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Foundations

The foundations of mathematical physics and its relationship with physical reality

Abstract

The Hilbert Book Model project produced this publicaon. The search for a reliable foundaon of physical

reality has had many setbacks and is slow. As a result, mainstream physics got at a sidetrack. Quantum

Field Theory, Quantum Electro Dynamics, and Quantum Chromodynamics use the minimal acon

principle as their base. The Hilbert Book Model shows that connuums belong to the third phase of a

special set and cannot work as a foundaon of mathemacal physics. This document shows how the

three phases of the special set lead to a vector space and number systems, which apply to a system of

Hilbert spaces in which the local universe and a parallel mulverse can pose. Also, the document shows

that science must not consider the Higgs parcle or the Higgs eld as part of the Standard Model.

Instead, the Standard Model of experimental parcle physicists should restrict to elementary fermions.

Most physicists interpret photons as excitaons of the electric eld. In contrast, the HBM interprets

photons as chains of dark energy objects, and the dark energy objects are shock fronts that excite the

eld, represenng the local universe. Hop landings of the state vectors of the fermions produce spherical

shock fronts that move with light speed away from the locaon of this landing. This conicts with the

ideas of conservaon laws that play in mainstream physics. According to the HBM, a big bang never

occurred. The model considers two episodes, and at the beginning of the second episode, me starts

running together with an ongoing creaon of fermions.

2

• Table of Contents

1 Justification ........................................................................................ 7

2 Introduction ....................................................................................... 8

3 Explanation ........................................................................................ 8

4 Clarification ...................................................................................... 10

5 Vector space .................................................................................... 11

5.1 Independent directions .............................................................. 13

6 Number systems .............................................................................. 13

6.1 Real numbers ............................................................................. 13

6.2 Phase transitions ........................................................................ 14

6.3 Spatial numbers .......................................................................... 16

6.4 Division rings .............................................................................. 17

6.5 Confusing calculation rules ......................................................... 18

7 History ............................................................................................. 18

8 Set theory. ....................................................................................... 19

8.1 Collections in space .................................................................... 19

9 Coordinates ..................................................................................... 21

9.1 Hops and symmetries. ................................................................ 21

10 Mainstream Science ...................................................................... 23

10.1 Warning ................................................................................... 23

11 Hilbert spaces ................................................................................ 23

11.1 Function space ......................................................................... 25

11.2 Quantum logic ......................................................................... 25

11.3 Other features of Hilbert spaces .............................................. 26

3

11.3.1 Subdividing into Hilbert spaces .......................................... 27

11.3.2 Subdividing into parameter space and target space .......... 27

11.3.3 Adding change with time .................................................... 31

12 Potentials and forces ..................................................................... 31

12.1 Center of influence of actuators .............................................. 32

12.2 Forces ...................................................................................... 32

12.3 Actuators ................................................................................. 33

13 Stochastic processes ...................................................................... 34

13.1 Optical Transfer Function and Modulation Transfer function ... 37

13.2 Photons .................................................................................... 38

13.3 Light ......................................................................................... 38

13.4 Refraction ................................................................................ 38

13.5 Holographic imaging ................................................................ 39

13.6 Electron optics ......................................................................... 39

14 Social influences ............................................................................ 39

15 Ongoing investigation .................................................................... 40

16 New insight .................................................................................... 41

17 A System of Hilbert spaces ............................................................ 43

17.1 A System of separable Hilbert spaces ....................................... 43

17.2 A modeling platform ................................................................ 45

17.2.1 Conglomerates ................................................................... 48

17.2.2 Interaction with black holes ................................................ 49

17.2.3 Hadrons .............................................................................. 50

17.2.4 Atoms ................................................................................. 50

17.2.5 Molecules ........................................................................... 52

4

17.2.6 Earth ................................................................................... 52

17.2.7 Particles and fields .............................................................. 54

17.2.8 Modular system communities ............................................ 54

17.3 A System of non-separable Hilbert spaces ............................... 55

18 Conclusions ................................................................................... 57

19 FormulasEquation Chapter (Next) Section 1 ................................. 63

19.1 Relativity and curvature ........................................................... 63

19.2 Physical units ............................................................................ 63

19.3 Vector arithmetic ..................................................................... 64

19.3.1 Base vectors ....................................................................... 65

19.4 Arithmetic of real numbers ...................................................... 66

19.5 Arithmetic of spatial numbers .................................................. 67

19.6 Mixed arithmetic ...................................................................... 68

19.7 Arithmetic of change ................................................................ 69

19.7.1 Differentiation .................................................................... 69

19.7.2 Enclosure balance equations .............................................. 83

19.8 Dirac’s’ bra-ket procedure ........................................................ 86

19.8.1 Countable number systems ................................................ 87

19.8.2 Uncountable number systems ............................................ 93

19.9 Lattice theory ........................................................................... 96

19.9.1 Relational structures .......................................................... 96

19.9.2 How quantum logic got its name ........................................ 96

19.9.3 Lattice structure ................................................................. 97

19.9.4 Orthocomplemented lattice ............................................... 98

19.9.5 Weak modular lattice ......................................................... 98

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19.9.6 Atomic lattice ..................................................................... 99

19.9.7 Logics .................................................................................. 99

19.9.8 Rules and relational structures ........................................... 99

19.10 Fourier transforms............................................................... 100

19.11 Uncertainty principle ........................................................... 103

19.12 Center of influence of actuators .......................................... 103

19.13 Forces .................................................................................. 105

19.14 Deformation potentials ....................................................... 107

19.14.1 Center of deformation ................................................... 107

19.15 Pulse location density distribution ....................................... 109

19.16 Rest mass ............................................................................ 111

19.17 Observer .............................................................................. 111

19.17.1 Lorentz transforms. ........................................................ 112

19.17.2 Minkowski metric ........................................................... 113

19.17.3 Schwarzschild metric ...................................................... 114

19.17.4 Event horizon ................................................................. 114

19.17.5 Time dilatation and length contraction .......................... 116

19.18 Inertial mass ........................................................................ 116

19.19 Inertia .................................................................................. 117

19.20 Momentum ......................................................................... 120

19.20.1 Forces ............................................................................ 123

20 Postscript..................................................................................... 124

20.1 The initiator of the project ..................................................... 124

20.2 Trustworthiness ..................................................................... 124

20.3 The author ............................................................................. 125

7

1 Justification

During my study, I used my scienc books to quietly read the formulas

and make notes between the text. In that way, I got a beer

understanding of the content of the book. This book reects most of the

content of a mulpart PowerPoint presentaon of the Hilbert Book

Model. PowerPoint presentaons are not suitable for adding notes.

hps://www.youtube.com/playlist?list=PLRn2RuujW3IJsZPxh7iNvlajFW

Y12G2Af

The HBM is a private research project I started more than a decade ago

when I was 70 years old to invesgate the foundaon of theorecal

physics and the relaon between mathemacal physics and physical

reality. As a long-rered physicist, I had ample me to rethink the

physics I used most of my career. In the university, I discovered that the

lectured physics was incorrect and held aws and omissions. However, I

am convinced I found and repaired most defects and signicantly

reduced the remaining mysteries.

Building on the discussion results between Hilbert, von Neumann,

Cantor, and Zermelo that stopped in the fourth decade of the tweneth

century, I accidentally discovered a special set that features phases and

phase transions. Phases of the special set cannot pass the phase

transions step-by-step. This set becomes the foundaon of the Hilbert

Book Model (HBM).

My targets are students and young sciensts that, like me, are curious

about the universe in which they live.

I keep the price of this book as low as possible. The price of paper and

the cost of pressing and managing the book decide this price. I use

Pumbo.nl for managing the book. Pumbo oers the book as a print-on-

demand service and in massive quanes as oset. A freely accessible

8

paper will reect the content of the book and will contain the acve

versions of the URLs to which the book refers.

2 Introduction

With some arrogance, I dare to say that the Hilbert Book Model

now exposes the essenal part of the foundaons of physical

reality. Some mysteries stay, but the model describes these

clearly. For me, these mysteries exist because my knowledge of

mathemacs does not allow me to explain the origin of these

mysteries. It is also possible that this mathemacs does not yet

exist. The mulpart PowerPoint presentaon oers suggesons

for solving these mysteries via the modular structure of the

universe's content. A single sentence can shrink the essence of

the structure and behavior of the observable universe. "The

universe that manifests itself to researchers is one connuous

lm of the possible coverages of space with versions of

number systems belonging to the associave division rings."

The HBM shows that each Hilbert space applies this version in

the archival of the members of the division ring that the Hilbert

space uses.

3 Explanation

The observaon that humans cannot think and communicate

about things without providing these things with idencaon

in the form of a name or pointer and a short compact

descripon establishes a brief explanaon. Indirectly, the

Hilbert spaces provide the ideners and descripons humans

require. The curious thing is that physical reality can funcon

9

without these limitaons. Yet physical reality also appears to

adhere to strict rules and exisng structures. Many researchers

have come to know these rules and structures substanally and

formulate them in what they call mathemacs and physics.

Several researchers doubt whether people can discover the

calculaon rules that physical reality uses. However, the author

of this publicaon does not belong to that group.

My arrogance results from my convicon that those with

educaon at the level of a bachelor in the exact sciences of

mathemacs or physics should easily be able to follow the

argument given here and check it as desired. With less prior

knowledge, much of the debate is easy to follow. I, as an author,

have done my best to make as many as possible of the by me

retrieved details freely accessible via the included URLs. The

text points in enumerated brackets to the URLs that make the

subject accessible online. The book publishes the URLs in the

references chapter. On the internet, the free accessible pdf

paper oers the URLs acvely. The book displays the text

primarily in grayscale because colors increase the costs of

prinng books. The corresponding pdf le shows the full-text

colors. The author publishes both the book and the related pdf

for this reason.

Because formulas scare o several readers, they house in

separate places. This applies to the calculaon rules, the bra-ket

procedure of Paul Dirac, and essenal equaons. The formulas

locate in an individual chapter. Previous papers already

10

published the formulas. Many of these formulas retrieve from

publicly accessible resources such as Wikipedia. Compared to

the publicaon “Setbacks of Theorecal Physics,” this trease

adds the formulas for lace theory and puts more emphasis on

the special set. This publicaon introduces the name

“special_set” for the menoned special set. This new name

produces fewer problems with spelling and grammar correcon

tools.

4 Clarification

When people focus their research on space, they quickly realize

that an empty space stands for the ulmate nothingness.

There is nothing in this space to which one could orient oneself.

There is no center, and there are no boundaries. It is not hard to

imagine that the space could hold many anonymous locaons.

However, for humans, tracking the behavior of these locaons

without giving them idencaon and a precise descripon is

impossible.

Locaons are point-shaped objects that can occupy a posion

in space. That posion diers per locaon. Applying number

systems provides the required idencaon. The values of the

number system elements show the locaons' posions.

Without the locaons, the container is empty. What results is a

simple space that can funcon as a container. It is possible to

interpret this simple space as the ulmate nothingness. As a

container in which locaons reside, the simple space funcons

11

as a vector space. Two locaons and their connecng direcon

line form a vector.

The vector space owns a simple arithmec. That arithmec

enables the speciaon of the more complicated arithmec of

number systems. Hilbert spaces apply these number systems.

They select a private version of a number system.

This paper introduces a structure that harbors a system of

Hilbert spaces that all share the same underlying vector space.

Moreover, that system puts number systems in a well-dened

interrelaonship.

5 Vector space

There is sll no possibility to point to the posion. The pointer

can consist of a base locaon and a poinng locaon

connected by a direcon line. Sciensts call this pointer a vector

and a space in which vectors occur a vector space. A simple

scalar number characterizes the length of the vector. The HBM

applies the name “vector space” and manages the vector space

accordingly.

Physicists give reality the extra adjecve “physical” to indicate

that this noon concerns the structure and behavior of what

experimenters can observe from what they experience about

the university. The HBM copies that habit by using the name

physical reality for this noon. This publicaon will apply

“physical_reality” to ease the spelling and grammar-checking

tools.

12

The HBM will use the name “direcon_line” for the noon of a

direcon line. This renaming happens for the same reason that

the HBM uses the name special_set. Direcon_lines obey

simple arithmec.

The direcon_line and the length fully characterize the vector.

The integrity of the vector does not change when it shis

parallel. The parallel shi can occur on the direcon_line but

may also occur in another direcon. Direcon_lines can

therefore move parallel in the vector space. They have no

beginning and no end. This situaon at once provides the

operaon with which two vectors can add. If the base point

shis from one vector to the pointer of the other vector, then

the non-overlapping points form a new vector called the sum

vector. If the direcon_lines dier, the sum vector uses a new

direcon_line.

The two possibilies form a parallelogram in which the sum

vectors are parallel and have equal lengths.

By mulplying the vector by a scalar, the length mulplies by

that scalar. This acon creates a new vector. When the scalar is

negave, the base and pointer point change funcon and the

vector gets the opposite direcon. At the same me, its length

may change. These simple calculaon rules allow vectors to

pinpoint all locaons in the vector space. The secon Vector

arithmec in the chapter Formulas contains the formulas.

13

5.1 Independent directions

Vector arithmec enables a scalar product of two vectors. The

scalar product can show the independence of the

direcon_lines of vectors. The scalar product of independent

vectors is equal to zero. This way, several mutually independent

basic direcon_lines exist in the vector space. Since

direcon_lines can shi in parallel, a raster of direcon_lines

can cover the vector space. The raster can form a primive

coordinate system.

6 Number systems

The HBM applies the arithmec of its vector space to derive the more

complicated arithmec of the number systems that it uses.

6.1 Real numbers

With their calculaon rules, vectors can help to construct

number systems. For example, an ongoing addion of a starng

vector and vectors equal to the starng vector and located on

the same direcon_line yields an ordered series of designated

locaons collecvely represenng the natural numbers. Using

the natural numbers as a label, we can count collecons of

locaons. The subtracon procedure appears by removing

locaons from the collecon and introduces the countdown

procedure. Finally, we meet the number zero on the base point

of the original starng vector and subsequently follow the

negave integers. The method for mulplying numbers appears

by adding groups of vectors frequently. That does not supply

new integers. The name of the reversal of mulplicaon is

14

division and delivers fracons. Fracons can be new numbers.

The integer numbers, together with the fracons, form the

raonal numbers.

6.2 Phase transitions

Scholars have shown that there are as many raonal numbers as

natural ones. David Hilbert and his followers knew this. This size

equality means all raonal numbers can label with a natural

number. However, this procedure only works if both number

sets hold innite elements. The transion from nitely many

elements to innitely many elements implies a change in state

for the special_set. In the new phase, the collecon shows

dierent behavior. For this parcular set, achieving this phase

transion step-by-step is impossible. Also, the way back does

not go in a step-by-step manner. Sciensts do not oen use the

terms phase and phase transion when concerning number

systems. This paper uses these terms to show the change in the

status of the number system that derives from the special_set.

David Hilbert used the parable of the Hilbert hotel to show that

countable innity introduces another behavior of the

considered set. He did not use “phase transion” to classify the

set's behavior change. He and his followers did not consider the

dierent behavior as a dierent phase of the set. The HBM

assumes the dierent behavior as a disnct phase and the

behavior change as a phase transion. Accepng the

parcularies of the special_set has this consequence.

15

Adding or removing elements does not change the state of the

innite special_set. The innite set of well-ordered raonal

numbers lls a large part of the same direcon_line. A raonal

number can arbitrarily close approach any locaon on this line.

Nevertheless, there are sll many locaons on this line that

raonal numbers cannot appoint. We call the numbers that

these places show irraonal numbers. Irraonal numbers

include transcendental numbers, and raonal numbers include

prime numbers. The third phase of the set consists of both. The

third phase is innite and not countable.

Thus, the set of raonal and irraonal numbers again form a set

that can show as another phase of the special_set. The phase

transion happens again in one go and cannot occur step-by-

step. Counng the elements of the special_set in its third phase

is no longer possible. In this phase, all series of converging

members end in a limit that is a set member. The phase

transion adds several new calculaon rules that manage the

change of cohesive parts of the collecon. We obtained a

special_set that features parcular behavior by adding the

irraonal numbers. Mathemacians call the extra calculaon

rules dierenal calculus. The author applies this name for the

addional arithmec rules of the third phase of the special_set.

Dierenal calculus is closely related to the calculaon rules of

raonal numbers. The calculaon rules can even mix. Without

disturbing actuators, nothing will change in the new phase of

the special_set. If something disrupts, this collecon phase

16

tends to remove the disturbance as quickly as possible by

sending away the consequences of the disrupon in all

direcons unl the eects eventually disappear into innity. We

know this because dierenal calculus shows this. As

menoned, the disturbance never reaches disappearance step-

by-step. The result is that the number-covered area expands.

The dierenal calculus tells precisely how that happens. On

the so-far-considered direcon_line, the response acts in a

single dimension.

When mulplied by themselves, the raonal numbers treated

so far yield a posive number on the direcon_line of the

natural numbers. We call the numbers that behave in this way

real numbers. We use this name for all numbers on this

direcon_line and, therefore, for all phases of the numbers on

this direcon_line. Squaring is the name for mulplying by

oneself. The secon Arithmec of the real numbers holds the

formulas.

The phase transions cause the underlying set to be parcular.

In this way, this set diers from standard sets. The set exists

because simple space holds it and only consists of point-like

locaons. The author discovered this set by accident. He never

found a set with these features published beforehand.

6.3 Spatial numbers

There also appear to be systems of numbers that yield a

negave number that shares the direcon_line of the real

numbers when mulplied by themselves. We call these spaal

17

numbers. Oen the name used for these numbers is imaginary

numbers. The HBM does not use that name because the

qualicaon imaginary also has dierent meanings. The spaal

numbers no longer t on the direcon_line of the real numbers.

They occupy one or three dimensions. Suppose spaal numbers

fall outside the rst spaal dimension. In that case, the

calculaon rules of the spaal numbers ensure that a third

spaal dimension covers with spaal numbers in addion to the

second spaal dimension. The result of the product of two

spaal numbers consists of an internal product that supplies a

real number and an external product that is zero or produces a

result in a direcon that is independent of the direcon_lines

of both factors. The internal product is the reason for the

negave square. Therefore, the spaal numbers' calculaon

rules dier from the calculaon rules of the real numbers. The

reacon to a disturbance of the third phase of spaal numbers

is more spectacular in the three-dimensional spaal number

system than in the one-dimensional spaal number system. The

secon Arithmec of spaal numbers holds the formulas.

6.4 Division rings

Nevertheless, real numbers can add with spaal numbers, and

spaal numbers can mulply with real numbers. This addion

creates new number systems. The real and one-dimensional

spaal numbers form the two-dimensional set of what the

model calls complex numbers. The HBM shares this name with

common mathemacs. The real and three-dimensional spaal

numbers form the four-dimensional set of what the model calls

18

quaternions. Again, the HBM shares this name with common

mathemacs. This name sharing shows that the HBM applies

exisng names for its number systems where no conicts arise

and sucient similarity exists. The Mixed Arithmec secon of

the chapter Formulas holds the corresponding formulas.

6.5 Confusing calculation rules

Two vectors can together deliver a scalar product. That scalar

product is zero or posive, and for two equal vectors, the scalar

product supplies the square of the length of the vector. This

length is the norm of the vector. The almost idencal eect of

the inner product of spaal numbers has led to confusion

among many mathemacians and physicists, so these sciensts

somemes confuse spaal numbers with vectors. This confusion

happened, among other things, with the discoverer of the

quaternions. This confusion led to a public scandal that caused

the quaternions to fall into oblivion aer the sixes of the last

century. As we will see, this had signicant consequences for

mathemacs and physics. [2]

7 History

Before Christ, Egypans discovered simple fracons. Cantor

found the second and third phases of real numbers around

1870. Cantor did not use the designaons phase and phase

transion. Instead, he and others turned their aenon to

various kinds of innies of sets. Cantor called them transnite

numbers. Together with natural numbers, they form the

cardinal numbers. The Hilbert Book Model deals with only two

19

forms of innity. These are the countable innity of the second

phase of numbers and the uncountable innity of the third

phase of numbers.

Gerolamo Cardano discovered the complex numbers as early as

1545. In 1854 Sir William Rowan Hamilton discovered the

quaternions. He formulated his discovery using the four base

numbers. The base numbers are one real base number and

three spaal base numbers. The external product appears in the

outcome of the product of the rst two spaal base numbers.

Hamilton discovered this formula while walking with his wife

over a sandstone bridge in Dublin. Out of joy, he scratched the

formula into the bridge's wall. The rain quickly erased the

inscripon. Hamilton's students immortalized the formula on

the bridge through a bronze commemorave plaque. [3]

8 Set theory.

8.1 Collections in space

Around the turn of the nineteenth to the tweneth century, a

group of mathemacians and mathemacal physicists led by

David Hilbert had an intense discussion about set theory. [4] [5]

David Hilbert intended to establish an axiomac theory of both

mathemacs and physics. Unfortunately, he rered before he

could nish that target.

The discussion focused on the various forms of innity and

countability. The discussion partners also paid signicant

aenon to the phases and phase transions of the collecon.

20

For example, they paid aenon to the connuum hypothesis.

[6] However, they never used the words phases and phase

transions. The HBM applies these names for the special_set to

disnguish this set from other sets.

The menoned discussion ignored the container of the set and

paid no aenon to the type of objects that formed the set.

These choices are signicant in physical_reality and the Hilbert

Book Model. By choosing space as a container and locaons as

elements of the set, the number systems the HBM uses to

discover the locaons obtain added properes that human

researchers and physical_reality must consider. These added

properes are the symmetries that stand for the freedom of

choice that the calculaon rules of the number systems do not

dene. As a result, in the HBM, the number systems exist in

several versions that their symmetry disnguishes. For example,

the locaon of the geometric center of the number system can,

in principle, be anywhere in the vector space. Also, the

arrangement of the numbers can occur along the direcon_lines

in one or the opposite direcon. Physical_reality must adhere to

the calculaon rules and will use as many symmetry choices as

possible. A dierent choice of symmetry yields a dierent

version of the number system. The word symmetry has various

meanings. These disnct meanings also occur in this

publicaon. In the HBM, geometric symmetries play a

prominent role. Dierences between geometric symmetries are

essenal.

21

9 Coordinates

Three associave division rings exist. [7]

These are the real numbers, the complex numbers, and the

quaternions. Each of these number systems exists in several

versions that dier in their symmetry. Recording the symmetry

is possible with coordinate markers. These markers use the

locaon that shows the value of the number. In the HBM, a

Cartesian coordinate system records all the selecon freedoms

of a version of a number system. The record removes the

selecon freedom and helps establish the version of the

number system.[8]

In this way, the HBM connects the selected version to the

geometric symmetry of the number system and the symmetry

of everything that exclusively applies that version.

The limitaons imposed by the vector space create geometric

symmetry. Therefore, if a model designs number systems

without these limitaons, then that model does not meet

geometric symmetries.

9.1 Hops and symmetries.

A hop can split in paral hops that occur only along the

cartesian coordinate lines. The rst part jumps along a selected

coordinate line. The second part jumps along a perpendicular

coordinate line, and the third part occurs along a coordinate

perpendicular to both the rst and second. This procedure takes

a choice at each of these jumps. These selecons concern the

up or down direcon along the coordinate line. These selecons

22

correspond to the symmetries that we discussed before. The

paral jumps lead to the Frenet–Serret formulas. These

formulas form the base of dierenal geometry.

Willard Gibbs promoted dierenal geometry, and Oliver

Heaviside advanced vector calculus. Both used complex

numbers rather than obliviated quaternions. Mainstream

physicists quickly embraced the suggested approaches, and

many of these sciensts rejected quaternionic eld theory. The

mainstream physicists spent lile aenon to the symmetries of

versions of number systems. Instead, symmetry groups and Lie

groups draw their aenon. Universies wanted to coordinate

their lectures on theorecal physics and wanted to avoid

confusion. That is why most universies follow what they now

consider mainstream physics. Also, the part of the press that

treats science tends to follow mainstream physics and ignores

new developments in theorecal physics. This history explains

why theorecal physics appears to have entered a dead end.

Invesgate:

hps://www.researchgate.net/publicaon/363541991_The_set

backs_of_theorecal_physics

Sll, Gibbs and Heaviside smulated the development of

muldimensional dierenaon technology and indirectly

promoted mathemacal quaternionic dierenal analysis

development. The introducon of me as a progression

indicator produced the quaternionic dierenal analysis that

the Hilbert Book Model advocates. This development preceded

23

and took place independent of the discussion of Hilbert, von

Neumann, Cantor, and Zermelo on set theory. The HBM

combines and exploits the results of dierenal calculus and set

theory.

10 Mainstream Science

10.1 Warning

Mainstream science sll plays a crucial role in promong a

standard reference for teaching and comparing science. This

role limits confusion for students and scienc instuons.

However, being promoted by mainstream science is not

synonym with granng the truth.

This warning especially holds for mathemacs, theorecal

physics, and mathemacal physics.

11 Hilbert spaces

David Hilbert discovered an extension of the concept of vector

space. His assistant John von Neumann provided the name

“Hilbert space” to this widened vector space. The Hilbert spaces

have the surprising property that they can archive elements of

the version of the number system used by the Hilbert space.

Aer the archival in an abstract structure, the stored

quaternions retrieve in an orderly manner. A dedicated

operator manages the archival and the retrieval.

Sciensts oen describe the Hilbert space as a vector space that

owns an internal product. However, as previously argued, each

24

vector space has a scalar product, not an internal product.

Moreover, it is dicult to imagine that a vector that depicts

itself via the scalar product yields a complex number or

quaternion as an eigenvalue.

Instead, Paul Dirac discovered a trustworthy procedure for

converng a vector space into a Hilbert space. This procedure

combines covariant ket vectors and contravariant bra vectors.

These are not vectors but are closely related to them. One

problem is that Dirac only showed this for real and complex

numbers. In that period, sciensts showed lile interest in

quaternionic Hilbert spaces. However, a small eort can adapt

the procedure to apply for quaternions. Hilbert spaces can thus

work with any of the associave division rings.

The HBM restricts the archival to the second phase of the

special_set. This choice limits the dened archival capability to

the separable Hilbert spaces.

Each separable Hilbert space chooses a private version of one

of these number systems. As menoned, the separable Hilbert

space can archive collecons of elements of this version and

retrieve them in an orderly manner. This capability also applies

to the enre chosen version of this number system. There is a

devoted operator who manages this collecon. The HBM calls

this operator the reference operator. This assignment means

that each Hilbert space has a private parameter space. The

HBM gives that parameter space the name natural parameter

space of the Hilbert space. The natural parameter space of a

25

separable Hilbert space is countable. It also means that the

symmetry of the version of the selected number system

characterizes the Hilbert space. The rst version of the bra-ket

process works with countable number systems and yields

Hilbert spaces that use a countable number of independent

base vectors. Therefore, the HBM calls them separable. Secon

Dirac’s bra-ket procedure treats the formulas.

11.1 Function space

The private parameter space turns every Hilbert space into a

funcon space. Through the funcons, Dirac's bra-ket

procedure denes new operators who manage the target space

of the sampled funcon as eigenspace.

11.2 Quantum logic

To the surprise of many mathemacians, the set of the closed

subspaces of Hilbert space appears to have a lace structure

that is slightly dierent from the lace structure of classical

logic. Some sciensts suggested that this deviaon could be the

cause of the quantum structure of the energy exchange seen in

small parcles and atoms. Therefore, they assigned the name

quantum logic to this new lace. [9] A closed subspace of a

Hilbert space is again a Hilbert space. Dierenal calculus oers

a more obvious explanaon. Dierenal calculus only comes

into eect in the third phase of number systems. Funcon

theory and dierenal calculus describe the third phase of

number systems. The Arithmec of changes secon describes

the formulas that govern the third phase of number systems.

The formula chapter treats lace theory in a separate secon.

26

The countable parameter space of the separable Hilbert space

concerns the rst two phases of the number systems, or it is

uncountable and concerns the undisturbed third phase. In that

case, the Hilbert space is no longer separable. The non-

separable Hilbert space provides operators with uncountable

eigenspaces or can manage mulple phases of the chosen

number system. The non-separable Hilbert space uses a

modied version of Paul Dirac's bra-ket procedure that uses

integrals of funcons instead of sums of series. This changed

version supplies insight into the workings of uncertaines and

the expectaon value of a stochascally spread series of

numbers.

The extension to non-separable Hilbert spaces uses Dirac

distribuons rather than standard funcons.

Not all features of standard funcons hold for Dirac

distribuons which are generalized funcons. This disncon is

why non-separable Hilbert spaces do not behave like separable

Hilbert spaces. This disncon becomes actual in the system of

non-separable Hilbert spaces.

11.3 Other features of Hilbert spaces

Several unique features reveal by playing with subspaces of the

Hilbert space. First, subdividing into subspaces does not prohibit

the content of the subspace from funconally relang to the

content of other subspaces.

27

11.3.1 Subdividing into Hilbert spaces

Every closed subspace of a Hilbert space is a Hilbert space. The

set of closed subspaces of a Hilbert space is lace isomorphic

with quantum logic.

The version of the number system that denes the private

parameter space subdivides into other number systems with a

lower number of dimensions. For example, the quaternionic

number system holds a complex number system for every

direcon_line in the spaal part of a quaternionic number

system that crosses the number 0. The complex number system

contains a real number system. Thus, the quaternionic Hilbert

space holds complex-number-based Hilbert spaces as

subspaces. These complex-number-based Hilbert spaces have

real-number-based Hilbert spaces as a subspace. These Hilbert

spaces support their own funcon space.

11.3.2 Subdividing into parameter space and target space

When visualizing funcons, humans intuively put the

parameter and target spaces into separated independent space

parts. The HBM shares that habit.

The parameters relate to the target values. In non-separable

Hilbert spaces, funcons usually act in the third phase of the

number system. However, the model applies sampled funcons

in separable and non-separable Hilbert spaces.

The subdivisions require extra dimensions. The vector space

owns ample space to harbor these extra dimensions. We call

the subspace space that holds the target spaces of all funcons

28

the common target space. In a separable Hilbert space, an

orthonormal set of base vectors represenng a target value of

one or more funcons can span the common target space.

11.3.2.1 How and why the HBM creates time

The Hilbert Book Model applies the real part of the parameter

space to implement the indicator for the progression of change.

It uses the common target space to harbor a collecon of target

spaces of stac funcons that each belong to the values of the

corresponding progression indicator. We will call the value of

the progression indicator a mestamp. This replacement of the

real parts of the quaternions by a progression indicator

introduces the noon of me into the model. This subdivision

acts as the funconality of a book in which each page stands for

an instant of the history of the usual target space. Thus, me is

an arcial parameter. The hop landings never coincide.

Therefore, me can intercalate, and the model can sequence

the real parts of quaternions in the archived hopping paths.

The model applies this opportunity by exchanging the real parts

of the hop landings against the arcial progression steps that

the HBM introduces as instances of me.

Humans created dierenal calculus as part of mathemacs.

The creaon of the arcial me concept allows humans to

apply dierenal calculus.

29

11.3.2.2 Keeping the relation between parameter value and target

value

The original arrangement of locaons in the parameter space

can be demolished in the target space. This demolishment

would occur when oscillaons or rotaons are involved. The

demolishment endangers the relaon between parameter value

and target value. In the model, embedding other Hilbert spaces

or clusters of Hilbert spaces into the target space resolves this.

The embedding plots the image of the Hilbert space or the

cluster of Hilbert spaces into the target space. The embedded

Hilbert spaces or Hilbert space clusters will implement the

oscillaons and rotaons. Secon A system of Hilbert spaces

treats this. Embedding oang Hilbert spaces, or clusters of

Hilbert spaces, disrupts the relaon with the background

parameter space.

Consequently, these objects own a dierent me sequence than

the elementary oang Hilbert spaces. That me sequence

depends on the local gravitaonal potenal in the embedding

eld. See the presentaons of Carlo Rovelli about the noon of

me and gravitaonal me dilataon. The following secon

explains how the HBM introduces me.

11.3.2.3 The Hilbert Book model

11.3.2.4 Separating the target space into a mirror-symmetric and an

anti-mirror-symmetric part

Along direcon_lines on each page of the usual target space,

superposions of cosine funcons can stand for the mirror-

30

symmetric funcons. Likewise, the superposions of sine

funcons can stand for the an-mirror-symmetric funcons.

At the geometrical center of the parameter space, the cosine

funcons have a maximum. At the geometrical center of the

parameter space, the sine funcons switch from negave to

posive. The an-mirror-symmetric target spaces realize in a

separate subspace. In the formulas, the imaginary factor

i

shows this. In Hilbert space, this imaginary factor stands for a

split into another subspace.

A cosine funcon can combine with a sine funcon with the

same frequency into a complex number-valued exponenal

funcon. This combinaon is allowed because the imaginary

factor

i

belongs to the direcon of that same direcon_line. The

resulng complex exponenal funcon has the remarkable

property that it relates to the paral dierenal change

operator that belongs to the selected direcon_line. The secon

Fourier transform in the formula chapter presents the details.

The sine and cosine funcons use spaal frequencies as their

parameters. This applicaon introduces a frequency parameter

space parallel to the spaal posion parameter space. The

frequency parameter space covers three spaal dimensions in

the quaternionic Hilbert space. The frequency parameter space

serves spectral funcons that populate the common target

space. We also call this representaon the change space.

31

The HBM does not restrict frequencies to a single direcon_line.

It enables spaal frequencies up to three dimensions and

quaternionic frequencies that cover four dimensions.

11.3.2.5 Separating the target space into scalar function targets and

spatial function targets

The split into mirror symmetric target space and an-mirror

symmetric target space can occur separately for the scalar and

spaal funcon targets.

11.3.3 Adding change with time

If the change with me also includes the split into mirror-

symmetric and an-mirror-symmetric dependency, then the

frequency parameter space will cover four dimensions. Fourier

series show that the base vectors that span the locaon

parameter space are superposions of the base vectors of the

frequency parameter space with all coecients having the same

amplitude. This statement also holds vice-versa.

12 Potentials and forces

In physics, potenal energy is energy held by an object

because of its posion relave to other objects.

The potenal at a locaon is equal to the work (energy

transferred) per unit of actuator inuence that physics requires

to move an object to that locaon from a reference locaon

where the value of the potenal equals zero.

The Hilbert Book Model considers the potenal to be zero at

innity. Suppose the model selects innity as the reference

locaon. In that case, the potenal at a regarded locaon is

32

equal to the work (energy transferred) per unit of actuator

inuence that involves moving an object from innity to that

locaon. In that case, the potenal at a locaon stands for the

reverse acon of the combined actuator inuences that act at

that locaon.

12.1 Center of Influence of Actuators

The inuence of similar actuators can superimpose. Thus, a

geometrical center of these inuences denes the locaon of

the virtual locaon of a representant of the considered group of

actuators. In physical_reality, virtual locaons do not exist. It is

a theorecal concept.

This virtual representant has a potenal that has the same

potenal that a point-like actuator of the same inuence type

would possess. In the Hilbert Book Model, stac point-like

actuators other than charges do not exist because the

embedding eld tends to remove them as quickly as possible.

However, a model can dene stac virtual point-like actuators.

12.2 Forces

The rst-order change holds ve terms, two scalar terms, and

three spaal terms. In each of these subgroups, the terms can

compensate for each other. For example, in the group of spaal

terms, the gradient of the scalar part of the quaternionic eld

can compensate for the me variaon of the spaal component

of the quaternionic eld. If we neglect the curl of the part of the

quaternionic eld, then the gradient of a local potenal can

cause a me variaon of a spaal eld that describes the

33

movement of inuenced objects. If these are uniformly moving

massive objects, then these objects will accelerate. So, the

spaal eld will stand for a force eld.

12.3 Actuators

We list the actuators of spherical responses discussed in this

paper in the table below.

Actuator

Description

Influenced

objects

Symbol

Symbol

Actual electric

charge

Electric charges are the sources or sinks of

electrical fields and cause potentials in the

electrical field. The influenced objects are

other electric charges. In the HBM, these

charges exist at the geometrical centers of

floating Hilbert spaces.

Other

electric

charges

Q

q

Virtual electric

Charge

Virtual charges stand for a collection of

electric charges

Other

electric

charges

Q

q

Isotropic pulse

Isotropic pulses are embeddings of hop

landings of the state vector of floating

Hilbert spaces into the dynamic universe

field. These pulse responses are spherical in

the form of spherical shock fronts.

Other

massive

objects

M

m

Floating Hilbert space

Virtual mass represents a collection of

isotropic pulses that a floating Hilbert space

generates.

Other

massive

objects

M

m

Virtual mass

Virtual masses stand for a collection of

masses of floating Hilbert spaces.

Other

massive

objects

M

m

The Hilbert Book Model also explains the noons of aracng

and repelling by introducing progression as me.

34

Electric elds and gravitaonal elds dier fundamentally in

their start and boundary condions.

Electric charges can aract or repel each other.

Masses will always aract each other.

Spherical pulse responses in the form of spherical shock fronts

are dark maer objects. However, the qualicaon “dark” not

juses when vast numbers of these objects cooperate such

that they become perceivable.

13 Stochastic processes

Replacing the real parts of archived quaternions with progression

indicators introduces a stochastic process. The HBM suggests that this

stochastic process is a combination of a Poisson Process and a binomial

process. If we consider this process as a combination of a Poisson

process and a binomial process, and if a location density distribution

that owns a Fourier transform in the form of a frequency spectrum that

describes the effect of the binomial process, then the stochastic process

holds a characteristic function. In the HBM, the frequency spectrum

can cover up to four dimensions.

The characteristic function of a stochastic process in the change space

can control the spread of the location density distribution of the

produced location swarm in position space.

A dedicated footprint operator archives the production of the stochastic

process in its eigenspace. After reordering the timestamps, the

footprint operator stores its eigenvalues in the quaternionic storage

bins. The storage bin contains a real number valued timestamp and a

three-dimensional spatial number value for the archived hop landing

location. After sequencing the timestamps in equidistant steps, the hop

35

landing locations stand for a hopping path of a point-like object. The

hopping path regularly regenerates a coherent hop landing location

swarm. The location density distribution describes this swarm.

If this location density distribution is a Gaussian distribution, then its

Fourier transform decides exactly the location density distribution of

the swarm. The Fourier transform is again a Gaussian distribution but

has distinctive characteristics.

The author dares to suggest that the stochastic process combines a Poisson and binomial

process because he measured the spatial frequency characteristics of many imaging spots and

line images in images produced by lenses and image intensifier devices.

36

The optical transfer function is the Fourier transfer of the point

spread function (PSF), shown in the second picture (b,c).

The modulation transfer function (MTF) is the modulus of the optical

transfer function. Each cut through the center of the MTF is

symmetric. Therefore, it suffices to specify half of that curve.

Often, a peak appears at the center of the MTF. Optical experts call

the cause of the peak veiling glare. Picture (d) shows this peak.

Analyzing the Fourier transfer of the line spread function (LSF) is more

manageable because it covers more contributing imaging objects and

corresponds with a cut through the central axis of the MTF.

The central axis of the MTF shows the distribution of the imaging objects in the image. The HBM

states that photons are one-dimensional chains of shock fronts. Thus, if the imaging objects are

photons, then according to the HBM, the central axis of the MTF shows the distribution of the

energy that the shock fronts carry. In the peak, the shock fronts are less spatially related than in

the broader part of the MTF. In analyzing the image of a galaxy, the veiling glare might stand for

the halo that cosmologists see around these galaxies.

The notion of the MTF does not restrict to photons. The imaging objects can form a mixture of

photons, elementary fermions, and conglomerates of elementary fermions. In that case, the

MTF is a function of these contributors' angular, chromatic, and phase distribution. The author

participated in developing world standards for specifying and measuring the OTF and the MTF.

It started with a STANAG standard, the ISO and IEC standard, and included country-wide

standards such as the German DIN standard accepted these worldwide standards. At low dose

rates, the relative contribution of noise will increase. The Detective Quantum Efficiency (DQE)

objectively measures this influence. The author also participated in standardizing the DQE for

IEC and DIN.

The described stochastic process can deliver the actuators that

generate the pulse responses that may deform the dynamic universe

field. In some way, an ongoing embedding process must map the

eigenspace of the footprint operator onto the embedding field. As

previously argued, the footprint operator's eigenspace corresponds to a

dynamic footprint vector that defines a location density function and a

probability amplitude. The footprint vector exists in the underlying

vector space and has a representation in Hilbert space via the footprint

operator. The footprint vector acts as the state vector of the separable

37

Hilbert space, and the probability amplitude corresponds to what

physicists call the wave function of the represented moving particle.

13.1 Optical Transfer Function and Modulation Transfer function

Some stochastic processes own a characteristic function. This

characteristic function is the Fourier transform of a location density

distribution. Experimenters commonly use such stochastic processes to

qualify imaging excellence via the Optical Transfer Function of an

imaging process or imaging equipment. The Optical Transfer Function is

the Fourier transform of the Point Spread Function. For spatial

locations, the PSF acts as a location density distribution. The

Modulation Transfer Function is the modulus of the Optical Transfer

Function and is a symmetric function. The vertical axis of the MTF

shows the energy distribution of the spatial spectrum. In the case of

light, it is the chromatic distribution of the PSF. A central peak in the

form of a rapid decrease of the MTF at low spatial frequencies shows

the existence of a veiling glare or halo. It is energy that is less correlated

to location.

The Line Spread Function (LSF) equals the integral over the Point Spread

Function in the direction of the line. The Fourier transform of the Line

Spread Function equals the cut through the center of the Optical

Transfer Function. The cut runs perpendicular to the direction of the

line. The LSF can be a function of the direction of the line. In that case,

the PSF has a non-isotropic angular distribution. The Fourier transform

of the convolution of two functions equals the product of the Fourier

transforms of the functions. The result of the Fourier transform

conforms to the convolution of the OTF with the Fourier transform of

the blade sharp pulse that corresponds to the Fourier transform of the

integral along the line.

38

A phase distribution will also occur if an ongoing dynamic process

generates the PSF. The Optical Transfer Function combines the

Modulation Transfer Function and the Phase Transfer Function. In

complex number-based descriptions, the Phase Transfer Function is the

argument of the Optical Transfer Function.

A system of Hilbert spaces that share the same underlying vector space

can perform the job of the imaging platform. In this system, the

embedding process is the alternative name for the imaging process.

However, this explanation still says nothing about the essence of the

underlying stochastic selection process. That stays a mystery.

The concept of the Optical Transfer Function also makes sense for

dependence on time. For time dependence, the name of the tool is also

Fourier analysis. Together the two tools perform a four-dimensional

spectral analysis.

13.2 Photons

Photons are not electromagnetic waves. Instead, photons consist of

chains of equidistant one-dimensional shock fronts that travel along a

geodesic. The one-dimensional shock fronts are shock fronts that often

get the name dark energy objects. However, when cooperating in huge

quantities, the objects become observable, and then the name “dark

object” becomes confusing; see the section on differentiation.

13.3 Light

Light is a distribution of photons. A beam of light can have an angular

distribution, a chromatic distribution, and a phase distribution. A

homogeneous light beam holds a single frequency and usually a narrow

angular distribution.

13.4 Refraction

Refraction occurs at the borders of transparent media in which

information transfer occurs with constant speed. The information

39

transfer can take place through chains of absorption and reemission

cycles. In free space, nothing exists that absorbs or emits photons, but

photons can travel through free space along geodesics [10].

Refraction enables the construction of lenses, fiber plates, optical fibers,

prisms, and mirrors.

A separate part of optics covers refraction. [11]

13.5 Holographic imaging

Transparent optical lenses and tiny apertures can function as Fourier

transformers. They map distributions of photons in position space into

distributions in frequency space. The name of these distributions is a

hologram. [12]

Photographs can capture holograms. Also, metal mirrors imprinted with

phase patterns can generate holograms when the imprinted mirror

reflects a coherent beam of light.

13.6 Electron optics

Electron optics concerns imaging charged particles by artificially

constructed electric or magnetic fields or electromagnetic fields

[13][14]. Construction elements are metallic electrodes at a given

voltage or coils that carry electric currents.

Radio transmission is a special kind of electron optics.

14 Social influences

The rise of National Socialism in Hitler's Nazi Germany disrupted

the promising discussion about set theory and number systems.

Nazism threatened key discussion participants, or they had to

flee to safer places. Many fled to the United States of America,

where the government morally obliged them to cooperate in

the fight against Nazism by taking part in the development of

40

new weapon systems, such as the atomic bomb. Sets and

number systems no longer attracted their attention. The

success of the complex functional analysis, which can treat

singularities, worsened this effect. [15]

Joshua Willard Gibbs and Oliver Heaviside led the physicists

toward geometric differential theory and vector analysis. [16]

[17]

In this way, many scientists thought the spatial functions would

be sufficient to explain physical phenomena. However, this

choice is at the expense of the relationship with the real

functions, which quaternionic function theory regulates more

clearly. Many physicists no longer understood the reason

Hilbert spaces attracted their attention. The complex Hilbert

spaces became a toy of the mathematicians who developed all

kinds of fancy complex Hilbert spaces.

15 Ongoing investigation

At CERN in Geneva, sufficiently far from the Nazi sphere of

influence, a small group continued with quantum logic and

Hilbert spaces. The book "Foundations of quantum mechanics"

by Josef M. Jauch guided my attention to quaternionic Hilbert

spaces. [18]

Due to too few results, this research languished and died out in

the sixties.

41

16 New insight

Now we are taking a giant step. This step concerns a significant

difference in understanding between me and mainstream

theoretical physics. The curious shortlist of properties of the

electric charges of the first generation of elementary fermions

prompted this difference. This list covers charges with values -1,

-2/3, -1/3, 0, +1/3, +2/3, and +1. This list is part of the Standard

Model of the experimental particle physicists who have

summarized their main observations in that Standard Model.

[19]

Multiplying with 3 turns the list into a list of integers -3, -2, -1, 0,

+1, +2, and +3. This series is the list of differences between a

reference symmetry and other symmetries of versions of

quaternionic number systems when the coordinate axes restrict

to be parallel.

We limit our use of the Standard Model to a subset and exclude

the bosons and the gluons. We exclude theoretical theories like

Quantum Field Theory, Quantum Electro Dynamics, and

Quantum Chromo Dynamics. Opportunistic theoretical

42

physicists introduced QFT, QED, and QCD that spoiled the

experimental results with these not-so-well-founded theoretical

ideas by inserting them into the Standard Model. The minimal

action principle from which a Lagrangian derives forms the

foundation of these theories. These concepts play in the third

phase of number systems. The calculation rules and restrictions

of the third phase exist in the first and second phases.

Therefore, these theories cannot explain the existence of

electric charges and diverse types of fermions. Furthermore,

these theories have no reasonable explanation for the presence

of the wave function, and their rationale for the existence of

conglomerates is questionable.

The similarity with the symmetries of versions of number

systems stimulated me. However, it is not the similarity with

the symmetries themselves that provides the reason. Instead,

one of the Hilbert spaces plays the role of a background system.

All other system members float with their geometric center

over the parameter space of this background system. Especially

the difference between the symmetries of the versions of the

number systems that float with their separable Hilbert space

and the symmetry background platform control the situation.

This opportunity occurs in a system of separable Hilbert spaces

that all apply the same underlying vector space.

43

17 A System of Hilbert spaces

The author calls the system of Hilbert spaces the Hilbert

repository because it stores all data of a multiverse. Two types

of systems of Hilbert spaces exist.

The first type is a system of separable Hilbert spaces.

The second type is a system of non-separable Hilbert spaces.

Both systems hold a member that acts as a background

platform.

17.1 A System of separable Hilbert spaces

The background platform owns a companion non-separable

Hilbert space that embeds its separable companion. This

companion archives a dynamic universe field. The floating

separable members can harbor an electric charge at their

geometric center. A dark hole holds the countable parameter

space of the separable Hilbert space that functions as the

background platform. The HBM employs the name “dark hole”

because continuous objects cannot penetrate this countable

subset and cannot leave the encapsulated region. It is a second

phase contained in a third phase surround.

We limit ourselves to Hilbert spaces derived from the same

vector space. Furthermore, we choose four mutually

independent directions in the underlying vector space. The axes

of the Cartesian coordinate system of the number system shall

be parallel to one of the chosen direction_lines. This choice,

therefore, leaves only a few different symmetry types. The

exact reason, which enforces this restriction, is not apparent.

44

However, the limitation makes comparing symmetries and

computing symmetry differences easier. To understand the

consequences of the limitation, we put the symmetries of the

remaining versions of the quaternionic number system in a

table whose lines we arrange with binary written hexadecimal

rank numbers. We choose one of the sixteen remaining versions

as a frame of reference platform and place this version at the

front of the queue. The table mentions the fitting fermions by

name.

You will notice that the anti-attribute raises a conflict between

symmetries and the electric charges of the Standard Model. The

reason might be that the anti-attribute is not measurable.

No

R

G

B

real

Difference

charge

type

Rgb

0

0

0

0

0

0

0

background

1

1

0

0

0

1

-1/3

down

R

2

0

1

0

0

1

-1/3

down

G

3

1

1

0

0

2

-2/3

anti-up

B

4

0

0

1

0

1

-1/3

down

B

5

1

0

1

0

2

-2/3

anti-up

G

6

0

1

1

0

2

-2/3

anti-up

R

7

1

1

1

0

3

-3/3

electron

8

0

0

0

1

0

0

neutrino

9

1

0

0

1

-1

1/3

anti-down

B

A

0

1

0

1

-1

1/3

anti-down

G

B

1

1

0

1

-2

2/3

up

R

C

0

0

1

1

-1

1/3

anti-down

R

D

1

0

1

1

-2

2/3

up

G

E

0

1

1

1

-2

2/3

up

B

F

1

1

1

1

-3

3/3

positron

B

G

R

45

All these Hilbert spaces are separable and use number systems

that belong to the first or second phase.

The remaining system of Hilbert spaces holds a Hilbert space

that can serve as a background platform. Therefore, the HBM

assumes that the reference version functions as a background

platform.

The background platform must have an infinite number of

subspaces. An infinite number of subspaces means that the

version of the number system chosen by this Hilbert space has

an infinite number of elements.

17.2 A modeling platform

A system of Hilbert spaces that all share the same underlying

vector space can function as a modeling platform that not only

supports dynamic fields that obey quaternionic differential

equations. The model can, in principle, capture all phenomena

in a dynamic universe.

The system of separable Hilbert spaces applies the structured

storage ability of the Hilbert spaces that are members of the

system. The requirement that all member Hilbert spaces must

share the same underlying vector space restricts the types of

Hilbert spaces that can be a member of the system of separable

Hilbert spaces. In the change chapter, we already restricted the

definition of partial change along the directions of the Cartesian

coordinate system. It appears that the coordinate systems that

decide the symmetry type of the members of the system of

separable Hilbert spaces must have the Cartesian coordinate

46

axes in parallel. The Cartesian coordinate system is due to the

existence of the primitive coordinate system in the underlying

parameter space. The restriction enables the determination of

differences in symmetry. The Model selects the sequence along

the axis only up or down. It also means that partial change has a

systemwide significance. Thus, the model tolerates only a small

set of symmetry types. One of the Hilbert spaces will function as

the background platform, and its symmetry will serve as

background symmetry. Its natural parameter space will act as

the background parameter space of the system. All other

system members will float with the geometric center of their

parameter space over the background parameter space. These

features already generate a dynamic system. The symmetry

differences cause symmetry-related sources or sinks that will

exist at the geometric center of the natural parameter space of

the corresponding floating Hilbert space. The sources and sinks

correspond to symmetry-related charges that generate

symmetry-related fields. In physics, these symmetry-related

charges are electric charges.

Not the symmetries of the floating Hilbert spaces are essential.

Instead, the differences between the symmetry of the floating

member and the background symmetry are crucial for showing

the type of the member Hilbert space. The counts of the

differences in symmetry restrict to the shortlist -3, -2, -1, 0, +1,

+2, +3.

47

It is possible to understand the existence of symmetries and

symmetry differences. However, the presence of corresponding

symmetry-related charges is counterintuitive. The Model does

not yet explain the realization of these charges as sources or

sinks of symmetry-related fields.

All floating Hilbert spaces are separable. The background Hilbert

space is an infinite-dimensional separable Hilbert space. It owns

a non-separable companion Hilbert space that embeds its

separable partner.

The system of separable Hilbert spaces supports the containers

of footprints that can map into the quaternionic fields. The

vectors that stand for the footprint vectors originate in the

underlying spatial field. They function as state vectors for the

Hilbert spaces that serve as containers for the footprints. The

state vector stands for the vector from the underlying vector

space that aims at the geometric center of the floating Hilbert

space. This picture enables the maps of these state vectors and

the corresponding footprint in the dynamic universe field. The

state vector stands for a vector from the underlying vector

space that tries to find the position of the floating platform's

geometric center in the background platform's parameter

space. State vectors are particular footprint vectors. Together

this entwined locator installs an ongoing embedding process

that acts as an imaging process that maps the geometric center

of the floating platform onto the background parameter space.

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Finally, the eigenspace of a dedicated footprint operator maps

this image into the dynamic field that stands for the universe.

In this way, the image maps a vast number of ongoing hopping

paths onto the embedding field. Physicists call this dynamic

field the universe. On the floating platforms, the hopping paths

close. The movement of the floating platforms breaks the

closure of the images of the hopping paths.

17.2.1 Conglomerates

Elementary fermions behave as elementary modules. The

conglomerates of these elementary modules populate the

dynamic field that we call our universe. All massive objects,

except black holes, are conglomerates of elementary fermions.

Therefore, all conglomerates of elementary fermions own mass.

This mass ownership of modules means that massive modular

systems cover the universe.

A private stochastic process decides each elementary fermion's

complete local life story. The fermion controls that stochastic

process in the change space of its private Hilbert space. The

private stochastic process produces an ongoing hopping path.

This hopping path corresponds to a footprint vector that

consists of a dynamically changing superposition of the

reference operator's eigenvectors. The section of the formula

chapter that treats the arithmetic of change explains this. Each

floating platform of the system of separable Hilbert spaces

owns a single private footprint vector. The footprint vector acts

as the state vector of the elementary fermion, and the

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probability amplitude corresponds to what physicists call the

particle's wave function.

This picture invites the idea that stochastic processes whose

characteristic functions define in the change space of the