory also
the directions in space and the lengths and also time and momentum are quantized. Time is in
Standard model a parameter and the model does not explain the nature or time. The space
model of D-theory describes them at the level of quantum effects. The Standard model includes
several different substances for the physical world. According to the D-theory only one
substance is needed. The only substance explains in principle all physical phenomena.
Standard model includes the ideas ”accident” and ”probability”, but D-theory does not need
them. According to the D-theory the world seems to be completely deterministic. The Standard
model does not offer so far any tested model for gravitation. So the gravitation has not been
able to be combined with quantum mechanics. D-theory anyhow presents a model to combine
gravitation with quantum mechanics. The model describes appearing of gravitational field with
help of a particle model. The model also describes quantitative the properties of the three basic
quantities, time, length and mass, in a gravitational field.
The Standard model does not help at all to understand or to interpret the measurement
problem in quantum mechanics, the wave function collapse or the non-locality in context of
entangled pair. The physicists argue if the world is non-local or indeterministic or both. D-theory
presents an interpretation and explanation to these old questions of physics.
Quantization of space and time is missing from the Standard model, but it is an essential idea
in D-theory. Energy, particles and fields are already quantized in physics. Next the space and
time are quantized. It is the third quantization and a new paradigm. When we think against the
Standard model that the space is cell-structured, we seem to meet a problem. The space is
observed isotropic or similar in all directions. How could the space then be cell-structured? We
get an answer, when we define the structure of space and matter in a certain way. So let’s
define the cell-structured space in a way that makes it seem isotropic in macroscopic scale.
This definition leads us, for example, to understand quantum effects in a new way based on
geometry. The definition is also the hypothesis of the theory.
6
Background of D-theory:
Geometry
General relativity:
When the existence of gravitation
depends on the frame of reference,
gravitation is the feature of the
geometry of physical space.
Quantum mechanics:
- non-locality
- uncertainty principle
- statistical
- problems in interpretation
- abstract algebra
Extension:
Everything, which can exist, is the
feature of space. The space is the
only substance.
Pekka
Virtanen
D-theory
- local Manhattan-metric
- deterministic
- space and time are quantized
- observer’s space is only an image
Physical
reality
Change of paradigm:
According to modern physics the absolute space does not exist. According to
D-theory only the absolute space with its several properties can exist.
7
The unreasonable effectiveness of mathematics in physics
Many effects of the nature are described effectively by means of mathematics. Mathematics
seems to have a direct contact to the basic effects of the nature and the reason is unknown.
The basics of math, like the sets of numbers, appear as internal abstract feature of the world
and they can not be chosen high-handed. So it can be presumed that they have a connection to
the internal structure of the world.
Hypothesis:
What kind of physical space - that kind of algebra. What kind of physical space - that kind of
geometry or What kind of physical space - that kind of mathematics.
This means that our physical space has affected to the result of developing our mathematics
and logic. The space is an essential factor in all physical effects and the space is also a
mathematical concept. We can think that an abstract mathematical theory tells about the nature
of the physical space. We get information about our physical space by examining the basic
ideas of mathematics.
One example is the imaginary numbers. Let's consider a strange number i, which is not from
this world. It has no size and it can not be negative or positive. This number anyhow lies at its
straight axis of numbers, which has in the space of numbers an imaginary direction. This
straight axis is perpendicular to the axis of real numbers. An imaginary number is possible to
get visible or real by adding a new perpendicular direction to it or by squaring it. So this number
i can be understood as a number, which has in the space its own direction, which we never can
observe, but its square has a real value.
The imaginary numbers appeared into mathematics a long time ago, but they were not fully
understood until the idea of a complex space was born. Our three-dimensional space of real
numbers gets one more dimension in this way. The appearing of imaginary numbers into our
mathematics expresses that our physical space is four-dimensional, and also that in principle it
is not possible for us to observe the fourth, imaginary direction in our space. Still we can use
the complex numbers to handle phenomena in direction of the fourth dimension.
By writing a transform
X = x² , Y = y² , Z = z² and I = i²
we transfer from (x,y,z,i)-observer’s space to a quadratic 4-dimensional space (X,Y,Z, I). This
kind of space (X,Y,Z, I) is called an absolute (or invariant) space, because all 4 orthonormed
directions in space are there observable or real. For example, a square
±X ± Y = 1
is transformed from the absolute quadratic space to a circle
x² + y² = 1
into the non-quadratic observer's (x,y,z)-space. The previous transformation, although it is
mathematically uncontrolled, can really be made with certain preconditions and certain
consequences, which are told later.
8
In absolute 4-dimensional space the Pythagoras' theorem is written, for example,
ds = a + b + c + d , where a  b  c  d.
In the observer's space the same theorem is ds² = a² + b² + c² + d².
The expanding of the sets of numbers is described with the next diagram:
Natural
numbers
Integers
Negative
Integers
Rational
numbers
Fraction
numbers
Real
numbers
Irrational
numbers
Complex
numbers
Imaginary
numbers
The range of numbers is not possible to expand any more! So the set of the complex numbers
is the widest possible set of numbers , which includes certain algebraic basic features
(commutativity, associativity, distributive law, neutral element, negative and inverse element). It
is also algebraic closed set of numbers.
All these sets of numbers exist in the observer's world or in n-dimensional Euclidean space or
surface. The existence of the complex numbers there means that in absolute physical space
the number of dimensions (basis) is n+1. We presume that the observer's n-dimensional space
is closed and n = 3.
Group theory ia applied successfully in quantum mechanics, especially the Lie’s algebra. It is
abstract algebra, which examines the features of rotations in different spaces. The Lie’s groups,
which depict the properties of the real particles, are U(1), SU(2) and SU(3). They all three are
based on complex spaces. Mathematician Felix Klein proposed that the geometric objects do
not characterize and define the geometry, but rather the group transformations, which keep the
geometry unchanged, or the symmetries. The different spaces have different symmetries. We
can say that the geometry of the space is defined best by its symmetry group. The use of these
symmetry groups in physics hints that the particles stand in complex space.
Finding the formula for solutions of the fifth or the more order equation is proved to be
impossible. The proof is based on group theory and on the symmetry properties of assumed
solutions or in the last analysis on geometry. In four-dimensional absolute space the fourth
power of a variable determines a volume, which still fits into the space. If our space would have
one dimension more, its symmetry properties would be different. Also our math would be
different and obviously the general fifth order equation would have in that space a formula for
the solutions.
Mathematics includes the idea of infinite. We can add any number with an another and the
result fits always into the axis of numbers. The axis never ends. The idea infinite means that
the space has no end. Then the physical space must be a closed structure, which is possible to
travel around, but not in any observable way. (It is possible to travel an infinite way round a
closed circle.)
9
The mathematician and logistic Kurt Gödel showed that it is not possible to prove watertight
true or false all theorem in any axiom system of mathematics, which is finally based on the fact
that the space is closed and it is not possible for the observer to exit outside to see, what is true
and what is not. So we can never know, where our physical space stands in relation to
something else.
D-theory shows that the structure of space in large and in small scale is an essential factor in all
effects of physics. Therefore physics can not reach its final form without finding first this
structure. The structure of space leads us to some later logically and geometrically derived
issues as, for example, the constancy of the light speed.
The structure of space is not a hypothesis in Relativity Theory (but a mystery). The two
hypothesis in Relativity Theory are according to Albert Einstein:
1. The speed of light in vacuum is the same in all moving sets of coordinates.
2. The laws of physics are the same in all evenly moving sets of coordinates.
These both hypothesis can be derived logically from the hypothesis of D-theory concerning the
structure of space. (The hypothesis was written at the beginning of this document.) Many
observations support the hypothesis of D-theory. For example:
- Lorentz's transformations for time and length are observed at high speeds.
- The spin of fermions gets the values ½ and -½.
- Bell's inequality is violated in experiments.
- A wave function collapses in measurement.
- Observer’s consciousness seems to have a role in measurement process.
- The observations support the idea that the space is Euclidean or flat in large scale.
- Michelson-Morley's experiment shows that the speed of light is the same in all directions.
- In double slit experiment the electron seems to move through both slits simultaneously.
- The magnetic field is curled and sourceless and it is perpendicular to the electric current.
It is told later, how these results are linked to the hypothesis of D-theory. In addition the ideas of
mathematics support the hypothesis of D-theory.
Mathematics is an abstract issue. Abstract is also the physical absolute space, which is
impossible to observe, as soon is proved. Absolute space is in physics an abstract limit, which
is not possible to cross in understanding the nature. The absolute space is the shared base of
mathematics and of physics and it explains the unreasonable effectiveness of mathematics in
natural sciences.
The School Of Athens
Paul Benioff: “The final Theory of Everything should not only unify physics but also offer a common
explanation for physics and mathematics”.
10
The cell-structured absolute space
The expanding space is described as a space spaned by set of orthonormal base vectors so
that the number of dimensions (or of bases) increases with the dimension number N = 1, 2, 3…
At the beginning N=1 and increases with the expanding space.
The space is defined simple as possible by starting from a 1-dimensional line segment. The line
segment is an abstract model for an unknown substance of nature. The line segment is
background independent and will span or create the space. The line segment has 2 ends and
its length is one unit. The line then turns 90 degrees to a new dimension and we get a square,
which has 2 diagonals or two main axes. The diagonals will cross each other, which divides the
both diagonals into two segments of lines. Manhattan-metric is valid in the space.
Y
y
X
x
When N = 2, the absolute space can be described as a square in set of coordinates (X,Y)
lXl + lYl = 1 , when lXl,lYl <= 1.
The imagined sides of the square are at distance X+Y = 1 from the centre of square, when the
distance is measured only parallel to the main axes as the lengths X and Y. The absolute
space (X,Y) is according to the hypothesis of D-theory presented at the beginning quadratic in
comparison with observer's space in the set of coordinates (x,y). In transformation
±Xx², ±Yy²
, we get for a square in observer's space
x ² + y ² = 1.
That is the unit circle. The previous transformation can be made with certain preconditions and
consequences, which are told later. (The observer's space is described later in D-theory.)
The square then turns 90 degrees to new a dimension and we get an octahedron with 3
diagonals. We can say that the space has now 3 basic vectors or main axes and N = 3.
Octahedron is a regular polyhedron, which includes 3 diagonals and 6
vertex. The number of faces is 8. The diagonals are of equal length and
perpendicular to each other. Every point of the 2-dimensional imagined
surface of an octahedron lies at the same distance from the centre,
when the lengths are measured parallel to main axes only or
lXl + lYl + lZl = 1 , when lXl,lYl,lZl <= 1.
The diagonals of an octahedron defines the distances of space in
directions of the 3 main axes. The diagonals will cross each other,
which divides the diagonals into two segments of lines.
11
The absolute space (X,Y,Z) is quadratic in comparison with observer's space in the set of
coordinates (x,y,z). In transformation
± X  x ² , ± Y  y ² , ± Z  z ² , we get for an octahedron in observer's space
x ² + y ² + z ² = 1.
That is the unit sphere in observer's space.
The octahedron then turns 90 degrees to new a dimension and we get an hyperoctahedron (or
hexadecachoron) with 4 diagonals. We can say that the space has now 4 orthogonal basic
vectors or main axes and N = 4. The hyperoctahedron includes 16 tetrahedra. The surface of a
hyperoctahedron is 3-dimensional and it can be filled with 3-dimensional irregular tetrahedra. All
4 diagonals or dimensions are equal in the hyperoctahedron and one can not differ from the
others.
Every point of the 3-dimensional surface of a hyperoctahedron lies at the same distance from
the centre, when the lengths are measured parallel to the main axes or
lXl + lYl + lZl + lUl = 1 , when lXl,lYl,lZl,lUl <= 1.
In transformation
± X  x ² , ± Y  y ² , ± Z  z ² and ± U  u ² , we get in the observer's space
x²+y²+z²+u²=1 ,
which is the Riemann's hypersphere. In the hypersphere the directions of the main axes have
disappeared and the surface of the hypersphere is 3-dimensional.
In simplified picture the hyperoctahedron has eight vertex.
Visualizing of a 4-dimensional object in 3D-space is
impossible. When a hyperoctahedron is cut by a plane, which
is perpendicular to any diagonal, the result is an octahedron.
Because there are four diagonals , the results are named as
Ox, Oy, Oz, and Ou. The surface of a hyperoctahedron is 3dimensional and cell-structured. On this surface it is possible
to set in any way a local 3-dimensional orthonormed set of
coordinates (x,y,z). Then the fourth spatial direction u is
always in space perpendicular to the surface.
12
When the observer travels on a surface and transfers from one face to another, changes the
fourth dimension to another so that each of dimensions X,Y,Z and U are in their own face
perpendicular to the surface.
The local 4-dimensional set of coordinates forms a space, where the fourth coordinate has a
special status at 3-dimensional surface in comparison with the three others. Locally it is called
"fourth dimension" or "4.D" and it is impossible to observe directly at an Euclidean 3D-surface.
The fourth dimension is always edged, when the three others are closed through the surface
and have no end or edge.
The 3-dimensional surface of a hyperoctahedron can be partly filled with 3-dimensional
tetrahedra. The tetrahedra are not regular. Eight irregular tetrahedra form a regular octahedron.
We can define innumerable number of regular octahedra to build the 3-dimensional surface of
a hyperoctahedron. They build there also layers, which are as thick as the diagonal of an
octahedron. The thickness of a half of regular layer is the smallest useable length unit. We can
think the thickness of 3D-surface in direction of 4.D to be zero (or equal to so called Planck’s
radius as later is told). The octahedra fill only a part of the 3D-space. The rest are filled by the
reversed octahedra or antioctahedra, as soon is told.
Two irregular tetrahedra. Eight tetrahedra are
needed to build one regular octahedron.
+
+
_
+
_
_
The cell-structured 3D-space. Each cell is as
far from the centre of 4-dimensional space in
direction of 4.D.
The location of each origin in the net of
diagonals is determined.
In absolute space the lengths exist only parallel to the diagonals of octahedra or to the main
axes of space. On the 3D-surface of the hyperoctahedron the axes stand in three directions.
The metric of this kind of space is called for Manhattan-metric.
The centre of every octahedron forms an origin so that on the one side of the origin the half
of the diagonal is positive and on the opposite side it is negative. Then the location of the
origin in the net of diagonals is determined. The positiveness and the negativity are possible
to define so that their absolute value is bigger than zero but their sum is zero. The issue is
considered more later in D-theory.
13
The octahedra do not fill the 3-dimensional space completely but only 2/3-part of it. Outside the
octahedra stays regular tetrahedra T, which are each divided into four irregular tetrahedra t.
We can define for an octahedron its "inside out"-object or an antioctahedron made of eight
tetrahedra t. Together the octahedra and their antioctahedra fill completely the 3-dimensional
space. Their parallel but separate diagonals are of equal length and form there layers and
antilayers. When the octahedra are regular, also the tetrahedra T are regular and form the
antispace.
The physical cell-structured space is
2 irregular tetrahedra t
built of the diagonals of octahedra
and antioctahedra. This division to
two different spaces means for the
elementary particles the division to
spin-up- and spin-down-particles
according to their location (but not the
division to particles/antiparticles,
because a particle and its antiparticle
have the same spin).
Regular
tetrahedron T
Octahedron and the red
diagonals of antioctahedra.
When we connect in antioctahedra the centres of the opposite edges of a regular tetrahedra
T, we get 3 line segments x’, y’ and z’ (in the next picture). The length of each line segment is
1. The length is the same as the halves of the diagonals in octahedra or x, y, z = 1. In addition
the line segments are perpendicular to each other like x  y  z. The line segments x’, y’ and z’
are also parallel to x, y and z in octahedron. They are thus the halves of diagonals of the
antioctahedron in the same sense as the x, y and z are halves of diagonals in an octahedron.
The existence of antispace does not, however, expand the observer's space but doubles the
size of absolute space.
We observe in the picture that the diagonals of the antioctahedra form their own separate net
between the diagonals of the octahedra. The nets of diagonals are identical. So any of the nets
can be thought as diagonals of octahedra, and the other net as diagonals of antioctahedra.
The diagonals are thus the real substance of space. (The edges or faces of octahedra are not.)
The diagonals are background independent or they are not assumed to stand in any
background but they create themselves the room or the space.
T
z
y'
x'
y
z'
x
An regular tetrahedron stands between the two halves of
octahedra.
The diagonals form 2 separate identical nets,
space and antispace or two Manhattan-metric.
14
a
d
Va
Vo
z
y
x
The unit vectors in an octahedron and the "inside out"-unit
vectors in an antioctahedron define the same point in
observer's space.
a
In the picture the half of octahedron has been separated from the regular tetrahedron. The
volume of the half of the octahedron is Vo, when x,y,z = 1 and a = √ 2
Vo = a² z / 3 = 2/3.
The area of the face of regular tetrahedron (red one in the picture) in an antioctahedron is
A = ½ a d, when the central line segment of triangle is d = a √ 3 / 2. The volume V a of the
tetrahedron is, when the height is h
Va = Ah/3 = ½ a d ( 2 √ 3 / 3 ) / 3 = 1 / 3.
Altogether the volume of the halves of octahedron and of antioctahedron is V = V o +Va = 1.
The diagonal form also cubes. However, the quadratic absolute space does not come up by
considering only the cubes.
The surface of a hyperoctahedron is 3-dimensional and cellstructured (quantized space, grainy space or granular space.) All
diagonals of the octahedra at 3D-surface are connected to the next
ones to build a large loop. Thus through every point (octahedron) of
the surface goes 3 loops perpendicular to each other. The loops are
at the surface of equal lengths and go around the whole 3D-surface.
Complex space
The observer’s space seems to be isotropic or in other words it is similar in all directions.
Rotations of a macroscopic rigid stick are there length-remaining. In order to get the cellstructured space to work isotropically, a fundamental part needs to be added to the model. It is
a complex space outside the 3D-surface extending to a limited distance from the surface. The
complex space is cell-structured and Manhattan-metric is valid there. Complex space is 4dimensional. It is built by 3-dimensional octahedra, which stand perpendicular to each others (
see next page). The three octahedra diameters stand at an 45º angle to the 4.D or to the
imaginary axis and are projected to the planes xy, yz and zx of the 3D-surface at an 45º angle
to the main axes x, y and z of the 3D-surface. Together the complex space and the 3D-surface
make the observer’s space seem isotropic as later is told.
The complex space is necessary in the model for many reasons. One reason is electromagnetism. A macroscopic stick is hold in one piece by electromagnetism. Gravitation and
other forces have not any important role in that.
15
Let’s consider the structure of complex space, when the real space is a 1-dimensional straight
line like in the next picture. The segments of lines, which have the equal length and stand
perpendicular to each other, are standing outside the line at an 45º angle to it and their
vertexes are connected like in the picture. The 1-dimensional segments of lines create there
diameters of squares and also a complex 2-dimensional surface made of the squares.
Correspondingly, if the real space is a 2-dimensional surface, several squares are added
outside it so that the diagonals of the squares are connected as in the picture. The 2dimensional squares create together a 3-dimensional complex space. It is possible to travel
through the vertexes of the squares in 3-dimensional complex space. Only 2 diagonals are
crossing in the centre of the squares and it is possible to travel through a square only in 2
directions.
When the real space is a 3-dimensional surface, the 3-dimensional octahedra perpendicular to
each other are added outside it so that the diagonals of the octahedra are projected to the
planes xy, yz and zx of the 3D-surface at an 45º angle to the main axes x, y and z of the 3Dsurface. The 3-dimensional octahedra create now together a 4-dimensional complex space.
The octahedra, which stand perpendicular to each other, are not possible to visualize. It is
possible to travel through the vertexes of the octahedra in a 4-dimensional space. However,
only 3 diagonals are crossing in the centre of the octahedra and it is possible to travel through
an octahedron only in 3 directions.
Real space
Y
Z
X
Outside a 1-dimensional real space stands the 2dimensional complex space, which is built of 1dimensional line segments. The line segments
create squares.
Outside a 2-dimensional real space stands the 3dimensional complex space, which is built of diagonals
of 2-dimensional squares perpendicular to each other.
The subspaces or planes (X,Y), (Y,Z) and (Z,X) are
standing in the space (X,Y,Z).
The real 3D-surface and the 4-dimensional complex space outside it are both made of
octahedra. The difference is that on the real 3D-surface or in the (x,y,z)-space the octahedra
are not standing perpendicular to each other. The main axes have there 3 different directions.
In the complex space (X,Y,Z,W), however, there exist 4 directions for the axes. Still in the
complex space it is possible to travel inside an octahedron only in 3 directions. The symmetry
group of the octahedron in the complex space is SU(3).
The complex space (X,Y,Z,W) includes four 3-dimensional subspaces made of octahedra.
They are (X,Y,Z), (Y,Z,W), (Z,W,X) and (W,X,Y). Each subspace consists of its own elements,
octahedra. The four subspaces are not projected into 3D-space as perpendicular to each
other.
16
The 4 main axes of the complex space are marked by the letters X, Y, Z and W. Their
projections on the 3D-surface, or on the (x,y,z)-space, are at angle of 45º to the planes xy, yz
and zx.
y
Xp

Zp
Xp

 = 45º
Wp
Yp z
 = 45º
x
The four projection directions of the main axes X, Y, Z and W
are called for the main projection directions Xp, Yp, Zp and
Wp. Each main axes X,Y,Z and W are projected on the 3dimensional 3D-surface in the direction, which stands as far
from the main axes x, y and z of the 3D-surface. There exist 4
directions of the projections. They are shown in the picture
Positive and negative directions are marked by the colors.
The projection angle of the main axes X, Y, Z and W to the main axes x, y and z of the 3Dsurface is  = 54.74º.
cos  = 1 / √ 3 .
As told the space outside the 3D-surface is cell-structured. The cells are 3-dimensional
octahedra perpendicular to each other. Their diagonals create a 4-dimensional lattice. The
diagonals create, like on the 3D-surface, layers of two line segments. Outside or above the 3Dsurface the length of the main axes of the complex space is 137 line segments or 68,5
diagonals of octahedra and below the 3D-surface 136 line segments or 68 diagonals. The
reason for this share is told later. The limited main axes of the complex space are called also for
the lattice lines. The complex lattice space is fixed to the 3D-surface. Two separate complex
lattice space are interspersed with each other, the space and the antispace
137 line segments long 1-dimensional
lattice lines outside the 3D-surface
create on the 3D-surface the so called
projection ratio
 = 1/137.035999,
4.D
137 line segments
0
136 line
segments
which is described later. Projection ratio
is called also by the name fine structure
constant.
3D-surface
Complex lattice is
made of lattice lines
Hyperoctahedron
17
The cells outside the 3D-surface create a positive and negative lattice or the space and its
antispace. When observed in the 3D-space the axes of the lattice space are complex or all
points on the main axes are described by complex numbers. The lattice lines and the complex
lattice made of them are edged or they do not reach far into the centre of the space
(hyperoctahedron). Thus the whole physical space is built of the 3D-surface and of the space
close by (or the 4-dimensional complex lattice). The space has not any cell-structured
physical radius. Together these both spaces create a whole.
The surface alone is enough to define the size of the space.
The observer's space is the 3D-surface of the Riemann's hypersphere. The curvature of the
surface of hypersphere is positive. The hypersphere is, however, only "the illusion" from the
real absolute space got by a mathematical transformation and is not the same as the real
physical space. The directions of the main axes have disappeared on the surface of the
hypersphere and the cell-structured 3D-surface made of the octahedra has changed to the
surface of unit spheres.
All points at the side of a square are as far from the centre of the square measured only in
directions of diagonals or main axes. Other directions does not exist in an absolute space of
the squares or in the Manhattan-metric. In the same way all points of the surface of an
hyperoctahedron are as far from the centre of the space. It is possible to define for a surface
the idea "curvature" and "radius", which is the distance of the surface from the centre. The
curvature of the surface in hyperoctahedron is zero. It means that the surface is Euclidean.
Such a space is impossible to visualize. In order to understand the space and its effects it is
needed to use simplified laws and rules, which do not alone tell the whole truth. The space
can be understood mathematically, but the results must still be concrete and able to be
connected to the observations in 3D-space.
One important result of the mathematical analyse is that in a large scale the local Gaussian
curvature of the 3D-surface or the surface of the hyperoctahedron is zero. (The local
gravitation fields are not taken into account.) The other important result is that it is possible to
travel on this surface around the space in all directions of the 3D-space and return back to
start point. The space is limited and 4-dimensional and a body can travel around it clockwise
or anticlockwise. Because the Gaussian curvature of 3D-surface is zero, the imaginary radius
or the dimension 4.D is not possible to be observed. The surface resembles in this respect
the surface of a cylinder.
When the new dimension or the new base 4.D were added to the Universe, then the so called
Big Bang started. When the space will expand great enough, a new dimension 5.D is added.
Then the symmetry of the space changes and, for example, time like our time does not any
more exist.
18
The 3D-surface is made of so called d-layers, where d = 2.8179403 fm, which is the same as
the classical radius of electron. The length d is there the octahedron diagonal. Let’s consider
next the structure of the complex lattice. The segments of line form outside the 3D-surface are
2D-layers like in the next picture so that the length of diagonal is 2D and D = d. The size
(= 2 x D) of the lattice layer is different than on the layer of 3D-surface ( d = 2 x d/2 ) or the
size of the cells are different. The layers of the complex lattice stand at 45º angle to 3Dsurface. So in an even space d = √ 2 D, when projected to the direction of a main axis of 3Dsurface like in the picture.
Note! In the picture the lengths of
the lattice are shown as a projection
in direction of one main axis of 3Dsurface. Else d = D would be valid.
N68 = 2D-layer
projection
Note! The layers outside the 3Dsurface are the same layers as the
electron layers in atom. The main
quantum numbers of electron
corresponds to each layer.
N2 = 2D-layer
projection
1-dimensional
cells
2D-layer
N1 = 2D-layer
projection
D=d
2D-layer
d/2
d/2
lattice box
or an
octahedron
d
d/2
d/2
d-layer
D
3D-surface is d-layer
No= 2D-layer
projection
d
d-layer
The 3D-surface is located at the distance of
½-layer below the layer N1.
The complex lattice and the 3D-surface
have a fixed connection to each other.
-N1 = 2D-layer
Lattice box of the 3Dsurface
projection
All 3 diagonals of the complex lattice box are projected to the
xy-, yz- and zx-planes of the 3D-surface at an 45º angle to the
main exes of the surface and stand at an 45º angle to the
direction of the fourth (imaginary) base.
-N68 = ½ - layer
projection
19
In an even space d = √ 2 D in direction of the main axis of the surface (see the previous
picture). However, the complex space determines all the observer’s lengths. It determines also
the light speed and time passing. Therefore we define for the complex space the horizontal
length d’ = d in an even space or when the lattice lines stand at an angle of 45º to the 3Dsurface. The length d’ will change in contraction of the complex space in relation to d in an
even space. But d’ is observed always as a constant, because its length is not possible to
compare with any length in an even space. So the complex lattice space is always an even
space for the observer. After this the length d will mean here the length d’, which is a constant
for the observer and which is calculated to correspond to the value in an even space. The
constant value is one factor to make the observer’s space seem isotropic.
d’ = P or Planck’s length
on the 3D-surface
d’
d’
d’ = constant
d’
D
=45º
3D-surface
d
Smooth complex space Contracted complex space
Contraction of the complex space in relation to itself (not in relation to some background) is
not possible to observe, because there does not exist any stable object to compare with.
Instead in a smooth Manhattan-metric the absolute length of a body will change on the 3Dsurface, when the complex space is contracting. When the complex space is contracted to its
limit, the width of a lattice box is equal to Planck’s length P in direction of the 3D-surface. At
the contraction limit the half of diagonals of octahedra stand side by side parallel to each
other and their common width must be bigger than zero. (previous picture.)
The length of a body S in an complex space in direction of the 3D-surface is
S = X · d + Y · d + Z · d
, when X  Y  Z
where d is a constant segment of line and X = a, Y = b and Z = c are the number of
segments of line in directions of the main axes of the complex space.
Unobserved local contraction of the line segment d distorts the absolute space to become
nonlinear when observed in the observer’s space. The equivalent nonlinear length s in the
linear observer’s space is a scalar and gets its value
s = (a² + b² + c²) d ,
where d is the same and which means that the absolute space is quadratic in relation to the
Euclidean observer’s space. The numbers a, b and c parallel to main axes are not observed
but they are theoretic.
The quadraticness means the transformation of coordinates ± X  x² , ± Y  y² ,
± Z  z² . More about the subject in the chapter ”Calculating a distance in cell-structured
quadratic absolute space”.
20
Transformation gives a linear
correspondence between the
spaces x² and X.
Transformation
x
x²
a²
a²
X
X
a
a
Nonlinear correspondence
Linear correspondence
Let’s presume that a half of the diagonal of a lattice box is projected to the 3D-surface at an
45º angle to the length d in such a case, where the space is completely smooth and not any
force field or energy exist. The lattice lines stand there at an 45º angle to the 3D-surface. In
this case a value for the length d is calculated with help of four measured constants. This kind
of case is however impossible. An undefined scalar field affecting everywhere in the space
will decrease the value of 45º inclination of the lattice lines and broaden all lattice boxes and
the length d in direction of 3D-surface. In the next formula the deviation of the term
137.03599911 from the value 137 in the divisor will decrease the result to correspond to the
theoretical length d in the even space.
d=
ħ
= 2.8179403 fm
137,035999174 mec
, where me is the mass of electron, ħ is Planck’s constant and c is the light speed.
The field is not observed directly because it appears equally everywhere. The field changes
the projection of the length d longer than the length calculated in an even space with help of
other constants. The effect is observed for example in the projection length of 137 line
segments long lattice line. It should be 137d at an angle =45º, but in the scalar field it is
based on the measured values
R = 137,035999174d =
ħ
mec
.
The length of the linear projection of a lattice line can be transformed into observer’s space by
squaring and in this way the radius r1 of a hydrogen atom is got. The unit length d is not
squared.
r1 = R² = 137.035999174² d = 0.5291772 x 10-10 m
The structure of hydrogen atom is depicted later in D-theory in context of the geometric atom
model. All quantum numbers of an atom get in that context a geometrical and also
quantitative description.
The dimensionless projection ratio  describes projection of one lattice line to the 3D-surface
to the length 137.03599911d. The projection ratio  would be exactly 1/137, if the scalar field
would not affect as an ‘offset’.
21
The complex lattice space determines all lengths in the observer’s space. As already is told,
the quantities d and  are observed as constants also when the space is contracting. At very
high energies the projection ratio has got in measurements bigger values. The previous
scalar field is not accurately equally strong everywhere and the value of  can vary locally.
When the Universe was being born over 13 billion years ago, the 3D-surface did not first
exist. There existed only the complex 4-dimensional lattice space depicted before, which
consisted of 274 segments of lines long main axes. The number 274 can be shared into two
factors or 274 = 2 x 137. The number 137 is a prime number. The whole space can thus be
shared symmetrically into a lower (or an inner) part and into a upper (or an outer) part, which
both consists of 137 cells long axes. The upper part is called for Higgs’ upper doublet field
and the lower is correspondingly called for Higgs’ lower doublet field. This space consists of a
space and of an antispace interspersed with each other.
There happened a spontaneous symmetry violation. The inner part of the space or the Higgs’
lower doublet field changed irreversible so that the halves of the octahedra at the upper edge
of the space and in the antispace rotated 45º creating the 3D-surface made of octahedra. As
a result of this the length of the main axes of the inner part of the complex space is one
segment shorter or 136 segments of lines. This share has a crucial significance for
functioning of the model.
137
137
Outer part of the space
w+
w-
The spontaneous
symmetry violation
created the 3D-surface
Inner part of the space
137
136
The space (2x137) before the spontaneous symmetry
violation. Only a small set of the main axes is shown in
the picture.
Zo
The space after the spontaneous symmetry
violation. The 3D-surface has appeared.
Scalar field
Lattice box
The interaction quantums of the upper Higgs’ field are its positive and negative lattice lines
made of electrons e+ and e- called here for W + and W-, which are one segment of line
longer than the lattice lines Zo of the lower Higgs’ field. The difference in length means the
charge difference e in the lattice lines occupied by the electrons as later is told. So the
charge difference of electrically neutral Zo-quantums to the charged quantums W+ and W- is
e.
In the spontaneous symmetry violation changed (but not disappeared) part of the Higgs’
doublet field is the 3D-surface with its Manhattan-metric. It creates in the space a hidden
scalar field, which includes the Higgs’ potential.
22
The spontaneous symmetry violation created the 3D-surface. At the same time appeared the
gravitation into the Universe and the particles got their mass and momentum. A scalar field
caused by the 3D-surface appeared to affect everywhere. The scalar field is also called for
Higgs’ field. The field includes a potential, because the space proceeded in violation to a
lower energy state. The potential tries to decrease the angles between the main axes of the
complex space and the 3D-surface. It operates as the scalar field mentioned before. The
symmetry is hidden in this kind of violated space, because the 3D-surface still exists as a
separate fixed part of the space. It is said that it’s a question of so called hidden symmetry
and not a real symmetry violation.
The 3D-surface creates mass and momentum and transmits the gravitation potential. Also the
color force or the strong nuclear force interacts only in the 3D-surface. Standing on the 3Dsurface gives the color charge for a particle.
The 3D-surface forms an exceptional structure in space. The electrical force and the weak
force can be united theoretically as a part of the complex lattice. For those forces the
symmetry spaces U(1) and SU(2) are valid as a part of the structure of the complex lattice
space as soon is shown. Instead the strong force is a part of the 3D-surface. It works so
closely in connection to the complex space that its symmetry group is complex SU(3).
Gravitation is included only to the 3D-surface and it has no direct connection to the complex
lattice space. So the gravitation is not a part of the Standard model of quantum mechanics.
However the influence of the 3D-surface on appearing of mass of particles is already included
to the Standard model by the Higgs’ potential.
When we discuss about one single particle/body and its place, we need in principle always to
express, whether the place belongs to the observer’s space or to the absolute Manhattanspace. The alternatives are exclusive of each others and their relation is not unique. A point of
the absolute space is not local in the observer’s space. So we can say that a point of the
absolute space spreads as a smudge when looked in the observer’s space.
The quantum numbers of a particle depend only on its location in the Manhattan-metrics of
the space. In the D-theory, for example, all quantum numbers of an electron in atom can be
expressed with help of the location of the electron. But because the location, when observed
in the observer’s space, is not unique, also the quantum numbers of a particle are not unique.
So, for example, the spin of a particle can have simultaneously a positive and a negative
value.
When two spin-½-particles can not stand in the same place in the Manhattan-metrics, their
quantum numbers can not be the same. As a result we get so called Pauli’s rule, which
denies, for example, the equal quantum numbers of two electrons.
23
The complex lattice is made of lattice boxes like in the picture. The
lattice box is 3-dimensional and it is made of three diagonals of
octahedron or six cells. The lattice boxes are described so that its
diagonals are shown at 45º angle to horizontal line.
e-
Empty cell
Each lattice box contains one ½-layer long spin-½-particle e+ or e- as a part of the complex
lattice. The particle is called for a lattice particle and it can be positive or negative. The other 5
cells of the lattice box are empty cells. An empty cell means that in the cell exists not any
wave with a certain curving amplitude. The regularly packed lattice particles e+ and e- (or the
curved line segments) form together into the lattice the shapes of positive and negative lattice
lines. All lattice particles stand in their boxes in such a position that the shapes of the lattice
lines form in complex space 2-dimensional planes of nets, which are called for electron
planes. The directions of the planes are equal among themselves in each of the four
subspaces and they all will change simultaneously in the recurrent rotations of the lattice
particles. The recurrent rotations create in the lattice boxes the quantized circulation motion.
The planes are complex and form the symmetry spaces SU(2).
At the beginning there existed only the complex lattice space, which was made of 2 x 137 =
274 segments long main axes. All the main axes were at first parallel to each other or
perpendicular to the later appearing 3D-surface. The octahedra were then flattened to the
width of Planck’s length. Soon the octahedra, however, expanded rapidly and the space
expanded strongly faster than light. The angle between the lattice lines and the 3D-surface
decreased to a bit less than 45 degrees. This kind of phenomenon is called for “cosmic
inflation”. During the cosmic inflation the space transferred into a lower energy state and the
released energy were transferred into each lattice box of the complex space as energy of
lattice particles e+ and e-. The lattice particles started their everlasting circulation motion to
and fro in their lattice box and the time passing of the observer’s relative time started. In some
phase during the cosmic inflation appeared the 3D-surface (as already told), gravitation and
also the symmetry violation 137/136 of the complex lattice space.
When a lattice box is contracted in one direction, a phase shift appears into its rotations. The
phase shift  is zero, when the lattice lines are at an 45º angle to the 3D-surface. Always in
other cases    > 0. The phase shift is local and there exists always a force field. A force
field will thus change the shape of the lattice box and causes the phase shift.
In the wave equation of a particle the phase of a wave function can be always changed
globally and it causes not any observed effect. Instead a local phase shift insists adding a
potential function to the wave equation and that means existence of a force field.
 = 0
   > 0
   > 0
24
The lattice lines form a vacuum, which has so called zero point energy and also other
quantum mechanical features. Rotations or the motion of the lattice line shapes in the
complex 2-dimensional electron plane gives the phase for a wave function. Next we consider
the rotations and appearing of the lattice line shapes.
The local phase invariance insists appearing of a local interaction field. The interaction field is
quantized. Interaction happens with help of the interactions particles like virtual photons.
When a field is quantized, must also the local phase shift be quantized. The phase shift is
described with help of an angle, so the angle must also be quantized. The phase of a wave
function is complex and it is not possible to measure. So there does not exist any quantized
quantity, which could be measured, corresponding to the phase shift angle. We can talk
about hidden quantization. Only the quantum of interaction field can be observed.
Contraction of the complex space and the change of the angle connected to it is quantized,
but not in any observable way. The quantization of curving of the space from the point of view
of momentum of a particle is considered later.
Momentary rotation of a lattice particle:
- -
-
+
+
+
-
+
The opposite rotation of an antiparticle at
the same moment:
-
-
-
-
+
+
+
+
auxiliary line
A lattice particle differs from an empty cell because of its energy.
Energy is described as curvature of the cell. Energy is always
linked to curvature of space. A lattice particle is an energy
package rotating around in the lattice box containing kinetic
energy and potential energy. (It is like a balance wheel in the
clock.) The motion to and fro means that the quantum
mechanical time direction changes regularly at microscopic
scale.
Curvature appears in the 2-dimensional electron plane, in which
the so called elementary rotation (depicted later) is going on. The
positive direction of axes is in the picture downwards or to the
3D-surface. Curvature has an amplitude. Its direction in relation
to the axes and also to the rotation direction gives the sign, plus
or minus, for a state of particle ( color in the picture). So on the
right side of an lattice box stands a green lattice particle and on
the left side a red one like in the picture or vice versa. The colors
will change cyclically, when the direction of quantum mechanical
time changes (more later). The number of curvature is quantized.
All the lattice particles turn during elementary rotation similarly and move into the next empty
1-dimensional cell in their lattice boxes. In the same time the lattice line shapes move
depending on the rotation direction forwards or backwards on the 3D-surface in direction of
projection of one complex main axis. The speed of this motion is the same as the speed of
light. A spin-½-particle needs to rotate in its lattice box through all main axes two full circles or
720 degrees (X,Y,Z,X,Y,Z) before the lattice has returned back to start state. These rotations
are described in detail later.
The total energy of rotation of a particle is constant and represents zero-point energy of
vacuum. During rotation the kinetic and potential energy change to each others so that the
total energy remains. A wave equation describes the motion of a particle. Similar wave
equations can be in principle written for all different oscillation types of the space.
25
All properties of the quantized space affecting on the local oscillation of space are considered
in those equations. So the wave equations describes also the interaction fields of the
Manhattan-space and their local charges.
The building elements of the complex space (X,Y,Z,W), the 3-dimensional octahedra, form
four 3-dimensional subspaces, which are (X,Y,Z), (Y,Z,W), (Z,W,X) and (W,X,Y). The
subspaces stand at an angle of 90º to each other in the 4-dimensional Manhattan-metric. In
each subspaces stand electron planes, which stand parallel to each others. The electron
planes stand at every moment in four different perpendicular directions. All electron planes
turn in every elementary rotation in their own subspaces. All events of the spin-½-particles or
the state transitions are possible only at the 2-dimensional electron planes. No events exist in
the perpendicular direction.
We have before described space and time with help of line segments and with rotations. A
line segment is an abstract, background independent model for quantized space.
Correspondingly elementary rotation is an abstract, background independent model for
quantized time. So the line segment is a quantum of space and rotation is a quantum of time.
The line segments and the rotations have fundamental properties, which are not possible to
explain but only describe.
The direction of the curvature of electron depends also on the momentary rotation direction in
the lattice box. The antiparticle is curved into opposite direction and rotates into opposite
direction. The rotation direction changes in all lattice boxes at the very moment, which means
a global phase transformation for the phases of the wave functions of particles. (The global
phase invariance means in quantum mechanics that also the rotation direction of a phase can
be turned overall in space opposite and the change is not possible to observe.) In the lattice
exist the same number of lattice particles and of its antiparticles. They form overall in the
space the so called zero-energy level.
Changing the direction regularly in a lattice box creates the quantum mechanical time, which
differs from the time of the macroscopic observer. Quadratic time in a lattice box is got by
multiplying the number of positive rotations by the number of negative rotations into opposite
direction. This kind of time can progress only into one direction, when observed in the
macroscopic space.
On the 3D-surface does not exist any elementary rotations in a similar way as in the complex
space, which means that the spin of Higgs’ boson describing the interaction of the 3Dsurface, is zero.
Quantum mechanical time is quantized and bidirectional and appears from the electron
rotations in the lattice boxes. The wave function as a result of the wave equation can be
written in simple form (t) = e-iEt .
For antimatter the wave function is (x,t) = e-i(-E)t = e-iE(-t) .
Antimatter can thus be interpreted to have negative energy or to move back in time on
grounds of the signs of quantities -E and -t.
26
According to the D-theory there exists a quantum mechanical bidirectional time, which works
in the scale of uncoarsened quantum effects. Before is already depicted the bidirectional
rotation of the electrons in the complex lattice space. The rotation creates the time in all
points of the space. The momentary direction of the rotation determines globally the direction
of the quantum mechanical time. The direction of rotation, forward or backward, depends on
the position of the electron in space and changes everywhere in space at the very moment
according to the measure principle or to the phase invariance of the wave function. The
change of the rotatíon direction turns the charges of all particles to opposite as well the sign
of the quantum mechanical time. Because the direction of the quantum mechanical phase of
a particle is not an observable quantity, it is also not possible to observe the global change of
the signs of charges and of time. Multiplying a negative charge by a negative time gives a
positive result.
When a so short moment is depicted that the direction of the quantum mechanical time does
not have time to change, the quantum mechanical time must be considered instead of the
macroscopic time. This appears for example, when annihilation of electrons and positrons are
depicted. The bidirectional quantum mechanical time does, however, not appear as direction
change of the orbital angular momentum of an electron in atom as later is told. In order to get
the macroscopic time from the bidirectional quantum mechanical time a rectifying divisor by
two is needed. The principle of such a thing is shown in connection with the four-dimensional
atom model. They are needed also to form the observer’s R(3)-symmetry space from the
complex SU(2)-symmetry space or the rotations of 720 degrees changes to rotations of 360
degrees.
t
1
Positron travels in the quantum mechanical time
backwards before its annihilation with an electron.

When the scale grows or when the motion of particles is
observed in in the macroscopic space they bot are
traveling only forward in macroscopic time.
0
e+
e-
x
The positron and the electron are extra lattice particles
moving in the regularly packed lattice. They have the
equal structure with the lattice particles. They are depicted
later in D-theory.
The quantized time is not possible to observe, because there is not any smoothly flowing
other time to compare with. Correspondingly the quantized space is not possible to observe,
because there is not any real smooth or continuous space, to compare with. The Euclidean
observer’s space exists only as an image created by the observations.
Calculating of time passing and the relativity of time are considered later in D-theory.
27
4.D
c
137 cells
c
3D-surface
The lattice line shapes made of lattice particles
e+ and e- on the 2-dimensional planes ( or the
electron planes) are moving outside the 3Dsurface in space and in antispace at speed of
light to and fro into opposite directions. A
momentary direction of this motion is shown in
the picture.
c
Lattice line shape made of electrons e+ and e-.
The motion of the lattice line shapes past each other in phases determines the wave function
phase, which is not possible to measure. If the lattice is not locally homogenous, a local
change appears into the phase of a wave function. In the change there exist a force field or a
potential. The change describes the nature and strength of the field.
The next picture presents a plane parallel to XY-axes in the complex lattice in the sub space
(X,Y,Z). In the picture the lattice particles e+ and e- form together a plane and the shapes of
the lattice lines. In the next rotation the lattice particles parallel to X-axis turn parallel to Y-axis
and particles parallel to Y-axis turn parallel to Z-axis. Shapes of the lattice lines leave the XYplane. They transfer to YZ-plane and the interactions of rotation appears there. Next time the
rotation interactions appear on ZX-plane. After a full cycle the sign of a lattice line shape (or the
color in the picture) is changed opposite and an extra cycle is needed in the planes XY, YZ and
ZX in order to return to the start case. Equal rotations happen in all four subspaces (X,Y,Z),
(Y,Z,W), (Z,W,X) and (W,X,Y).
Negative lattice line
Neutral lattice line
Positive lattice line
The lattice particles in the lattice boxes form in the
lattice a 2-dimensional electron plane. The curved
e-
lattice particles are depicted here as straight.
e+
The directions of rotations are marked in the picture.
After rotation the plane has changed to an other
direction.
e-
e+
X
Auxiliary line to perceive
octahedron
Y
The interactions of the lattice particles are considered in 2-dimensional electron plane of the
lattice line shapes. The axes of the plane stand outside the 3D-surface and are in an even
space at an 45º angle to it. The axes are thus complex and the symmetry space of rotations
for spin-½-particles is SU(2).
28
When Paul Dirac developed his relative wave equation for electron, he understood that a wave
function links the point defined by two complex axes (or dimensions) to every point of space
and time. According to the model of D-theory these axes are the shapes of the lattice lines and
electron interacts with them.
According to Dirac a wave function must be a vector including 4 components or it is a so
called spinor. Two of its components are linked to the states of positive energy and two to the
states of negative energy. In the states of both positive and negative energy one of the spinor
components means the spin-up state and another the spin-down state. This is understood so
that the electrons of positive energy e+ will rotate in their lattice boxes at one moment
forwards and electrons of negative energy e- backwards and later the directions are changed
opposite. They have together the same spin but they are antiparticles of each other. Both
electrons will rotate both in the space, where the spin is spin-up and in the antispace, where
the spin is spin-down. There exist thus four components or cases.
According to Dirac the vacuum is not empty but it is evenly full packed by the electron states
of negative energy. This kind of vacuum is called for Dirac’s see, and it is not possible to
recognize in the real vacuum, where the total momentum, total charge, total spin and total
energy are all zero. For every invisible electron in the Dirac’s see exist an equivalent particle
(in the antispace), of which momentum and spin are opposite. In addition it is not possible to
define the electrostatic potential of the homogenous see (and the total charge) or the total
energy, because all the measurements are always done in relation to the vacuum.
The electron planes and their rotations are one property of Dirac’s see. Bound electrons in
atoms and free electrons are considered later in D-theory.
The complex lattice space differs from 3D-surface in relation to the rotations in its octahedra.
The rotations create in space a motion, which makes the lattice line shapes move in relation to
the 3D-surface. The rotations create also elementary time in every point of space. When
observed on the 3D-surface the rotations happen in the complex space. Depending on the
case the rotation spaces are U(1), SU(2) or SU(3). When we for example consider a photon,
the symmetry space is U(1), but if we consider a spin-½-particle, which feels the color force,
like a proton, the symmetry space is SU(3).
4.D
c
c
P2
P1
The picture shows moving lattice line
shapes, which move in the picture to the
right and to the left. They penetrate the
3D-surface in two points P1 and P2. The
square appears, when the lines are looked
perpendicularly from side.
In observer's space a square of absolute
space is observed as a circle.
c
c
29
When the line segments are transformed into observer's space, we observe that the line
segments are arcs of circle quarters. They are perpendicular to the 3D-surface only near the
3D-surface.
137 cells
Ln
Ln
n
n
In absolute space the length of a line
segment is linearly Ln + n = 137,
where Ln and n are the components
In observer's space the circle Ln ² + n² =
137² defines the line segment to a quarter of
a circle.
parallel to the main axes.
The planes created by the lattice line shapes travel through the whole hyperoctahedron forming
there equally long loops in the sub spaces XYZ, YZW, ZWX and WXY in the complex space.
Later in D-theory is proved with help of a loop space model that time is not a substance and that
Lorentz’s transformation equations work in this space model.
4.D
3D-surface
In observer's space the line segments
are perpendicular to the 3D-surface and
form there circles.
A layer parallel to
4.D
In the picture the cells outside the 3D-surface are
perpendicular to each others, which means that they do not
have any mutual interactions. The interactions appear,
when a cell turns in the space. The lattice of cells aims
always to be homogenous through interaction.
We might think now that the lattice lines would form a dimension closed to itself in very small
scale. A similar thought is presented in the famous Kaluza-Klein-theory and in the string theory.
The diagonals of octahedra could be here called for strings, but because of the geometric
reasons it is avoided. Correspondingly the circle formed by the complex main axes into the
observer’s space could be called for compact dimension.
The complex lattice space looks here fully symmetric in all directions of the 3D-space. So it,
however, is not as later is shown. But also the physical laws of the world are not fully
symmetric. For example, the matter/antimatter-symmetry does not realize, as neihter the
parity-symmetry of the weak interaction.
30
When the 3D-surface appeared from the positive halves of octahedra diameters of the lower
part of the complex space (137  136), appeared also simultaneously Dirac’s electron field
and the mutual phase differences of its electrons. The next picture shows the structure of
Dirac’s electron field at one electron plane and the phase differences of electrons and
positrons. The field contains electron-positron-pairs, which form together positive and
negative shapes of lattice lines into the complex space.
Pair: Positron e+ and electron e- .
The particles rotates into opposite
directions.
e+
e-
The negative maximum of lattice
Lattice box
T+
T-
In the Dirac’s field of the picture each complex
lattice line is made of pairs of parallel electron
and positron. The pairs appear and disappear
in the successive rotations in the limits of the
uncertainty principle.
The positive maximum of lattice
c
Negative lattice line
Positive lattice line
Color sets
The picture shows that the structure of the complex Dirac’s field in not symmetric in regard to
the positive and negative shapes of the lattice lines. The same fundamental asymmetry
prevails in the amounts of matter and antimatter.
This structure formed by the lattice line shapes moves upwards in the picture at the electron
plane by steps of one layer during the positive quantum mechanical time T+ and downwards
during the negative time T-. The structure does not change its form in rotations but only
moves depending on the direction of the quantum mechanical time. When the time direction
changes, the signs of the lattice lines or the colors, green and red, in the picture will change
between themselves. The potential maximums in the picture are reflected after time T+ from
the upper edge of complex space and after time T- from the lower edge. This structure is
found also in the complex antispace interspered to the space, which shares the Dirac’s field to
the spin-positive and spin-negative particles. On the side of the antispace the lattice line
shapes move into the opposite directions.
During the positive quantum mechanical time T+ a single lattice line shape seems to move in
rotations step by step to the right or left depending on its inclination direction, and during Tinto opposite directions. The speed of motion is the light speed c.
31
In the previous picture the electrons e+ and e- are rotating in their octahedra into opposite
directions. When the quantum mechanical time direction has changed, they both have
changed their rotation directions and still have opposite directions. Also their charges have
changed to opposite as well the sign of time. What does now make the difference between
the electrons? The symmetry is not perfect, because the electrons e+ and e- can be
separated from each others with help of the previous asymmetry of the lattice structure. The
complex lattice and its asymmetric structure, which is a wider whole than a single electron,
determines the sign of the electric charge of an electron and also the sign of the electric
charge of a proton. D-theory does not so far tell in details, how the sign of the electric
charge is determined by the asymmetry of the lattice structure.
e-
e-
e+
e+
The electrons e+ and e- in the picture share the space into right
and left side in directions of the 3D-surface absolutely. The
structure remains in lattice rotations, but the location of structure
moves in up-down-direction (4.D) one layer in each rotation.
When the direction of rotations changes, the symmetry remains,
only the colors and charges change between themselves. Thus
a wider whole must be considered to find the asymmetry.
In a more large scale into the lattice appears a positive and a
negative maximum, which stand in the lattice as a stable
asymmetry. Changing the rotation direction opposite does
not change the structure but only changes the colors in the
picture between themselves.
The asymmetry of the lattice depicted before is an obvious reason to the matter/antimatterasymmetry and a possible reason to the violation of parity symmetry in a weak interaction.
According to the Noether’s theorem for every symmetry must exist an observable quantity,
which remains. So the space model of D-theory needs to include a group of symmetries.
That is needed, for example, for the energy conservation. A symmetry for conservation of
energy is the bidirectional quantum mechanical time. The symmetry for the electric charge
conservation is a certain symmetry property of the complex Dirac’s field.
32
When the cell-structured space is the only substance in the world, a question
wakes up, what are the cells or what exists between the cells. A similar
question could be done also by a clever software creature, who lives in the
digital world of a computer. After examining its own world it may come to ask
"What are the bits?" or "What is between the bits?".
We know that the bits exist in transistors, in vacuum tubes or even in relays.
The software creature in the digital world can never find out the nature of bits
by his own examinations or conclusions. As well the cells of our physical
space stay incomprehensible for us. They are for us the fundamental
abstraction in our world. We can never understand, what they are.
Note! We can not think that between the bits would exist any room like space, where the bits of memory
space are standing. The order of bits and their relation to each other give them meaning and no space
between bits is needed. A bit is a model for something, which can not be known more. The same is valid for
the segments of line of our space. They are, like bits, models independent on the background. Their order
and relations, like length and angle, are meaningful. Geometry has appeared to describe these relations
and therefore geometry is a useful way to describe the basics of physics.
Observer’s space means a space, which appears through coarsened observations made by
the observer. The observer’s space is isotropic and Euclidean. The Euclidean space is defined
with help of the validity of Pythagoras’ theorem. According to the Relativity theory the
observer’s space is different for every observer depending on the way of motion. In the
observer’s space every observed body gets its location on grounds of observation in relation
to other bodies, but not in relation to any absolute background. If not any observation can be
done, the observer’s space does not exist. There exists then only the non-observed absolute
space. An undetected particle is not localized in the observer’s space and does not belong to
it. Not until a measurement or an observation gives a location in the observer’s space for the
particle standing in nonlinear absolute space, and the wave function is said to collapse
similarly.
The observer’s space is a coarsened image of the real physical space, which is called here for
an absolute space and which will exists regardless of observations. The reason for several
interpretation problems of quantum mechanics is that the observer’s space is incorrectly
thought to be the real physical space.
Let’s consider next, why the absolute space is quadratic in comparison with the observed
space as it is presented in the hypothesis of D-theory, and what kind of consequences the
quadraticness or the nonlinearity has, and what makes the cell-structured Manhattan-metric
space seem isotropic.
33
Calculating a distance in the quadratic absolute space
A length in the absolute space (X,Y,Z) can be transformed into the observer’s space (x,y,z) by
a mathematical transform (± X  x² , ± Y  y² , ± Z  z²). When a quantity calculated in the
absolute space, for example the length N, is linear, and we know that the absolute space is
quadratic in comparison with the observer’s space, the quantity needs to be squared when
transformed into observer’s space. When the quantity is squared, the length N² units, is
nonlinear.
Transformation gives a linear
correspondence between the
spaces x² and X.
Transformation
x
x²
a²
a²
X
X
a
Nonlinear correspondence ± X  x
a
Linear correspondence ± X  x²
Using a high-handed measure unit is not always sensible in the transform because the
absolute space is quadratic. In scale of quantum effects the measure unit needs to come out
directly from the structure of space. A suitable unit is the smallest indivisible length d (classical
electron radius) or some other in the structure of space repeated absolute length, which is
connected directly to the measured quantum effect or to its quantity. First the linear lengths of
the absolute space are added. The result, for example N units in direction of X-axis, is squared
and the nonlinear result N² in the observer’s space is got. In this way we get, for example, a
value for the radius [m] of the hydrogen atom of Bohr’s atom model and the value of Rydberg’s
constant [1/m]. The same technique is used also in quantum mechanics, when the squared
amplitudes are calculated to realize in the observer’s space.
When we consider the mutual relations of the basic quantities, length, time and mass, the
quadratic quantities need to be used also according to the Relativity theory. The examples of
this are the length contraction and the time dilatation. The all other quantities can be derived
with help of these three ones. This leads to the conclusion that all laws of nature observed in
the observer’s space appear from the nonlinear effects of the quadratic absolute space.
When the distance between two points in 3D-space is small, we must take account of the cellstructure of the space. Also motion needs to be taken into consideration. The observed length
is different for every observer because of the length contraction.
34
d
s
In the cell-structured space only integers are used to express the
distance. A distance for example from the centre of a cell-structured
axis can be calculated. In the next picture the distance r is not the same
as the linear lengths d or s. In quadratic space at 3D-surface a
geometric average is calculated as
r ² = ds.
When s = d + 1, then r ² = d(d+1)
The two-way distance is in quadratic space 2 r ².
r
The geometric average is used in absolute space to calculate also the length of a moving
body. The motion will happen in the next chapter in regard to the cell-structured space, but
could happen also in regard to some other set of coordinates as later in the chapter “Time is
not a substance” is depicted in detail.
The motion of bodies during the measurement of distance affects on the values of the lengths
s and d. The linear length of the body in the inertial frame of reference is n cells. The
measurement is done by sending a light pulse from one end to the other of the body. If the
body moves during the measurement k cells, the lengths for the light pulse are in opposite
directions d = n - k and s = n + k or r² = ds = n² - k². The relative change r² / n² = (n² - k²) / n²
= 1 - k²/n². Ratio k/n = v/c, so it is written
r² = 1 - v²
n²
c²
or r = n
1 - v²
c²
So, if n = k or the body travels at speed of light (v=c), the length of the body is zero. The
quadratic length is in measurement always the length of outward way multiplied by the length
of return way.
When the length is calculated in the previous way, appears an observer’s space, where the
lengths depend on observer’s motion and which is a different space for every observer. There
exist not any global common observer’s space.
If several consecutive lengths need to be added together, they all are first added and after
that they are transformed into observer's space or r ² = d x s.
We know that on the 3D-surface the length or thickness of an octahedron diameter is
d = 2.817940325(28) x 10-15 m.
The value can be calculated with help of the known constants of physics. This length is called
for the “classical radius of electron”. It is roughly in the same scale where renormalization
becomes important in QED.
Later is shown that a particle with a spin ½, has a length of ½-layer or of a half of octahedron
diameter . Then a particle with spin 1, has the length of one layer (for example, the photon).
35
Calculating a speed in cell-structured quadratic space
Length and time needed for calculating a speed are both quantities parallel to 3D-surface.
Relative and absolute speeds occur in absolute space. Absolute speeds are not possible to
observe except for the speed of light, which has always the same value c, as later is depicted
in detail, and which is the maximum speed. We get for a relative speed v with help of absolute
speeds
v² = c² - w² or
w² = c² - v² ,
where w is an unobservable absolute speed. It has a physical meaning as we can understand
for example in the length contraction of relativity theory for the length s
s1 = s √ 1 - v ² / c ²
,
which becomes
s1 ² c² = s² ( c² - v²) = s² w² , where w² = c² - v².
The relative speed v has its direction in 3D-space, but its square v² will express the relative
sinking in direction of the fourth dimension 4.D. When v is the escape velocity of a field, its
square v² will express the number of absolute sinking of the point of field in direction of 4.D.
When w² = c² - v² , it is also valid
w² = (c – v)(c + v) = w1 w2 ,
when w1 = c – v and w2 = c + v.
The lattice line shapes travel as a result of elementary rotations into opposite directions at
speed c in relation to 3D-surface as already is told. When a particle moves at speed v on the
cell-structured 3D-surface into one of these directions, its speeds in relation to the lattice line
shapes are w1 = c – v and w2 = c + v. The speed w is a geometric average of the speeds w1
and w2. This kind of motion in relation to the lattice makes a particle absolutely asymmetric as
it is told later in D-theory. The number of asymmetry is depicted with help of an ellipse.
The formula v² = c² - w² describes an ellipse, which has a focal length v. For an ellipse is
generally valid
f ² = a² - b² , when a  b and
P
PF + PF' = 2a, when in (x,y)-plane x²/a² + y²/b² = 1.
b
For the speeds is valid
v
c
F
c
a
w
f
F'
v² = c² - w², when c  w . Then
a c and bw and fv.
Point P describes the instantaneous state of a particle in phase space.
“The centre of gravitation” of the particle lies in one of the focus points
depending on the direction of relative speed.
The eccentricity e = v/c of ellipse is used later in D-theory to describe the asymmetry of a
particle and of space in relative motion and in different force fields.
36
Left in the next picture the vector g, which depicts a particle, rotates in an inertial frame (x,ct).
Its rotational motion is depicted by a circle drawn by the head of the vector. Rotational motion
of an other particle moving at speed v is depicted by an ellipse in the (x’,ct’)-frame. The frame
(x’,ct’) is transformed by Lorentz-transformation from the (x,ct)-frame. The transformation
makes the particle asymmetric and its time passing will slow and its length will shorten. When
the speed v will increase, the length units on the axes x’ and ct’ are scaled hyperbolically.
ct
cT
Light cone
ct’
ct
Light cone edge
w=c
c
c
cT’
g
w
g
x’
c
x
Symmetric case v = 0:
cT is the distance, which light
travels in (x,ct)−frame in time T.
x
Asymmetric case v  0:
cT’ is the distance, which light travels
in (x’,ct’)−frame in time T ’.
The motion happens in relation to the cell-structured space, but could happen also in relation to
some other frame, as later in the chapter “Time is not a substance” is shown.
In the picture the absolute speeds c and w are parallel to the light cone. In this representation
in phase space the longer axis of ellipse is always at an 45 degrees angle to x-axis. In a speed
vector representation the relative motion v however turns the ellipse so that the speed vector c
pointing to the focus of ellipse in the right picture is always perpendicular to the horizontal plane
or to the 3D-surface as later is told.
Calculating a time in cell-structured quadratic space
Before is told that a length was calculated as the geometric average of two-way distances d
and s or r² = ds. Correspondingly the speed w were calculated as geometric average of
opposite speeds (c – v) and (c + v). Both the length and the speed are at biggest, when the
quantities in opposite directions are equal.
In a similar way the time is calculated as a geometric average. Elementary time appears as
elementary rotations of the lattice particles in the complex lattice boxes so that six 90º rotations
are done into forward direction and then the same number into opposite direction. Quadratic
time in a lattice box is got by multiplying the number of positive rotations by the number of
negative rotations into opposite direction. This kind of time can progress only into one direction,
when observed in the macroscopic space.
37
left
right
lattice box
Rotations in a lattice box will determine the phase
of a wave function. In quantum mechanics the
global phase invariance means that also the
rotation direction of the phase can be changed
globally opposite and the change is not possible to
observe. The phase of a wave function is
imaginary.
A segment of line is an abstract background
independent model for space. Correspondingly
elementary rotation is an abstract background
independent model for time.
In the picture the red vectors are a part of a lattice line shape. When the vectors will rotate, the lattice line
shape moves left or right in a plane depending on the direction of rotation (line made of points). In this way
appears the motion of the lattice line shapes at speed of light into two opposite directions. In fact the lattice
boxes are 3-dimensional and they are shown here on the plane for clarity.
When the space is quantized as cells, must also the time be quantized.
Otherwise it would be possible to set time intervals or moments, when the
particle should be moving somewhere between two cells. These moments
does however not exist because of time quantization and the particle always
stands in one cell and never between the cells.
Elementary time T is defined to mean the duration of one elementary rotation R. It is the
smallest indivisible time unit. At the end of an elementary rotation appears an elementary event
T1 and the interactions between the cells are then possible. Duration of the elementary event
T1 is zero unit. A new elementary event T2 is possible after an elementary time T. The time
between the events T1 and T2 is thus the duration of the elementary rotation R, which is T.
Time is measured with help of consecutive events T1...Tn by counting their number. Time
measurement in this way is interior in the world and independent of any background.
The time T lis calculated from the formula t = s/v, where s = d is the classical radius of
electron and v = c is the speed of ligth or
T = d/v =
2.8179403 fm
299792458 m/s
= 0.9399637065  10-23 s .
To understand the issue we can think the internal time in a computer used by the programs.
The time appears from a clock signal of a computer. The frequency of the clock determines the
speed of program execution and thus also the speed of the internal time passing. It is not
possible in the computer to observe the change of the clock signal frequency without any
external signal. We know that outside the computer exist an other time. But does any time exist
outside the physical world? It is not possible to get any answer to that question.
38
Motion of a particle affects on time passing of a particle. If a particle moves during rotation in
the lattice to the next lattice box, where the rotation phase is 90 degrees behind, the time of the
particle does not pass at all. When the motions continues into the same direction and when the
direction of rotations changes, the particle soon moves into a lattice box, where the phase is 90
degrees ahead. So the number of rotations of a moving particle increases into one direction
and decreases into opposite direction. The time passing of a particle calculated as number of
rotations changes asymmetric because of motion, and time passing calculated as geometric
average then slows. More about rotations of the lattice is told later.
In scale of quantum effects the time is symmetric, its has two directions, positive and negative.
The macroscopic time calculated as a geometric average of elementary events has only one
direction.
In the absolute space the time of a body passes fastest, when the body does not move in
relation to the absolute Manhattan-metric or the relative speed v = 0. In addition the body may
not stand in the gravitation potential of an other body. The neutral gravitational wave emitted by
the other body would cause for the body an acceleration and motion to-and-fro in direction of
the field in relation to the absolute Manhattan-metric. The motion or the extra traveling would
slow the time passing of the body. When the time passes fastest, also the lengths are at the
longest. The absolute mass of the body is in this case at the smallest. The wave motion
emitted by the body to the space around is now symmetric in all directions. The absolute
speeds of the body are w = c, and v = 0.
Dualism
Dualism of the particles or Bohr’s principle of complementarity has been in physics a difficult
issue to understand.
Particles seem to behave in a dualistic way. On the one hand they behave as particles,
because they have a certain location and a speed, on the other hand they behave wavelike as
a widespread phenomenon. From the point of view of classical physics these ways of
description exclude each other. Dualism needs a suitable explanation.
In D-theory a limited size particle is a part of the absolute space. The absolute space is
however nonlinear and not unique for the observer. Absolute location of an unobserved particle
spreads out like a wave, when observed in the observer’s space. Measurement gives a
location or makes the particle locate in a certain location in the linear observer’s space.
39
A particle can thus be described with help of a wave package. The wave package does not
however mean that a free unobserved particle would be a widespread wave. The space is
only understood in two parallel ways, or dualistically. The more the known points and the
distances between them will exist, the more “wave lengths” will exist and the shorter is the
wave package and the more accurate is the image of the observer’s space, which appears in
this way. Locating a particle into the observer’s space needs observations or known points in
the space. Also appearing of the observer’s space needs observations and existence of the
known points. More about the wave package and localization of a particle later in D-theory.
Classes of phenomena
Let’s consider next the four classes of phenomena. They all appear from the features of a
cell-structured space. Three of them belong traditionally to quantum mechanics and the
fourth is gravitation.
Quantum mechanics : 1. U(1)-rotations or electromagnetism, 2. SU(2)- rotations or weak
interaction and 3. SU(3)- rotations or color force.
When we consider a quantized space, we may come to ask if there exists any phenomenon,
which really refers to the absolute space mentioned in the hypothesis of D-theory. Before is
already shown that the absolute space is not possible to observe directly. So there is not any
known technique to observe it. There exists, however, a well known statistical phenomenon,
which is a strong evidence in favour of cell-structured absolute space. The result can be
measured for single particles, which do not know the coarsened observer’s space but live in
Manhattan-metric. It’s question of quantum correlation, which diverges from the classical
correlation. The phenomenon does not produce any direct observation, because correlation is
an abstract mathematical concept, which must be calculated from the measured data. The
phenomenon is called for EPR-paradox.
The explanation for the differences between quantum correlation and classical correlation
offered by D-theory is based on the geometry of space as on the next pages will be told. In
practice the explanation scraps the idea of quantum mechanics about the real quantum
system formed by entangled particles. In addition the explanation offers one more thing, which
is an evidence against the non-locality of quantum mechanical reality. This should not,
however, be understood so that also the direct observations would prove against the nonlocality, because the observations are always considered in the observer’s space, which is not
so far understood in the right way in physics. On the background of the observations,
however, stands the absolute and abstract reality, which the hypothesis of D-theory describes
and for which the next phenomenon proves.
40
The smallest scale
The absolute space is quadratic in comparison with the observed space in all scales. The light
and matter, however, select between two points a path, which leads by coarsening to
appearing of the linear macroscopic space for the conscious observer only. The
consciousness is connected here with observing and with the ability to coarsen observations
to macroscopic. If light and matter are missing, the observer’s space does not exist. There
exists only empty absolute space and no observations. The conclusion is that the absolute
space is not possible to observe. It means that the location of a moving body in the observer’s
space exists only in relation to an other observed body but not to any absolute background.
The observed space can exist and be linear for an observer only as macroscopic. The
innumerable amount of cells of the 3D-surface and of the complex space are needed for
appearing of the observer’s space. Therefore it is not possible to have any dividing line
between the macroscopic linear space and the quadratic space at quantum level.
All possible events of the nature happens in minimum scale of the cell-structured space or
they are based on the quantum level. This kind of view is called in physics reductionism.
Examining the physical events more thorough leads us to use the quadratic basic quantities in
calculations. When all the basic quantities, length, mass and time, are the quantities of the
linear observer’s space, must the reality be described by quadratic quantities.
The Minkowski’s four dimensional space-age used in Relativity theory is invariant or it is
equivalent for all observers. Its geometry is made of space-age points. The distances
between the space-age points are observed to be equal in all sets of coordinates. The
invariant space is in certain terms the same as the absolute space. The idea of distance
between the space-time points, which Minkowski introduced, is based on the expression (s)²
- (ct)². The next result can be derived from Lorentz-transformation for the square of the
distance
(s)² - (c t)² = (s’)² - (c t’)².
The left expression is valid in set K of coordinates and the right expression is valid in set K’ of
coordinates. The sets of coordinates may be in even motion in relation to each other. We can
see that the quantities in the expressions are quadratic and therefore they represent the
quantities in the absolute space of D-theory.
Let’s define the quantity u = ic t, and we can write the distance
(s)² + u² = (s)² - (c t)² .
Here i means imaginary unit. Using the quantity u means that the invariant Minkowski’s space
is complex. The imaginary unit i in expression u = ic t does not refer to time but to the speed
c. In geometry of four dimensional absolute space the absolute speed c represents the
direction of the fourth base or of the imaginary base. That direction is edged and the quadratic
maximum speed c² describes now the distance to the edge and it can be used as a constant
vector.
41
The light speed c is the biggest speed and it is used in D-theory to describe indirectly the
maximum values of several quantities in direction of 4.D. The quadratic relative speed v²
describes then the relative location of a body in direction of 4.D.
Total energy of a body is E = mc². When the quantity c² is parallel to the fourth dimension, also
the energy, which is an abstract quantity, is parallel to the fourth dimension.
By taking a square root of the coordinates (X,Y,Z) of absolute 3D-space we get positive and
negative coordinates ±√ X , ±√ Y and ±√ Z . This result can be interpreted so that there exists
two opposite signed spaces overlapping each other, space and antispace. By adding the
negative and positive coordinates we get as a result zero, which means symmetry.
We can write U = 0, where U describes everything in the world. A better way is to write it in
form
U–U=0,
which means that there exist two opposite worlds U and –U. (The quantities U and –U should
not be confused with matter and antimatter, because they are more fundamental or they
represent the only substance, which the world is made of.)
When we do not observe the two worlds U and –U, we can write for the observer
u² = lUl  0, where u is observer’s quantity and always real,
which tells that the quantities of absolute reality are quadratic in comparison with observer’s
quantities u and positive.
In addition to observer’s quantities there exists the theoretical wave function , which gets its
physical meaning only as quadratic ². Note that the wave function (x,y,z,t) is defined in the
complex space, which is a linear (x,y,z,i)-space. The wave function does then not exist as
such in the worlds U and –U, which are not linear for the observer.
42
Let's consider next the realization of Pythagoras’ theorem in absolute space.
The stick s is a rigid macroscopic body in space so that it is not parallel to any of the main axes.
Let's use the stick as hypotenuse of a rectangular triangle like in the next picture.
Y
By
The stick s is a fraction line in absolute space, as
also the sticks a and b, which are the legs or
catheti of the rectangular triangle, and a  b.
s
b
Bx
In observer's space is valid
a
Ay
Sy
s² = a² + b² when a  b.
Ax
Sx
X
The main axes of absolute space are X and Y. In
the picture the sticks a and b are divided into
components parallel to these main axes.
For the stick a is valid in absolute space, when Ax and Ay are its components parallel to the
main axes, a = Ax + Ay.
Correspondingly for the stick b is valid b = Bx + By.
In absolute space the components of the length S of stick are added together before
transforming to observer's space. We get for S in the previous picture
S = Ax - Bx + Ay + By.
When in the picture Ax - Bx = Sx and Ay + By = Sy, we get
S = Sx + Sy.
By transforming to observer's space, we get for the stick s
s² = Sx² + Sy² = a² + b².
We can suppose for any hypotenuse s of a rectangular triangle the legs S x and Sy parallel to
the main axes corresponding the legs a and b, and Pythagoras’ theorem is valid for them all.
So it is not necessary to use only the components parallel to the main axes to express the
length of a macroscopic stick with help of Pythagoras’ theorem.
Pythagoras' theorem
B
c
a
A
a
b
b
In the picture the lengths a,b, and c are gauges in observer’s space. Distance
between points A and B is in absolute space a + b = c as sum of vectors (but in
the observer’s space for the scalars c  a + b). In observer’s space is valid
c² = a² + b². The line segment AB does not exist in reality but it appears by
coarsening a fractional line or we can write it as a sum c² = (a)² + (b)², where
a = a and b = b. All the distances must in principle be calculated with help of
components parallel to the main axes although they are not known. Only they will
exist.
We can see that Pythagoras' theorem appears from the features of quadratic
absolute space.
43
Isotropic and quadratic space
When observed in the observer’s space, the length of any rigid stick remains in rotation. So
the observer’s space is isotropic. This does not however mean that an empty space as such
would be isotropic. The mechanism of isotropicness is considered more soon in this chapter.
The directions of the main axes of cell-structured space exist only in the absolute space and
other directions do not exist there. Other directions appear geometrically by coarsening a
fractional line made of the components parallel to the main axes. The direction of a coarsened
fractional line is an emergent property of its components. The direction of a fractional line is a
new property, which the components parallel to the main axes do not have. Emergence leads
to appearing of the observer’s space. Because of coarsening the observer’s space does not
exist in the same sense as the coarsened fractional line is not a real line. A special ability is
needed for coarsening. It is called here for “consciousness”. Each observer makes different
observations depending on the speed of their motion. Therefore the observer’s space, which
appears from the observations, is not the same for everybody. The lengths and time passing
are observed to be different depending on the observer’s motion. A global observer’s space or
global time, which are equal for everyone, does not exist.
The coarsened observer’s space must be isotropic or the so called sphere space, because
only there all directions are equal. The space of octahedra becomes exactly the space of
spheres, when all directions are made equal. An octahedron transforms into a sphere by a
mathematical transform or by squaring the set of coordinates. When observed in observer’s
space, the absolute space must appear as a quadratic space in all scales.
y
Y
x²  X
x
X
y²  Y
A coarsened direct stick creates in space its own direction.
When the stick rotates, its length remains in observer’s
space and changes in the smooth Manhattan-metric of
absolute (X,Y,Z)-space. The track of stick ends creates a
sphere surface. The space is called a sphere space.
In the smooth Manhattan-metric of absolute
space the stick length exists only in directions of
the main axes. The space is created by octahedra
and the ends of an imagined stick stand on their
faces. This kind of stick does not exist.
When observed in the smooth Manhattan-metric of the absolute space, the stick length must
change in rotation to make the track of stick ends approach the sphere surface instead of the
faces of octahedron. The change of the absolute stick length is caused by geometric properties
of light and macroscopic matter especially in the complex space. The locally contracting and
expanding complex space determines, as already told, all the observer’s lengths, propagation
of light and speed of time passing.
44
Single elementary particles move either as elements on 3D-surface or as elements in complex
space. Neither of these cell-structured spaces is isotropic as an empty space, so they both are
quadratic in comparison with the isotropic observer’s space. A quantum mechanical particle
does not rotate in space length-remainingly and its space is not isotropic. Only when the
elementary particles form together an entangled whole big enough ( = macroscopic body ), the
whole turns in space length-remaining, as soon is proved. Also light behaves as the space
would be isotropic. More about it later.
Y
Z
X
The picture presents octahedra and their quadratic
forms or spheres. The spheres in the picture are
compressed into shape of octahedra so that the
scales of a sphere distort. The scale does not make
here any difference. A straight line inside a sphere is
not any more a straight line after compressing. Its
shape depends for example on its location in relation
to the centre of the sphere. The quadratic space is not
unique for observer, as later is told.
We can consider, what would be the equivalent of the observed physical bodies like a
macroscopic circular ring in the absolute space. The circumference of a ring is first divided
into small line segments, which are then transformed one by one to components parallel to
main axis of the absolute space. The result is a fraction line, which resembles roughly a circle.
Correspondingly for a sphere of macroscopic world we get in absolute space a surface, which
roughly resembles a sphere.
The previously described mathematical transform from absolute space to observer’s space is
mathematically uncontrolled. The side of a square maps in transform to the arc of the quarter
of circle. But where does a certain point of the side of square map? That is not possible to
define uniquely. The observer’s space does not actually exist. It is an illusion from an other
reality. But because we understand our observed space to be real, the absolute space is in
our opinion an illusion, which appears for example as non-locality in the locations of
unobserved particles. The locations spread and change wide like waves. The unobserved
particles does not belong to observer’s space but they move in quadratic absolute space.
Mathematically uncontrolled is also the so called ”wave function collapse”, in which the square
of a widespread wave function seems in measurement to appear in an uncontrolled way (”at
random”) in one point of the space. Some eigenstate of a wave function will realize in
measurement to a real particle in one point of the observer’s space, but it is impossible to
know where. The accident does not however determine the location of the particle. The
particle has always its exact location in absolute space, which is not unique for the observer.
Note! Also the length of a coarsened fractional line is an emergent property, which the one
unit long components of the fractional line does not have, neither the multiples of the
components.
45
The next picture shows on the 2-dimensional plane the relations between the absolute space
and the observer’s space. The line segment AC in the observer’s space corresponds to the
line segment BC in the absolute space, when the middle point of observing is the point C. At
an angle  = 0 BC = AC. The line segment BC can be shown as a sum a + b of the line
segments a and b, when a  b. In the picture also the line segment AC can be shown as a
sum a + b, when a and b are scalar quantities, or they are parallel between themselves. So
any length ℓ in absolute space can be written with help of components parallel to the main
axes or vectors
S=a+b+c
when a  b  c.
The corresponding length ℓ in the observer’s space can be written with help of scalars
ℓ = a + b + c.
It is not possible in the observer’s space to know the lengths or the directions of the vectors a,
b and c, but their scalar sum ℓ is known. The gauge ℓ is a scalar, because in the isotropic
observer’s space its direction has no importance. Thus we can mention that any length S in
the absolute space can be transformed to a gauge ℓ in the observer’s space by changing the
absolute vector components to the scalar components of the corresponding length.
y
tan 
.
1 + tan 
a=
b=
cos 
= 1 - tan 
.
sin  + cos 
1 + tan 
a+b=
tan  + cos 
= 1
1 + tan 
sin  + cos 
A
a
B
[BC] =
1
= S.
sin  + cos 
b
a
S

x
C
b
ℓ=1
The relation between the gauge ℓ in the observer’s space and the corresponding length S in
the absolute space depends on the angle  and is
ℓ = S (sin  + cos ) .
46
Let’s consider the way, how the absolute space is contracted by a body. Considering is done
first in a static case, where the body does not move in relation to the Manhattan-metric and
later is more generally considered the local dynamic changes in the space caused by a body,
which moves in Manhattan-metric. The observations are in these both cases equal.
The principal picture besides shows a red square
prepared of matter into an even 2-dimensional
absolute space (Manhattan-metric). It has after
contraction changed into a disc resembling a green
circle in the picture and contracted the Manhattanmetric with it. The absolute number of contraction
depends on the quality matter. The contraction is
biggest in the direction of the main axes of
Manhattan-metric and reaches to infinity as
weakening.
When observed in the contracted space, the square
is still a square.
Only a part of of the curved (blue) axes of the contracted space is shown in the picture. The picture is
misleading in the sense that the points of the unobserved Manhattan-metric are not unique in the
observer’s space or they have not any unique location there.
Turning a macroscopic body, for example a stick, in absolute space made of octahedra
insists the change of absolute length of the stick to prevent the points of the body to follow
in rotation the surface of octahedra. On the other hand the stick length can not change for
the observer. Also the time, which light needs to travel through the stick, can not change.
Therefore the stick length changes only in relation to the smooth space, but the stick
length remains in the contracting and expanding complex space, which is also the path of
the light.
When the absolute length changes in rotation in relation to the even space and the
directions of the main axes are unknown, also the instantaneous amount of absolute
change of the length is unknown. So the absolute gauges are not useable, but the gauges
of observer’s space are used. The calculated gauges in absolute space used later are
announced for clarity always as lengths of projections parallel to the main axes of 3Dsurface, for example the length d, which is always a constant for the observer.
The contraction of the space always has a centre (of gravity), where the force vectors
causing the contraction in the Manhattan-metric are directed to. The directions of the
vectors are new directions and also an emergent issue created by contraction in
Manhattan-metric.
47
The fields are quantized, as later is shown.
Let’s consider next the way how matter contracts the space in relation to an even space. Mass
is equivalent to energy and energy is known to curve the space. Every spin-½-particle interacts
with the complex space creating there a contraction potential, which is the reason for
contraction of Manhattan-metric in relation to an even space.
Presence of matter or an interaction field in the space will change locally the angle of the
lattice lines in complex space. Also the density of the lattice lines will increase locally. A lattice
box of the complex space presented before will change its shape by stretching or contracting.
Matter will in this way change locally the shape and the density of complex space.
lattice lines
The density of the lattice and the angle of the lattice
lines will vary locally inside a body causing also a
local change in the phases of wave functions. The
change requires the presence of some force field.
Macroscopic body
Let’s consider next the rotation of a macroscopic body in the cell-structured space, which is
not isotropic, but a body turns there length-remaining and makes the observer’s space seem
isotropic. The lattice lines or the main axes of the complex space stand in empty space at an
45º angle to 3D-surface and 4.D (the fourth dimension). Let’s consider a homogenous thin
stick, which turns in space around its middle point. Turning causes a change in density of the
local lattice lines and also motion of the lattice in relation to an even space. The absolute
amount of the change has no importance here. The relative differences of the change in
various directions of the stick are important. These differences can be described with help of
geometry. The space does not contract or expand in relation to any existing background but to
the space itself, which changes the angles of Manhattan-metric.
y
Yp
x
Xp
This simplified picture shows how the presence of a stick in
space causes a change into the complex lattice space in
different directions. The axes Xp and Yp are the projections of
the complex space main axes on the 3D-surface. The length of
the arrows in the picture express the amount of contraction of
the complex lattice space, when the stick stands in 4 different
directions. We can see that the contraction is relatively
smallest in direction of the main axes x and y of the 3D-space.
48
In the lower picture on the left exists an empty even 3D-space and the lattice lines (red ones)
outside it in the complex space. Below it the picture shows the space beside a body. The
centre of the body stands in origin. The lattice lines in the picture stand there at bigger angle.
The lattice space is contracted besides the body in direction of Xp-axis. The macroscopic body
contracts itself and also the Manhattan-metric locally in the directions of the main axes
projections of the complex space.
y
y
Yp
Yp
1
x

Xp
y
x
1
-1
Yp
x
Xp
-1
Xp
Local contraction caused of a body in different directions.
In the direction of the vector the number of relative
contraction of the complex space is bigger than 1 as later
is shown.
49
Behaviour of light and matter in the contracted Manhattan-metric affect on appearing of the
observer’s space. According to Newton the momentum of a body is a conservative quantity
in a even observer’s space. Conservation of momentum is a fundamental property of space
and means that when observed in a contracted Manhattan-metric of the 3D-surface the
momentum does not remain locally. The next picture shows a curved 3D-surface, which is
even in the direction of 4.D. On this surface travels directly a body when observed in an
even observer’s space, but in Manhattan-metric it windings. This property of momentum,
which actually is a property of space, affects on appearing of the Euclidean observer’s
space. There appears an image of an even Euclidean space.
In addition to the above some other (dynamic) things have an influence on the appearing of
the observer’s space. They are observed later in D-theory.
The influence of the body itself on the space is missing in the next picture. The momentum
of the body appears in the direction of the curvature in the local 3D-surface around the body.
So here the body can be understood as a directed and localized wave package propagating
in the space. Instead the curvature or the sinking of the space in direction of 4.D means a
static acceleration field, which changes the momentum of the body because of the
gravitation acceleration. Gravitation is however a very weak force in comparison with other
basic forces.
y
4.D
x
y
x
On the locally contracted 3D-surface the
momentum remains, when observed in an
even space. The picture show a 2dimensional surface. With help of the angles
and of the lengths of the line segments the
curved surface can always be returned back
to an even space.
The curvature of the 3D-surface in direction of 4.D
affects on the momentum of the body by a gravitation
force.
50
Let’s consider the two Manhattan-metric in the sets of coordinates (X,Y) and (Xp,Yp). Their
origos are overlapping like in the next picture. The first one (X,Y) is a part of the 3D-surface
and the second one (Xp,Yp) is a projection of the axes of the complex space on the 3Dsurface. The sets of coordinates stand at an angle of 45º to each others. A rigid stick is set on
the 3D-surface to rotate in relation to the origo. Let’s consider the change of the stick length in
both sets of coordinates.
The stick is absolutely contracted in the direction of the Xp-axis because of the properties of
matter. The minimum value of the contraction can not be determined likewise the limit for the
smallest scale of macroscopic effects can not be determined. When the amount of contraction
changes in rotation of the stick, the space seems to be isotropic as at the next page is shown.
The stick contraction in relation to the even space caused by contraction of the complex lattice
space is both absolute and relative. In rotation only relative contraction has importance for the
isotropicness of space. When a stick turns, its length component Lx in direction of Xp-axes will
change. The length Lx is, as the picture shows.
Lx = √ 2 L cos(45º - )
Y
Xp
= L (sin  + cos )
A
B
Yp
L
F
E
C
X
L
G

L
√2L
D
In the picture a stick turns around the point D. The main
axes X and Y of the absolute 3D-space are in the picture
vertical and horizontal.
The contraction of the stick happens in the direction of
Xp-axis from the locus ABC to curve AEC. The stick
shortens by the length FE. Alternatively the stick seems
to lengthen in the direction of Xp-axis from the line
segment AC to curve AEC.
If the stick would not contract, the stick ends would in the picture travel in rotation linearly
either (1.) through the points A, B and C or (2.) along the straight line between the points A
and C depending on if the length is determined (1.) only by the non-contracted complex
space (Xp,Yp) or (2.) only by the uncontracted 3D-surface (X,Y). However the stick length in
relation to the even space is determined by the local contraction of the absolute space in
direction of the main axes Xp ja Yp of the complex space from the locus ABC to the curve
AEC. (Correspondingly on the locus AC the stick seems to lengthen to the curve AEC.)
There does not exist any contracting force parallel to the main axes of the 3D-surface. A
color force exists in the 3D-surface but its carry is very short.
51
In order to make the cell-structured space seem isotropic the contraction of the stick must work
in rotation as told before! Let’s see next, if that is so. In the next picture the length of a line
segment AD describes the amount of contraction in the observer’s space at different values of
the angle  and 0 <=  <= 90º. . The end points of AC stand on the red arc of circle. Also the
length of the AD is in the picture equal to the length of Lx
√ 2 L cos(45º - ) = L(sin  + cos ) = Lx = AD
Result: AD depicts for contraction of a stick (ED) in direction of the Xp-axis at different .
Y
Xp
Yp
√ 2L
Lx
1
L=1
E
A
X

D
√ 2L
L=1
Y
Xp
Yp
1
X

1
-1
-1
The local contraction caused of a body in
different directions relatively. In the direction of
the vector the number of relative contraction of
the complex space is bigger than 1 as later is
shown. Contraction weakens in square of the
distance.
52
According to the previous picture the line segment AD describes the change of a stick length
in direction of Xp-axis at an angle  in comparison with any constant length. In this direction
the constant length is represented by line segment BD, because its projection on Xp-axis is a
constant. The change, which is depicted by the line segment AD in direction of Xp, is
proportioned to projection of BD at all . The length of BD is multiplied by the length of AD
and the squared length SD² is got. By taking a square root we get SD, which is shown in the
next picture.
√ AD x BD = SD
X
The picture shows rotation angle  in the observer’s
space in relation to the main axes X and Y of 3Dsurface. The point S on the stick moves during
rotation on the blue arc of a circle.
Xp
Geometric average of line segments AD and BD is
of equal length to the line segment SD at all values
of . So the observer’s space seems to be isotropic
or it is so called sphere space.
The same is valid also in other scales. For example
line segment KD appears as geometric
average of the line segments AD and HD
at all values of 
H
K
A
S
B

Y
D
1
AC = √ 2 cos(45º - )
= sin  + cos 
BC =
1
sin  + cos 
SC = √ AC x BC =
√ 2 cos(45º - )
sin  + cos 
= 1
at all values of 
HC =
2
sin  + cos 
KC = √ AC x HC =
2√ 2 cos(45º - )
sin  + cos 
=√2
at all values of 
The points S and K travel in rotation of a stick on the 3D-surface along a fractional line in
Manhattan-metric.
When the complex space contracts in directions of its main axes projections, the 3D-surface
resists the transformation in every point of the near space in directions of its main axes. The
number of contraction finds a balance. The size of the body then stays to oscillate around the
balance as described later.
53
Let’s consider next the balance of the forces contracting and expanding the space in some
point around the contraction centre. The next picture shows the contraction centre and a
force vector Fc pointing to it in the set of coordinates (x,y) of the 3D-surface. The force vector
depicts a potential contracting the space. It is created into the complex space by interactions
of an energetic body with mass. It is a sum vector, which is built of components parallel to the
projections Xp and Yp of the complex space main axes , and points to the contraction centre.
The directions of the sum vectors create new directions into the Manhattan-space
(emergence).
The forces a and b, which resist the force vector Fc, are in the picture opposed to the
transformation of the 3D-surface. They are in the picture parallel to the main axes x and y of
the 3D-surface and the location of the considered point P determines their mutual length. We
can see in the picture that in balance the sum of all the vectors is zero.
Fc = a + b
y
a
a = yo Fc sin 
b = xo Fc cos 
Fc
Vectors xo and yo are unit vectors.
b
a
-x

Fc
b
a
Fc
b
Contraction centre
The directions of the sum vectors Fc create
new directions into the Manhattan-space.
a + b = Fc (sin  + cos ) ~ AD = Lx
We can see that the forces resisting contraction
and the contracting force itself have in every point
of space the same format as the segment of line
AD, which depicts the proportional number of
contraction at different angles as considered
before. It means that the contraction appears
because of the balance of forces at all angles .
At a certain angle  the force Fc(,r) becomes weaker in square of the distance r seen in the
observer’s space.
The size of the body or the number of contraction oscillates around the balance location, as
also the space around it, at a frequency, which depends on the size of the body, and at an
amplitude, which depends on the mass of the body. Oscillation creates into the space a
neutral gravitational wave, which is depicted more in details later in D-theory. The
gravitational wave has an effect on appearing of the observer’s space.
The change, which the line segment AC depicts, is proportioned to the line segment BC,
which has a constant length (=1) in absolute space at all values of . Therefore the line
segment BC is called for a normed line segment. Its length can now be multiplied by the
number of relative change and the result is the length, which changes in rotation.
54
A stand in detail has not been taken in this connection on the forces of a rigid stick in rotation,
like electromagnetism or gravitation. Electromagnetism is in a macroscopic body an essential
factor of internal structure of a body in scale of quantum effects. Contraction of complex
lattice space is described later in detail.
When a spin-½-particle contracts the space towards a centre point there appears a
contraction potential U(r). The potential becomes weaker in inverse proportion to the distance
in the absolute space. The derivative of potential dU(r)/dr is parallel to 4.D, so there is no
observable force field but only the energy E = mU(r). The potential appears when the angles
between the lattice lines and the 3D-surface approach to 90 degrees. The potential includes
the total energy E = mU(r) = mc² of a particle, where the mass m is the scaling factor
characteristic for the particle/body.
The contraction potential is not the same as gravitational potential, which has for a single
particle the proportional value 10-38 in comparison to the contraction potential. When the size
and mass of a body increases, the share of the gravitational potential of the total potential
increases and approaches a half of it in the black hole as later is told.
r

Space contraction creates into the complex space a
potential U(r). The potential of a particle contains a
standing transverse wave called Compton-wave,
where the space is waving. The particle emits it
around. The Compton-wave damps out and is
reflected from the potential back to the centre. A
standing wave does not carry energy with it, mut
contains in its amplitude the total energy E of a
particle.
E = hf = hc = mc²

= h ,
mc
Lattice line
- U(r)
which is the wave length of Compton-wave of the
particle. Compton-wave is a transverse wave
parallel to 4.D.
Later is shown how the Compton-wave of a particle is observed during interaction as de
Broglie-matter wave, which wave length depends on the momentum of the interaction. The
wave is also depicted with help of Schrödinger’s wave equation. The wave function as a
result of the wave equation describes the cyclic motion of the absolute Manhattan-metric in
relation to the even observer’s space.
In a macroscopic body the sum of waving of numerous Compton-waves appears as neutral
gravitational wave, which is not possible to observe directly. The neutral gravitational wave
has a longitudinal and transverse components. The longitudinal components appears as
gravitational potential, which slows time passing and shortens the local distance as later is
told in details. The length of the neutral gravitational wave is determined by the quantity,
which depicts the local curving of space or by so called proportional length Rs/R. Here Rs is
the Schwarzschild’s radius and R is the radius of the body.
55
The next picture shows the motion of a particle standing at 3D-surface in direction of 4.D and
the transverse standing wave motion and also appearing of local longitudinal motion. The
green points in the picture present the cells of Manhattan-metric of the 3D-surface. At the
bottom of the wave the cells have contracted more near to each others than at the top. This
causes around every node in a wave to appear a local shift x of Manhattan-metric in relation
to the even space.
If we consider only the motion parallel to 3D-surface, we observe the cells of 3D-surface to
move fro and to the way x in relation to the even space . The amplitude x of the motion is
local and so the gravitational potential V(x)  0 . We get x(x,t) = A cos (kx -t ), when the
transverse wave is (x,t) = A sin (kx -t ). The speed part of the shift is
(x) = ² (x,t) = - A k² sin (kx -t )
x
x²
So the second derivative of the transverse wave (x,t) corresponds to the horizontal speed
part of the space wave in relation to the even space. The contraction potential U(x)
corresponds to the potential part. The wave function (x,t) travels in a standing wave into
opposite directions. The picture does not depict any certain particle but is a common principal
model for all spin-½-particles.
(x,t)
4.D
c²
x=0
x
x
x
4.D
Particle
Motion paths of cells
The picture shows the standing Compton-wave of a particle. It damps out with contraction potential U(x). The
wave of an spin-½-antiparticle is at an 360 degrees phase shift of 720 degrees or there is no difference in the
picture. To drive the animation use PageUp- and PageDown-keys in SlideShow-state (F5).
The amplitude of a Compton-wave is always c². According to the model of D-theory a
quantity parallel to 4.D is depicted unlinearly as squared speed v². In that direction the space
has an edge and the maximum value. The maximum is c², which is also the maximum of
speed.
In macroscopic bodies the sum wave of numerous Compton-waves of particles is not a
standing wave and includes also a longitudinal wave parallel to 3D-surface so that the
longitudinal wave gets its maximum value in black hole. More about the wave later in Dtheory.
56
Schrödinger’s equation for elementary particles
The equation depicts a matter wave of a single particle. In the wave an absolute space is
waving in relation to the even space. The equation is written in 1-dimensional form for a
particle limited by some potential U(x) in its inertial frame of reference
- ħ² ² (x,t)
2m x²
+ U(x) (x,t) = i ħ  (x,t)
t
The solution of this equation is a complex wave function (x,t). The quantity m is the mass of
the particle and ħ = h / 2 is Planck’s constant.
The equation can be written also in form
Ek + Epot = E
where Ek is kinetic energy of the particle Epot is the potential energy and E is a constant
energy. In Schrödinger’s equation the term Ek corresponds to the kinetic energy of the
horizontal speed of the absolute space or the negative second derivative of the transverse
part T(x,t) = i A sin(kx-t+) of the complex wave function (x,t)
- ħ² ² T(x,t) = A k² sin (kx -t + ) = Ek.
2m
x²
, when ħ²/2m = A and k = 2/ .
Potential function U(x) expresses the value of potential in point x. Potential U(x) Is presumed
to be localized into a limited area. Inside the potential the location of an unobserved particle
spreads out because of nonlinearity of the absolute space. If the particle is free, its wave
function spreads out from the same reason everywhere to the Euclidean observer’s space.
A
In the picture the horizontal speed part Ek of a particle and
perpendicular potential part Epot are added. The sum is the total
energy E. The shape of potential part U(x) determines the shape of
the wave in the observer’s space.

Ek
E
Epot
0
/2
x
x
If there is no external potential U(x) to localize the particle into the
observer’s space, the spreading of the matter wave is determined in
the Manhattan-metric by its own contraction potential U(r). The shape
of U(r) in Manhattan-metric is determined by the geometry of each
particle. However because of the unlinearity of the absolute space
the wave function of a free particle spreads everywhere into the
linear observer’s space as later is shown.
The complex wave function as a solution of the wave equation depicts the motion of the
points of the absolute complex space in an even Euclidean observer’s space, which does not
exist as a substance. Quadration transfers the amplitude of the wave function to the
observer’s space.
The physicists have been uncertain, if does the wave function have a physical equivalent in
reality or is it only a mathematical object.
57
Neutral gravitational wave
Let’s consider next a wave emitted by a macroscopic body. The motion of the absolute space
in relation to an even space causes a phenomenon, which is called for a gravitational wave.
The phenomenon appears in connection with all particles with mass.
An evenly moving body or a body in rest emits constantly neutral gravitational waves around.
The waves do not transfer energy but they are an instrument of interaction like the virtual
photons in electromagnetic field. In a gravitational wave time passing slows down and a length
shortens. In systems under acceleration an energetic gravitational wave appears into the
space. When it is a question of a single particle, we define for this wave the Compton-wave
length and in interaction it is de Broglie-wave length.
When a body is accelerated, it emits into the space an energetic gravitational wave, which has
a different polarization as the neutral gravitational wave, as later is told.
Let’s consider a rotating stick. The stick can be shared into two equal parts and then
concentrate the masses of the parts into their both centre. So there is a rotating system, where
two equal masses travel around a common centre of gravitation. According to the relativity
theory this kind of rotating system radiates energetic gravitational waves into its surroundings.
Next we however consider only the neutral part of the wave motion.
Rotation of the stick makes the complex Manhattan-space
contract and expand and thus move cyclically to-and-fro in
relation to the imagined even space. Then all particles
around it will make an extra traveling in the complex lattice
space. They do not move along the complex lattice but their
momentum remains in a way described later. According to
the relativity theory the time passing of a traveling particle
becomes slower and lengths are shortened. The
phenomenon is weak and therefore very high speeds and
big masses are needed to observe it indirectly. However
also a small body emits gravitational waves.
To drive the animation use
PageUp- and PageDown-keys in
SlideShow-state (F5).
A neutral gravitational wave contains motion of the complex lattice space in relation to an
even space but not in relation to any background. The Euclidean space is here the same
as the observer’s space, which actually does not exist. The wave is in the four based
space longitudinal and transverse. The bodies do not move in a gravitational wave with the
complex space fro and to but keep their momentum when observed in the observer’s
space. Then they do an extra traveling in the absolute space. Extra traveling slows down
their time passing and shortens a length, which is considered in the next chapters. The
influence of a gravitational wave propagates at speed of light.
58
When a body moves in relation to Manhattan-metric, near the body in a point of space
appears contraction and expanding of the space and also motion and curving of an empty
space in relation to an even space. The transverse and longitudinal motion of the space fro
and to near a moving body refers to a wavelike behavior of the body. It looks like a wave
would propagate through the space. A wave like this interacts depending on the speed with
other bodies, because the wave causes acceleration such as an empty curved space does in
an acceleration field according to the relativity theory. An other name for a gravitation field is
“acceleration field”.
The next picture shows the transverse and longitudinal wave motion of space and the space contraction
and expanding connected to it. Waving space contains always an inclination in relation to the horizontal
plane. Inclination is biggest in a point where the change of the motion speed is biggest. (the 2th
derivative of transition = acceleration)
level of the surface in a transverse wave
A sin x
Expansion
Contraction
Transverse transition. A circle
depicts the motion and location of
one cell of the surface.
0
Horizontal transition of the space
A cos x
0
/2
Longitudinal transition or the
wave at 90º phase shift to the
transverse wave.
Horizontal speed of the space
- A sin x
0
first derivative of horizontal
transition
Horizontal acceleration of the space
- A cos x
0
second derivative of horizontal
transition
The motion of space in relation to an even space always contains asymmetry of space and
inclination from the horizontal plane. Where the space contracts or expands because of
passing of a body, there the space is locally inclined and the inclination means always an
acceleration field. So the contracting or expanding space interacts always to the bodies
around through an acceleration field. The bodies in absolute Manhattan-metric meet
acceleration fro and to, when an other body passes them. In the observer’s space the bodies
instead seem to stay on their locations or to continue their motion. They keep their
momentum in the observer’s space. Let’s consider next the appearing of the observer’s space
through conservation of momentum.
59
Conservation of momentum in Manhattan-metric
A symmetric wave propagating on the water surface affects on momentum of a floating body
only momentarily. If friction is presumed to zero, the body gets first on the rising edge of the
wave an acceleration to the propagation direction of the wave because of gravitation and soon
on the falling edge an acceleration to opposite direction. After this the body stays on its
location or continues former motion.
v
g
In the previous example both the body and the wave have their own momentum. They can be
considered as different cases.
In the model of D-theory to every body or particle, which has a mass (=energy), is connected
a wave in the 3D-surface and its potential. A body itself consists only of a group of waves ( =
elementary particles), which curve the space cyclically and locally as later is told. In a body
those wave components, which are parallel to the 3D-surface, determine the momentum of a
body. A perpendicular component to the 3D-surface determines energy. The more asymmetric
the wave is in some direction of the 3D-surface, the bigger is the speed of the body in relation
to the surface. The asymmetry is described with help of an ellipse. The momentum of a wave
remains, if no interaction like a reflection does not change it. The wave motion on the 3Dsurface and its conservation are the property of a space.
An asymmetric body/particle as a wave
4.D
c² = v² + w²
w
c=w
w
c

v
v
c

v
v
v=0
c
e=v/c
Principled picture about an asymmetric wave and its phase space. Rotation of the set of coordinates of a
body in relation to the observer’s set of coordinates is relative because the light speed c is the same
constant for all. The eccentricity of ellipse e = v / c. The absolute speed w is defined later.
When a body moves in relation to the 3D-surface at speed v, the wave of a body becomes
elliptic and the set of coordinates of the body is rotated by the angle . The slope of rotating is
v/w and v  w. When always c² = v² + w², the speed vector c is also in the frame of a body
always perpendicular to the 3D-surface. So the speed vector c shows the direction of 4.D in
both sets of coordinates. Every body has its own set of coordinates, which rotates depending
on the relative speed v. The set of coordinates is relative and determines the relative
differences, which can be observed.
60
In the next picture a body with diameter of L/2 emits neutral gravitational wave to the right and
left. The moving points in the picture depict the single cells of the 3D-surface. The cells move
in an inertial set of coordinates of the body along a track like circle or ellipse. The size of the
body changes with contraction and expanding in relation to the even space. The size L/2 in
the picture is thus a medium. The cells of the body move in a gravitational wave in directions
of 3D-surface and 4.D. If the body is considered only in 3-dimensional space, it would be
expanded and contracted in direction towards the centre of the body in relation to the even
Manhattan-space.
4.D
Body
Wave direction
L/2
To drive the animation use PageUp- and PageDown-keys in SlideShow-state (F5).
Gravitational wave contains transverse and longitudinal component. A wave is a sum of its
components. A body, which is in rest to absolute Manhattan-metric, emits a symmetric
gravitational wave. The both components of the wave are in phase shift of 90º. In this kind of
wave a point of the absolute space is moving in a circle or ellipse track in a wave. Antimatter
behaves like in time backwards waving matter or the rotation direction of the points of the 3Dsurface is opposite and the wave travels into opposite direction.
4.D
Body
The motion directions on the opposite sides of a body are opposite. Note that the wave of a static gravitation
field does not transfer energy with it. The parts of a wave are bosons of interaction field as for example
photons in electric field. The surface in the picture is in reality 3-dimensional.
Compton-wave and de Broglie matter wave depict in quantum mechanics one single particle
but can be expanded to depict a macroscopic body by taking in use a concept of proportional
length R/Rs, as later is told. It is needed to combine quantum mechanics and Schwarzschild’s
metric made for gravitation. Rs is Schwarzschild’s radius.
61
If the emitting body however moves in relation to the absolute space at some speed v, the
gravitational wave changes asymmetric.
4.D
w
c=w
c
w

v
v
c
v=0
c² = v² + w²
Gravitational wave emitted by an
asymmetric body.
p = mv

v
A point of the space travels in the wave on a circle or on an
ellipse in phase space.
level at transverse wave
0
L/2
level at transverse and longitudinal wave
Expanded space
Contracted space
0
L/2
The next animation shows the motions of the single points of the space in a half of sequence of a gravitational
wave. The motion creates first a negative half of sequence then the positive. A single point travels a circle or
ellipse track anticlockwise at angular velocity +.
+
Wave direction
L/2
To drive the animation use PageUp- and PageDown-keys in SlideShow-state (F5).
62
Let’s consider next a wave emitted by a macroscopic body and the gravitational potential
created by longitudinal component of the wave.
v
4.D
ve(x,t)

A = c²
0º
90º
R
x
c
w

L/2
In the picture the speed v is the horizontal amplitude of longitudinal wave on the surface. A is the amplitude of
the transverse potential of the wave. The maximum values are shown. ve(x,t) is the escape velocity on the
inclined surface. The wave interferes normally.
Gravitational wave of a static gravitation field V(r) can be represented as transverse and
longitudinal waves UT  UL in the inertial set of coordinates in the absolute space (x,t).
UT(x,t) = i A sin (kx -t + ) = i √ 2GM/r sin (2x/  - 2ct/  + ) , transverse part, where
A=c² is a quantity parallel to the fourth base 4.D. A quantity in that direction is generally
depicted in D-theory as quadratic speed A² [m²/s²], and 2c/  = 2/T, T = /c ,  = 2R and k =
2/ = /R,  is a phase shift, which is 0º for matter and 180º for antimatter and 2R is a size
constant of emitting body [m]. The quantity r>>x is the distance to the gravitation field centre.
UL(x,t) = v cos (kx -t + ) = √ 2GM / r cos (2x/  - 2ct/  + ), horizontal longitudinal
part, where v is the longitudinal speed of the wave. So UT(x,t)  UL(x,t).
The escape velocity ve(x,t) = ve cos (kx -t + ) so that ve² = c² - w², describes the local
inclination of the surface dUT(x,t) /dx. The scalar quantity ve = √ 2GM / r determines the
amplitude v = ve of the gravitation wave UL. The constant R or the size of the emitting body
determines the wave length . Outside the body the wave length (r) increases inversely to
the amplitude, (r)  1 / A²(r).
The speed vector c is always perpendicular to horizontal plane and vew. The angular
velocity for antimatter is -  = - 2c/.
The field potential V(r) = -GM/r = -(ve/√ 2 )² = -ve²/2. So V(r) is the effective value of the
quadratic longitudinal speed part UL²(x,t) or V = -ULeff² = -ve²/2. The acceleration in a potential
field is dV(r) / dr = MG/r². Potential is an abstract quantity of the absolute space.
The propagation velocity of the wave is the constant c. The amplitude of the gravitational wave
UL is equal to the escape velocity and the wave causes for other bodies in the field an extra
traveling in space and thus affects on time passing of the body and on the length parallel to the
motion direction. The escape speed ve determines the influence of the gravitation potential V =
-ve²/2 on time passing and on the length parallel to the motion direction or to the field.
63
The contraction of absolute space makes the 3D-surface change locally inclined. During
contraction the space is moving through other bodies or on the other hand the bodies move
through the contracting space. Appearing of motion needs acceleration, which is caused by
inclination of the 3D-surface for a body standing there. As is well known in an acceleration
field the acceleration is the same for all bodies independently on their mass.
a
a
0
A wave causes through accelerations to a body an extra traveling in space. According to the
relativity theory the time passing of the body then slows down and a length shortens. Wave appear
continuously in all acceleration fields as function of the field potential and is observed indirectly.
Neutral gravitational wave is an instrument of interaction in a gravitation field. A single wave
can be called for a “graviton”, which is the boson of gravitational interaction. The lower picture
shows gravitons emitted by matter and antimatter. The angular velocities have opposite signs.
The same difference exists also for example between proton and antiproton as later is shown.
Matter
Antimatteri
+
-
Neutral gravitation wave
emitted by matter
Neutral gravitation wave
emitted by antimatter is
like backwards in time
moving gravitational wave
When the rotation direction depends on the observation direction, a gravitational
wave shares the directions in the 3D-space absolutely to positive and negative
directions. The same is valid in electromagnetism as later is told.
64
In a static gravitational field the gravitational wave shortens a length in the direction of the
field and slows the time passing, too. Lengths are shortened also in other directions because
of contraction of the space. So, when these both factors affecting on length are considered,
length is shortened in all directions of the acceleration field by the equal number determined
by the field potential.
In the next picture the cells of the surface maintain their length in gravitation potential in
direction x but are shortened in direction y because of the space contraction caused by
potential V. The length in direction x is a sum of two components a and b so that a ≈ y.
Relation b/a is determined by derivative of potential V or dV/dr and in Manhattan-metric
x = a + b = constant and y ≈ x - b = a. However the gravitational wave caused by potential V
shortens the observed length ( = xv ) in direction x exactly by the quantity b, which is equal to
x - y ≈ b. Then the length in direction x is observed equal to y or equal to lengths in all other
directions. The extra traveling in direction x caused by the wave is correspondingly the
reason for slowing of time passing.
a
y
x
x
b
V
Field direction
Field direction
r
In direction x the observed length is shortened by the extra traveling caused by a gravitational wave. In direction y
length is shortened by the space contraction. As result all observer’s lengths are shortened by the equal number
in all directions. Note that geometric the length x and the observed length xv are different things.
When b² ~ V, a relative change of length in potential Vx at the escape velocity is ve,
b² = Vx = ve²
xc²
Vc c²
=>
b² = xc²ve² , where xc is length x, when Vx = Vc
c²
or b = xc, when the potential Vx gets its maximum value Vc at the escape velocity c. The
ratio is calculated as squared, because the lengths and velocities are quantities of absolute
Manhattan-space. In potential Vx the observed length xv gets a value shortened by the
gravitational wave or xv² = xo² - b² = a² and xo = xc
xv = xo √ 1 - ve²
=a
, where xo is length x, when V = 0.
c²
Thus the change of length x is equal to the length change caused by the extra traveling at
speed ve. The traveling resembling a sine wave is done at efficient velocity v = ve/ √ 2 ,
when the maximum efficient velocity is c/ √ 2.
The gravitational wave emitted by a static gravitation field shortens the length parallel to the
field direction, which is also the propagation direction of the wave. Instead the energetic
gravitational wave emitted by an accelerated body shortens length in direction, which is
perpendicular to the propagation direction as later is told.
65
Gravitational wave has been tried to observe through different
arrangements, like interferometer. They do however not
consider that time passing becomes slower in the wave when
length shortens. That may be the reason for zero results.
Observing gravitational waves
If we presume that the Earth emits neutral gravitational waves at 50 Hz frequency, can the observer far in the
space observe the effect of gravitational waves on the frequency of a radio transmitter at Earth? The gravitational
waves sometimes slow the time passing of the transmitter and sometimes speed up it but also change length
shorter or longer. The changes eliminate each other so that the light speed is always observed the same. The
effect of gravitational waves can not be observed directly. Gravitational waves slow and speed up the observer’s
time and change length far in the space so that the radio frequency remains the same.
Or does the radio frequency, which comes far from the space, change in gravitational potential of Earth at 50 Hz
frequency? No, because the radio signals will contract and expand in gravitational waves so that the observer at
Earth measures a constant frequency for the radio signals even though observer’s time passing and length at
Earth changes at 50 Hz frequency.
Gravitation or the effect of gravitational waves is eliminated. So there is no way to observe the gravitational
waves directly but only indirectly by measuring their effects on time passing in different gravitation potentials. So
far not any direct observation has been done. It seems that the equivalence principle of Einstein is valid for all
observer’s regardless of the observer’s location in the gravitational wave.
Strong equivalence principle: Gravitation is eliminated in local inertial system in all physical
interactions. So the special relativity theory with all laws and constants of nature is valid in
local inertial system.
A static gravitation potential curves Manhattan-metric also staticly. Here is no difference
between matter and antimatter, because the phase shift has no meaning here.
The curvature direction in static gravitation potential determines the
share 137/136 of the complex space. Above the 3D-surface lies 137
cells and below 136 cells. Therefore the interaction in the gravitation
wave in space contraction is bigger above the surface when the both
sides interact into opposite directions nonlinearly.
66
The gravitational wave in a static gravitation field is neutral and does not transfer energy as
for example the electric field of a charge. The wave is not polarized. Instead a mass in
accelerated motion emits energy, as also an accelerated electric charge. In the next picture
two bodies travel round each others and emit a spirallike gravitational wave. The wave is a
prognosis of relativity theory and transfers energy with it. The wave is transverse polarized.
An accelerated body emits gravitational waves and emits energy. The wave
is spirallike and differs from the neutral gravitational waves normally in a
static gravitation field.
Also in this case the gravitational wave slows time passing of bodies, which
is a general property in all gravitation fields.
A spirallike gravitational wave creates into
space an elliptic acceleration vector on the
rotation level of the source. Rotation direction
of the vector is the same as rotation direction
of the emitting source. The picture shows the
rotation directions in both cases.
Propagation direction
of the wave
A neutral gravitational wave creates into space an acceleration vector, which has only
two momentary directions. All bodies emit neutral gravitational waves.
An electric field and a gravitation field resembles each others also so that their amplitudes
weaken in square of distance. They both propagate at light speed.
A body, which stands in a spirallike gravitational wave with an elliptic acceleration vector,
absorbs energy from the gravitational wave. Absorbed energy becomes rotational energy of
the body into same direction with energy source to remain the total impulse moment.
4.D
4.D
3D-surface
Neutral gravitational wave
Energetic gravitational wave. Note the rotation plane!
Polarization differs from the neutral gravitational wave.
Neutral gravitational wave emitted by static gravitation field shortens length in direction of the field, which is
also the propagation direction. Instead the energetic gravitational wave emitted by an accelerated body
shortens length in a direction, which is perpendicular to the propagation direction.
67
When we consider appearing of the observer’s space, curvature of space caused by the
moving bodies has there an effect. The body A, which is moving in relation to an absolute
Manhattan-metric, causes at every moment in space local contracting and expanding. As
result of that the motion of space in relation to an even space is biggest in directions of the
main axes projections of the complex space, as already is told. A result of contraction or
expanding of the space is always inclination of space from the horizontal plane. A body
staying on its location in Manhattan-metric does not cause any inclination. The next picture
presents the principle of curvature in direction of 4.D or of the inclination caused by a body
moving at different velocities. The curvature in the picture is caused only by the space
contraction caused by the motion of a body and not for example curvature of a static
gravitation potential or of neutral gravitational wave is not considered there.
A body moving in relation to the
Manhattan-metric causes at different
absolute speeds a different contraction
and expanding and also a different
curvature. The number and direction of
the kinetic energy is included in the
curvature in relation to an even isotropic
space.
A
v=0
v1
v2 v3 v1>v2>v3
The absolute complex space (X,Y) is contracted in the next picture in directions of its main
axes X and Y when observed in the observer’s space. The circumference and the vector
describe the number of contraction in different angles  as already is told. The space (x,y) is
in the picture the 3D-surface and Xp and Yp are the projections of the complex space main
axes on the 3D-surface.
x
Xp
Yp
1

1
-1
y
-1
Local contraction caused by a body. In direction of the vector the number of
proportional complex space contraction is bigger than 1.
68
In the next picture is shown that the inclination in curvature of space causes an acceleration
outwards for an other body B. The acceleration on the other hand causes for the body B a
speed, which is exactly the same speed as the speed, at which the contracting space moves
in relation to an even space past the body B. So the speed of the body B does not change in
relation to an even space.
a
B
A
s
Contraction starts / acceleration
A
-a
B
The distance s between the bodies A and B in the contracting Manhattanmetric changes sideways because of the accelerations a and -a of the
body B but remains in the observer’s space and in an even Manhattanmetric. Therefore it is not possible to observe contraction or expanding
directly.
Two bodies moving together to the same direction interact trough the
accelerations +a and -a, which eliminate each other regardless of their
absolute velocities.
s
Contraction ends / braking
When the moving body A has passed the place the space begins on the backside of the body
to expand and curvature appears as opposite as also the acceleration of the body B in
comparison to contracting. The accelerations will realize now as opposite.
The number of the accelerations are at their height, when the derivative of the transition
speed of the space is at its height. Thus the acceleration is equal to the second derivative of
transition distance
a = d²s / dt , where s(t) = distance in Manhattan-metric.
The bodies have not experienced any acceleration in the observer’s space. For example a
free observer in an acceleration field does not feel any acceleration. The contraction and
expanding of the space are not observed. The effect is however measureable in principle,
because the body B has done in the space an extra traveling and experienced the
accelerations during it. Its time passing has slowed down for a moment and the length parallel
to the motion has shortened. The effect is however weak. Note that the source of this effect is
different than the slowing of time passing in a static gravitation field and it slows the time
passing in addition.
Motion of a body in an acceleration field resembles the motion of the floating body in the
previous example. The difference however is that the acceleration is now caused by
inclination of the space and not the gravitation. Inclined space means always a local
acceleration field in a curved space. It has its own role in appearing of the observer’s space.
Before only the affect of other body was considered. Both bodies however interact with each
other as the wave mechanics depicts. Einstein’s equations E = hf and E = mc² give de Broglie
equation  = h / mv , which depicts interaction between the particles.
69
The total energy of a body is equal to the number of curvature of space in direction of 4.D
caused by the body. The speed v included to a certain kinetic energy E k = ½mv² does not
remain same when a body moves in Manhattan-metric into different directions, because the
number of curvature is different there in different directions. Instead the speed of the body in
the isotropic observer’s space remains the same at a certain kinetic energy in all directions.
Let’s consider next the kinetic energy of a body in the Manhattan-metric and the number of
contraction of the space caused by the motion. The number of contraction in a point of space
is proportional to a current included to the contraction. The current i expresses how many
cells of the space moves past a point of an imagined even space in a time unit. In the next
picture the change of the space is the biggest in the directions of the main axes X p and Yp of
the complex space.
Even Manhattan-metric
The sum  i of the currents i, which cause the contraction, has
an opposite direction to the motion direction of the body in the
own set of coordinates of the body. The sum depends on the
speed v of the body. The sum  i is simply equal to the current of
the cells caused by the motion of the body past the body in a
time unit. The single currents i1, i2, i3 and i4 are the biggest. The
currents are depicted in the observer’s space. In their directions
the curvature and the energy included to the curvature are the
biggest. Also the kinetic energy is the biggest in those directions.
x
i1
y
i2
i
i6
i5
v
i3
i4
Xp
Yp
If the motion direction of the body is changed equal to for
example the direction of the current i3 and its speed in
Manhattan-metric remains, increases the sum  i and also the
kinetic energy.
When a body moves in Manhattan-metric, should the sum of the currents  i = Ek be the
same in all directions to conserve the kinetic energy Ek. It, however, does not happen so. The
reason is that the complex space contracts and also curves in some directions more that
others. The bigger curvature insists bigger kinetic energy and energy is a remaining quantity.
So, to keep the kinetic energy the body travels in an even Manhattan-metric slower in those
directions, where the space is contracted mostly. It means that in all directions of the
observer’s space the speed of a body is the same. An image of an isotropic space appears.
Still the speed of a body in relation to an even space in Manhattan-metric is not the same in
all directions as also the length of a body. The length changes as already is told.
If in one point of 3-dimensional even space bodies with equal
mass and equal kinetic energy starts to fly at one moment into all
directions of the space, the surface created by those bodies
resembles roughly a sphere surface. An image of an isotropic
observer’s space appears.
70
Disappearance of interference
According to quantum mechanics things like wavelike behaving and interference are
connected to elementary particles. One example about the interference is the Young’s double
slit experiment. Before is already depicted the wave function of elementary particles or the
cyclic motion of the absolute space in relation to the even space. The wavelike behavior and
also the interference however will disappear, when the scale is changed from the scale of
quantum effects to the macroscopic scale.
Let’s consider next the disappearance of interference, when the scale changes as far as to
the scale of a black hole. The next principal picture shows the three scales of matter.
o
V
Single elementary particle
Macroscopic body
V
A macroscopic body differs from others, because the
distance between its components side to side is much
bigger than the size of the components. Therefore it is
possible to calculate for them in Schwarzschild’s metric the
critical size or radius Rs, which zeroes the distances. The
corresponding length for a single particle is its size or o.
When the energy (=mass) of of a body is a constant, the
energy E = mc² is proportional to the product of amplitude
A of the wave and the proportional length
E  A²  = A² R = constant.
o
Rs
R
V
Black hole
The amplitude A² decreases, when the size R of body
increases and space curvature decreases. Ratio Rs/R
depicts in Schwarzschild’s metric the number of local
curvature of space.
ve² = A² = c² =
2Gm/Rs
Differing from the proportional length the absolute lengths
o and Rs will increase the number of energy when they get
shorter according to the formula E = hf = hc/.
Rs
The next picture shows a potential sag caused by the average of matter wave. It is called
also for the gravitational sag.
As an average of matter wave appears a potential
sag V(r) so that the wave damps out depicted by
the potential
V(r)
V = ve²
V(r) = Gm / r .
The acceleration caused by the field instead
damps out by quadratic distance
vn²
c²
dV(r) / dr = - Gm / r².
r
For the amplitudes (the speeds) of the cycles is valid
vn²
= ve²
c² - ve²
c²
or
vn² = ve² (1 - ve² )
c²
The wave is transverse and longitudinal. The
horizontal longitudinal part includes kinetic energy
Ek and transverse part includes potential energy
Epot.
The picture shows the potential and the wave in
two different phases with phase shift of 180º.
71
The part Rs/2R from the total energy E = mc² can be shown geometrically as an amplitude A
emitted by the longitudinal gravitational wave of a body like in the next picture. The picture
shows a horizontal longitudinal wave although the amplitude A has been written transverse.
c²
The energy Ek of a horizontal longitudinal wave is
in its amplitude A². The wave stands in a potential
sag Epot = mV and Ek + Epot = E Rs/R.
U=c²
4.D
m

A
Amplitude A is a quantity of absolute space and
 = 4R is a quantity of the observer’s space. In the
picture the size of the body is 2R.
A²
2R = /2
r
A² = Rs , where Rs is Schwarzschild’s radius
c²
R
In the picture the ellipse depicts motion of a point of space. The transverse motion is here
bigger than longitudinal or horizontal. Potential U is the contraction potential of a macroscopic
body and A²/2 is the gravitational potential of the longitudinal wave. The contraction potential
U corresponds to the absolute Compton-wave and its value is always c²/2 and the wave
length is 2R. The eccentricity of the ellipse is ve/c, where ve = A is the escape speed of
gravitational potential.
According to the model of D-theory the quantity parallel to 4.D is depicted indirectly as
squared speed v². In that direction the space has an edge and the maximum value. The
maximum is c², which also is the maximum of the speeds. We got before for the amplitude A
of the longitudinal wave
A² = ve² = V(r) = Gm
2
2
r
, where ve is the escape speed.
The potential V(r) determines the efficient amplitude A²/2 of the wave outside the body. Let’s
presume the mass of a body m = E/c² to be concentrated into its centre of mass and the wave
length /2 equal to body size 2R as in the picture. The kinetic energy Ek of the wave is equal
to the horizontal kinetic energy of the longitudinal wave
Ek = ½mv² = E A² = E ve² = E 2Gm = E Rs ,
2c²
2c²
2c² R
2 R
and gravitational potential energy
Epot = mV = m 2Gm = m c²Rs = E Rs
2R
2R
2 R
or
Ek + Epot = E Rs
R
, when Schwarzschild’s radius Rs = 2Gm/c². So we get for the body of size R
Rs = A² .
R
c²
A²  1/ = 1/4R .
The quantity Rs/R depicts the square of ellipse eccentricity shown before ve²/c², because
A = ve. The same quantity depicts also the curvature of space in a body. In a black hole the
ellipse is changed to a circle.
72
Let’s consider first the wave properties of a macroscopic body, when the relative size is
changing, and especially the longitudinal wave emitted by the body. The amplitude A of a
horizontal longitudinal wave and its gravitational potential V(r) decreases, when the relative
size R/Rs increases. At the same time the wave properties and the interference decreases
sharply. When the body is macroscopic, the interference appears from the horizontal
longitudinal wave A.
A², , m
When the size r = R of a body increases (or the distance
between the atoms gets longer) and when the mass remains
unchanged, the wave amplitude A² correspondingly decreases
and with it gets the wavelike behavior weaker. As a result the
interference disappears.
(r) = r
m(,A) = E/c²  A² 
30
A²(r)  m / r
r
Interference
Macroscopic space
Schwarzschild’s metric does not be valid in the whole scale of
R. For example it is not meaningful to calculate it for proton
For a lead pellet R/Rs = As²/A²  10 , which shows the great
change of the amplitude and interference disappearing. In
black holes the matter is collapsed and they may interfere as
elementary particles.
In scale of quantum effects the amplitude A of a Compton-wave of a particle is expressed
with help of light speed c. Energy is quantized and represents the share of one particle in a
sum wave of the body. Energy of the sum wave is got by adding all wave parts.
E =  ( hc / n )
, when n is the number of elementary particles.
Calculating the sum is however impossible in practice and this way is not used. With help of
wave mechanics it is possible to show that the local sine waves emitted by single
elementary particles produce a sum of different wave shapes or a wave package.
The wave properties of a particle are depicted in scale of quantum effects by the
proportional size / o, where o corresponds to the Schwarzschild’s radius Rs and  to the
size R of a body. The wave is now transverse wave or the Compton-wave of a particle and
not a longitudinal wave as in macroscopic body. Correspondingly as before the increase of
the proportional size / o makes the amplitude of the transverse wave to decrease and the
wave properties and interference will disappear. / o  R/Rs
In a black hole ve = c and V = ve²/2. The average energy of the longitudinal wave is
Ek = mV = E
c²
ve² = E c² = E
2
2c²
2
or on average a half of the total energy exist in the wave amplitude A. We get for a tranverse
wave in a black hole, when R = Rs and V = Gm/Rs
Epot = mV = m Gm = Gm² c² = mc²
Rs
2Gm
2
, where Rs is Schwarzschild’s radius Rs = 2Gm .
c²
= E or Ek + Epot = E Rs/R = E.
2
73
For a black hole is valid
Ek + Epot = E
, which is the same as the content of the Schrödinger’s equation of a particle. A single particle
and a black hole resemble in this sense each other. In a black hole the particles are narrowly
side by side otherwise as in ordinary body. A normal macroscopic body is thus an exception.
Its wave properties are weak and interference is missing.
Compton-wave length
The whole energy of elementary particles is in the wave, which they emit around. The wave
length can thus be calculated by the total energy or by the mass of the particle
E = hf = hc = mc²
c
which gives a Compton-wave length
c = h
mc
.
Compton-wave length has a direct connection to the length d, which is the classical radius of
electron
d=
ħ
=
c
,
137,035999 mc
2  137,035999
and to Bohr’s radius
R = 137,035999 c .
2
The Compton-wave length is an absolute quantity in the same sense as the basic length d of
the absolute Manhattan-metric. Spin-½-particle oscillates in tempo of the elementary
rotations in the way required by its geometric structure and by its location and emits its wave
motion around. The wave causes observable phenomena in interaction. Those are often
depicted by de Broglie-matter wave. De Broglie’s matter wave appears only in interaction and
therefore its wave length depends on the relative speed as soon is shown.
Let’s consider next a particle, which approach to an obstacle, which includes two slits side by
side. During approaching the particle emits its Compton-wave into all directions. The wave
then reflects back from the obstacle and that wave length is now shorter because of Dopplereffect. When the particle has traveled through the slits, the emitted wave again reflects from
the obstacle back and passes the particle. The wave length of the passing wave is now
longer because of Doppler-effect. The passing wave interferes with the emitted wave. The
new wave length is calculated next.
74
½d
fc
fr
fc
v
In the picture the particle approaches from left and travels
through two slits. Compton-wave length of the particle is
c. After the slit the wave emitted backwards by the
particle reflects from the wall and then reaches the
particle. Because of Doppler-effect the wave length of the
reflected particle r is longer than c. As a consequence
of interference of the waves c and r there appears a
wave packet, which has de Broglie-wave length d. The
precondition for the interference is the interaction with the
obstacle
d
=
h .
mv
In the picture the distance between the slits is
exaggerated. Its size is about the same as d.
The frequency fr of the reflected wave is less that the frequency f c of Compton-wave. The
distinction frequency after the slits is got
f = fc - fr.
Correspondingly, when  is the wave length of the appearing wave package
c = c _

c
 =
c
,
c +
c (c - )

When c = cT and  = vT, where v is the speed of the particle in relation to the obstacle, we
get
 = c T (c - v)  c c , when v<<c. When c = h / mc, we get
vT
v
 =
h
mv
, which is the wave length of de Broglie’s matter wave.
We can see that de Broglie’s matter wave exists only in interaction. Its wave length is at
relativistic speeds much longer than the Compton-wave length of a particle.
75
Deduction chain to understand quantum physics
Probability density of a particle position  Particle position simultaneously here and there
 Spreading of the real position of the particle to the even observer’s space  There
exists an even Euclidean space and in regard to it an nonlinear and changing absolute
space  The latter space is real and the former is only an image created by the
observations  An macroscopic observer is needed to create the image or the observer’s
space  The consciousness seems to have an effect on the measurement and to share
the world into microscopic/macroscopic world
Particles or matter
Einstein’s equation or the motion equation of general relativity theory can be written in form
GEOMETRY OF SPACE( ) = MATTER( ) .
The equation is in practice a second order differential equation. The equation describes the
influence of space on matter and the influence of matter on the space.
The equation can be interpreted also in other way. We can think that matter is made of the
same fundamental substance as the space and matter appears as local waving of the
substance. We can present a geometric model, where the locally wavy space appears as a
particle or as matter. Oscillating makes the space contract and curve locally. Mode of the
oscillation, the phase and the location, will expose, what charges and other quantum
properties the particle has. Asymmetry of curving on the 3D-surface appears as a relative
speed in one direction of the surface. Curving perpendicular to the surface gives for a particle
a scalar type mass and its rest energy. On the other hand when a locally oscillating object
moves to a curved space, its asymmetry or the relative speed will change at some
acceleration as influence of the space. So the mass and the motion of a particle determine the
local geometry of the space (curvature) and the space will on its side determine the motion of
the matter.
Next we consider three elementary particles, proton, neutron and electron.
When the space is mainly defined, what are the particles, which move in the space?
(According to quantum field theory the particles are quantized fields.)
An elementary particle is not a separate substance of its own! A particle is fundamentally a
part of the space and so the space is the only substance (base of reality), which is needed in
the world. The space is also the only abstraction of the world.
A part of the particles in the four-dimensional space are four-dimensional. Proton, for example,
is composed of 3 quarks parallel to 3D-surface. One of the quarks is always folded in its
middle point parallel to one lattice line shape and thus forms the complex 4.D-component of
proton. Let’s consider next, how does the cell-structured space form the 3 quarks of proton.
76
Quark
Gluon
3D-surface
lattice box
The three quarks, which feel the color force, stand
on the 3D-surface as diagonals of its octahedron.
There are in all 8 gluons and everyone has 2 colors.
In the 4-dimensional space the three quarks of proton
stand inside a complex lattice box. The lattice box
determines the size of the proton.
A particle or a quark can be described as a cyclic curving or as a space wave in a 1dimensional cell. The direction of a quark is the same as the direction of one main axis on the
3D-surface. It means that on 3D-surface 3 quarks are needed (for example, for a proton). When
a color code, blue, red and yellow, has been selected for all three main axes of absolute space,
we also get a color code for each quark according to its direction in space. In Quantum theory
the color of a quark is an abstract feature of a quark. The colors are unobservable like the
main axes. A particle made of three quarks, like a proton, is then neutral or colorless. When the
main axes are isotropic, must the colors be isotropic as well.
The color charge of particle means that a particle is a part of 3D-surface in direction of one main
axis. If a particle has no color charge, it stands in the complex lattice outside the 3D-surface like
electron.
The diagonal of octahedron bends in the middle to a form, where the space is contracted or
curved to a minimum point, which has the radius called Planck’s length P. The diagonals of
octahedron fold up in turn in their centre to the direction of a complex lattice line like in the next
picture. At the same time appears on the 3D-surface a longitudinal motion of the space towards
the centre. The longitudinal motion is limited in a small area as the color force too.
P
3D-surface
Folded quark
The complex diagonal of a lattice box has in a
proton folded out to one side of the 3D-surface.
There exist 6 different positions.
3D-surface
A folded quark forms a length longer than zero in
direction of 3D-surface. The length is the shortest
possible length and it is also a half of the thickness
of the 3D-surface. (Planck’s length) A quark is even
u- or d-quark depending on whether the diagonal is
contracted or not, as later is shown in detail.
Planck’s length P is 1,6 · 10 -35 m.
The different phases of proton and the properties of quarks u and d are considered later in Dtheory. Let’s consider next the size of proton and let’s search a theoretical prognosis for the
size of proton diameter.
77
The size of proton is determined by the size of complex lattice box in different directions of the
3D-surface. The length of a lattice box diagonal in even space is always a constant
for the observer although the lattice box would be contracted in relation to the 3D-surface.
However, the length of a cell on the 3D-surface is a half of the length of the cells in the complex
space. On the 3D-surface the length of the diagonal of a layer is d, which is the same as the
diagonal of proton.
t0
t1
t2
t3
P
=
L=d
Octahedron +
antioctahedron
P
The size of a particle is in absolute space L, where L = d is the size of a layer or the
maximum size of a particle. When L is the measure unit, the size of the particle is
L = 2.8179403 · 10-15 m = d,
which is the proton diameter and the classical radius of electron. ( The theoretical prognosis
for the measurement result of proton diameter is published on the next page.) Both the
halves of cycles in a quark cause the space around to be contracted and we say that the
space is curved.
When each of the proton quarks is contracted, they interact by curving the diameters of
octahedra of the 3D-surface around the proton. Those interactions of proton appears in
environment as cloud of particles around the proton. If the curved diameters are projected
to the proton itself, the proton seems to contain more than 3 quarks and 8 gluons. So the
model of proton looks more complicated, but still there are two more u-quarks than dquarks.
The quark, which is folded parallel to a complex lattice line, interacts in 2-dimensional electron
plane with the lattice line shapes, which gives for the particle its electro-magnetic features, as
later is shown. When there is 3 quarks and they all take one after another a part in interaction,
the electrical charge e+ of proton can be shared into 3 parts. The contraction of the 3D-space
gives for a particle its mass.
d
d
p+
R
p+
e-
p+
In electrodynamics the classical radius of electron R = d is calculated by
presuming that nucleus or proton p+ is pointlike and that the charge e- of
electron stands at a distance R from the proton. By presuming now that
the rest energy E = mc² of electron is of equal size to the electrical
potential energy E = -ke² / r , where r is the distance, we get as a result
r = d. The same electrical potential energy is got by setting side by side 2
protons, of which diameter is d, and by presuming that their charges are
pointlike.
78
Let’s consider next, how near to each other can two protons or octahedra stand. The distance
has an effect on measurement result of proton diameter. In the next picture the centres of
protons 1 and 2 stand at a distance d from each others. Their edges do not touch each others
but the vertexes are united. The centres of octahedra 1 and 3 stand at a distance √ 2 d/2 or
much more closer. The spins of these protons have the same sign in the picture.
√ 2 d/2
3
√(½d)²/2 + (½d)²/4 = ½d √ 3 / 2 = 0.8660d/2
2
1
z
y'
x'
y
x
d/2
Three protons side by side. The spins
have same signs.
Two halves of protons side by side stand at a distance
0.8660d/2 from each other. The spins have opposite signs.
In other picture at right stand two protons, which have opposite spins or the other stands in
space and other one in antispace. Their centre stand at a distance 0.8660d/2 from each other.
The octahedra in the picture can be imagined also as spheres of observer’s space. The
spheres overlap each other and their distances are the same as presented before.
The protons can pass each other at these distances. If we think that the protons with the same
spins can pass each other in minimum at a distance √ 2 d/2, and that the protons with the
opposite spins can pass each other in minimum at a distance 0.8660d/2, we get for the
distances as an average
L = ( √ 2 d/2 + (√ 3 /2) d/2 ) / 2 = 1.1401 d/2
The average can be used here, because spin-up- and spin-down-protons exist on average the
same number.
L= 1.1401 d/2 = 0.5701 d = 1.61 · 10-15 m.
This quantitative prognosis appears from the geometric features of proton and matches to the
newest measurement results ( proton radius = 0.805 ± 0.011 fm, diameter = 1.61 ± 0.022 fm
link: http://scienceworld.wolfram.com/physics/Proton.html ).
We use later the length L = d as a proton diameter, because it also has geometric arguments.
Structure of proton is described more in detail later in D-theory.
79
The contracted space around a particle means that the particle has a mass. On the other hand
the wave can be understood a pure energy so that the square of the amplitude of the wave is
proportional to the amount of energy. So the mass and energy are in a particle the same thing,
as the well known formula of physics E = mc² insists.
The local diagonals of an octahedron on 3D-surface are contracting frequently in different
phases. Appearing of proton quarks (uud) and neutron quarks (udd) from cyclic rotations is
described later in D-theory. The direct component of a wave is directed to the centre of a
particle and spreads out to the environment and gets the space around a particle to be curved.
The mass of a particle means ability to contract (curve) the space around. The space model
helps us to describe the most fundamental properties of the elementary particles like mass,
electric charge and spin.
(The ancient Greeks understood that there exist four substances; earth, water, air and fire. The
substances of today are not written up to the holy articles of modern theoretical physics.)
When a particle moves freely in space, it has a certain direction when observed in the
observer’s space. The direction is a coarsened concept, which is based on coarsening a
fractional line standing in Manhattan-metric. So the coarsened direction does not exist in the
same sense as the fractional line does not exist. Instead of the direction of motion the whole
path must be defined for a particle by defining with help of integers all those points, through
which the particle travels.
Quantization of momentum
The momentum (or the mass, the speed and the direction) of a particle is determined only by
the projections of the particle on the 3D-surface. The projections appear as curvature of the
cells of the surface. According to Newton the momentum of a body remains forever, if no
interaction will exist. So the information about the direction or about the upcoming path of a
particle in Manhattan-metric is located in curvature of the cells of the 3D-surface. The
information will determine the path of the particle uniquely regardless of the path length. The
amount of information is thus enormous big. Let’s consider next curving of a single 1dimensional cell.
P
2 P
In the picture a segment of line is bended so that the heads of the line
segment stand side by side. The thickness of the line segment is not zero
and the distance between the heads is greater than zero. The distance is
the Planck’s length P. The distance is a constant and it is also the smallest
possible distance. The distance is also quantized. If the bending of the cell
is decreased a minimum number, the distance is now two Planck’s lengths.
The change of the distance can not be continuous, because then the
change should also be infinitesimal and the nature does not contain
infinities. Quantization of curvature is so called fundamental property of the
cells.
The relation between the classical radius d of electron to the Planck’s length is
d / P = 0.5 x 1020. So the line segment d can be bended to numerous different positions and
the curvature amplitude can thus have numerous values. Still all the values are fully
determined or quantized.
80
Let’s presume that the momentum of a particle is determined by the N octahedra of the 3Dsurface belonging to the projection. They all contain 6 1-dimensional curved cells, which all
can get 0.5 x 1020 different values. When we calculate, how many different values the
momentum of the particle can get, the number is so incomprehensible big that it is not worth
writing here. The number is marked by the letter M.
The number M includes the path information of a particle for the almost infinite long path in
Manhattan-metric. The path is precisely determined and the particle without any interaction
travels along the path maintaining its momentum. The speed of a particle is contained in the
amount of asymmetry of curvature and it is possible to be depicted with help of eccentricity of
ellipse as later is told.
The curvature considered before means the curvature inside the 3D-surface to different
directions of the surface. So the relative speed and the motion direction of a particle are
quantities of the 3D-surface. Some curvature appears also perpendicular to the 3D-surface.
The mass of a particle or the rest energy E = mc² gets its value from the longitudinal and
transverse waving of space and at the same time from the absolute sinking in direction of
4.D, as later is told. Also mass is a quantized quantity.
4.D
y
x
y
x
The internal curvature of the 3Dsurface creates the momentum of
a particle. In the picture 2dimensional depiction.
The curvature of the 3D-surface and waving to the direction of 4.D
gives a mass for a particle or the rest energy E = mc².
81
Non-locality in quantum mechanics and the ”spooky action-at-a-distance”
The well known paradox in quantum physics is the EPR-paradox including action-at-a-distance.
Einstein, Podolsky and Rosen proposed in 1935 a thought experiment to measure entangled
pairs of particles. Later in 1951 David Bohm proposed that correlated spins of particles would
be measured. John Bell proved theoretically that so called classical correlation and quantum
correlation would differ. Later in the measurements physicists observed that the quantum
correlation really violates the Bell's inequality and differs from the classical correlation. The
difference means that there seems to be a non-zero correlation between the particles. The
classical correlation means phenomena in macroscopic world or in observer's space. The
quantum correlation instead appears at quantum level. Let's consider next reasons for the
correlation differences. It is also shown that the action-at-a-distance, which progresses in space
unlimited fast between the entangled particles, is an incorrect but natural conclusion.
In the experiment made by Alain Aspect the polarization correlation C of entangled photons at
the distance of 15 meters from each other were measured. In the experiment a pair of photons
appears and the photons fly to opposite directions. The linear polarizations of the photons are
equal. Both of the photons are driven to polarizer or crystal. If the optic axis of both polarizers
have the same direction, each photon will always do the same as its twin in probability P = 1
and a complete correlation is got or C = 1. If a polarizer is set at an angle of 45 degrees to the
other one, the photons behave equally in 50 % of the cases. For a single photon passing
through the crystal is fully random and there is no correlation in behavior of the photons. This
means that they will do the same thing in 50 % of the cases or C = 0 and P = 0.5.
When the optic axis of both polarizers are perpendicular to each other, all pairs of results in the
experiment are opposite or P = 0 and complete anticorrelation is observed or C = -1.
+
a
light source
polarizer
b
+
detector
penetration
-
-
no penetration
correlator
An interesting case appears, when polarization crystals are rotated at an angle between
complete correlation ( 0 º) and complete anticorrelation ( 45º). For example at an 30 degree
angle the experiments show that the photons behave in the same way in 75 % of the cases
(P=3/4). In the remaining quarter of the cases they behave in opposite way. This is against the
classical or linear behavior, of which probability is P=2/3, because from the angle corresponding
the probability P=1 is decreased 30 degrees or 1 - 30º/ 90º = 2/3. We can see that 3/4 > 2/3,
which means violation of Bell's inequality.
Quantum correlation thus differs from the classical correlation. Let's consider next the reason of
the difference on grounds of the space model of D-theory.
82
A photon travels through the crystal, if its polarization angle to the polarization axis is smaller
than 45º, and the result is +. If the angle is bigger, the result is - . In both crystals the
penetration angles will fully overlap, when the crystals are parallel or  = 0º. The results are
only (++) or (- -). (See the picture below.)
If the angle between the crystals is  = 90º, the penetration angles do not overlap. The results
for two photons are now (+ -) or (- +). When 0º<  <90º, penetration angles will partly overlap
and the results are random. The correlation C of the results should according to the classical
view change linearly with the angle 
C = 2 (90º - ) / 90º - 1 = 1 -  / 45º.
 = 0º and C = 1
 = 45º and C = 0
 = 90º and C = -1
According to the space model of D-theory the quantum effects will happen in the smallest
possible scale or in Manhattan-metric of the absolute space. In this kind of space all angles at
the level of cells are right angles. However macroscopic bodies and the macroscopic angles
between them have their effects on the behaviour of photons.
A macroscopic stick is rotated in the next picture from the direction of the main axis X to an
angle . Its length in Manhattan-metric is a cells. The stick will shorten in the direction of the
main axis X as the function cos . The length of the stick in the observer’s space is quadratic or
L = a² and the length of the component parallel to X-axis is correspondingly Lx = a² cos²  and
parallel to Y-axis Ly =a² sin² .
The absolute space is quadratic in comparison to the
observer's space. It means that quadratic quantities are used
to describe these quantum effects. With help of them it is
possible to get in use the components parallel to the
assumed directions of main axes. The components are added
up, for example
Y
a² sin² 

a² cos² 
a²
X
a² cos²  + a² sin²  = a².
The quadratic functions cos²  and sin²  will describe
directly the quantum correlation. (See the next page.)
The directions of the main axes will disappear in transformation to the observer's space (X 
x², Y y², Z z² ) and no effect can reveal their directions. The direction of the angle  can be
what ever in 3D-space.
83
According to the hypothesis of D-theory a circle in observer's space is a square in absolute
space. In the next picture a line segment AB is a side of the square and the circumference of
the circle passes points A, P and B. The angle  between polarizers must be realized on the
level of cells in absolute space.
In the picture the quantum correlation is described in absolute space
with help of the line segment AC and BC according to Manhattanmetric. Length of the line segment OC is a constant. The corresponding
classical correlation is described in the observer's space with help of
the circumference APB. Length of the line segment OP is a constant.
A
When the angle between the polarization crystals is , the
distances AP and BP on the circumference describe the
classical probability. We get by normalizing
(AP+BP)2/R = (90º - )/90º + /90º = 1.
OC √2 cos 
C
P
 = 30º
O
R
H
OC √2 sin 
B
The squares of AC and of BC describe the quantum
probability. When in Manhattan-metric OC = OH+HC = R,
then AC² = 2R² cos²  and BC² = 2R² sin² . We get by
normalizing
(AC² + BC²) / 2R² = cos²  + sin²  = 1.
Note! The line segments AC and BC belong to the absolute space and they need to be squared in transformation
to the observer’s space.
When  = 30º, a quadratic normalized line segment AC² = cos²  describes the quantum
probability of the identical behavior of the two photons, which is P = cos² 30º = 3/4. The result
is the same as in the previous experiment for the quantum probability. The length AP on the
circumference describes the corresponding classical probability P = (90º - )/90º = 2/3.
The experiment gives for each pair of photons one of the results (++), (- -), (+ -) or (- +). For
the probabilities of these results is valid P(++) + P(- -) + P(+ -) + P(- +) = 1
The factor cos²  describes the probabilities P(++) and P(- -) or the identical behaviour of the
photons. When  = 0, both photons of the pair do always the same thing and
P(++) + P(- -) = cos² 0 = 1 and correlation C = 1. We can now write
P(++) + P(- -) = cos²  .
Correspondingly we get for the results, which will increase the anticorrelation,
P(+ -) + P(- +) = sin² .
At an angle  = 90º the photons behave in crystals always in the opposite way. In the
experiment the expectation value of the result is
E() = P(++) + P(- -) - P(+ -) - P(- +) = cos²  - sin²  = 2 cos²  - 1.
Expectation value E() gets the values -1<= E() <= +1 and it corresponds to the correlation
C in this experiment or
E() = C() = 2 cos²  - 1 ( = cos 2 ).
The result is the same as the correlation product of Quantum Mechanics.
84
P(++) + P(- -) = cos² 
1.0
quantum probability
f() = cos² 
The probabilities P are calculated for the identical
behavior in polarization crystals or for the results
(+ +) and ( - -).
0.75
0.67
Other results are (+ -) and (- +) .
classical probability
0.5
The results correspond completely to
the prognosis of quantum mechanics.
0.25
For the angle 30º the quantum probability is
cos² 30º = 0.75. The classical probability is
0.67.

0º
30º
60º
90º
E() = C()
1.0
quantum correlation
f() = 2 cos²  - 1 = cos 2
The results correspond completely to
the prognosis of quantum mechanics.
0.5
classical correlation

0
30º
60º
90º
We can see that the classical correlation
and the quantum correlation differ from
each other. Thinking classically (=in the
observer’s space) it seems that there is a
non-zero correlation or some interaction
between the particles.
-1.0
So it is shown that in quadratic Manhattan-metric the correlation of photons appears as the
measures and quantum mechanical theory have shown.
D-theory is so called local hidden variable theory. It has been alleged and mathematically
argued that violation of Bell's inequality proves that local hidden variable theories are not
possible. The non-local "action-at-a-distance" between entangled photons would make all local
hidden variable theories impossible. It is not, however, paid attention to the possibility that the
space at the level of quantum effects is not similar to the observer's space. The scale and the
structure of space have their significance and therefore as a consequence of these the
quantum correlation differs from the classical correlation. We can actually think that the
violation of Bell's inequality proves that the space in scale of quantum effects has a different
structure as the observer's space seems to have.
85
As before is proved, no action-at-a-distance needs to be related to quantum correlation. In
addition it is very obvious that the action-at-a-distance needs not to be related to any other
phenomena of physics.
Modern physics is based on two great theories; Relativity theory and Quantum theory.
According to Relativity theory nothing can move faster than light. Quantum theory on the
other hand contains as a conclusion the unlimited fast action-at-a-distance. Both theories are
proved right in numerous experiments. With help of a new space model these theories are
able to get unified and remove the obvious contradiction. A new space concept is an element,
which bot of these theories need.
Chaos and determinism in cell-structured space
A nonlinear system behaves chaotically, when its internal feedback is strong enough.
Sensitivity for the initial values is typical for that kind of systems. The system travels in a
limited time into different states depending on the initial values at the beginning. The
differences in the initial values may be infinitesimal. One example is the so called butterfly
effect.
The cell-structured space is the space of big integers. The initial values are given as integers
and thus it is always possible to set the initial values precisely. Then the system behaves
always accurately in the same way. The development of system is in principle possible to
predict completely in all details.
In four-dimensional cell-structured space the principle of uncertainty does not mean any real
inability to predict as the physicists have interpreted. At the level of quantum effects the
quantity x p has always an exact value, which depicts the exact geometric structure of the
cell-structured space. So we can think that at the starting moment of the universe, perhaps
about 15 billion years ago, progressing of the universe were exactly determined in all details
until today. An other world, which started with the same initial values, would have progressed
exactly in the same way.
This means, for example, that such an ideas than ”free will” or “accident” are illusions
according to the model of D-theory. It looks like that the physical world especially in scale of
quantum mechanics is totally different than the macroscopic world known by people. The
model of D-theory makes it possible to perceive the world and the human himself purely by
means of deterministic events. The events are fundamental and quantum mechanical. Matter
needs to be understood as a group of parallel and serial events in space as the particle
model assumes. A human has “lived” in the events of the world already before his birth. The
events have necessarily led to human’s birth. A human can imagine to live forever in the
events of the world. It is comforting to know that it is not possible to really affect on the
events; ”May your will be done...”
86
Rotations and gauge principle in cell-structured space
The elements in Lie groups used in connection with the rotations of quantum mechanics are
real numbers. In absolute space the elements of rotation groups can however be pure integers.
The elements are rotations. The length always remains in rotation. Let’s consider next 2dimensional space of integers N(2), which is a level in absolute space.
In the picture the length of vector v is 4. After first rotation to
clockwise the length is 1 + 3 = 4, after second rotation 2 + 2 = 4,
then 3 + 1 = 4 and at the end the length is 4. A rotation or an element
of group can be expressed as an integer instead of an angle. The
integers 1, 2, 3 and 4 are here the first terms of the previous sums.
The term 4 means here a rotation of 90 degrees. In this way different
kind of Lie algebras are possible to define in absolute space. The Lie
algebras used in quantum mechanics can thus be applied also in
absolute space. The difference to observer’s space is that the length
of vector v is not unique in observer’s space, as also the amplitude
of a wave function is not unique. (1² + 3² ≠ 4²)
v
Next we consider rotations of the lattice particles in the lattice boxes. The rotations create the
periodic and regular motions of the lattice line shapes. All the phenomena of quantum
mechanics appear from these background independent rotations.
The lattice line shapes move periodically in directions of the projections Xp, Yp, Zp and Wp of
the complex main axes at four electron planes in phases. From a point of view of such a point,
which does not move with the lattice, the motions of lattice line shapes are rotations. This kind
of a series is called here an elementary rotation series. Elementary rotation series is a series
of consecutive rotations in the smallest scale and forms in each of the four subspaces, like
(X,Y,Z)-space, one regular (X,Y,Z)-period of 360 degrees. Then the same is done to the same
direction.
When one 360 degrees elementary rotation series like this is done, the lattice is almost similar
as before the rotation. Only the positive and negative lattice lines seem to have changed their
places between themselves or a phase shift of 180 degrees has appeared in the lattice. An
other elementary rotation series is needed, or 720 degrees altogether, to make the lattice
seem similar again. More about the elementary rotations in detail later.
Y
Z
The lattice line shapes travel on the axis into opposite directions parallel to
each axis.
X
The movements of the lattice line shapes in phases always in the same order
past every point of 3D-space create rotations in every point outside the 3Dsurface in complex space. The rotations are the same at every moment
everywhere or globally in empty space.
87
As later is told in detail, the same thing is valid for the spin-½-particles as for the lattice; Two
full elementary rotations is needed before they seem similar again.
The 1-dimensional lattice line shapes moving on the 3D-surface into opposite directions and
spin-½-particles stand mathematically in 2-dimensional electron plane in the complex space so
that the axes, which span the 2-dimensional space, are in these rotations parallel to the lattice
lines. Both axes are complex or they both are at an 45 degree angle to 3D-surface The fourth
spatial direction outside the 3D-surface is defined as imaginary. This kind of 2-dimensional
complex space or an electron plane is called in rotations for SU(2)-symmetry space. The
SU(2)-rotations have 3 so called generators and physically they have connection to the
elementary rotations in R(3). More about SU(2)-rotations later in D-theory.
Elementary rotations happen everywhere in space simultaneously or globally in the same
phase. The elementary rotations determine the phase of all wave functions of the system. So
the phase of the wave function used in quantum mechanics is globally and locally the same or
invariant. If the phase of a wave function is changed in wave equation, it will change everywhere in space by the same number at once. The speed of change is apparently against the
principle of relativity. Changing the phase or so called gauge transformation corresponds in the
space model of D-theory to changing the phases of the lattice line shapes. Changing the phase
has no observable physical meaning. This property of a wave function and of the cellstructured space is called with the name gauge principle. The gauge principle is considered
more in connection with electromagnetic field and acceleration field.
The lattice line shapes have no other chance
than move globally into opposite directions.
The length of the motion is exaggerated in
animation. To drive the animation use PageUpand PageDown-keys in SlideShow-state (F5).
Note that in quantum mechanical measurements both the measurement object and the
observer himself with his observation devices are made of the same elementary rotations.
Therefore it is not possible to observe the phase of an elementary rotation in the object. From
the same reason there can not also exist any measurement, which would reveal the phase of
wave function. (In solutions of Schrödinger wave equation the phase is an imaginary quantity.)
Only a local phase shift is possible to be observed indirectly in force fields, for example. More
about them later in D-theory.
Let’s consider next, what does the space model tell about the phenomena called for ”wave
function collapse” and for ”observer’s consciousness”. They are both linked to the interpretation
of quantum mechanics and to the unresolved measurement problem of quantum mechanics .
88
The uniqueness of space and the "wave function collapse"
It is already mentioned that empty absolute space without noticeable light and matter is not
unique for the observer.
Let’s presume that a macroscopic stick S stands in empty space. The stick includes in
observer’s space the well known points S1, S2, .... Let’s consider a point P outside the stick in
absolute space and its distance from the known points S1, S2, .... We can see that the location
of the point P is not unique observed from these points.
S3 S2 S1
r r
P
r=a
P2
P3
Rn
rn
Let’s presume that the distance of point P from point S1 is in observer’s space r = a and in
absolute space R = a². Now we set r = a, so that we can use it as a measure unit. Let the
measure unit be the same for both spaces. The measure unit needs not to be squared because
1² = 1.
The distance rn of point P from point Sn is calculated in observer’s space, which is linear, as
sum of the lengths or
rn = r + nr = a + na = a(1 + n)  an,
, where n = 1,2,3...
In the absolute space the lengths are first added and then the sum is squared into the
observer’s space or Rn = (a + nr)². When it was chosen r = a, and the measure unit or the
value a is not squared, we get
Rn = a(1 + n)²  an²
We can see that rn is linear and Rn is quadratic. If we are now looking from poínt S2, the point P
in absolute space stands in point P2 in observer’s space and not in P. Correspondingly looking
from point S3 the same point P stands in point P3. When the known points S1, S2,...S n exist
plenty in space, the point P seems to stand as a set of points or it spreads out to empty
observer’s space. If in point P would stand an unobserved particle, its location would spread in
similar way and the wave function describes now the spreading of a particle. The particle must
be unobserved, because else it would have an unique location in linear observer’s space.
Collapse of the wave function is said to be mathematically uncontrolled. In addition it happens
simultaneously everywhere in space. The reason of the collapse is the change of space image
from nonlinear to linear by the previous transformation of coordination set. The change of space
does not mean any concrete transfer. The change is always connected with observing.
89
Let the known points A and B exist in space so that observer stands in point A. A particle, which
is not yet observed, travels in empty space between these points from A to B. The particle has a
well defined place, which is not known. According to quantum mechanics a particle, which is not
observed yet, is in indefinite state and travels via all possible paths at the same time. According
to D-theory the reason for the indefinite state is that an empty absolute space is not unique for
the macroscopic observer. Only a measurement makes the place of a particle unique. The wave
function collapses simultaneously everywhere in space. The wave function collapse means
localisation of the particle to observer’s space caused by the measurement.
B
The colored area in empty absolute space is not unique for
the observer (in the picture). An unobserved free particle has
an exact place in absolute space, but in observer’s space its
location spreads everywhere.
a
y
b
x
A measurement during travelling of a particle will change the
case.
In the picture the place of the particle is found out with help
of Pythagoras’ theorem, but the wave function will "collapse".
x² + y² = s²
In Manhattan-metric the way AB = a+b can be
traveled in numerous different ways. The path is
not unique.
If the observer’s space is thought to be real and the location of an unobserved particle
spreads there, we must presume that the particle travels in observer’s space simultaneously
via all possible paths and the wave function describes the particle. This assumption is of
course against common sense, but just that does the non-uniqueness of absolute space
mean and quantum mechanics confirms the presumption to be right. In 4-dimensional
absolute space the particle has however only one location and one path.
Observing needs always to be connected to a "consciousness". With help of consciousness
the observer understands the macroscopic space unique, Euclidean and macroscopic.
Behaviour of light and matter makes it possible, as already has been told. The observer’s
space does not exist without a macroscopic consciousness. The consciousness needs to be
macroscopic and able to coarsen the observations, but no other requirements are needed.
The consciousness or the ability to coarsen will produce the emergent properties like
directions and lengths of the observer’s space from the components of absolute space. So a
proton, for example, can not have consciousness. This space concept finally solves the
measurement problem in quantum mechanics.
90
By adding to the space the known points C,D,E,.. and so on the uniqueness of space increases.
When finally there are points plenty enough, the space can be understood fully unique. The
known points are particles and they form together a macroscopic body. Localization of a body is
considered at the next page.
Henri Poincaré: “If rigid bodies would not exist in nature, we would not have geometry.”
When only one known point exists, the space around is everywhere non-unique. The nonuniqueness of space means spreading of the location of a free particle to observer’s space as a
quantity called for a wave function
(x) = A1 e
+ikx
+ A2 e
- ikx
.
The position of a particle is defined by the density of probability:
(x) = *(x) (x) = A1 ² + A2 ² = constant.
This expression describes the non-uniqueness of the absolute space. Because (x) is a
constant everywhere, a free particle can not be localized. At the same time it describes the
wave function, which spreads everywhere in space as a cause of non-uniqueness. Also a pure
location in absolute space without a particle spreads in the similar way into observer’s space.
Physical nature of the wave function is unknown in Standard model regardless of its central
meaning. Instead the square of the wave function is postulated there as the mathematical
probability density of the location of a particle. According to the model of D-theory the square
of wave function is in the observer’s space the spreading figure of the pointlike location of a
particle in absolute space. The spreading figure appears because of nonlinearity (= nonuniqueness ) of absolute space when observed in observer’s space. Squaring of a wave
function transfers it to the observer’s space.
When the number of known points in space around the particle is big enough, the particle is
tied. Then free motion of a particle is limited by a potential field and the wave function of a
particle is limited inside the potential.
The wave function as the solution of a wave equation is complex or it includes a real and an
imaginary component. According to the space model of D-theory energy of the wave equation
is perpendicular to the 3D-surface as abstract quantities and can thus be described as
complex.
We consider next appearing of the observer’s space by means of Fourier transformation.
91
Localisation of body or how does the observer’s space appear
The existence of the observer’s space is based on existence of observed points. An interaction,
of which type is not important, is always needed in observing. The simplest way is to describe
the interaction as an regular sine wave, which is here an abstract description of interaction. Two
known points or particles are needed for interaction, for example P1 and P2. The distance
between the points is the wanted abstract wave length in absolute space. The wave is
considered to continue past the points and to fill the whole space. In observer’s space the wave
length and amplitude needs to be quadratic.
Observed bodies are needed for appearing of observer’s space and for its linear geometry. The
bodies are made of numerous points and interactions between them. Increasing the number of
the points means that also the abstract waves between the points will increase. We presume
that the waves will interfere. In some direction the distances between the points form as
coarsened a continuous limited function. The locations of every point in relation to all others are
significant. This function is possible to present with help of sine waves or with Fourier
transformation. When the distances between the points or between the abstract waves
correspond to the sine waves in Fourier transformation, so called wave package appears in
space besides the points. The wave package means the localisation of the body made of the
points in observer’s space and also appearing of the observer’s space. So we can use the
Fourier transformation as a mathematical model in description for appearing of the observer’s
space.
An unobserved particle does not belong to the observer’s space. It has no observable
interactions and therefore not any wave is connected to it.
P1
L
P2
The closer the distances of the points are to the smallest scale of cell-structured space, the
wider and smoother is the spectrum of sine waves. In smallest scale the spectrum is fully
smooth and the amplitudes of the sine waves or of interactions are fully homogenous.
Lets consider first empty space and the only free particle in it. The size of the particle forms
then the only known length in the whole space. The whole space can be then divided into
multiples of this length or gauge. Lets consider the space only in one direction. The observer’s
space is then described as a sine wave broaden into space.
Description of space is not good enough because the observer’s space
can not be defined with help of one gauge more exact or not at all. This
case describes also an unobserved particle, which travels only in
absolute space, when the observer’s space does not exist. The abstract
wave connected to the particle spreads evenly everywhere and the
particle has not localized.
92
Let’s add into space an other particle of equal size. Its distance from the known point is x. A
new length x appears now into space. These two lengths creates so far the whole observer’s
space. The space is now described with help of two waves, which interferes as in the next
picture. The interference term defines the frequencies w1 – w2 and w1 + w2 of sum wave.
The observer’s space of two gauges is still quite ambiguous
By adding new particles into space we get new lengths, which will help to create the observer’s
space, or a set of particles will be localized. The space can now be described by a set of
waves, which contains more wave lengths. When still more particle are added, more wave
lengths appear and the location of the set of particles or of macroscopic body is more exact.
The body creates the observer’s space around itself through the interactions or is localized.
A big set of points creates the observer’s space and the set is
localized there. The observer’s space can not exist without
macroscopic bodies or light.
When only one known length exists, the local observer’s space is totally indefinite everywhere.
A big number of points and lengths between them creates already more accurate observer’s
space besides the body. In physics localization of wave package means appearing of more
accurate observer’s space.
We have described the lengths or gauges of space as waves with a certain wave length.
Together the waves with different wave lengths form a wave package. Traditionally in physics
localization of wave package has been used to describe a particle. However in D-theory
localization means appearing of observer’s space and sharpening of the location there.
One attempt to explain localization is so called decoherence or continuous interaction of
environment to a particle. On the background of this attempt is an incorrect space model and a
need to search interpretation of quantum mechanics.
93
Normal space and reciprocal space
The 3-dimensional surface of a hyperoctahedron closes off the normal space inside the surface
and the reciprocal space outside the surface. In normal space the distance parallel to 4.D is
expressed with a number n and in reciprocal space the corresponding distance is expressed
with a number 1/n. Let's define the position of the 3D-surface in direction of the radius of space
with help of a straight line of numbers.
0
Reciprocal space
1/n
137 cells fill an area -1...0
3D-surface
-1
R
Normal space
n
136 cells fill an area 0...1
-2
Straight line of
numbers
-3
-N
To the centre of
the space
The theoretical physics for solid state
materials is based almost totally on the
reciprocal space! The atoms interact with
each other through their electrons in
reciprocal space. The 4-dimensional
atom model is presented in the 3. part of
D-theory.
The negative straight line of numbers parallel to 4.D begins from zero (0) and then continues
as reciprocal space to number one (-1). On the line in a point (-1) exists the 3D-surface and
there the space changes to a normal space and continues as the normal space to the centre of
the hyperoctahedron. The straight line of numbers is negative, because i² = -1 (i = imaginary
unit).
In reciprocal space between the numbers (-1, 0) each number is written as a rational number
-1/n, where n is a number of the normal space. So the straight line of numbers and the
structure of space have an evident connection.
It is shown later in D-theory that according to the 4-dimensional atom model the electrons of an
atom travel always in the reciprocal space with quantized speed vn = k / n, where n= 1,2,3...
and k = constant. The numbers n lies in range 1 ... 68.
The number 137 is a prime number. When in space in direction of the straight line of numbers
appears waves, of which wave length is a multiple of the length units, must the wave length be
equal of one unit or  = 1. The prime number 137 is not divisible by any integer and thus no
other standing wave lengths can appear in this direction. The line of waves parallel to 4.D is
called a lattice line and it contains 137 waves.
94
When the lengths are quadratic or r ² = x ² + y ² + z ², we can include the lengths parallel to
4.D or to the imaginary axis in the addition of the path length. We can thus write
r ² = x ² + y ² + z ² + k i ², where i is the imaginary unit and k is a constant. No complex
numbers are found in quadratic absolute space. We can think that in absolute space it is
possible to observe all 4 dimensions. In 3-dimensional observer's space the 4.D is impossible
to observe. Therefore 4.D is described with the complex numbers. When i ² = -1, we can
consider the negative sign to point inside the closed space, where the normal and reciprocal
spaces are found.
The cell-structured absolute space can be considered as the space of integers. The integers
are indivisible like the unit cells of the space. In arithmetic calculations taking a square root of
the length of a quadratic space means transition from the quadratic space to the linear space
and the appearance of the irrational numbers to the calculations as well. No circle and no
sphere exists in the absolute space, so the number  is not recognized there. The irrational
number  belongs to the observer's space. The irrational numbers are not "realistic" or
"rational" numbers in the same sense as the observer's space is not the real space but only
an "illusion".
The number √ 2 is also an irrational number and it is the diagonal of the unit square. In the
Manhattan-metric the diagonals of the unit square does not exist, what offers a hint of the
nature of the physical space at background.
Calculating a square root of a negative number means in similar manner the transition to
observer's space and the appearance of complex numbers. Complex numbers are also not
"realistic" numbers in the previous sense. The absolute reality is thus quadratic in
comparison with the observer's reality.
Walter R. Fuchs: KNAURS BUCH DER MODERNEN MATHEMATIK:
"Irrational numbers are thus strongly tied to the square root operation."
95
Uncertainty principle in cell-structured space
We have proved before that the lengths and directions are quantized in cell-structured space
(called also for discrete space or for quantized spacetime). This causes several phenomena,
which are observed only in the smallest scale or in scale of quantum effects. The complex
space and the 3D-surface have both their own shortest measure unit for length. They are D
and d, so that d = D. The unit D of the lattice determines together with elementary rotations
the limit for the accuracy of measurements and all lengths.
D=d
Principle of uncertainty is written as an average mean error
in form x p  ħ / 2 .
d
d
The quantity x is inaccuracy of place or length and p is inaccuracy of momentum. They are
so called conjugated quantities of each other. An other pair of conjugated quantities are t and
E, time and energy. It is characteristic for the conjugated quantities that they both are not
possible to measure accurately simultaneously. Planck’s constant ħ determines the limit of
accuracy according to the previous formula. Explanation for the question, why place and
momentum can not be measured accurately, is found in elementary rotations of the lattice.
Let’s consider an elementary rotation first at the plane of X- and Y-axis during its elemenrtary
rotation. A particle on X-axis moves during elementary rotation in direction of X-axis and
interacts with the lattice in the same direction. The speed component parallel to X-axis affects
directly on the result (momentum p) of interaction. The speed of the particle can now be
measured accurately but the place not. An interaction component perpendicular to the XYplane is needed for accurate measurement of place at X-axis. Then the particle does not move
during measurement in direction of X-axis.
Y
The accurate place of the particle on X-axis is found only
Z
Rotation X-Y
during rotation at the plane of Y- and Z-axis perpendicular to
x
X
X-axis, for example in direction of Y-axis. Then the particle
can not move in direction of X-axis and it has no observable
v
speed there. In this rotation the particle again interacts with
Rotation Y-Z
the lattice and a perpendicular ”mark” is got as a result on
the X-axis. The interaction gives now the accurate location of
particle in direction of X-axis.
We need two elementary rotations at the planes perpendicular to each others to observe
accurately the momentum and the location of a particle. The two rotations mean two separate
measurements at different time. The both conjugated quantities are not possible to be
measured accurately in one measurement.
When the conjugated quantities need perpendicular measurements, the conjugated quantities
are themselves perpendicular to each other in the 4-dimensional absolute space. So the
conjugated quantities define in space a 2-dimensional complex surface. The area of that
surface is the multiple of minimum effect area ħ.
Let’s consider next the conjugated quantities in absolute space and let’s derivate with help of
them geometrically the mass of electron.
96
In absolute 3D-space exists a length x. An other length perpendicular to it is p. For a
particle, such as an electron, two perpendicular components, which are the length x and
momentum p on 3D-surface, are defined in 4-dimensional space. They are conjugated
quantities. The abstract internal property of a particle, absolute momentum
p = mc, is always perpendicular to 3D-surface.
The quantities x and p can not be added together, because they have different dimensions
when observed in 3D-space. Instead their area or the quantity A = x p has a constant value.
The area is
ħ = x p , ( = t E ) ,
A=ħ
where ħ is Planck’s constant.
D=d
In the lattice of the cell-structured space this area A/2 =
ħ/2 / 2 is the smallest separate area in the lattice and it
determines the limit for accuracy of all measurements
(principle of uncertainty). The area A/2 is mathematically
an average mean error in principle of uncertainty .
x
E
t
The area A is the smallest interaction
area. When A is rotated perpendicular
to the 3D-surface, we get the
conjugated pairs of quantities E ja t.
For electron x = R = 137,035999d and p = mc. Formula ħ = x p becomes
137,035999dmc = ħ ,
where m is the rest mass of electron. For the rest mass of electron is valid
me = ħ / 137.035999 dc = 9,10953 · 10-31 kg
The both sides of formula ħ = 137dmc are multiplied by the speed c
ħ c = 137d mc² = 137d E
, where mc² = E. And we get
h c = 2 137d E
When 2137d / c = t, where t is the time, which the light needs for the way
c = 2 r = 2137d, the previous is written
E t = h.
Time t is parallel to 3D-surface. The quantities t and E are conjugated quantities of each other.
We have derived before the mass of electron with help of its absolute length R = 137d.The
used formula was x p = ħ or Rmc = ħ . The formula is based on the structure of space. With
help of the same formula is possible to derive also mass of proton, when the size and structure
of proton in the absolute space is known. Derivation is shown st the next page.
97
Geometric derivation of proton mass
We have before derived geometrically the electron mass starting from the formula
Area = ħ
137dmec = h / 2 = ħ or
Eh =mc²
2 · 137d mec = h
, where me is mass of electron and 2 · 137d is
the circle length of the projection of electron in absolute space.
Area = h = ħ x 2
The projection of electron forms a circle, of which area is h. The area of rectangle drawn
inside the circle is ħ. In order to derivate geometrically the mass of proton with help of the
mass of electron must its geometric structure be defined in comparison with electron.
Proton contains 3 perpendicular components and 2 of them is always contracted (uud). The
sum of two vectors d + d in absolute space is S = √ 2 d. The sum is parallel to the main axes
of complex space. The components of proton do not form a circle. In previous formula the
factor x = 2 · 137d is the length of projection of electron and it is replaced in formula with
the length S of a quark of proton. Let’s calculate first the mass mq of one quark of proton and
then multiply it by 3. The previous formula becomes by broadening to form
4.D
√ 2  137 d √ 2 me c = h or
D=d
√2d
mq S c = h
, where the mass mq of a quark is
d
d
mq = √ 2  137 me.
Let’s multiply the mass mq by 3 and we get the mass mp
of proton
mp = 3 · √ 2  · 137,035999 me = 1826,5 me
The simplified picture shows 3 quarks of
proton perpendicular to each other.
Proton contains a positive electric charge e+ with its potential energy or mass m e, which
needs to be added to mp
mp = (1826.5 + 1) me = 1827,5 me.
The measured mass of proton is 1836,15 m e and relative error is 0,0047. Reason for the error
may be the fact that the 3 quarks of proton contain together an internal potential ( in shape of
gluons), which corresponds to the error converted as mass. The potential has not been taken
into consideration during derivation and during multiplying by 3. Electron instead is indivisible
and its derivative mass is more accurate.
The used formula was x p = ħ or x m c = ħ . The formula is based on the cell-structured
space and it defines the area h, which describes at least electrons e-, e+ and proton p+. By
substituting Planck’s length x = √ Għ / c³ in the formula, the formula gives for mass the
Planck’s mass m = √ ħc / G .
98
The next picture shows two quarks of proton. One quark is parallel to the 3D-surface (black in
the picture) and another quark is folded parallel to a complex lattice line (red) and it is the 4.Dcomponent of proton. All three quarks of proton stand on the 3D-surface.
+
+
lattice box
lattice layer
quark of proton, maximum length = d
+
+
4D-component of proton (folded quark)
3D-surface
d
-
+
+
+
lattice lines
The maximum length a quark is d and minimum length is zero (or the Planck’s length).
Neutron n is 90º behind proton p+ as later is described. Therefore it, for example, interacts in
a different way with the 2-dimensional plane (electron plane) made of lattice line shapes and
has not any electric charge but only a magnetic moment. Isospin of proton is +½ and of
neutron -½ or in abstract isospin-space proton changes to a neutron, when it is turned 180º
in isospin-space and also 90º in the lattice space. So an abstract quantum mechanical
isospin can be understood as a certain phase shift between two oscillating system.
99
The absolute orbital motion in a loop-space
Let's now look at the loop-space, which is made of two 1-dimensional circles symmetric side by
side. The loop-space is cell-structured.
Two straight lines or coordinates K and K' stand on the two circles. Both
of them are parallel to the radius of the circles. The coordinates K and K'
travel on their circle around the centre at the speed c. The directions of
orbital motion are opposite like in picture.
K
K'
The observer, who stands at both circles in place or in rest with the cells
of the circles, can see the coordinates to come from all (both) directions
simultaneously and to fly into all directions. Physically the set of
coordinates K and K’ are the rest frame of light or of the photons and is
caused by the orbital motion (rotations) of the lattice line shapes in a
loop-space.
Two elementary particles
stand at this loop-space side by side. The rest frames K and K‘
travel in relation to the particles into opposite directions. The particles stand in different loops
and in cells side by side. When the particles do not move in relation to each others and to the
cells, they both have the same speed c in relation the set of coordinates K and K’. The system
is now symmetric.
The particles can move together in relation to the cells in the loop at some speed v. The orbital
motion in relation to sets of coordinates K and K’ becomes in this case asymmetric. The
absolute speeds in relation to the K and K’ are
w1 = c + v
and
w2 = c - v.
According to the speed calculation rule of D-theory the quadratic absolute speed of the particles
is the geometric average of w1 and w2 or
w² = w1 w2 = (c + v)(c – v) = c² - v².
This speed describes the asymmetry of the particles and it has in the Relativity theory a
physical meaning.
Let's consider next the 4-dimensional physical space, where the sets of coordinates K and K'
are replaced with the space lattice made of the lattice lines.
100
On the 3D-surface of a 4-dimensional hyperoctahedron the orbital motion happens in almost
similar way as before in a 1-dimensional circle. The lattice lines made of positive and negative
particles stand on both sides of the surface at an 45 degree angle to the 3D-surface. The lattice
line shapes move in tempo of elementary rotations around the loop in opposite directions. The
speed between the lattice line shapes and the cells of a 3D-surface is the speed of light c.
In the picture an asymmetric body moves almost at speed of light in
relation to the lattice and an other body stays as symmetric on its
place. To drive the animation use PageUp- and PageDown-keys in
SlideShow-state (F5).
A particle standing on a layer can be thought to meet in turn the lattice lines moving at the
speeds w1 and w2. The speeds w1 and w2 will determine the absolute symmetry of the particle.
101
Let's consider, how does the previous symmetric orbital motion affect on observations. The
observer can send a light beam with the flying lattice line shapes and the beam can be
reflected back with the incoming lattice lines at the next loop.
s
Let's presume first that the orbital speeds into the
opposite directions are equal or w1 = w2 = c. The
light returns back from mirrors in the picture in the
same time after travelling the distances s.
w1
w2
s
Let's presume then that the observer moves with the mirrors to the right at speed v, when the
orbital speeds in relation to the observer are w1 = c + v and w2 = c - v and w1 + w2 = 2c .
When the distances (=s) to the mirrors stay equal for the observer, the times used for both
backward-and-forward travelling are the same or
t = s / w1 + s / w2 .
The observer, who in this way measures the speed of the “ether”, gets for the speed the same
value in both directions. The observer can not observe directly that w1 and w2 are not equal,
which is important. We get for the speed of “ether” from the previous expression
2s = 2 w1 w2
t
w1 + w2
= w1 w2 = c² - v² = w²
c
c
c
, where w² = c² - v². When c is a constant, the observer’s speed in relation to the absolute
resting points is proportional to the quadratic speed w² measured in this way. This speed is not
possible to observe. For the observer is always valid w = c. When we now substitute w = c or v
= 0 into the speed expression, we get for the speed of light
2s = w² = c
t
c
, which is the same for all observers.
This result is the same as the first hypothesis of Relativity Theory. This result has an effect on
the speed of observer’s time passing and the length in direction of motion.
The observer can thus not observe the speeds w1 and w2 as separate. When for the observer
is valid w1  w2, the observer himself (his wave function) is absolutely asymmetric and he is not
able to observe it. For an other body, for which is valid, for example, w1 = w2 = c, the observer's
wave function is asymmetric. Correspondingly for the observer the wave function of the body is
asymmetric as well, as later is told in detail. This means that the sets of coordinates are rotated
in relation to each other, which is described with Lorentz's transformations. This space model
produces the Lorentz's transformations, as soon is proved.
We find out that there exists no absolute place for the observer. The distance s from mirror to
the observer can be defined in this example only in relation to the observer. The absence of
absolute place means also the absence of absolute speed in the observer's space. All observed
speeds are declared as change of relative place in a certain time.
All those points in space, where w1 = w2 = c, can be defined to an absolute place in theory.
102
Let's look at the speed relations of the lattice line shapes geometrically. When a body moves
in a loop at some speed v in relation to the loop, its speed changes in relation to the lattice
line shapes in the other loop between 0 < v < 2c and in other loop correspondingly 2c > v >
0. We can draw the curves for these speeds in set of coordinates and then calculate their
geometric average ( c = sqrt(ab)).
The straight lines y = 2c - x and y = x
describe the speeds of a body in relation to
the lattice lines in two opposite loops and x is
the relative speed in one loop. We get as
square of geometric averages of the straight
lines
y
2c
y = 2c - x
y=x
y² = (2c - x)x = 2cx - x²
or
y² + x² = 2cx ,
c
which is a c-radius circle. By substituting here
x = x' + c we transfer the origin to the centre
of the circle and we get
w
c
y' = c - x' and y' = c + x' and
x
v
0
c
y'² + x'² = c² .
2c
We can now substitute into the circle the relative speed x = v and we get the absolute speed
y = w or w² = c² - v².
When we define in a loop an absolute rest point so that in that point w1 = w2 = c, the relative
speed v describes the speed in relation to this absolute point.
In the next chapters we talk about two basic quantities, time and length, when the absolute rest
frame is the frame of the moving lattice lines or of the moving light. Lorentz's transformations
are based on this model.
103
Asymmetric particle and time
An asymmetric (elliptic) particle travels in the picture to the right at speed v in relation to the
cells of the 3D-surface. The centre of gravitation of the particle stands in one of the focus of
ellipse. The particle travels k layers to the right in its way. During the way n lattice line
shapes will pass the point, which does not move (or for which w1 = w2), and n>k. There is
so n events. In the picture exist more lattice lines, (n + k) pieces on the left side in forward
direction, because they travel towards the particle or to the left in the picture, and the particle
passes thus more of them than the other lattice lines on the other side, (n - k) pieces. So the
mutual distances between the lattice lines are different on the opposite sides seen in frame
of the particle! The distances depend linearly on the numbers n and k and the distances are
quadratic, when observed in the observer’s space. So also the number of events depends
quadratic on the numbers k² and n².
v
The picture is drawn from the direction of 4.D. In the
picture the 4.D-component of a particle travels first
on the side of the upper lattice lines and transfers
then to the side of the lower lattice lines and then
back again.
The lattice lines have been set for clarity on two
sides according to the motion direction.
w1
(n + k) lattice lines
(n - k) lattice lines
w2
Changing from one side of the layer to an other side creates a positive and negative half of
cycle for the particle. The cycle forms the time of the particle, because there exists no other
time in all points of the space. The cycle appears in the projection of a particle on the 3Dsurface and it is described by the Schrödinger's imaginary wave equation.
In the picture is shown that the particle is asymmetric, because of its motion, or the set of
coordinates is rotated in comparison with a particle, witch does not move. The asymmetric
particle can be described by an ellipse and by its eccentricity e = k / n.
A long enough cycle of measure is needed to measure the time, or the number n must be big
enough. The quadratic number of the events in quadratic space is for an asymmetric particle
as quadratic geometric average
T² = (n - k)(n + k) = n² - k²
or the time passing is proportional to the speed of a body. The number k depends on the
speed v of body. Time passing and the wave function of a particle is considered more
thorough next. It proves that the lattice line shapes and their motion are the essential factor in
all observing and in the appearing of the time. Any information can not progress in the space
faster than the lattice lines do.
104
Time is not a substance
According to modern physics the time is a substance and it is connected to the part of the
space. According to the D-theory that is not quite true. Time is a consequence of absolute
orbital motion of lattice line shapes and time passing can be measured as a number of passed
lattice lines during the absolute motion.
Let's look closer at the previous case of the moving lattice line shapes in a loop-space. First the
orbital speed of the observer in relation to the light in both loops is c. The observer measures
the time or counts the events. The basic events in nature are the transitions of the positive and
negative lattice line shape past the observer. The distance between two lattice lines is 1 layer.
Because the time does not exist as a substance, the observer has no way to measure, for
example, the speed but count the accumulated events during the absolute way.
In the previous example the light moves the distance s at speed c on both loops into both
directions and is then reflected back from the mirrors. The distance s is n basic units long. The
effective quadratic distance, which the light travels from the observer to the mirror is s x s =
s² = n². The quadratic length of the way to-and-fro is S² = 2 n². During the one-way of light the
observer counts his own events or the transitions of the lattice line shapes, which accumulates
in both loops n units. The number T of the events is the passed time or T² = 2n². The quantity
T² is the quadratic time in quadratic, even and cell-structured space. The quadratic speed on
the way was
w ² = S² / T² = 2n² / 2n² = 1.
A body moves at relative speed v in relation to the observer. Let's presume that a body has as
before two moving mirrors with it at the same distance s² = n². The new value k tells the length
of the way, which a body moves in relation the observer, when light travels the way of n layers
and k<n. The events accumulates for the body in different loops during one-way of light (n-k)
and (n+k) events. The time passing is calculated as geometric average of the events. We get
T² = 2(n - k)(n + k) = 2(n² - k²). (Time is now asymmetric and also passes slower.)
The distance from the body to the mirror is n units, but the way that light needs to travel, is
because of moving mirrors s1 = (n - k) or s2 = (n + k) depending on the direction of light. We
get for the to-and-fro way of light S² = 2s1s2 = 2(n - k)(n + k) = 2(n ² - k ²). Then the speed is
w ² = S² / T² = 2(n ² - k ²) / 2(n ² - k ²) = 1.
The result tells that the speed of light in all directions is w² = 1 for both the observer and for the
body in their own set of coordinates. Thus for the speed of light is always measured the same
value, which does not depend on the observer's motion. In addition we consider that the
quadratic time T² = n² - k² of a moving body in relation to observer's time T² = n² becomes
slower and correspondingly the quadratic length S² becomes shorter.
In the previous example both the body and the observer are in certain position in relation to
each other. Let's change the roles of the body and the observer and modify the case so that the
body travels to-and-fro instead of the light. It proves that still the time of a moving body passes
slower.
105
The observer and the body travel first side by side in a loop-space so that for both of them is
valid w1  w2. The body is then accelerated to a relative speed v so that for it w1 = w2 = c. The
body travels now a distance of k units from the observer. The quadratic time of the symmetric
body passes n² events. The observer's time, however, passes n² - k² events. The body is now
accelerated back towards the observer so that the body moves to observer at the relative speed
v. When the observer has not experienced the change of speed, his time passes the same
number n² - k² more before the body will catch him. The time of a body passes now the number
(n + 2k)(n-2k) = n² - 4k². We can add the numbers of events for both of them.
Observer
body
absolute
n² - k²
+ n² - k²
n²
+ n² - 4k²
n²
+n²
2(n² - k²) = t1²
2(n² - 2k²) = t2²
2n²
rel. speed
v
v
outward
return
t1² - t2² = 2 k²
We can see that the accumulated number of events for the body t2 is less than the number
t1 for the observer. The difference for the to-and-fro way is 2k² during the absolute quadratic
time 2n² events. Absolute time means here the time of such observer, for whom it is valid
w1 = w2 = c. It is thus not important for observation, what is the absolute speed of the body
and of the observer together in loop-space. Instead the relative speed v between the body and
observer is crucial for mutual time passing.
In the previous example the observer and the body can measure relative speeds of each other
by sending a light beam with the lattice lines towards each other and by measuring the time for
reflected light. Both of them gets as a result the same relative speed v. For the distance s is
valid s = vt. When the time of a body t2 is shorter than the observer's time t1, must the
distance in set of coordinates of a body have been shorter than in observer's set of
coordinates. The speed v is equal for both.
If in an other example instead of a body for the observer is valid w1 = w2 = c, and the body
makes the way, we get the result or numbers of events:
Body
observer
absolute
n² - k²
+ n² - k²
n²
+ n²
n²
+n²
2(n² - k²) = t1²
2n² = t2²
2n²
rel. speed
v
v
outward
return
t2² - t1² = 2 k²
The difference t2² - t1² for the to-and-fro way is 2k² in this case as well. The observer's time
passed different number in these cases, because the observer's situation was absolutely
different. The results do thus not depend on absolute speeds, but only relative speeds.
The time is not a substance. Each observer has his own time or his absolute speed in space.
(qed)
106
The system of two mirrors and light forms a clock. Let's turn it sideways to its direction of
absolute motion. The light beam is reflected from a mirror to another and the number of
reflections expresses time passing. The distance of the mirrors is still n units. It is valid for the
clock in both directions of loop-space w1 = w2 = c. Travelling of light from a mirror to another
needs n² events of the quadratic time .
The clock now travels to the right in the picture so that the light goes once from a mirror to
another during the measurement. The light must then go diagonally to the main axes and the
length of the fractional way is m² = n² + k². The length of the way increases, because the
mirrors move k² units. The observer in absolute rest sees the "clock" to slow down, because
m>n.
The other observer moves with the system and his
time passes t² = m² - k² events during the
measurement. We substitute to it the expression
m² = n² + k² and we get
k
w1
t² = n² + k² - k² = n².
w2
w1
m
n
w2
The observer's time passing has became slower so
that the observer sees the clock to run still at the same
speed, or the light needs still n² units of the other
observer's time to travel the distance between the
mirrors.
In picture w1= w2 = c.
We notice that measuring the time does not depend on directions of absolute space. So there
exists for a body only one uniform time and its passing depends only on the motion of a body.
When two bodies stand in a loop side by side at the same absolute speed and then they are
accelerated apart, it is not possible to know which one's absolute speed decreases or
increases. The change depends on the direction of absolute motion. The change is absolute
and different for the bodies.
What does this all mean in observer's 3D-space?
In 3-dimensional space the observer can send a light beam to all directions from a centre of a
sphere and light is then reflected from the surface of the sphere back to the centre. The light
comes to the centre from all directions simultaneously. This happens even though the absolute
speeds w1 and w2 are not the same in all directions. For example, in GPS-system it is not
possible to observe the speeds w1 and w2 separately.
107
The principle of simultaneous
In the next case the observer H synchronizes the clocks A and B with his clock by using a light
beam. The clocks A and B do not move in relation to the observer.
A
H
w1
w2
w2
B
The distances of the bodies A and B from H are
equal and w1 > w2.
w1
S
At the moment T0 the observer H sends a light beam to A and B to synchronize the clocks. The
A gets a pulse first at moment Ta and B gets it later at moment Tb. A and B synchronize now
their clocks to the observer's clock and may now wait a second T and then send their own
time as a coded light beam to H. The observer H gets them simultaneously at moment T 1 and
finds out the times equal. In reality the time of A is fast in comparison with B absolutely, but A
must send his time to H earlier so that signals would be simultaneously at H (w2 < w1). The
event corresponds completely a reflection and T can be zero as well.
Let's presume that the coded light pulses continues their travelling past H toward A and B. They
both passes H at moment T1. After receiving the times from each other with the light A and B
realize that the differences between the received times and the present are equal. The
difference of times is then T1 - T0 - T. This example shows that at the same distance from
observer H standing bodies only seem to be in the same time. In reality their position in
absolute space determines their absolute phase difference of time with help of speeds w1 and
w2.
We have noticed that the difference between the speeds w1 and w2 can never be observed.
What sense does this difference in practice have? The sense is there that this invisible
difference means that the Lorentz's transformations are needed and that they will be realized in
this space model. The observations proves that in physical space the time and the length follow
these transformations. Albert Einstein, however, wrote in year 1905:
"It, however, is not possible without any extra assumption to compare the times of event in A and the event in B.
We have until now defined only "time at A" and "time at B". We have not defined any common "time" for A and B,
because this kind of time is not possible to define, until we set as a definition that "the time", which the light needs
to travel from A to B, is the same as "the time", which it needs to travel from B to A."
This "definition" is, however, a decision without any physical or logical foundation. Without that
assumption the absolute space should obviously be forced to assume. The realization of
Lorentz's transformations in absolute space model of D-theory, proves that this definition was
wrong. The meaning of the error is for the observer infinitesimal in practice, because the error is
not possible to observe in any measurement. The issue is only theoretical and based on the
space model.
Note! The previous model of time works also in an acceleration field and proves the time
passing to depend on the potential of the field.
108
Lorentz's transformations in loop-space
We have got for the length of a moving body S² = n² - k² at relative speed v. When for the
observer the same length is So² = n², we get a ratio for quadratic lengths S²
S² / So² = (n ² - k ²) / n ² = 1 - k ²/ n ² = 1 - v ²/ c ².
The contraction of the quadratic length S² in moving set K' of coordinates is described in set K
of the observer's coordinates with help of only x-coordinate. We get
x' = x - vt
1 - v ²/ c ²
Note! The reader can find detailed deriving in
textbooks of physics.
Correspondingly for the decreasing number of events or for the slowdown of quadratic time T²
we got T² = n ² - k ². When for the observer the same time is To² = n ², we get a ratio
T² / To² = (n ² - k ²) / n ² = 1 - k ²/ n ² = 1 - v ²/ c ².
When sets of coordinates are (x,t)-coordinates, we get correspondingly for the time slowdown
t' = t - vx / c ²
1 - v ²/ c ²
The next relation is identically valid for Lorentz's transformations in sets of coordinates K and K'
because of linearity of absolute quadratic space:
 x' ² +  y' ² +  z' ² - c²  t' ² =  x ² +  y ² +  z ² - c²  t ².
We have got in a loop-space for the speed of light w ² = 1 for all observers, in which case
c ² = w ² = 1. In set K' of coordinates is valid  x' ² = n ² - k ² and  t' ² = n ² - k ² and in set K of
coordinates is valid  x ² = n ² and  t ² = n ² as well in both y, z = 0, when we get
n ² - k ² - 1 x (n ² - k ²) = n ² - 1 x n ²
or
0 = 0.
The model of the loop-space matches to Lorentz's transformations. The loop-space then
describes the physical space and meets the observations as a model of the Universe. (qed).
Simulation
It is possible to build a simulator to count time passing and lengths of the moving bodies in loopspace and count the travelling of light and, for example, change of frequency (Doppler's effect)
to measure relative speeds. The simulator will produce the same results as by calculating with
Lorentz's equations for the moving bodies.
109
Spin – rotations
The next picture shows two octahedra. In both octahedra one red line segment stands on a
diagonal. On the left-handed octahedron the line segment length is ½ diagonal and on the
right it is the whole diagonal. For the left line segment exist in rotations 6 different positions in
directions of the main axes. On the right-handed octahedron exist only 3 positions. When
rotations happen synchronously, the longer line segment on the right rotates in octahedron
always 2 cycles, when on the left the shorter line segment rotates only one cycle.
The left-handed ½-layer long line segment plays an
electron in a lattice box. Spin of electron is ±½. Spin
gets negative value in the lattice box of antispace.
Let’s mark the centre of octahedron as origin and the names of axes with letters x, y and z.
On the left-handed octahedron we get globally a cycle T = x, y, z, -x, -y, -z for the line segment
rotations. When only the position of line segment on some main axes is considered, we get
locally a cycle T = x, y, z, x, y, z. On the right-handed octahedron appears during the equivalent
90º rotations two cycles, which are t1 = x, y, z and t2 = x, y, z. Thus a half of the diagonal
needs to rotate locally two full cycles or 720º before it looks globally the same again. Rotation
symmetry is for it similar as it is in the symmetry space SU(2).
In the next picture a ½-layer long lattice particle rotates 6 phases in its lattice box into both
directions. First the rotation angle is +90º and in return -90º. Result is two cycles or waves.
+90º
y
x
z
-x
-y
-z
-90º
Spin-1-diagonal represents in the lattice the size of one lattice layer and spin-½-line segment
represents a half of layer. Spin-1-rotations are shown in the picture below.
x
4x
y
z
110
Spin-2-rotation is got by unifying 1-layer sized octahedra
to a bigger octahedron. It includes 6 different positions for
spin-1-rotations and 6 x 6 positions for spin-½-rotations.
For the diagonal of bigger octahedron is then got spin-2rotations, which has 3 different positions. Full cycle needs
there less rotations in comparison with spin-1- and spin½-rotations.
Any rotation can in principle be chosen to spin-1-rotation
or to a unit rotation. The layer element of cell-structured
space is however the smallest regular element, so it is
chosen.
A
C
B
In the picture above a vector represents a lattice particle. Vector rotates in turn on each plane
A,B and C forwards 90 degrees. The 3D-space is not commutative for the rotations so it is not
possible to rotate the planes back in the same order (A,B,C) backwards to start point but order
of the planes must be change opposite or (C,B,A). Then the vector returns back through the
same but negative rotations.
Previous rotations happen in complex planes outside the 3D-surface. The line segments have
there two components, imaginary and real. The real component stands on the 3D-surface at an
45º angle to all main axes of the 3D-surface. How do the real components parallel to the 3Dsurface locate on the 3D-surface? Spin-½-particles have in a lattice box 6 different positions.
We can think that each of them is always projected on all the planes xy, yz and zx.
y
y
x
z
z
x
The shapes of lattice lines stand always at
an 45º angle to the 3D-surface. We can
think that so do also their projections on
the 3D-surface. So the projections of 6
different positions on the planes xy, yz and
zx can be shown like in the picture.
On the xy-plane the projection of a complex spin-½ line segment during positive phase (green line segment) hits
between positive x- and y-axes and during negative phase (red line segment) between negative x- and y-axes.
Note that for example in xy-plane there is in every state (6 states) always one projection!!!
111
Rotations in complex lattice space
Let’s consider next the smallest regular unit of the complex lattice outside the 3D-surface, which
is a lattice box. The size of lattice box is one layer. It is built of diagonals of octahedron. They
are at an 45º angle to the 3D-surface and 4.D. Let’s consider next the rotations on one 2dimensional plane of a lattice box.
lattice box contains one lattice particle. Its length is a half of diagonal of octahedron. In the next
picture the lattice particle is red or green depending on its sign. In its rotation the particle turns
90 degrees at a time and its sign changes twice during a half of cycle (=T). The particle rotates
through all 6 positions in 3-dimensional lattice box and returns back, when direction of time
changes, and cycle is then done.
Even parity
T
t
0
A lattice particle turns in a lattice box and its phase changes. lattice particle is described also by a curved
arrow. It describes a curved cell to make difference to an empty cell, which is direct or not curved. In turning
point of cycle at time T the sign of curvature amplitude and rotation direction will change. The lattice box is 3dimensional, so in fact there is 6 phases but only 4 is shown in the picture. The shapes of the lattice lines on
the electron plane form a 2-dimensional plane like in the picture. To drive the animation use PageUp- and
PageDown-keys in SlideShow-state (F5).
Start point (0 in the picture) determine the phase of the turning point (T in the picture) . Time T
is a half of the elementary time.
Odd parity
Start point
t
Turn point
0
T
The motion of a lattice particle resembles
motion of a balance wheel in the mechanical
clock.
In every lattice box the rotations happen at the very moment. Rotation direction and the phase
however depend on the location of the lattice box in the lattice. In the lattice boxes besides on
the same layer the phases of the lattice particles get 4 different values, 0º, 90º, 180º and 270º
on the 2-dimensional plane of the lattice line shapes.
Together all lattice particles form in their lattice boxes the shapes of lattice lines. The shapes
seem to move during rotations forwards and backwards in direction of one main axis of the
complex lattice. The lattice lines do not move, but their 2-dimensional shapes made of lattice
particles do.
112
The picture shows a lattice particle in its lattice box to rotate in different phases at one
moment. lattice particle is a 1-dimensional ½-layer long cell, which differs from empty cells
because of its curvature. Curvature means a curvature amplitude and energy.
In the picture a lattice particle rotates in its lattice box clockwise and maintains its curvature
direction during a full cycle in relation to rotation direction. Two of diagonals are drawn in the
lattice box. The particle has a curvature amplitude in relation to the diagonals. It determines
the sign of the particle together with the rotation direction. Sign is marked by the colors green
(+) or red (-).
_
_
_
_
-
+
+
+
+
lattice particle in its different phases
+
Anti lattice particle
A lattice particle is in the picture green at the right side of the box and red at the left side
during a positive time direction. When a lattice particle in the lower picture rotates in its lattice
box into opposite direction or into negative (-) direction of time, also the sign of curvature
amplitude changes. For the anti lattice particle the colors and the directions of axes are at the
both directions of time opposite than for a lattice particle.
_
_
_
_
+
+
+
+
+
lattice particle in its different phases, when time direction changed.
Anti lattice particle
A lattice particle has angular momentum, which describes its amount of rotation or the spin.
Its value is s = ½ ħ.
Before is told, how a distance is calculated in quadratic cell-structured space. When the
length of a radius r, which is rotating around an perpendicular axis, needs to be calculated and
rotation happens around the centre of a lattice box, length of radius is r ² = de. When
e = d + 1, so r = √ d (d+1)
d
e
Correspondingly absolute value for the total spin of electron is
S = √ s (s+1) ħ, which is the same as geometric average and
s = ½.
r
113
The picture below shows the rotation directions and the phase differences of the lattice
particles. The lattice particles form the shapes of positive (green) and negative (red) lattice
lines.
Negative lattice
line
Layer
lattice box
auxiliary line to
perceive octahedron
Positive lattice line
Let’s consider next the shapes made of several lattice particles together. They are called for
lattice lines.
In this simplified picture the lattice particles are described in their
lattice boxes by green or red arrow. The arrows are rotating in the
boxes forming together the shapes of green and red lattice lines.
When the arrows are rotating, the shapes move in a plane right and
left. When the rotation direction changes (or when the direction of
the quantum mechanical time changes), also the motion directions of
the shapes will change. So the lattice lines or rather their shapes
move in a plane to and fro in opposite directions at a constant
speed. Time appears from motion of these shapes.
The crossing points of the lattice line shapes move in the picture
down and up depending on the direction of the quantum mechanical
time
To drive the animation use PageUp- and PageDown-keys in SlideShow-state (key F5).
The speeds of the shapes of the lattice lines into one direction during the time T or during 6
elementary rotations are c and -c. During the next half of cycle (=T) the speeds are opposite
or -c and c. The speed in both directions will be then c. It is the maximum speed for any
interaction in the lattice.
114
A particle on a plane will thus meet in turn the motion of both shapes of lattice lines to the
right or to the left at speed c. Still the only real motion is the rotation in the lattice boxes,
which creates time in every 3-dimensional lattice box. The particle does not change from one
side to the side of the lattice lines moving into opposite direction, as once before were told for
clarity, but the lattice line shapes itself move to and fro in opposite directions at speed c.
We have considered only one 2-dimensional plane of the lattice line shapes. We can see that
in the lattice boxes besides in directions of diagonals the rotation directions are opposite. The
lattice is so shared into two parts (or interspersed zones) according to the rotation direction.
In one lattice box stands a lattice particle and in next one stand its antiparticle rotating into
opposite direction. Antiparticle has however the same spin as the particle but opposite parity.
Reason for the same sign of spins is that a half of elementary time cycle T of the antiparticle
is negative and amplitude too. Product of two negative is positive. Opposite spins instead are
found on the side of antispace, which is interspersed with octahedra of the space.
Electrons e- and e+, which have equal spins, stand in lattice boxes besides in their common
space. But for example two positrons e+, which have opposite spins, stand in space and in
antispace. According to the quantum mechanics we know that a particle and its antiparticle
have equal mass and equal spin, but electric charges are opposite.
On a layer of the complex lattice exist in an regular order zones, where the rotation direction
in the lattice is forwards and in the next zone backwards. Electrons e- and e+ are
antiparticles of each other. They rotate in lattice into opposite directions. Thus we can think
that they rotate also in time into opposite directions. Electron e- or e+ can never transfer into
a place, where it had to rotate into wrong direction.
As a summary we can mention that the complex space is shared into a space and antispace,
which are interspersed to each others. This shareout gives for the particles opposite signed
spins, for example +½ and - ½. Sharing the lattice according to the rotation directions into
interspersed zones, shares the lattice electrically into positive and negative part or into zones,
where the charge of a particle is +q or –q. Thus we got a geometric interpretation for two
fundamental polarized quantity of physics; spin ± and electric charge ±.
How do the particles e+ and e- diverge from each other, when the rotation direction regularly
changes? Here the gauge principle and the phase invariance of the wave function come up
as next is told.
Charge symmetry
The lattice outside the 3D-space consists of the lattice boxes, which contains each a lattice
particle rotating clockwise or counterclockwise. Let’s define that at a certain moment the
particles rotating clockwise are positrons and rotating counterclockwise are electrons. When
rotation direction will change the positrons became electrons and vice versa. This is against a
common sense. When we however understand that rotation direction will change globally
everywhere and simultaneously, there exist not any absolute charge to compare with.
115
So when rotation direction of positron e+ turns and the particle becomes an electron e-, there
will not exist any unchanged charge to compare with. The values of all charges to compare
with have also changed. When the particles e+ and e- differs only by their momentary rotation
direction, how do we know which is which?
The charge of atom nucleus is positive and it is surrounded by negative electrons. That is
however only a result of definition. Essential is that the charges are always opposite and for
example the charges of the nucleus have the same sign. When they both simultaneously
change opposite everywhere, the change is not possible to observe. So although the particles
will all the time change to each other, we can talk about separate particles e+ and e-. The
reason, why the nucleus is positive, does not any more exist. The choice has happened, when
the world was born.
In quantum mechanics the global phase invariance of a wave function means that the rotation
direction of the phase can be rotated to opposite so that all particles become their antiparticles
and the change is not possible to observe. The phase of a wave function is imaginary.
Invariance means also that there exist a symmetry, for example time rotation symmetry and
conservation of a physical quantity.
Electron has, as before is told, a negative and a positive half of cycle during the time T, when a
rotation happens in lattice box into only one direction. That makes it possible for electrons e+
and e- to differ from each other.
The phases of electron on the electron plane
e+
0
e-
Gauge principle appears in the model of
electron; There exist not any global fixed zero
point. Only the differences are important, in this
case the phase differences.
T
Rotation directions of
electron
T
Rotation direction will change at the moment T.
0
When rotation direction changes in space and in antispace globally at the same moment, also
the signs of the particle spins will change. The +½-particles in space became -½- particles and
vise versa.
116
Electron in lattice box
The electrons are ½-layer long lattice particles. As a part of the lattice they are not observable.
Next we consider ½-layer long electrons, which move in the lattice and which are not a solid
part of the lattice and they can be observed.
1-dimensional electron e- or e+ stands outside the 3D-surface in a lattice box and takes part in
the rotations of the lattice. Electron also interacts with the lattice lines causing a lattice current,
which is called for virtual photons. The lattice current will then polarize the lattice and causes
there an electric and magnetic potential. The electric potential is parallel to the 3D-surface and
the magnetic potential is perpendicular to the 3D-surface so that it is zero at the level of the
surface and does not penetrate the surface. So the magnetic field is not observed, if
observer’s set of coordinates is at the same 4.D-level with the charge or there is not any
relative speed.
Electrons e- and e+ do not rotate in a lattice box alone but with a lattice particle.
Electron e- interacts with the lattice and creates
there a the lattice current Id. Positron e+ causes
an opposite lattice current. In the picture the
lattice currents overturn each other.
Id
e+
Id
Id
Electron e+
Electron e-
ee+
e-
lattice
particle
e-
e+
Id
Id
Electrons e+ and e- affect on the lattice lines in
opposite ways and rotate in the lattice boxes into
opposite directions.
Id
positron
electron
The symbol of electron shows the direction of interaction on the lattice particle. Electron
“kicks” the lattice particle out of lattice box for a moment to a virtual photon.
In the picture stand four electrons. The location in space determines their sign and phase.
Electrons e+ and e- create as result of their opposite rotation directions and phases a lattice
current into opposite directions. The phase of an electron determines its interactions. Change
of rotation direction or change of the direction of quantum mechanical time changes the
electrons to their antiparticles.
117
When the time direction changes, the signs (colors in the picture) and the curvature
amplitudes of the lattice particle will change. Then the lattice will be polarized by electron in
opposite colors. The symmetry will realize.
Let’s consider the rotations of an electron and of an invisible electron (lattice particle) in a 3dimensional lattice box. The picture presents the electron e and the lattice particle g in three
different positions or phases. The left picture presents 3 first phases and the right picture the
3 next phases or together one half of cycle. In different phases on each axis appears the
positive and negative lattice current Id, but polarizing of the lattice happens on the axis only in
one direction. During the next half of cycle the lattice currents will polarize the lattice in
opposite way.
Id
Id
5.
3.
e
e
e
g
e
Id
e
Id
g
e
1.
4.
g
g
g
g
6.
Id
2.
Id
Phases 4., 5. and 6.
Phases 1., 2. and 3.
The directions of the axes in the picture are projected to the 3D-surface at an angle of 45
degrees to the main axes of the surface.
118
Electrons e+ and e- interact with the lattice lines in opposite ways. The difference between their
phases is thus 180 degrees. Reason for this phase shift is that when an electron is a curved
line segment, electrons e+ and e- are curved into opposite directions.
Electron e+
Id
Id
90º later:
e+
e+
e+
Id
e+
Id
lattice particle
lattice particle
lattice particle
lattice particle
Electron e-
Id
lattice particle
90º later:
Id
ee-
Id
eId
e-
The lattice particle and an electron are in a lattice box similar spin-½-particles. Electron forms
together with the lattice particle (invisible electron) the diagonal of the lattice box or a rotating
whole. The curvature direction of the particles tells their real rotation direction. For the
electrons e- and e+ it is the same as the normal rotation direction of the lattice particle.
Positron is in quantum theory a particle, which rotates backwards in time.
Spin angular momentum of electron is quantized and it is not possible to change in any ways.
Rotation speed of electron depends directly on the same elementary rotations of the lattice
lines, which create the time on the 3D-surface.
When an electron rotates in the complex 3D-octahedron or in a lattice box, someone could
think that it is described by SU(3)-symmetry space. That is not done because electron
interacts with the shapes of the lattice lines and they form in space a 2-dimensional complex
plane. Interactions on this plane is described by symmetry space SU(2). The 2-dimensional
shapes of the lattice lines stand in the complex space and they insist like electron 6 phases to
one half T of cycle. The whole cycle contains rotations clockwise and counterclockwise,
together 12 rotations.
119
Electron e+ and e- polarize the lattice absolutely to a certain direction until the rotation direction
changes.
Id
Id
e+
e-
Electron and positron will polarize the lattice
absolutely into opposite directions. The
polarization parallel to the 4.D is missing from the
picture. When the quantum mechanical time
direction changes, polarization changes opposite.
The phase of neutron n is 90º behind electron e+. The
polarization of the lattice is thus similar as in the picture. The
neutron does not polarize the lattice in direction of 3D-surface
but only parallel to 4.D. Therefore it does not have an electric
charge but only a magnetic moment. Neutron stands in a
lattice box of the 3D-surface and interacts there with an
electron plane.
Id
n
The picture below shows potentials of the lattice charges +Q and –Q parallel to 4.D appeared
from the lattice current Id of electron. (The lattice charges are created by the lattice current.)
The lattice will be polarized in the picture parallel to 4.D so that there exist not any potential
through the 3D-surface. The potential creates around the electron e- a magnetic field, which is
observed depending on the relative speed of the electric charge. The picture does not depict
the horizontal potential of electric field.
4.D
-Q
e-
+Q
+v
3D-surface
-v
+v
±0
-v
Passing a charged particle at speed +v or –v means the relative height difference in direction
of 4.D between sets of coordinates of the observer and of the particle, when the potential of
the lattice current comes out as magnetic field. When the field of the charge in the picture is
looked and passed on the opposite side of the picture plane, the speed +v is changed to the
speed –v and the potentials Q (  and ) of the lattice preserve their directions for the observer
or the case corresponds to the features of magnetic field. If v = 0, the lattice potential of the
lattice charge Q and magnetic field are not observed. Then also is not observed that the
charge e+ has polarized the lattice absolutely into a certain direction!
Electric field represents the component of the lattice charge parallel to 3D-surface. Magnetic
field represents the component parallel to the 4.D. Both components may have positive and
negative sign.
120
Magnetic field on the other hand represents the component of the lattice charge
perpendicular to the 3D-surface appeared from lattice current.
4.D
-Q
+Q
3D-surface
e-
v
In the magnetic field the lattice is polarized asymmetrically in relation to the 3D-surface and
the lattice charge Q creates a potential over the 3D-surface. The continuous change of an
electric field creates the asymmetry. The electric field changing at the speed v creates a
magnetic field corresponding the speed.
Let’s consider next how does a magnetic field appear as consequence of relative motion in an
electric field. Let’s consider the issue in Manhattan-metric and then transfer the result into
observer’s space as a macroscopic magnetic field.
Let’s presume first that relative speed in relation to the electric field is zero, when the
magnetic field is not observed. In case like this the sets of inertial coordinates are as
asymmetric and we can presume them both as symmetric. The surface, which we are
considering, is thicker than zero and perpendicular to the 3D-surface. Then the photons travel
in direction of the surface and form a photon current. In this case there is no photon current
parallel to 4.D through the surface, because the coming and going components overturn each
other. When the lattice current is zero, there does not exist any magnetic field. (The
components parallel to the 3D-surface represent here an electric field.)
-Q
Q
v = 0. The sum of lattice currents
parallel to 4.D:n is zero
121
In the next picture the observation surface of an observer moving at relative speed v = c / 2
has turned 45º and is parallel to the lattice lines and also perpendicular to the coming lattice
current.
-Q
The amount of the lattice current is now 1 unit per one
observer’s area unit. The amount depends on the
amount of the lattice charge.

v = ½ c. The sum of the lattice
currents parallel to 4.D is one unit.
-Q
When the observer’s speed gets near the light speed
c, the observer’s surface has turned almost parallel to
the 3D-surface. The amount of the lattice current is
now √2 unit per one observer’s area unit.
The lattice current as function of angle  per one observer’s area unit is got
Id = sin (+45º) - sin (45º- ) .
The lattice current is quantity of absolute space. It is transferred into observer’s space by
squaring
Id² = (sin (+45º) - sin (45º- ))² = 2 cos²(+90º) = cos(2(+90º)) -1.
This function depicts magnetic field. It is
similar as the previously function, which
depicted quantum correlation in context of
entangled photons. It depicts magnetic field
from a point of view of a single photon in
Manhattan-metric.
Id() = 2 cos² (+90)
2.0
1.5
Quantum
mechanical
In the observer’s space the macroscopic
magnetic field is observed as function of the
classical correlation corresponding to the
quantum correlation or as linear function of
the angle . The magnetic field is observed
also as a linear function of the speed v,
because the angle  is proportional to the
speed v. For the magnetic field is got
1.0
Classical
0.5

0
30º
60º
B = qv µ / 4r²
90º
122
Families of elementary particles
Leptons
Standard model contains three families of elementary particles. Let’s consider first the
geometric structure of lepton family. The family includes electrons e, myon  and tau  and
their neutrinos. Electrons e+, e-, proton p+ and neutron n have been described before.
Electron is a 1-dimensional cell or particle outside the 3D-surface. Its length is ½-layer and its
direction is equal to the direction of any lattice line. Electron has an uncharged lepton or
neutrino  and  of electron. When the curvature amplitude of particles is considered, neutrino
and antineutrino have 180 degrees phase shift to each other.
Electron e-
Neutrino of electron e
Antineutrino of electron
e

e+
e-
lattice particle
lattice particle

lattice particle
lattice particle
Electron neutrino  does not rotate in the lattice like electron, because the lattice particle in the
same lattice box repels it. Because of repelling it moves always at high speed.
Lepton family’s second particle myon  is described likewise electron, but as 2-dimensional. It
contains two ½-layer long cells perpendicular to each other. One of the cells stand always
parallel to a lattice line and another stands perpendicular to it. For this reason the properties of
myon are equal to the properties of electron except mass. Myon neutrinos are built like electron
neutrinos, but are 2-dimensional.
Myon
-
Neutriino of myon 
+
Antineutriino of myon 


+
lattice particle
lattice particle
lattice particle
lattice particle
The third particle tau  of the family is described likewise electron, but as 3-dimensional. It.
contains three ½-layer long cells perpendicular to each other. One of cells stand always
parallel to a lattice line and two other stand perpendicular to it. The properties of tau are equal
to properties of electron except mass.
Because of geometric reasons only 3 families of leptons can exist in 3-dimensional space
made of 3-dimensional elements. Only the members of the first family are stable.
123
Virtual photon
The next picture shows a virtual photon sent by an electron to progress through interactions in
the lattice. Photon will progress at speed of light or at the same speed as the shapes of the
lattice lines move in the lattice. Every rotation and interaction leads the photon forwards. A
single lattice particle will return back to its lattice box after interaction. Rotations made
perpendicular to the picture plane are not shown in the picture.
Interaction
A virtual photon will progress in the lattice
during one 90 degrees rotation the length of
one layer or the same length as shapes of
lattice lines. The length is ½-layer long, when
projected to the 3D-surface at an 45 degrees
angle.
To drive the animation use PageUp- and
PageDown-keys in SlideShow-state (F5).
Virtual photon continues its way at
speed of light.
Electron sends virtual photons during rotations into all 6 directions of 3-dimensional space.
Photons create a lattice current parallel to a lattice line and will so polarize the lattice. When a
virtual photon reaches the 3D-surface, it becomes for a moment a virtual electron, which will
send virtual photons to several directions. On the next pages is described, how an electron is
projected to the 3D-surface with help of virtual photons and how virtual electrons e+ and e- will
appear.
124
Electron projection on 3D-surface
Electron of atom stands outside the 3D-surface and is projected with help of lattice current or
virtual photons on the 3D-surface. The next picture shows in absolute space projecting of an
electron standing on the layers n = 1, 2 and 3. We can see that the distance r of the projection
increases linearly. The distance increases quadratic R = r² in the observer’s space.
137
3D-surface
Electron projection
on the 3D-surface
n=1
136
n=2
n=3
r = 137d
r = 3 · 137d
Contracting of the complex space is not shown in the picture.
Virtual photons progress in a lattice line parallel to the lattice line. Because in a lattice line
below the 3D-surface exists one cell less or there exists 68 layers, the location of a projection
depends on the electron layer n.
When the virtual photons meet each other on the 3D-surface, there appears the projection of
electron of atom. The location of the projection is the same as the Bohr’s atom model will give.
In the observer’s space the projection radius on the layers n is
rn = (n · 137,035999)² d = n² R1
, where n = 1,2,3...68 counted upwards from the 3D-surface, R1 = 137,035999²d.
When more accurate size of a half of one 2D-layer is d = 2.817940325 fm, the projection
radius is, when n = 1,
R1 = 137.03599911² d = 0.5291772 x 10
-10
m,
which is the same as radius of K-layer of hydrogen atom in Bohr’s atom model.
125
The picture below shows how a lattice current or the virtual photons caused by an electron of
atom will progress in space. The electron stands on the layer 2. and the lattice current moves
left as two separate currents on both sides of the 3D-surface as in the picture. The currents
or the photons meet each others perpendicularly in position, where a projection of the real
electron appears . Every second pair of photons traveling from the atom nucleus is reflected
back in Manhattan-metric at the projection and the rest continue past the projection. So the
projection operates as divisor by two. Thus the projection seems to emit photons into
opposite directions and it looks like an electron. The first photon pair has originally started
near the nucleus from the real electron during the positive quantum mechanical time and the
second during the negative one, which causes for them a variable phase difference at the
projection. For that reason the projection rotates constantly into the same direction
independently on the direction of the quantum mechanical time. The direction of orbital
angular momentum of the electron in atom does not change. The projection of the real
electron at the 3D-surface is an image and its properties have macroscopic nature. The real
electron above the nucleus is quantum mechanical and its time is bidirectional. This
phenomenon creates for its part the macroscopic one-way time and the macroscopic world.
A phase shift appears into the lattice current in the projection position of the electron, which
means that the lattice current transfers to the opposite signed lattice line. The lattice currents
are united again in the point P and form there the projection of a virtual positron e+. There
the photons are not reflected back because of the phase differences and a phase shift
appears again into the lattice current and later appears a projection of a virtual electron e- on
the 3D-surface. Virtual particles will eliminate each others. The picture shows that the lattice
will be polarized still so that on the upper and lower edges of the lattice exists on this side of
nucleus a negative (red) polarization and on the 3D-surface a positive (green) polarization.
Only in the point P at the virtual positron the polarization is different.
137 cells
Phase shift
Virtual positron
136 cells
133 cells
X = V-nd
V
Real electron
e- of atom
n=2
ep+
P
136 cells
137V
137V
Here appears 137 loops of lattice
current. The lattice currents meet in the
same point on the 3D-surface.
2nd
r = /2
V has the same length as 137 projections of
one lattice layer on the 3D-surface. V = 137d.
Location of electron projection,
when n = 2.
2 x 137V
To drive the animation use PageUp- and PageDown-keys in SlideShow-state (F5).
126
The virtual photons continue their traveling unlimited far and create electric and magnetic
field. The virtual photons, which reflected back from the projection, interact with the real
electron.
On the opposite side of the real electron (on the right side in the picture) appears also a
projection. There the lattice however is polarized as opposite signed or the space is polarized
absolutely into a certain direction during one elementary rotation cycle as before is already
told. Projections appear also elsewhere around the nucleus in positions, where the lattice
currents can meet as before is shown.
Projection
3D-pinta
To drive the animation
use PageUp- and
PageDown-keys in
SlideShow-state (F5).
137 d
The symbolic animation shows a rotating real electron at layer n = 1 (green arrow) and the virtual
photons flying around to the left. The photons create the projection of the real electron to the 3Dsurface. The projection operates as rectifying divisor by two. In the animation the arrow corresponding
to the real electron rotates exaggerated slow in regard to the speed of the photons. Radius of hydrogen
atom R = 137.035999174² d.
P(x)= *(x,t) (x,t) is the probability to find an electron in the place x at the moment t. Then
the wave function (x,t) describes the complex lattice current above the 3D-surface and
*(x,t) describes the complex lattice current below the 3D-surface.
In addition the picture shows that in the lattice exists a length 2 x 137V, which forms in the
lattice a symmetric diameter of a whole. This length corresponds to an inverse length [1/m],
which is called for the Rydberg’s constant. The multiples of the length form all the wave
lengths radiated by an atom. Next the value of Rydberg’s constant is introduced with help of
the space model.
127
Geometric introduction of Rydberg’s constant
We used before the diameter d of proton as a measure unit. We use onwards as a second unit
the length V = 137.035999d, which is one unit of the lattice. It is a unit of the repeated
structure created by lattice current. The extent and also the energy of photon is calculated with
help of this absolute unit V, which is repeated in the structure of photon. In absolute space we
get for the length or for the diameter from the previous picture
U = 2 · 137,035999 V
We get in the observer’s space the equivalent u for the length U by squaring. The measure
unit V is however not squared (1² = 1) or
u = U² = 4 · 137,0359² V.
When the structure u is repeated n times in the diameter h of photons circumference , we get
for the diameter in observer’s space
hn = n /  = n² U² = n² u.
Calculation gives, when n = 1 and V = 137.035999d = 386,159268 fm
1 =  · u =  · 4 · 137,0359² V = 911.267 · 10 - 10 m .
When Rydberg’s constant R is defined
1 / 1 = R / no ² and no = 1, we get
R = 1 / 1 = 1 / (4 · 137.035999² V)
R = 1,09737316 · 107 m -1.
Thus is got an accurate value for the Rydberg’s constant by starting from the geometry of
space and from the length d, which is used to fit the gauge units.
128
Quantum interaction
The reciprocal space is divided in direction of 4.D into cell-structured layers. Outside the 3Dsurface exists 68.5 layers and 137 cells, inside exists 68 layers. With help of the layers and the
cells it is possible to define any distance from 3D-surface as metric. When the distance in
absolute space is 137 cells, the same distance on quadratic 3D-surface is 137² cells.
N= 1/2
4.D
ħ = h / 2
ħ = h / 2
r = 137d
h = ħ x 2
e-
Eh =mc²
The smallest projection of a spin-½ electron.
In reciprocal space at every layer can stand an electron, which is described with vector. A
particle is a ½-layer high spin-½-particle. The area ħ corresponds now to the radius of the
projection. By multiplying the area ħ with a constant 2 we get an area h, which is a circle with
radius r. The area h describes symmetrically projection of ½-vector on 3D-surface. An electron
has a mass m and total energy Eh = mc².
An electron is a vector and its length is equal in one direction in 3D-surface and in direction of
4.D, because the angle between electron and the 3D-surface is 45º. When the directions of the
components of vector e (at an angle 45º) are both 4.D and one direction on 3D-surface, its
projection on 3D-surface is not energy but a quantum interaction, of which dimension is [Js].
The time [s] is a quantity parallel to 3D-surface and energy [J] is parallel to 4.D.
Let’s use now the total energy Eh = mc² of an electron as the height of the area h. When the
length of a projection circle is  = 2 r, light needs the time T = 2 r / c to go round it. We can
write for the area h at all layers in reciprocal space as the product of its side lengths
h = Eh T = Eh 2 r / c.
When r = 137d ,
hc / 2 = Eh r = mc² x 137d.
By reducing we get
ħ = 137dmc.
Note! If  is used as a length of a projection
circle, must the energy E be replaced by
the momentum p=mc.
Note! The Planck's constant ħ and the length d are both
needed only to adjust the macroscopic measure units. For the
mass of an electron is thus derived m = ħ / 137dc.
The area ħ = h / 2 is the Planck's constant. When d = 2.82 fm and m = 9.109 x 10 -31 kg
(= mass of an electron), we get a value h, which corresponds to observations,
h = 6.63 x 10 -34 Js .
129
We have got in reciprocal space for the radius of projections Rn = n²137²d, where n gets the
values 1,2,3...68. By substituting Rn = n²137²d to the radius r = 137d in the next formula
ħ = (137d)mc
, we get by expanding the formula
n ħ = (n²137²d)m x c /137n.
The speed vn = c /137n is the orbital speed of a projection on a layer n in 3D-space. The speed
v and the mass m means the momentum p = mv, when the projection of a particle surrounds
the projection centre on r-radius circle. The orbital speed v and radius r define with help of an
area ħ the projection of a vector. In reciprocal space we get the next table for different layers
n=1,2,3,...68:
n = layer v = speed
n=1
n=2
n=3
n = 68
c / 137
c / 137 x 2
c / 137 x 3
c / 137 x 68
The table describes the radius and orbital speeds of
an electron at different layers in a hydrogen atom
matching to the Bohr's atom model.
r = radius
137²d
4 x 137²d
9 x 137²d
68² x 137²d
Reciprocal
space
Note! The real electron does not travel around the
nucleus but an orbital speed can be defined for its
projection.
In normal space the same formula is valid
ħ = 137dmc.
We get by expanding the formula
ħ = n x 137dm x c / n
where n = 1,2,3...68. Note! The electrons does not move in normal space.
The speed v = c /n is the orbital speed of a projection on a layer n in 3D-space. The orbital
speed v and the radius r define with help of an area ħ the projection of a vector. In normal space
we get the next table for different layers :
n = layer
v = speed
n=1
n=2
n=3
c
c/2
c/3
n = 68
c / 68
The layer n = 1 in normal space is equivalent to the case
in 3D-space or at the absolute speed c of particles.
r = radius
137d
2 x 137d
3 x 137d
68 x 137d
Normal
space
We will see that at layer n = 1, where r = 137d and
speed v = c, the circle of the projection c = 137d 2 is
the shortest of all ones.. The circle c is called
"Compton's wave length". In addition the size of this
projection occurs as expanded at all layers. It is
according to the space model the smallest possible
(quantum) interaction. It defines the uncertainty
principle, which is known by the name "Heisenberg's
uncertainty principle".
With help of these projections we have looked at a projection of a ½-layer high vector at 3Dsurface. The vector may also be a photon, witch is a spin-1-particle. The projection thus
produces for a photon the formula of energy E = hf.
130
The lattice lines form the famous ether
It is already told that on both sides of the 3D-surface of a 4-dimensional hyperoctahedron stand
so called lattice lines, which are made of 1-dimensional elementary particles. They are
perpendicular to each other and at an 45 degree angle to the 3D-surface. All the lattice lines
form together the lattice space or the ether. The positive and negative elementary particles of
the lattice lines are called positrons and electrons. Outside the 3D-surface stand 68,5 layers
and inside 68 layers. The lattice lines are made of one-dimensional cells. The lattice line shapes
move at the speed of light in relation to the observer and the 3D-surface.
Id
V
68.5 layers
in
reciprocal
space
3Dsurface
+q
68 layers in
normal
space
A lattice line can move, like in the picture, as a
lattice current Id caused by a force, and at the
same time polarize the lattice or the ether. As the
result of a lattice current there accumulates a
lattice charge Qi in space. A potential V is
connected to a lattice charge Qi. The projection of
the potential V on 3D-surface is a magnetic field.
The lattice charge Qi is equal to an electric charge
q, which causes the lattice charge. The potential V
is observed entirely, when observer's speed to the
charge q is c or the speed of light.
In the picture the lattice lines are drawn
perpendicular to the 3D-surface or they are
looked from the direction of their motion.
The lattice current appears, when the lattice lines with the opposite signs move to area of each
other like in picture. The lattice current through the 3D-surface is zero, when equal but the
opposite currents are added together. Thus there exists no potential over the 3D-surface and
the integral of the potential over any closed 2-dimensional surface S in 3D-space is always
zero. That is also the feature of a magnetic field. To observe the potential a relative speed is
needed. Its direction has a special meaning in observing the magnetic field.
The potential has its maximum value or the intensity of magnetic filed in a certain point in the
space has its maximum value, which is defined by the features of the lattice. The lattice charge
and its potential will appear, when the lattice tries to stay homogenous in direction of 4.D and
that happens by means of a lattice current.
131
A lattice charge runs down in a moving lattice spontaneously, when the electric charge +q stays
after the moving lattice line shapes. Then the charge +q must move evenly. During acceleration
the lattice charge does not run down completely but a part of lattice charge stays left in the
lattice as photons or as electromagnetic wave. According to the observations accelerated
motion of a charge causes electromagnetic radiation.
The potential V of a lattice charge is quantized like an electric charge. The quantization can be
observed only in smaller scale than an atom. For the potential V and the lattice charge Qi is
valid in direction of 4.D
V  Qi = qv / c,
where v is the observer's relative speed in relation to a charge q, and v is proportional to the
observer's distance from the charge q in direction of 4.D.
The lattice current does not occur in similar way in direction of the 3D-space, but the lattice tries
to stay homogenous there as well by moving the electric charges q. Then a Coulomb's force is
observed in 3D-space between the charges.
The structure of photons
When the charge is in even motion, the lattice currents are identical in front and at the back of
the charge and the lattice charge appears and runs down symmetrically. During acceleration it
does not happen so and as a result several photons stays in the lattice. The several photons
together creates radio waves.
A photon is energy in the lattice and moves with the lattice. The photon has
in the lattice a vertical and a horizontal component, which means that it is
observed as an electrical and magnetic field moving at speed of light. It
however has no electric charge or rest mass.
The energy of a photon is quantized because of the features of the lattice.
Two rotated lattice lines
A part of the radio wave made of photons. In the
picture the lattice lines are vertical but in fact
their angle is 45º.
132
The cell-structured space outside the 3D-surface forms there a complex lattice. The lattice does
not consist of empty space without any properties. The particles are electrons and positrons.
They travel outside the 3D-surface in phases as a part of the lattice at the speed of light and
they interact with every point of 3D-surface. The interaction creates on the 3D-surface into its
every point a "clock pulse" or the time of the 3D-space. On the 3D-surface the speed of lattice
line shapes is measured to be the speed of light.
The Quantum field theory predicts the existence of the lattice or the ether. According to the
theory an electron is not a single object, but it is surrounded overall by the cloud of virtual
photons, of virtual positrons and electrons, vacuum is polarized. We talk about ”Dirac's field”.
Paul Dirac:
”In empty space exists an infinite number of electrons with negative energy and they are packed
together regularly and evenly. There should exist holes, too.” and
”The energy of a particle with mass can be positive or negative.”
The lattice must exist overall in the space. Because the lattice is not observed directly and not
any current or not any potential in the lattice has been measured directly, the lattice must exist
outside of powers of observation. The fourth spatial dimension gives the natural direction for the
lattice, because from the fourth dimension the measures can be done only indirectly and only
projections of phenomena can be observed.
No interaction can exist between two perpendicular cells in the 3D-surface or in the lattice ,
because they have not any component parallel to the other cell. When the perpendicular
lengths parallel to the main axes are a and b, their sum is S = a² + b². Interaction appears, if the
angle between two cells is  and   90º or the space is somehow curved. The sum is
S = (a + b)² = a² + b² + 2ab cos 
, where a and b are vectors and  is the angle between them.
The term 2ab cos  is called interference term, and it describes the number of interaction.
Interaction is strongest, when a and b are parallel. The term can be constructive (or positive)
destructive (or negative).
133
Properties of the lattice
Let’s consider next the lattice current loop created by a proton p+ standing on 3D-surface. The
lattice currents are reflected according to the picture from 3D-surface and from the edges of the
lattice. A part of the lattice current forms above the 3D-surface -shaped parts, which are ½layer longer than the corresponding V-shaped parts below the 3D-surface. The lengths L1, L2,
...Ln are growing up when the distance from the proton increases and in the picture they grow
up exaggerated fast. The picture shows only a part of the lattice lines.
The lattice currents at the opposite sides of 3D-surface are united to a loop, when they meet
each other at the 3D-surface and the loop is there closed. In that point appears the projection
of a virtual electron. 136 -shaped parts of the loop stand above 3D-surface and one more or
137 V-shaped parts stand below 3D-surface.
L4
2.
3.
4.
L3
L2
1.
L1
N
p+
N
L4 > L3 > L2 > L1 > 0
r = /2
The picture shows that the lattice current can be divided into two components; one is parallel to
main axis of 3D-surface, the other one is parallel to 4.D. Each shape of the lattice lines stands
in 2-dimensional mathematical space. One axis of the space is real and the other one is
imaginary. A rotation group U(1) is defined in this kind of space. The properties of the lattice
line can now be described in one point of 3D-space by a vector in U(1)-rotation space. The
length of the vector remains in rotations, but the phase will change. In U(1)-rotation space
exists only one generator and that is the angle of the phase.
134
Proton is not a 1-dimensional particle like an electron, but it is a 4-dimensional elementary
particle in complex space. Next we however consider the electric properties of proton.
The 4.D-component of the proton creates around itself in the lattice a potential, or an electric
field and its potential energy. Energy is always connected to mass. In this case the energy of
electric field appears as curvature of 3D-space in electric field of proton. Curving and appearing
of mass of electrically charged particles is considered later in D-theory. The mass of 3Dcomponents of proton is not taken into consideration.
The angle between the lattice lines and the 3D-surface is normally 45º. The electric charge or
the 4.D-component of proton makes the angle turn bigger so that the angle is biggest near the
proton. At the same time in electric field of the proton a circle drawn around the lattice lines
changes to an ellipse. The projection ratio  remains as a constant in the field, but the angle of
the lattice lines will change. Essential is that the phase of the lattice lines changes in the field
locally as the picture shows and the ellipse describes the change of phase. The change of the
phase is considered at the next page.
p+
3D-surface
Positive positron e+ causes parallel lattice current and similar changes in angle as a proton.
Schrödinger’s wave equation for a free particle is written for example in form
- h²
d² (x) = E (x) .
8²m dx ²
The wave equation is globally and locally invariant for change of the phase of the wave
function (x). However according to the previous picture the phase of the lattice and also of
the wave function (x) changes locally in electric field of a particle. The wave equation starts to
work, when a fixing term is added to it. The fixing term will change or fix the phase of the wave
function (x) locally by an equivalent number. This so called Yang’s and Mills’ fixing term
describes then the electric field of a particle in all points of the space and gets a form
q A(x) (x) ,
where q is the charge of the particle causing the field. Function A(x) is here the potential
energy function of the electric field. Later is observed that the same kind of change of the
phase occurs also in acceleration field and also there an equivalent potential energy function
A(x) can be written in the wave equation. When A(x) can describe different fields and fields
with different potential energy, it’s a question of so called gauge freedom, which is a
fundamental idea in Standard model.
135
The change of the phase of a wave function, which occurs in the lattice in electric field, is
described with help of an ellipse. For an ellipse is generally valid:
f² = a² - b² , when a  b and
P
b(=w)
PF + PF' = 2a.
Correspondingly it is valid for the speeds
F
F'
a ( = c)
f(=v)
v² = c² - w², when c  w and v is the escape velocity of
the field, for example the escape velocity of an electron
in the electric field of proton.
Then in ellipse
a c and bw and fv.
4.D
lattice line
c
v
w
c
c
An ellipse is drawn in the picture around the lattice lines.
The major axis of ellipse is in electric field parallel to 4.D or
perpendicular to 3D-surface as in the picture. A half of the
major axis is c or the speed of light in length. Without the
electric field the angle between the lattice lines and 3Dsurface would be 45º in even space and we would have a
circle.
The speed vector v describes the escape velocity of
electric field in a point of the field.
Ellipse describes the electric field of
proton and limits the lattice lines
The square made of lattice lines in even space changes in electric field to a rhombus
surrounded by an ellipse. The phase of a wave function in a point of the field changes
compared with the phase in some other point outside the field.
Before is shown an example of changing the phase of the lattice lines in a force field. The force
field is in the example the electric field created by electric charge. Then the force is directed
only to electrically charged bodies. In the force field the lattice changes asymmetric and the
asymmetry is described with help of an ellipse. Besides a macroscopic body the asymmetry of
the lattice means that the body is reduced in directions of the complex axes projections on the
3D-surface. Reducing makes the 3D-space seem isotropic as already is told.
The asymmetry of the lattice occurs also in gravitational field or acceleration field. The force
occurs there from the mass of a body and is directed to the masses of other bodies. The
acceleration field and electromagnetic field are separate force fields and the asymmetry in the
lattice caused by them is different. The fields affect however in the same areas of space and
their effects are accumulated, when a body has mass and electric charge. The electric force is
however much stronger between particles. Let’s consider next both force fields together and
their differences.
136
According to the gauge principle the phase of lattice lines needs to change also in an
acceleration field. When there exists no electric field, the angle between the lattice lines and
3D-surface is in acceleration field always smaller than 45º and no lattice current will occur in the
lattice. The major axis is in acceleration field parallel to 3D-surface and the surface is inclined
as in the picture.
In acceleration field the change of the lattice line phase has opposite sign as in electric field.
The next picture shows the asymmetry of the lattice lines on inclined surface. In acceleration
field the surface is inclined. We can see in the picture that vector c is always parallel to 4.D.
The speed vector v is now parallel to 3D-surface and points the direction of the field and shows
its relative strength compared with vector c. The speed v is the escape velocity of acceleration
field in one point of the field.
y(x)
3D-surface
w
c
4.D
v
x
Speed vector c is also in inclined
space always parallel to 4.D and v
is always parallel to 3D-surface.
The picture shows the changed phase of lattice lines in
acceleration field. For clarity the rhombuses are not
inclined with the 3D-surface.
In the picture the angle between the lattice lines and the 3D-surface is (x) and curvature of
surface is described by y(x). The 3D-surface may curve any way and ’ has in acceleration
field opposite sign as in electric field. Essential is that the local absolute size of angle  has no
meaning from the point of view of the field, as the phase of wave function too, but the change
of the angle is meaningful. So for example in acceleration field of the earth the angle of lattice
lines have a specific value (< 45º), but the local electric field will cause a change in it. The
change increases locally the value of . Then the speed vector c is not any more parallel to
4.D.
According to the model of D-theory the wave functions of a particle and its antiparticle have the
phase shift 180 degrees. It can be so when the lattice lines travels everywhere globally in the
same phase. Then for example the wave functions of all protons are in the same phase and
also the wave functions of antiprotons are in the same phase with themselves.
137
The next picture presents the lattice current loop of a free electron e-. The loop is quite
different in comparison with the loop of proton. The lengths L1, L2, ...Ln grow up at similar way.
The loop is similar with the loop of an electron at layer 68 in atom. 136 -shaped parts of the
loop stand above 3D-surface and one more or 137 V-shaped parts stand below 3D-surface. A
free positron e+ has similar lattice current loop, but its direction is opposite.
We can see that the loops of proton and of free electron are almost of equal length.
eN
L3
L1
L2
N
r = /2
An electron creates around it a potential or an electric field. The direction of the lattice current
caused by an electron is opposite to the lattice current of a proton or of a positron. Note that a
free electron travels at the outer edge of the lattice far away from 3D-surface. It has there a
zero-point energy, which does not affect to the observed energy levels of atom as later is told.
e-
3D-surface
4.D
lattice line
w
c v
Electron e- makes the angle between the lattice lines and
3D-surface turn bigger so that the angle is biggest near
the electron. At the same time the phase of the lattice lines
in electric field changes locally.
Next we consider the positron.
c
Ellipse describes the electric field of
electron and limits the lattice lines
138
e+
3D-surface
4.D
Positive positron e+ causes opposite lattice current in comparison
with electron e-. Change of the angle is similar.
c
w
v
Positron has the same mass as electron.
lattice line
c
Ellipse describes the electric field of
positron and limits the lattice lines
The model of the lattice with its lattice currents shown before is not a finished model.
Changes may occur in the upcoming versions.
Atom model
Electron and proton form together the simplest atom. Its structure in cell-structured space is
considered next.
139
Let’s consider in a 4-dimensional absolute space an atom, which has full electron layers. Let’s
then consider only the complex space (not the complex antispace interspersed with it) and only
the spin-positive part (red) of the atom. The picture presents the full electron layers of a heavy
atom. ( In the picture the green part of space contains the spin-negative electrons in antispace).
4.D
f
d
N
spin +
+4.D
spin -
p
s
d
p
n=1
M
s
3D-surface
p L
s
s K
-4.D
The profil of atom in the 4based space (red) and in the
antispace (green)
interspered with it.
When the stack of the electrons of a half of an atom is looked 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 vertical 4D-axis like in the picture. Every layer
corresponds to one plane or the surface parallel to 3D-surface. The planes lies outside of 3Dspace.
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 3Dspace. 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.
We can see in the picture that the electrons stand on their places and do not travel around the
nucleus in accelerated motion. They interact with the lattice lines around and as a
consequence appears an interaction field far from the nucleus, as before already is told.
140
The next picture presents a hydrogen atom and the left part of its lattice current loop. The
atomic charge is Z = 1 and when the atom has one electron, it is electrically neutral. In the
example of the picture the electron stands at layer 4. Its lattice current loop contains on one
side of nucleus n=4 pieces of -shaped parts of the loop. The loop is closed in point P. The
projection of an electron appears on the 3D-surface in point P, where the lattice currents of the
electron are united on the 3D-surface. The lattice current of proton p+ does not fit in the picture.
The both lattice currents do not meet each other from the opposite sides of 3D-surface.
Let’s prove next that with help of this structure it is possible to derive geometrically all the
radius rn of hydrogen atom to 7 numbers when n = 1,2,3... or
rn = 4 n² ħ² = n² ħ
Ze²m
Zmc
= n² c
Z
,
where m mass of electron and  = e² / 4 ħc = 1 / 137.03599911 is the fine structure
constant, and c is Compton’s wave length c = ħ / mc. In addition this geometric model is
used to derive the energy equation for hydrogen atom
E = ½mv² - Ze² = ½mc²
4r
Z
n
²=hf
en
L4
L3
L2
P
L1
N
P
p+
N
L4 > L3 > L2 > L1
r = /2
We can see from the picture that in absolute space the location of an electron on 3D-surface
lies in point P and it depends linearly on the layer n or on the distance between electron and
the proton p+ on the 3D-surface. When the electron leaves the layer n = 1 and reaches the
outmost layer n = 68, it is free from the potential hollow of nucleus and it does not anymore
surround the nucleus. It has at this layer the zero-point energy EZ = E1 / 68² of electron. The
zero-point energy does however not affect on the observed energy levels En, as later is
proved. The free electrons are thus travelling at the outmost edge of the lattice in reciprocal
space.
The lattice current loop around an electron corresponds to the geometric model of photon so
that energy of the photon is the same as potential energy of the electron at layer n. The length
of outer circumference of the loop  = 2r on 3D-surface is also the wave length of photon.
Note that the lattice current will polarize the lattice in directions of 3D-surface and 4.D.
141
When an electron transfers from layer n = 1 to the outmost layer n = 68, its zero-point energy
(EZ) stays and it is the energy of the layer n = 68. The electron never can be freed from it. The
zero-point energy would affect on all energy levels of atom by reducing the energy, if there
would not be any fixing factor of the same size, which is found of the structure of the lattice.
Electron on the layer n=1 stands in direction of the 3D-surface at the distance 136d from the
nucleus, when it according to the Bohr’s atom model and its electrodynamics must stand at the
distance 137d. Electron has then absolutely bigger energy than the atom model insists. Let’s
define therefore for the electron of atom absolute energy En*, which includes the zero-point
energy EZ. At the layer n = 1 in hydrogen atom the potential energy of electron is E 1 = E1* - EZ,
where E1 is equal to energy given by the Bohr’s atom model at the layer n = 1.
137 cells
133 cells
X = V-nd
V = 137d
e-
n=2
p+
136 cells
2nd
The electron in the picture
stands at the layer n = 2.
The zero-point energy EZ is not unique in quadratic absolute space but depends on the
distance to observe. The place and the energy of a particle are not unique. Only the
observation will uncover them. At the layer n the zero-point energy of electron is
EZ(n) = En / 68² = E1 / (n² 68²).
When n = 1, the energy of the lattice current loop corresponding to the layer length (U = 2D) in
direction of 3D-surface is ED = E1 /68² as told before. It is the same as zero-point energy or
ED = EZ. At layer n instead n-time bigger 3D-length in the lattice current loop means the energy
ED = EZ = E1 /(n² 68²) = En /68².
The diameter of the lattice current loop consists at layer n=1 of two parts (the right and left
part). Their both length is V. In the previous picture, where n=2, the exact value of Rydberg’s
constant were calculated with help of the length V. According to the picture from the length V
must however be subtracted ½-layer or X = V - d. The diameter of loop at layer n=1 is
2X = 2V – 2D or the diameter is shortened by one layer (= 2D) and its energy has increased
with number ED. The diameter 2X corresponds now the energy E1*. At layer n=1 is valid E1* =
E1 + ED = E1 + EZ. The addition ED will thus compensate the zero-point energy EZ of electron.
The previous picture shows that to get the exact length of the lattice current loop it is needed to
subtract from every radius nV of electron at layers n = 1,2,3,...68 the length nd in absolute
space. So we can write to all layers n: nX = nV – nd.
At the layers n the length nV is shortened by the number nd and the diameter of the lattice
current loop is shortened by the number 2nd. The proportion of the value 2nd to the diameter
2nX of the lattice current loop is a constant at all vales of n. The proportion between the energy
ED corresponding to the length 2nd and the energy En is also a constant or the same as at
layer n=1 or ED / En = 1 / 68².
142
E
En
ED
En
En*
So we can now mention that the zero-point energy EZ of
electron is compensated from all energy levels 1...68 of
atom, when the lattice current loop is shortened by the
proportional length beside the nucleus. The proportional
shortening increases the absolute potential energy of
electron by the number of zero-point energy.
The zero-point energy of electron is not unique as the
place of unobserved particle is not unique either.
EZ
The energy levels of electron at layer n.
The smallest energy of a photon emitted by atom is got when the electron transfers from layer
68 to layer 67. The wave length of photon is
max =
1
R (1 / 67 ² – 1 / 68 ² )
= 0.014 m.
It is the longest electromagnetic wave, which an electron can emit, when it changes the layer.
We have not paid attention to the atomic charge Z. Its effect on atom model is considered next.
143
e-
p+
3D-surface
The projection ratio of the space remains in curving or the unit U = 2 · 137,0359 d = constant.
When this kind of change happens in an atom, the places of projections of electrons will
change as well the sizes of the lattice current loops.
We got for the radius of projections of electrons in atom, when U = 2 · 137,0359 d,
rn = n² · 137,035999² d = n² · 137,035999 d = n²U

2
When atomic charge Z changes the inclination of lattice lines linear and U = constant, we can
use as projection ratio  the number Z, and we can write
rn = n² U .
2 Z
Later is derived a formula 137dmc = ħ, where m is the mass of electron. We get
rn = n² . ħ .
Z mc
We got for the lattice current loops
n = 2 n ² 137,0359² U = 2 n ² U
²
When atomic charge Z changes the inclination of lattice lines linear and U = constant, we can
use as projection ratio  the number Z, and we can write
n =
2 n ² U
Z²²
When the frequency of photon is fn = c / n and 137dmc = h / 2 , we get
fn = 1 . mc² Z ²  ²
2
n² h
and
E = hf = 1 . mc² (Z)² .
2
n²
144
End of part 1.
D-theory continues in part 2.
Sources:
W. R. Fuchs
KNAURS BUCH DER MODERNEN PHYSIK
W. R. Fuchs
KNAURS BUCH DER MODERNEN MATHEMATIK
B. K. Ridley
Time, Space and Things
Albert Einstein, Leopold Infeld
The evolution of Physics
Richard Feynman
QED
R.T. Weidner, R.L. Sells
Elementary modern physics
H. C. von Baeyer
Maxwell's Demon
Jukka Maalampi, Tapani Perko
Lyhyt modernin fysiikan johdatus
Raimo Lehti
A. Einstein - Erityisestä ja yleisestä suhteellisuusteoriasta
Malcolm E. Lines
On the shoulders of giant
David Ruelle
Chance and Chaos
P.C.W. Davies, J.R.Brown
The Ghost in the Atom
Bruce A. Schumm
Deep down things
Paul Davis
The Goldilocks Enigma
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Discovery: All the dimensions of the world are found in jazz!
145