1

D-theoryNew interpretation of quantum mechanics

based on quantized space and time

&

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 19.11.2007version 1.07 1.3.2003 version 1.17 10.3.2005 version 1.27 25.1.2008version 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 version 2.03 20.2.2010 version 2.04 20.4.2010version 2.05 24.10.2010 version 2.06 26.3.2011 version 2.07 11.11.2011version 2.08 4.5.2012 version 2.09 20.12.2012 version 2.10 6.4.2013version 2.11 3.1.2014 version 2.12 14.4.2014

email: virtanen.pekka1@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 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.

3

Pekka Virtanen

Studies of physics and mathematicsin University of Helsinki, Finland

D-Theory - Model of cell-structured spacePart : Space and time

Hypothesis of theory:

In large scale the physical space is a background independent, cell-structured, three-dimensional 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. Manhattan-metric 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 3-dimensional 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.

4

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.

5

Contents, part 1:

D-theory versus the Standard model of quantum mechanics 6The unreasonable efficiency of mathematics in physics 7Background of D-theory 10Cell-structured absolute space 11Complex space 15Calculating a distance in cell-structured quadratic space 34Calculating a speed in cell-structured quadratic space 36Calculating a time in cell-structured quadratic space 37Dualism 39Classes of phenomena 40 The smallest scale 41Isotropic and quadratic space 44Schrödinger’s equation for elementary particles 57 Neutral gravitational wave 58 Conservation of momentum in Manhattan-metric 60 Disappearance of interference 71 Compton-wave length 74Particles 76Quantization of momentum 80 Non-locality in quantum mechanics and ”spooky action-at-a-distance” 82Chaos and determinism in cell-structured space 86Rotations and gauge principle in cell-structured space 87The uniqueness of space and " wave function collapse" 89 Localization of body or how does the observer’s space appear 92 Normal and reciprocal space 94Uncertainty principle in cell-structured space 96Geometric derivation of proton mass 98 The absolute orbital motion in a loop-space 100Asymmetric particle 104Time is not a substance 105The principle of simultaneous 108Lorentz's transformations in loop-space 109Spin-rotations 110Rotations in the lattice space 112Charge symmetry 115Electron in a lattice box 116The families of particles 134Virtual photon 124Projection of electron on the 3D-surface 125Geometric derivation of Rydberg’s constant 128Quantum interaction 129 The lattice lines form the famous ether 131The structure of photons 132 Properties of the lattice 134 Atom model 139Sources 145

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 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.

7

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.

D-theory- local Manhattan-metric- deterministic- space and time are quantized- observer’s space is only an image

Physical reality

Background of D-theory:

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.

Pekka Virtanen

GeometryGeometryGeometryGeometry

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)-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.

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 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.)

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. - 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.

Paul Benioff: “The final Theory of Everything should not only unify physics but also offer a common explanation for physics and mathematics”.

The School Of Athens

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 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.

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 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.

Y

X

y

x

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.

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 <= 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 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.

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.

The location of each origin in the net of diagonals is determined.

++

_

_

_

+

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).

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.

An regular tetrahedron stands between the two halves of octahedra.

The diagonals form 2 separate identical nets, space and antispace or two Manhattan-metric.

2 irregular tetrahedra t

Regular tetrahedron TOctahedron and the red

diagonals of antioctahedra.

xy

z

z'

y'

x'

T

15

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.

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 Va 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 = Vo +Va = 1. The diagonal form also cubes. However, the quadratic absolute space does not come up by considering only the cubes.

The unit vectors in an octahedron and the "inside out"-unit vectors in an antioctahedron define the same point in observer's space.

VoVa

a

a

x

y

z

d

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 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. The complex space is cell-structured and Manhattan-metric is valid there. 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 observer’s space seem isotropic as later is told.

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 and other forces have not any important role in that.

16

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 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.

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, octahedra. The four subspaces are not projected into 3D-space as perpendicular to each other.

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).

Real space

X

Y

Z

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.

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 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 / √ 3 .

3D-surface

Complex lattice is made of lattice lines

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 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 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

4.D 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,

which is described later. Projection ratio is called also by the name fine structure constant.

Hyperoctahedron

137 line segments

136 line segments

= 45º

= 45º

XpXpy

z

x

Yp

Zp

Wp

0

18

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.

19

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 3D-surface. So in an even space d = √ 2 D, when projected to the direction of a main axis of 3D-surface like in the picture.

d-layer

N1 = 2D-layer projection

1-dimensional cells

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.

direc

tion

of p

rojec

tion

d

D

3D-surface is d-layer

N2 = 2D-layer projection

No= 2D-layer projection

-N1 = 2D-layer projection

N68 = 2D-layer projection

lattice box or an octahedron

-N68 = ½ - layer projection

Note! In the picture the lengths of the lattice 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.

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.

d

D=d

d/2 d/2

d/2

d-layer

2D-layer

d/2

2D-layer

Lattice box of the 3D-surface

20

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 3D-surface. 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.

proje

ction

dire

ction

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 3D-surface, 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”.

d

=45ºD

d’ = constant d’

Contracted complex spaceSmooth complex space

d’

d’ = P or Planck’s length on the 3D-surface

d’

3D-surface

21

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’.

X

x

a

a²

X

x²

a²

a

Nonlinear correspondence Linear correspondence

Transformation gives a linear correspondence between the

spaces x² and X.Transformation

22

136

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

Inner part of the space

The spontaneous

symmetry violation created the 3D-surface

The space (2x137) before the spontaneous symmetry violation. Only a small set of the main axes is shown in the picture.

The space after the spontaneous symmetry violation. The 3D-surface has appeared.

137

Lattice box

Scalar field

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.

w+ w-

Zo

23

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 3D-surface 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 Manhattan-space. 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.

24

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.

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.

Empty cell

= 0 > 0 > 0

e-

25

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.

The opposite rotation of an antiparticle at the same moment:

Momentary rotation of a lattice particle:

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.

+

-

+

-

- -

+ +

- -

+

+ +

- -

+

auxiliary line

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.

26

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 3D-surface, 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-iEt .

For antimatter the wave function is (x,t) = e-i(-E)t = e-iE(-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.

27

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.

e+ e-

x

t

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.

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.0

1

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.

28

The lattice line shapes made of lattice 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 to and fro into opposite directions. A momentary direction of this motion is shown in the picture.

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.

c

4.D

c

c

3D-surface

137 cells

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 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 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).

The lattice particles in the lattice boxes form in the lattice a 2-dimensional electron plane. The curved lattice particles are depicted here as straight.

The directions of rotations are marked in the picture. After rotation the plane has changed to an other direction.

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).

Negative lattice line Positive lattice line

Neutral lattice line

e+

e+

e-

e-

Auxiliary line to perceive octahedron

X Y

29

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).

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

4.D

c

P1

c

P2

c

30

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.

Ln

In observer's space the circle Ln ² + n² =

137² defines the line segment to a quarter of a circle.

nn

In absolute space the length of a line

segment is linearly Ln + n = 137, where Ln and n are the components parallel to the main axes.

Ln

137 cells

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.

In observer's space the line segments are perpendicular to the 3D-surface and form there circles.

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.

3D-surfaceA layer parallel to 4.D

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.

4.D

31

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 T- into opposite directions. The speed of motion is the light speed c.

T+

T-

The positive maximum of lattice

The negative maximum of lattice

Pair: Positron e+ and electron e- .The particles rotates into opposite directions.

Negative lattice line

Positive lattice line

Lattice box

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.

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.

c

e+

e-

Color sets

32

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/antimatter-asymmetry 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.

33

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.

34

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.

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 cell-structure of the space. Also motion needs to be taken into consideration. The observed length is different for every observer because of the length contraction.

X

x

a

a²

X

x²

a²

a

Nonlinear correspondence ± X x Linear correspondence ± X x²

Transformation gives a linear correspondence between the

spaces x² and X.

Transformation

35

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).

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 or r = n 1 - v² n² c² 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.

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.

d

s

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.

36

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

f ² = a² - b² , when a b and

PF + PF' = 2a, when in (x,y)-plane x²/a² + y²/b² = 1.

For the speeds is valid

v² = c² - w², when c w . Then

a c and bw and fv.

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.

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

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.

F F'

a

b

f

P

c w

vc

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.

37

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.

ct

x

Light cone

Symmetric case v = 0:cT is the distance, which light travels in (x,ct)−frame in time T.

Light cone edge

Asymmetric case v 0:cT’ is the distance, which light travels

in (x’,ct’)−frame in time T ’.

cT

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.

cw=c

g

ct

x

cT’

ct’

x’

c

gw

c

38

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.

rightleft

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.

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 = 0.9399637065 10-23 s . 299792458 m/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.

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.

39

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.

40

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 non-locality, 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.

41

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 - (ct)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.

42

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.

43

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.

s

a

b

X

YThe 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.

In observer's space is valid

s² = a² + b² when a b.

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 Sx 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.

SxSy

AxAy

Bx

By

Pythagoras' theorem

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.

a

bc

ab

A

B

44

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.

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.

x

y

x² X

y² Y

Y

X

45

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.

Y

XZ

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.

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.

46

A

B

C

y

x

ℓ = 1

b

a

a

a = tan . 1 + tan

b = cos = 1 - tan . sin + cos 1 + tan

a + b = tan + cos = 1 1 + tan sin + cos

[BC] = 1 = S. sin + cos

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 .

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 ) .

Sb

47

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 Manhattan-metric 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 3D-surface, 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.

48

Macroscopic body

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.

lattice lines

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.

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.

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.

y

x

Xp

Yp

49

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

x

Xp

Yp

y

x

Xp

Yp

1

-1 1

-1 Xp

Yp

y

x

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.

50

y

x

4.D

x

y

On the locally contracted 3D-surface the momentum remains, when observed in an even space. The picture show a 2-dimensional 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.

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.

51

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 3D-surface. 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.

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.

Lx = √ 2 L cos(45º - )

= L (sin + cos )

45º -

DX

Lx = √2 L cos(45º - )

√ 2 L

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.

Xp

L

St ickC

B A

L

Yp

Y

E

F GL

52

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 .

L=1

√ 2L

1

Lx

D

Y

X

L = 1

Xp

Yp

A

E

1

-1 1

-1

Xp Yp

Y

XThe 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.

√ 2L

53

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

The picture shows rotation angle in the observer’s space in relation to the main axes X and Y of 3D-surface. The point S on the stick moves during rotation on the blue arc of a circle.

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

45º -

A

B

D

X

Y

S

1

K

H

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.

Xp

AC = √ 2 cos(45º - ) BC = 1 SC = √ AC x BC = √ 2 cos(45º - ) = 1 = sin + cos sin + cos sin + cos

at all values of

HC = 2 KC = √ AC x HC = 2√ 2 cos(45º - ) = √ 2 sin + cos sin + cos

at all values of

54

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

a = yo Fc sin b = xo Fc cos

Vectors xo and yo are unit vectors.

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.

The directions of the sum vectors Fc create new directions into the Manhattan-space.

Fc

y

-x

a

b

Fca

ba

b

Fc

Contraction centre

55

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.

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

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.

r

- U(r)

Lattice line

56

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).

4.D

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

x

Motion paths of cellsParticle

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 D-theory.

c²

4.D (x,t)

x

x=0

57

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) + U(x) (x,t) = i ħ (x,t) 2m x² 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.

0/2

x

A Ek

E

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.

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.

x

Epot

58

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.

59

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.

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)

second derivative of horizontal transition

first derivative of horizontal transition

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”.

Longitudinal transition or the wave at 90º phase shift to the transverse wave.

Horizontal acceleration of the space

0

Horizontal speed of the space

0

level of the surface in a transverse wave

0

Expansion ContractionA sin x

- A cos x

- A sin x

Horizontal transition of the space

0

A cos x

/2

Transverse transition. A circle depicts the motion and location of one cell of the surface.

60

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

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 3D-surface and its conservation are the property of a space.

g

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.

v

w

c

c² = v² + w²

v

An asymmetric body/particle as a wave

e = v / c

c = w

v = 0vc

w

4.D

v

c

61

L / 2

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

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 3D-surface is opposite and the wave travels into opposite direction.

Wave direction

To drive the animation use PageUp- and PageDown-keys in SlideShow-state (F5).

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.

Body

4.D

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.

62

Contracted spaceExpanded spacelevel at transverse and longitudinal wave

0

L / 2

0

level at transverse wave

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 +.

If the emitting body however moves in relation to the absolute space at some speed v, the gravitational wave changes asymmetric.

v

w

c

c² = v² + w²

v

A point of the space travels in the wave on a circle or on an ellipse in phase space.

To drive the animation use PageUp- and PageDown-keys in SlideShow-state (F5).

Gravitational wave emitted by an asymmetric body.

L / 2

Wave direction

+

vc

w

4.D

c = w

v = 0

p = mv

63

Let’s consider next a wave emitted by a macroscopic body and the gravitational potential created by longitudinal component of the wave.

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 (2x/ - 2ct/ + ) , 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 2c/ = 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.

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.

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.

UL(x,t) = v cos (kx -t + ) = √ 2GM / r cos (2x/ - 2ct/ + ), 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 vew. The angular velocity for antimatter is - = - 2c/.

L / 2

A = c²ve(x,t)

wc

0º90º

v

x

4.D

R

64

a a

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.

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.

0

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.

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.

Neutral gravitation wave emitted by matter

Neutral gravitation wave emitted by antimatter is like backwards in time moving gravitational wave

+

-

Matter

Anti-matteri

65

Field direction

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.

xy

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.

x

a

bV

r

xv = xo √ 1 - ve² = a , where xo is length x, when V = 0. c²Thus 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.

When b² ~ V, a relative change of length in potential Vx at the escape velocity is ve,

b² = Vx = ve² => b² = xc²ve² , where xc is length x, when Vx = Vc xc² Vc c² 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

Field direction

66

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.

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.

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.

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.

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.

67

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.

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.

A neutral gravitational wave creates into space an acceleration vector, which has only two momentary directions. All bodies emit neutral gravitational waves.

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

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.

Neutral gravitational waveEnergetic 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.

4.D 4.D

3D-surface

68

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.

v1 v2 v3 v1>v2>v3v = 0

A

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.

Local contraction caused by a body. In direction of the vector the number of proportional complex space contraction is bigger than 1.

1

-1 1

-1

Xp Yp

x

y

69

a

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.

B

The distance s between the bodies A and B in the contracting Manhattan-metric changes sideways because of the accelerations a and -a of the body B but remains in the observer’s space and in an even Manhattan-metric. 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.

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

-a

B

A

Contraction starts / acceleration

Contraction ends / braking

s

s

70

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 Ek = 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 Xp and Yp of the complex space.

i1 i2

i3 i4

i5 i6 i

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.

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.

v

Xp Yp

xy

Even Manhattan-metric

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.

71

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.

V

V

V

Single elementary particle

Macroscopic body

Black hole

ve² = A² = c² = 2Gm/Rs

V = ve²

c² vn²

V(r)

r

For the amplitudes (the speeds) of the cycles is valid

vn² = ve² or vn² = ve² (1 - ve² ) c² - ve² c² c²

As an average of matter wave appears a potential sag V(r) so that the wave damps out depicted by the potential

V(r) = Gm / r .

The acceleration caused by the field instead damps out by quadratic distance

dV(r) / dr = - Gm / r².

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.

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

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.

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/.

The next picture shows a potential sag caused by the average of matter wave. It is called also for the gravitational sag.

Rs

R

o

72

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.

A²

c²

m

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.

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 = Rs , where Rs is Schwarzschild’s radiusc² 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 , where ve is the escape speed. 2 2 r

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 or Ek + Epot = E Rs 2R 2R 2 R R

, when Schwarzschild’s radius Rs = 2Gm/c². So we get for the body of size R

Rs = A² . A² 1/ = 1/4R . R c²

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.

2R = /2r

4.D

A

U=c²

73

r

(r) = r

m(,A) = E/c² A²

Macroscopic space

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.

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 1030

, 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.

A², , m

Inter-ference

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(r) m / r

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 ve² = E c² = E c² 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 = E or Ek + Epot = E Rs/R = E. Rs 2Gm 2 2

, where Rs is Schwarzschild’s radius Rs = 2Gm . c²

74

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 Doppler-effect. 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.

75

fc

frfc

½d

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 fc 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 v T v

= h , which is the wave length of de Broglie’s matter wave. mv

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.

76

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.

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

77

3D-surface

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.

A particle or a quark can be described as a cyclic curving or as a space wave in a 1-dimensional 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.

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.

3D-surface

P

lattice box

Folded quarkA 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 D-theory. Let’s consider next the size of proton and let’s search a theoretical prognosis for the size of proton diameter.

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

Gluon

Quark

78

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.

t0 t1 t2 t3

Octahedron + antioctahedron

P

P

d d

e-R

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.

p+ p+

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.

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 d-quarks.

=

L = d

79

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.

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 fmlink: 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.

1 2

3

d/2

√(½d)²/2 + (½d)²/4 = ½d √ 3 / 2 = 0.8660d/2

xy

zy' x'

√ 2 d/2

80

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 1-dimensional cell.

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.

P 2P

81

Let’s presume that the momentum of a particle is determined by the N octahedra of the 3D-surface 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.

y

x

4.D

x

y

The internal curvature of the 3D-surface creates the momentum of a particle. In the picture 2-dimensional 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.

82

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.

correlator

light source polarizer

detector

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.

penetration

+

-

a b +

-

no penetration

83

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 = 90º and C = -1 = 45º and C = 0

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 .

a²

a² cos²

Y

X

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

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.

a² sin²

84

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.

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 Manhattan-metric. 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.

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.

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.

OC √2 cos

A

B

PC

= 30º

RO H

OC √2 sin

85

90º

1.0

0.5

P(++) + P(- -) = cos²

30º0º

quantum probabilityf() = cos²

classical probability

0.25

0.75

60º

The probabilities P are calculated for the identical behavior in polarization crystals or for the results (+ +) and ( - -).

Other results are (+ -) and (- +) .0.67

90º

1.0

0.5

E() = C()

30º

0

quantum correlation f() = 2 cos² - 1 = cos 2

classical correlation

60º

-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.

The results correspond completely to the prognosis of quantum mechanics.

The results correspond completely to the prognosis of quantum mechanics.

For the angle 30º the quantum probability is cos² 30º = 0.75. The classical probability is 0.67.

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.

86

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...”

87

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 2-dimensional space of integers N(2), which is a level in absolute space.

X

YZ

The lattice line shapes travel on the axis into opposite directions parallel to each axis.

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 3D-surface in complex space. The rotations are the same at every moment everywhere or globally in empty space.

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.

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

88

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 every-where 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 cell-structured 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 PageUp- and 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 .

89

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.

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 + nr = 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 + nr)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,...Sn 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.

S3 S2 S1 P P2 P3

r r r = aRn

rn

90

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.

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 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

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.

In Manhattan-metric the way AB = a+b can be traveled in numerous different ways. The path is not unique.

B

y

x

a

b

91

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 (= non-uniqueness ) 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.

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 non-uniqueness 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 + A2 e . +ikx - ikx

92

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.

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.

P1

P2

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.

L

93

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.

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.

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

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.

94

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.

Normal spacen

Reciprocal space1/n

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.

3D-surface

Straight line of numbers

137 cells fill an area -1...0

R

0

-1

-2

-3

-NTo the centre of the space

136 cells fill an area 0...1

95

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 writer 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".

Walter R. Fuchs: KNAURS BUCH DER MODERNEN MATHEMATIK:

"Irrational numbers are thus strongly tied to the square root operation."

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.

96

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.

Principle of uncertainty is written as an average mean error in form x p ħ / 2 .

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 XY-plane is needed for accurate measurement of place at X-axis. Then the particle does not move during measurement 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.

The accurate place of the particle on X-axis is found only during rotation at the plane of Y- and Z-axis perpendicular to 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 speed there. In this rotation the particle again interacts with 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.

D=d

dd

X

v

YRotation X-Y

Rotation Y-Z

Z

x

97

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 ) ,

where ħ is Planck’s constant.

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 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 2137d / c = t, where t is the time, which the light needs for the way c = 2 r = 2137d, 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.

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 .

-31

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.

A = ħ

D=d

x

p

t

E

98

Geometric derivation of proton mass

We have before derived geometrically the electron mass starting from the formula

137dmec = h / 2 = ħ or

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.

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

√ 2 137 d √ 2 me c = h or

Area = h = ħ x 2

Eh =mc²

Area = ħ

mq S c = h , where the mass mq of a quark is

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

Proton contains a positive electric charge e+ with its potential energy or mass me, which needs to be added to mp

mp = (1826.5 + 1) me = 1827,5 me.

The measured mass of proton is 1836,15 me 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 .

The simplified picture shows 3 quarks of proton perpendicular to each other.

4.D

√ 2 dD=d

d

d

99

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.D-component of proton. All three quarks of proton stand on the 3D-surface.

4D-component of proton (folded quark)

3D-surface

quark of proton, maximum length = dlattice layer

lattice lines

++

+

+

-

-

+

++

The maximum length a quark is d and minimum length is zero (or the Planck’s length).

lattice box

d

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.

100

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.

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.

K

K'

101

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.

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.

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).

102

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 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.

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

2s = 2 w1 w2 = w1 w2 = c² - v² = w² t w1 + 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

2s = w² = c , which is the same for all observers. t c

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.

w1

w2

103

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)).

2c

2c

y = 2c - x y = x

c0

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

y² = (2c - x)x = 2cx - x² or

y² + x² = 2cx ,

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

y' = c - x' and y' = c + x' and

y'² + x'² = c² .

c

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.

v

w

c

y

x

104

(n - k) lattice lines

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.

(n + k) lattice lines

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.

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 3D-surface 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.

v

w1

w2

105

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 = 2s1s2 = 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.

106

Observer body absolute rel. speed

n² - k² n² n² v outward+ n² - k² + n² - 4k² +n² v return

2(n² - k²) = t1² 2(n² - 2k²) = t2² 2n² 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 = vt. 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 rel. speed

n² - k² n² n² v outward + n² - k² + n² +n² v return

2(n² - k²) = t1² 2n² = t2² 2n² 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)

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.

107

m

n

k

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

t² = n² + k² - k² = n².

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.

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.

w1w2

In picture w1= w2 = c.

w1

w2

108

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.

w1 w2 A H B The distances of the bodies A and B from H are equal and w1 > w2.

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 T1 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.

w2 w1

S

109

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

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.

Note! The reader can find detailed deriving in textbooks of physics.

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 loop-space 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.

110

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.

z

yx

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.

4x

-z-y-xzyx

+90º

-90º

111

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-2-rotations, which has 3 different positions. Full cycle needs there less rotations in comparison with spin-1- and spin-½-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.

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 3D-surface 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.

AB C

y

x

y

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!!!

112

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 2-dimensional 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.

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 3-dimensional, 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.

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.

The motion of a lattice particle resembles motion of a balance wheel in the mechanical clock.

Turn point

Start point

0

T t

Even parity

0

t

T

Odd parity

113

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 (-).

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.

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)

e

r

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.

d

Anti lattice particlelattice particle in its different phases

lattice particle in its different phases, when time direction changed. Anti lattice particle

_

++

+

_

-

_

+

_

+

-+

+

_ _

+

_

+ +

_

114

Laye

r

Negative lattice line

Positive lattice line

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.

Let’s consider next the shapes made of several lattice particles together. They are called for lattice lines.

lattice box

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.

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).

auxiliary line to perceive octahedron

115

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.

116

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

Rotation directions of electron

T

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.

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.

T

e+ e-

0

117

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.

Electrons e+ and e- affect on the lattice lines in opposite ways and rotate in the lattice boxes into opposite directions.

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.

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.

Id

Id

e+

e-

Id

e+

Id

e-

e+

Electron e+ Electron e-

e-

Id

Id

lattice particle

positron electron

118

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 3-dimensional 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.

ee

e g

gg

ee

e g

gg

Id

IdId

Id

Id

Id

Phases 1., 2. and 3.

1.

2.

3.

4.

5.

6.

Phases 4., 5. and 6.

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.

119

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.

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.

90º later:

90º later:e+

e-

lattice particle

e+

e-

lattice particle

e+

e-

lattice particle

Id

Id Id

Id

IdId

e+

e-

lattice particle

Id

IdElectron e+

Electron e-

lattice particle

120

Electron e+ and e- polarize the lattice absolutely to a certain direction until the rotation direction changes.

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.

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.

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.

- Q

+Q 3D-surface

+v

-v

e- +v

-v

±0

4.D

e+ e-

IdId

n

IdThe 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.

121

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.)

Magnetic field on the other hand represents the component of the lattice charge perpendicular to the 3D-surface appeared from lattice current.

- Q

+Q 3D-surfacee-

4.D

v

v = 0. The sum of lattice currents parallel to 4.D:n is zero

- Q

Q

122

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.

- Q

90º

2.0

1.5

Id() = 2 cos² (+90)

30º

1.0

60º

0

0.5

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.

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.

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

B = qv µ / 4r²

v = ½ c. The sum of the lattice currents parallel to 4.D is one unit.

- 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.

Classical

Quantum mechanical

123

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.

lattice particle

Electron e-

lattice particle

Myon -

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.

lattice particle

Neutrino of electron e

lattice particle

Antineutrino of electron e

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.

lattice particle

Neutriino of myon

lattice particle

Antineutriino of myon

e+

e-

lattice particle

+

+

lattice particle

124

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.

Virtual photon continues its way at speed of light.

InteractionA 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).

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.

125

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.

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 m ,

which is the same as radius of K-layer of hydrogen atom in Bohr’s atom model.

-10

Contracting of the complex space is not shown in the picture.

n = 1

n = 2

n = 3

r = 3 · 137d

r = 137d

3D-surface

Electron projection on the 3D-surface

137

136

126

137 cells

r = /2

133 cells

V has the same length as 137 projections of one lattice layer on the 3D-surface. V = 137d.

137V

Location of electron projection, when n = 2.

Here appears 137 loops of lattice current. The lattice currents meet in the same point on the 3D-surface.

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.

137V

Virtual positron Phase shift

2 x 137V

Real electron e- of atom

e-p+

n = 2V X = V-nd

2nd

P

136 cells

136 cells

To drive the animation use PageUp- and PageDown-keys in SlideShow-state (F5).

127

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.

To drive the animation use PageUp- and PageDown-keys in SlideShow-state (F5).

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 3D-surface. 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.

3D-pinta

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.

Projection

137 d

128

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.

- 10

7 -1

Calculation gives, when n = 1 and V = 137.035999d = 386,159268 fm

1 = · u = · 4 · 137,0359² V = 911.267 · 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 · 10 m .

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.

129

Quantum interaction

The reciprocal space is divided in direction of 4.D into cell-structured layers. Outside the 3D-surface 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.

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.

-34

-31

By reducing we get

The area ħ = h / 2 is the Planck's constant. When d = 2.82 fm and m = 9.109 x 10 kg (= mass of an electron), we get a value h, which corresponds to observations,

h = 6.63 x 10 Js .

ħ = 137dmc.

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.

Note! If is used as a length of a projection circle, must the energy E be replaced by the momentum p=mc.

h = ħ x 2r = 137d

Eh =mc²

ħ = h / 2ħ = h / 2

N= 1/2

e-

4.D

130

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 r = radius

n = 1 c / 137 137²d n = 2 c / 137 x 2 4 x 137²d n = 3 c / 137 x 3 9 x 137²d

n = 68 c / 137 x 68 68² x 137²d

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.

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 r = radius

n = 1 c 137dn = 2 c / 2 2 x 137d n = 3 c / 3 3 x 137d

n = 68 c / 68 68 x 137d

The layer n = 1 in normal space is equivalent to the case in 3D-space or at the absolute speed c of particles.

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".

Reciprocal space

Normal space

With help of these projections we have looked at a projection of a ½-layer high vector at 3D-surface. 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.

131

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.

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.

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.

3D-surface

68.5 layers in reciprocal space

68 layers in normal space

+q

VId

In the picture the lattice lines are drawn perpendicular to the 3D-surface or they are looked from the direction of their motion.

132

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.

Two rotated lattice lines

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.

A part of the radio wave made of photons. In the picture the lattice lines are vertical but in fact their angle is 45º.

133

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).

134

L3 L1L2 p+

L4 > L3 > L2 > L1 > 0

N

N

r = /2

L4

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 ½-layer 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.

1.2.3.4.

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.

135

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 3D-components 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.

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) . 8²m 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.

3D-surfacep+

136

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:

vw

4.D

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 3D-surface 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.

c

lattice line

F F'

a ( = c)

b ( = w )

f ( = v )

f² = a² - b² , when a b and

PF + PF' = 2a.

Correspondingly it is valid for the speeds

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 bw and fv.

P

c

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.

137

vc

w

3D-surface

Speed vector c is also in inclined space always parallel to 4.D and v is always parallel to 3D-surface.

4.D

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.

The picture shows the changed phase of lattice lines in acceleration field. For clarity the rhombuses are not inclined with the 3D-surface.

y(x)

x

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.

138

e-

L3 L1L2

N

N

r = /2

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.

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.

3D-surface

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.v

w

4.D

c

c

lattice line

Ellipse describes the electric field of electron and limits the lattice lines

e-

139

3D-surface

Positive positron e+ causes opposite lattice current in comparison with electron e-. Change of the angle is similar.

Positron has the same mass as electron.vw

4.D

c

c

lattice line

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.

e+

140

N

M

L

K

4.D

+4.D

The profil of atom in the 4-based space (red) and in the antispace (green) interspered with it.

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).

fd

ps

dp

s

pssn = 1

-4.D

3D-surface

spin +spin -

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 3D-space.

The nucleus stands on the 3D-surface and for every electron there exists one proton in the nucleus. In the half of the nucleus the protons stand in the same way at different layers of 3D-space. The layers are in 3D-space spheres inside each other. On the layers the numbers of protons increases with the radius of sphere 1,4,9,16,25...

The negative half of the atom stands in four-dimensional space on the surfaces of the spheres, which are turned inside out and have the same centre. It can not be distinguished from the positive half in any way but from the signs of the spins of the particles. In this model the mechanical structure of the auxiliary quantum number is found out. The auxiliary quantum number (or orbital angular-momentum quantum number) describes the distance of an electron from the vertical 4D-axis of the atom. The distance is parallel to 3D-surface. The auxiliary quantum number 0,1,2 are named with letters s,p,d,f. The main quantum number ( or principal quantum number) describes the distance in direction of 4.D.

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.

141

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 = 2r 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.

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.

N

N

r = /2

e-

p+

P

n

L2L4 L3 L1

L4 > L3 > L2 > L1

P

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 = n² c , Ze²m Zmc 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 Z ² = h f 4r n

142

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 E1 = E1* - EZ, where E1 is equal to energy given by the Bohr’s atom model at the layer n = 1.

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.

e-p+

137 cells

n = 2

133 cells

V = 137d X = V-nd

2ndThe electron in the picture stands at the layer n = 2.136 cells

143

ESo 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

ED En En*En

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 = 0.014 m. R (1 / 67 ² – 1 / 68 ² )

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.

144

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 Z ² ²

When the frequency of photon is fn = c / n and 137dmc = h / 2 , we get

fn = 1 . mc² Z ² ² and 2 n² h

E = hf = 1 . mc² (Z)² . 2 n²

3D-surface

e-

p+

145

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

Wikipedia

Discovery: All the dimensions of the world are found in jazz!