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Title: D-theory


1
D-theory New interpretation of quantum mechanics
based on cell-structured physical
space combining gravitation with quantum
mechanics version 1.01 1.4.2002 version
1.11 11.2.2004 version 1.21 8.11.2005
version 1.02 21.4.2002 version 1.12 16.4.2004
version 1.22 8.4.2006 version
1.03 31.5.2002 version 1.13 23.5.2004 version
1.23 12.5.2006 version 1.04 12.7.2002
version 1.14 04.6.2004 version 1.24
4.12.2006 version 1.05 31.7.2002 version
1.15 09.9.2004 version 1.25 8.5.2007
version 1.06 31.10.2002 version 1.16 26.11.2004
version 1.26 19.11.2007 version 1.07
1.3.2003 version 1.17 10.3.2005 version 1.27
25.1.2008 version 1.08 26.5.2003
version 1.18 28.5.2005 version 1.28
17.5.2008 version 1.09 10.10.2003 version
1.19 03.6.2005 version 1.29 25.6.2008
version 1.10 18.01.2004 version 1.20 19.7.2005
version 1.30 29.11.2008 version 2.02
8.12.2009 versio 2.03 20.2.2010 version
2.04 20.4.2010 version 2.05 24.10.2010
versio 2.06 26.3.2011 version 2.07
11.11.2011 This is the version v2.07 published
11.11.2011. email virtanen.pekka1_at_luukku.com
Pekka Virtanen
2
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 quantized space or cell-structured
space. In that case the space can be described
with the unit vectors, which span the cells. This
kind of space is absolute, but it is not the same
as Newtons 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 observers space appear
from the cell-structured space. The classical
observers space appears geometrically as an
emergent property of the absolute space. There
exist two different images of one space,
coarsened and noncoarsened. 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 the 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 70 years and
no satisfactory interpretation is found.
Observers consciousness seems to be a part of
the measurement process. The model of
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 the violation of Bells
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 absolute
space, which is not unique for a macroscopic
observer because of its structure. 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 observers space
or in other words the wave function of the
particle collapses simultaneously
everywhere. Different kind of 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
absolute space. When the space includes also the
quantized complex space, the rotations of a
macroscopic stick in the cell-structured space
are length-preserving. 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.
3
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 four-dimensional
hyperoctahedron. It has a cell-structured
three-dimensional surface and it is quadratic and
absolute in comparison to the Euclidean
observers space. Inside and outside the closed
surface exists a cell-structured complex space
extending to a limited distance from the surface.
Manhattan-metric is valid in the space. (The
observers 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
observers 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
observers 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-preserving 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.
Momentum of a particle were quantized. The
four-dimensional atom model and its all quantum
numbers and projections on the 3-dimensional
surface of the hyperoctahedron were defined
geometrically. The accurate values for proton
diameter, Rydbergs 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.
4
The new D-theory 2.07 is published D-theory
presents a new way to deal with all physical
effects. The theory is based on geometry, algebra
and logic. The space model of D-theory creates
the base for the Quantum Mechanics and for the
Relativity Theory. The model matches with the
result of Michelson-Morley's experiment,
Maxwell's equations and Lorentz's
transformations. At the beginning the geometry
of absolute space in large and in small scale is
defined. 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 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
Rydbergs 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. 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.
5
Contents, part 1 D-theory versus the Standard
model of quantum mechanics 6 The unreasonable
efficiency of mathematics in physics
7 Background of D-theory 10 Cell-structured
absolute space 11 Complex space 15 Isotrop
ic and quadratic space 28 Gravitational
wave 36 The smallest scale 37 Particles
40 Quantization of momentum 44
Calculating a distance in cell-structured
quadratic space 46 Calculating a speed in
cell-structured quadratic space 48 Calculating
a time in cell-structured quadratic
space 49 Dualism 51 Classes of phenomena
51 Non-locality in quantum mechanics and
spooky action-at-a-distance 53 Chaos and
determinism in cell-structured space 57 Rotation
s and gauge principle in cell-structured
space 58 The uniqueness of space and " wave
function collapse" 60 Localization of body or
how does the observers space appear 63 Normal
and reciprocal space 65 Local cell-structured
space 67 Uncertainty principle in
cell-structured space 67 Geometric derivation
of proton mass 69 The absolute orbital motion
in a loop-space 73 Asymmetric
particle 75 Time is not a substance 76 The
principle of simultaneous 79 Lorentz's
transformations in loop-space 80 Spin-rotation
s 81 Rotations in the grid space 83 Charge
symmetry 86 Electron in a grid
box 88 The families of particles 93 Virtua
l photon 98 Projection of electron on the
3D-surface 96 Geometric derivation of
Rydbergs constant 101 Quantum
interaction 102 The grid lines form the
famous ether 104 The structure of
photons 105 Properties of the grid 107
Atom model 112 Sources 117
6
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. D-theory
explains geometrically with help of its space
model what are the wave function and its 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 in quantum mechanics. The features of
the rotation group SU(2) are applied to the
geometric description of spin-½-particles
together with an abstract isospin-space. 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. 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 quantum correlation. 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 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 is quantized, which 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 lets 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.
7
Background of D-theory
Quantum mechanics - non-locality - uncertainty
principle - problems in interpretation
General relativity When the existence of gravity
depends on the frame of reference, gravity is the
feature of space.
Extension Everything, which exist, appears only
from the space and from its features. The space
is the only substance.
Geometry
D-theory - local - deterministic - space is
quantized - Manhattan-metric
Pekka Virtanen
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.
8
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)-observers 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 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.
9
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
Rational numbers
Real numbers
Complex numbers
Negative Integers
Fraction numbers
Irrational 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 n1. We presume
that the observer's n-dimensional space is closed
and n 3. Group theory ia applied successfully
in quantum mechanics, especially the Lies
algebra. It is abstract algebra, which examines
the features of rotations in different spaces.
The Lies 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 symmetry. 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.)
10
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. -
Observers 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.
Paul Benioff The final Theory of Everything
should not only unify physics but also offer a
common explanation for physics and mathematics.
11
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
N1 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.
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 lt 1. The sides of the
square are at distance XY 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 ? x ² , Y ? y ² , 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 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 lt
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.
12
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 lt 1. In
transformation X ? x ² , Y ? y ² , Z ?
z ² and U ? u ² , we get in the observer's
space x ² y ² z ² u ² 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 3-dimensional 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.
13
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 Plancks 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.
14
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 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).
2 irregular tetrahedra ?t
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'
z'
y
x
The diagonals form 2 separate identical nets,
space and antispace or two Manhattan-metrics.
An regular tetrahedron stands between the two
halves of octahedra.
15
a
d
Va
Vo
z
y
The unit vectors in an octahedron and the "inside
out"-unit vectors in an antioctahedron define
the same point in observer's space.
a
x
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 v 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 v 3 / 2. The volume Va of the tetrahedron
is, when the height is h Va Ah/3 ½ a d ( 2
v 3 / 3 ) / 3 1 / 3. Altogether the volume of
the halves of octahedron and of antioctahedron is
V Vo 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 cell-structured (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 observers space seems to be
isotropic or in other words it is similar in all
directions. Rotations of a macroscopic rigid
stick are there length-preserving. In order to
get the cell-structured 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. Complex space is
4-dimensional. 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 observers space seem
isotropic as later is told. The complex space is
also cell-structured and Manhattan-metric is
valid also there. The complex space is necessary
in the model for many reasons. One reason is
electro-magnetism. A macroscopic stick is hold in
one piece by electromagnetism. Gravitation has
not any important role in that.
16
Lets 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. 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
2-dimensional 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 3D-surface. 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 2-dimensional real space stands the
3-dimensional 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).
Outside a 1-dimensional real space stands the
2-dimensional complex space, which is built of
1-dimensional line segments. The line segments
create squares.
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.
17
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
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 3-dimensional
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
directions are not possible to observe.
Xp
Xp
?
?
Zp
? 45º
x
Wp
Yp
? 45º
z
The projection angle of the main axes X, Y, Z and
W to the main axes x, y and z of the 3D-surface
is ? 54.74º. cos ? 1 / v 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 grid. The
diagonals create, like on the 3D-surface, layers
of two line segments. Outside or above the
3D-surface 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 limited
main axes of the complex space are called also
for the grid lines. The complex grid space is not
connected to the 3D-surface and can move in
relation to it. Two separate complex grid space
are interspered with each other, the space and
the antispace
A certain length ratio ? exists between the
lengths of the diagonals in the complex grid and
of the diagonals in octahedra on the 3D-surface.
137 line segments long 1-dimensional grid lines
outside the 3D-surface create on the 3D-surface
the so called projection ratio ?
1/137.035999, which is described later.
Projection ratio is called also by the name fine
structure constant.
4.D
137 line segments
136 line segments
3D-surface
Hyperoctahedron
Complex grid is made of grid lines
18
Analogously to the 3D-surface the cells outside
the surface create a positive and negative grid
or the space and its antispace. When observed in
the 3D-space the axes of the grid space are
complex or all points on the main axes are
described by complex numbers. The grid lines and
the complex grid made of them 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 grid). 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. 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 Gaussian curvature of the 3D-surface or the
surface of the hyperoctahedron is zero. The
surface is flat and Euclides' geometry is valid
in this space. (The local gravity 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.
19
The 3D-surface is made of so called 2d-layers,
where d 2.8179403 fm, which is the same as the
classical radius of electron. The length d is
there a half of the octahedron diagonal. Lets
consider next the structure of the complex grid.
The cells of the grid form outside the 3D-surface
two cells wide 2D-layers like in the next
picture. The size ( 2D) of the grid layer is the
same as the layer of 3D-surface or d D. The
layers of the complex grid stand at 45º angle to
3D-surface. So in smooth space d v 2 D, when
observed in direction of a main axis of
3D-surface like in the picture.
N68 2D-layer projection
Note! In the picture the lengths of the grid are
shown as a projection in direction of one main
axis of 3D-surface. Else d D would be
valid. Note! The layers outside the 3D-surface
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
direction of projection
N1 2D-layer projection
Grid box or an octahedron
d
D
3D-surface
No 2D-layer projection
The 3D-surface is located at the distance of
½-layer below the layer N1. The complex grid and
the 3D-surface have not any fixed connection or
they are able to move in relation to each other.
d
-N1 2D-layer projection
All 3 diagonals of the complex grid 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
20
A half of the octahedron diagonal on the
3D-surface is in smooth space d v 2 D (see the
previous picture). However, the contraction (or
the expansion) of the complex space in relation
to the 3D-surface determines all the observers
lengths. It determines also the light speed and
time passing. Therefore we define for the complex
space the length d d in a smooth space. The
length d will change in contraction of the
complex space in relation to d. But d is
observed always as a constant, because its length
is not possible to compare with any length on the
3D-surface. So the complex grid space is always a
smooth 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
observers space seem isotropic.
d ?P or Plancks length on the 3D-surface
projection direction
? D/d length ratio d constant
d
d
d
D
?45º
3D-pinta
d
Smooth complex space
Contracted complex space
Contraction of the complex grid space is not
possible to observe, because there does not exist
any stable object to compare with. Instead the
absolute length of a body will change on the
3D-surface, when the complex space is
contracting. When the complex space is contracted
to its limit, the width of a grid box is equal to
Plancks 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.)
Lets presume that a half of the diagonal of a
grid 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 grid 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 45º inclination of the grid lines and broaden
all grid boxes 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 d to correspond to the
length in the smooth space. d
h 2.8179403 fm , where me is the
mass of electron. 137,03599911 mec The
field is not observed directly because it appears
equally everywhere. The field changes the
projection longer than the length d calculated in
a smooth space with help of other constants. The
effect is observed for example in the projection
length of 137 line segments long grid line. It
should be 137d, but it is in the scalar field
137,03599911d. The length of the linear
projection of a grid line can be transformed into
observers space by squaring and in this way the
radius R1 of a hydrogen atom is got. The unit
length d is not squared. R1 137.03599911² d
0.5291772 x 10-10 m
21
The dimensionless projection ratio ? describes
projection of one grid 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. The complex grid space
determines all lengths in the observers space.
As already is told, the quantities d and ? are
observed as constants also when the space is
contracting. The cells of the grid line are
projected on the 3D-surface according to the
projection ratio 1/?. (At very high energies the
projection ratio has got in measurements bigger
values. It is also possible that the previous
scalar field is not accurately equally strong
everywhere and thus the value of ? can vary
locally.) When the Universe was being born, the
3D-surface did not first exist. There existed
only the complex 4-dimensional grid space, 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 an inner and outer part, which
both consists of 137 segments long axes. This
space consists in addition of a space and of an
antispace interspered with each other. In this
happened a spontaneous symmetry violation. The
inner part of the space changed irreversible so
that the halves of the octahedra at the upper
edge of the space as well 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.
The spontaneous symmetry violation created the
3D-surface
137
137
Outer part of the space
Inner part of the space
137
136
The space after the spontaneous symmetry
violation. The 3D-surface has appeared.
The space (2x137) before the spontaneous symmetry
violation. Only a small set of the main axes is
shown in the picture.
The spontaneous symmetry violation created the
3D-surface. At the same time appeared the
gravitation into the Universe and the particles
got their mass. A scalar field affecting
everywhere formed by the 3D-surface appeared. 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 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
3D-surface gives the colour charge for a
particle. The symmetry is hidden in this kind of
violated space, because the 3D-surface still
exists as a separate part of the space. It is
said that its a question of so called hidden
symmetry and not a real symmetry violation. As a
result of the way of violation the 3D-surface
contains only the space but not the interspered
antispace. It differs in that from the complex
space.
22
The complex grid is made of grid boxes like in
the picture. The grid box is 3-dimensional and it
is made of three diagonals of octahedron or six
cells. The grid boxes are described so that its
diagonals are shown at 45º angle to horizontal
line.
e-
e-
Empty cell
Each grid box contains one ½-layer long
spin-½-particle e or e- as a part of the complex
grid. The particle is called for a grid particle
and it can be positive or negative. The other 5
cells of the grid box are empty cells. An empty
cell means that in the cell exists not any wave
with a certain curving amplitude. The grid
particles form together into the grid the shapes
of positive and negative grid lines. All grid
particles stand in their boxes in such a position
that the shapes of the grid lines form in space
2-dimensional planes of nets, which are called
for electron planes. The directions of the planes
are equal among themselves in each subspaces but
they all will change in the rotations of the grid
particles. The planes are complex. At the
beginning there existed only the complex grid
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 Plancks length.
Soon the octahedra, however, expanded rapidly and
the space expanded strongly faster than light.
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 grid
box of the complex space as energy of grid
particle. The grid particles started their
everlasting circulation motion to and fro in
their grid box and the time passing 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 grid space. The grid lines form a vacuum,
which has so called zero point energy and other
quantum mechanical features. Rotations or the
motion of the grid line shapes gives the phase
for a wave function. When a grid box is
contracted in one direction, a phase shift
appears into its rotations. The phase shift ?? is
zero, when the grid lines are at an 45º angle to
the 3D-surface. Always in other cases ? ?? ? gt 0.
The phase shift is local and there exists always
a force field. A force field will thus change the
shape of the grid 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.
? ?? ? gt 0
? ?? ? gt 0
?? 0
23
Momentary rotation of a grid particle
A grid particle differs from an empty cell
because of its energy. Energy is described as
curvature of the cell, because energy is always
linked to curvature of space. A grid particle is
an energy package rotating around in the grid box
containing kinetic energy and potential energy.
(It is like a balance wheel in the clock.) The
motion to and fro means that the time direction
changes regularly at microscopic scale. Curvature
appears in the 2-dimensional electron plane, in
which the so called elementary rotation (defined
later) is going on. The positive direction of
axes is in the picture upwards or outwards from
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 ( colour in the picture).
So on the upper side of an grid box stands a
green grid particle and on the lower side a red
one like in the picture. The colours will change,
when the time direction changes (more later). The
number of curvature is quantized.

-
-


-

The opposite rotation of an antiparticle at the
same moment

-

-
-
-


The building elements of the complex space
(X,Y,Z,W) or 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 each
subspaces of the octahedra are standing electron
planes, which stand in each subspaces parallel to
each others. The electron planes stand at every
moment in four different directions so that the
directions are perpendicular to each other. All
electron planes turn in every elementary rotation
in their own subspaces. In one subspace the
electron planes can turn in 3 directions
perpendicular to each other.
We have before described space and time with help
of segments of line and with rotations. A segment
of line is an abstract background independent
model for space. Correspondingly elementary
rotation is an abstract background independent
model for time. The line segments and the
rotations have fundamental properties, which are
not possible to explain but only depict. The
direction of curvature depends also on the
momentary rotation direction of a grid particle
in the grid box. The antiparticle is curved into
opposite direction and rotates into opposite
direction. The rotation direction changes in all
grid boxes at the very moment, which means a
global phase transformation for the phases of the
wave functions of particles. In the grid exist
the same number of grid particles and of its
antiparticles. They form overall in the space the
so called zero energy level. All the grid
particles turn in rotation similarly and move
into the next empty 1-dimensional cell in their
grid boxes. In the same time the grid line shapes
move depending on the rotation direction forwards
or backwards in direction of the projection of
one complex main axis. The speed of this motion
is the same as the speed of light. These
rotations are described in detail later.
24
4.D
c
137 cells
The grid line shapes made of grid particles e
and e- on the 2-dimensional planes ( or the
electron planes) are moving outside the
3D-surface in space and in antispace at speed of
light into opposite directions. A momentary
direction of this motion is shown in the picture.
c
3D-surface
c
Grid line shape made of electrons
The motion of the grid line shapes past each
other in phases determines the wave function
phase, which is not possible to measure. If the
grid 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 grid in the sub space
(X,Y,Z). In the picture the grid particles e and
e- form together a plane and the shapes of the
grid lines. In the next rotation the grid
particles parallel to X-axis turn parallel to
Y-axis and particles parallel to Y-axis turn
parallel to Z-axis. Shapes of the grid lines
leave the XY-plane. 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 grid
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).
Neutral grid line
Negative grid line
Positive grid line
e-
The grid particles in the grid boxes form in the
grid a 2-dimensional electron plane. The
directions of rotations are marked in the
picture. After rotation the plane has changed to
an other direction.
e
e-
e
X
Y
The interactions of the grid particles are
considered in 2-dimensional electron plane of the
grid line shapes. The axes of the plane stand
outside the 3D-surface and at an 45º angle to it.
The axes are thus complex and the symmetry space
of rotations for spin-½-particles is SU(2).
25
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 grid 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 undersood so that the electrons of
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