The foundations of mathematical physics and its relationship with physical reality

Abstract

This paper contains in an electronic pdf version the contents of a book that accompanies the multipart PowerPoint presentation of the Hilbert Book Model that appears as a playlist on YouTube.
See https://www.youtube.com/playlist?list=PLRn2RuujW3IJsZPxh7iNvlajFWY12G2Af
The paper and the book do not cover part 8 of the playlist. That part is very provocative and treats the consequences of the HBM for planet Earth.

Full text

1
Foundations
The foundations of mathematical physics and its relationship with physical reality
Abstract
The Hilbert Book Model project produced this publicaon. The search for a reliable foundaon 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 acon
principle as their base. The Hilbert Book Model shows that connuums belong to the third phase of a
special set and cannot work as a foundaon of mathemacal 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 mulverse can pose. Also, the document shows
that science must not consider the Higgs parcle or the Higgs eld as part of the Standard Model.
Instead, the Standard Model of experimental parcle physicists should restrict to elementary fermions.
Most physicists interpret photons as excitaons 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, represenng the local universe. Hop landings of the state vectors of the fermions produce spherical
shock fronts that move with light speed away from the locaon of this landing. This conicts with the
ideas of conservaon 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 creaon 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

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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

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20.4 Early encounters .................................................................... 126
21 References .................................................................................. 129

7
1 Justification
During my study, I used my scienc books to quietly read the formulas
and make notes between the text. In that way, I got a beer
understanding of the content of the book. This book reects most of the
content of a mulpart PowerPoint presentaon of the Hilbert Book
Model. PowerPoint presentaons are not suitable for adding notes.
hps://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 invesgate the foundaon of theorecal
physics and the relaon between mathemacal physics and physical
reality. As a long-rered 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 signicantly
reduced the remaining mysteries.
Building on the discussion results between Hilbert, von Neumann,
Cantor, and Zermelo that stopped in the fourth decade of the tweneth
century, I accidentally discovered a special set that features phases and
phase transions. Phases of the special set cannot pass the phase
transions step-by-step. This set becomes the foundaon of the Hilbert
Book Model (HBM).
My targets are students and young sciensts 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 oers the book as a print-on-
demand service and in massive quanes as oset. A freely accessible

8
paper will reect the content of the book and will contain the acve
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 essenal part of the foundaons of physical
reality. Some mysteries stay, but the model describes these
clearly. For me, these mysteries exist because my knowledge of
mathemacs does not allow me to explain the origin of these
mysteries. It is also possible that this mathemacs does not yet
exist. The mulpart PowerPoint presentaon oers suggesons
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 connuous
lm of the possible coverages of space with versions of
number systems belonging to the associave 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 observaon that humans cannot think and communicate
about things without providing these things with idencaon
in the form of a name or pointer and a short compact
descripon establishes a brief explanaon. Indirectly, the
Hilbert spaces provide the ideners and descripons humans
require. The curious thing is that physical reality can funcon

9
without these limitaons. Yet physical reality also appears to
adhere to strict rules and exisng structures. Many researchers
have come to know these rules and structures substanally and
formulate them in what they call mathemacs and physics.
Several researchers doubt whether people can discover the
calculaon rules that physical reality uses. However, the author
of this publicaon does not belong to that group.
My arrogance results from my convicon that those with
educaon at the level of a bachelor in the exact sciences of
mathemacs 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 oers the URLs acvely. The book displays the text
primarily in grayscale because colors increase the costs of
prinng 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 calculaon rules, the bra-ket
procedure of Paul Dirac, and essenal equaons. 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 publicaon “Setbacks of Theorecal Physics,” this trease
adds the formulas for lace theory and puts more emphasis on
the special set. This publicaon introduces the name
“special_set” for the menoned special set. This new name
produces fewer problems with spelling and grammar correcon
tools.
4 Clarification
When people focus their research on space, they quickly realize
that an empty space stands for the ulmate 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 locaons.
However, for humans, tracking the behavior of these locaons
without giving them idencaon and a precise descripon is
impossible.
Locaons are point-shaped objects that can occupy a posion
in space. That posion diers per locaon. Applying number
systems provides the required idencaon. The values of the
number system elements show the locaons' posions.
Without the locaons, the container is empty. What results is a
simple space that can funcon as a container. It is possible to
interpret this simple space as the ulmate nothingness. As a
container in which locaons reside, the simple space funcons

11
as a vector space. Two locaons and their connecng direcon
line form a vector.
The vector space owns a simple arithmec. That arithmec
enables the speciaon of the more complicated arithmec 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-dened
interrelaonship.
5 Vector space
There is sll no possibility to point to the posion. The pointer
can consist of a base locaon and a poinng locaon
connected by a direcon line. Sciensts 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 adjecve “physical” to indicate
that this noon 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 noon. This publicaon will apply
“physical_reality” to ease the spelling and grammar-checking
tools.

12
The HBM will use the name “direcon_line” for the noon of a
direcon line. This renaming happens for the same reason that
the HBM uses the name special_set. Direcon_lines obey
simple arithmec.
The direcon_line and the length fully characterize the vector.
The integrity of the vector does not change when it shis
parallel. The parallel shi can occur on the direcon_line but
may also occur in another direcon. Direcon_lines can
therefore move parallel in the vector space. They have no
beginning and no end. This situaon at once provides the
operaon with which two vectors can add. If the base point
shis 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 direcon_lines dier, the sum vector uses a new
direcon_line.
The two possibilies form a parallelogram in which the sum
vectors are parallel and have equal lengths.
By mulplying the vector by a scalar, the length mulplies by
that scalar. This acon creates a new vector. When the scalar is
negave, the base and pointer point change funcon and the
vector gets the opposite direcon. At the same me, its length
may change. These simple calculaon rules allow vectors to
pinpoint all locaons in the vector space. The secon Vector
arithmec in the chapter Formulas contains the formulas.

13
5.1 Independent directions
Vector arithmec enables a scalar product of two vectors. The
scalar product can show the independence of the
direcon_lines of vectors. The scalar product of independent
vectors is equal to zero. This way, several mutually independent
basic direcon_lines exist in the vector space. Since
direcon_lines can shi in parallel, a raster of direcon_lines
can cover the vector space. The raster can form a primive
coordinate system.
6 Number systems
The HBM applies the arithmec of its vector space to derive the more
complicated arithmec of the number systems that it uses.
6.1 Real numbers
With their calculaon rules, vectors can help to construct
number systems. For example, an ongoing addion of a starng
vector and vectors equal to the starng vector and located on
the same direcon_line yields an ordered series of designated
locaons collecvely represenng the natural numbers. Using
the natural numbers as a label, we can count collecons of
locaons. The subtracon procedure appears by removing
locaons from the collecon and introduces the countdown
procedure. Finally, we meet the number zero on the base point
of the original starng vector and subsequently follow the
negave integers. The method for mulplying numbers appears
by adding groups of vectors frequently. That does not supply
new integers. The name of the reversal of mulplicaon is

14
division and delivers fracons. Fracons can be new numbers.
The integer numbers, together with the fracons, form the
raonal numbers.
6.2 Phase transitions
Scholars have shown that there are as many raonal numbers as
natural ones. David Hilbert and his followers knew this. This size
equality means all raonal numbers can label with a natural
number. However, this procedure only works if both number
sets hold innite elements. The transion from nitely many
elements to innitely many elements implies a change in state
for the special_set. In the new phase, the collecon shows
dierent behavior. For this parcular set, achieving this phase
transion step-by-step is impossible. Also, the way back does
not go in a step-by-step manner. Sciensts do not oen use the
terms phase and phase transion 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 innity introduces another behavior of the
considered set. He did not use “phase transion” to classify the
set's behavior change. He and his followers did not consider the
dierent behavior as a dierent phase of the set. The HBM
assumes the dierent behavior as a disnct phase and the
behavior change as a phase transion. Accepng the
parcularies of the special_set has this consequence.

15
Adding or removing elements does not change the state of the
innite special_set. The innite set of well-ordered raonal
numbers lls a large part of the same direcon_line. A raonal
number can arbitrarily close approach any locaon on this line.
Nevertheless, there are sll many locaons on this line that
raonal numbers cannot appoint. We call the numbers that
these places show irraonal numbers. Irraonal numbers
include transcendental numbers, and raonal numbers include
prime numbers. The third phase of the set consists of both. The
third phase is innite and not countable.
Thus, the set of raonal and irraonal numbers again form a set
that can show as another phase of the special_set. The phase
transion happens again in one go and cannot occur step-by-
step. Counng 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
transion adds several new calculaon rules that manage the
change of cohesive parts of the collecon. We obtained a
special_set that features parcular behavior by adding the
irraonal numbers. Mathemacians call the extra calculaon
rules dierenal calculus. The author applies this name for the
addional arithmec rules of the third phase of the special_set.
Dierenal calculus is closely related to the calculaon rules of
raonal numbers. The calculaon rules can even mix. Without
disturbing actuators, nothing will change in the new phase of
the special_set. If something disrupts, this collecon phase

16
tends to remove the disturbance as quickly as possible by
sending away the consequences of the disrupon in all
direcons unl the eects eventually disappear into innity. We
know this because dierenal calculus shows this. As
menoned, the disturbance never reaches disappearance step-
by-step. The result is that the number-covered area expands.
The dierenal calculus tells precisely how that happens. On
the so-far-considered direcon_line, the response acts in a
single dimension.
When mulplied by themselves, the raonal numbers treated
so far yield a posive number on the direcon_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
direcon_line and, therefore, for all phases of the numbers on
this direcon_line. Squaring is the name for mulplying by
oneself. The secon Arithmec of the real numbers holds the
formulas.
The phase transions cause the underlying set to be parcular.
In this way, this set diers from standard sets. The set exists
because simple space holds it and only consists of point-like
locaons. 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
negave number that shares the direcon_line of the real
numbers when mulplied by themselves. We call these spaal

17
numbers. Oen the name used for these numbers is imaginary
numbers. The HBM does not use that name because the
qualicaon imaginary also has dierent meanings. The spaal
numbers no longer t on the direcon_line of the real numbers.
They occupy one or three dimensions. Suppose spaal numbers
fall outside the rst spaal dimension. In that case, the
calculaon rules of the spaal numbers ensure that a third
spaal dimension covers with spaal numbers in addion to the
second spaal dimension. The result of the product of two
spaal numbers consists of an internal product that supplies a
real number and an external product that is zero or produces a
result in a direcon that is independent of the direcon_lines
of both factors. The internal product is the reason for the
negave square. Therefore, the spaal numbers' calculaon
rules dier from the calculaon rules of the real numbers. The
reacon to a disturbance of the third phase of spaal numbers
is more spectacular in the three-dimensional spaal number
system than in the one-dimensional spaal number system. The
secon Arithmec of spaal numbers holds the formulas.
6.4 Division rings
Nevertheless, real numbers can add with spaal numbers, and
spaal numbers can mulply with real numbers. This addion
creates new number systems. The real and one-dimensional
spaal numbers form the two-dimensional set of what the
model calls complex numbers. The HBM shares this name with
common mathemacs. The real and three-dimensional spaal
numbers form the four-dimensional set of what the model calls

18
quaternions. Again, the HBM shares this name with common
mathemacs. This name sharing shows that the HBM applies
exisng names for its number systems where no conicts arise
and sucient similarity exists. The Mixed Arithmec secon 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 posive, 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 idencal eect of
the inner product of spaal numbers has led to confusion
among many mathemacians and physicists, so these sciensts
somemes confuse spaal 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 aer the sixes of the last
century. As we will see, this had signicant consequences for
mathemacs and physics. [2]
7 History
Before Christ, Egypans discovered simple fracons. Cantor
found the second and third phases of real numbers around
1870. Cantor did not use the designaons phase and phase
transion. Instead, he and others turned their aenon to
various kinds of innies of sets. Cantor called them transnite
numbers. Together with natural numbers, they form the
cardinal numbers. The Hilbert Book Model deals with only two

19
forms of innity. These are the countable innity of the second
phase of numbers and the uncountable innity 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 spaal base numbers. The external product appears in the
outcome of the product of the rst two spaal 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
inscripon. Hamilton's students immortalized the formula on
the bridge through a bronze commemorave plaque. [3]
8 Set theory.
8.1 Collections in space
Around the turn of the nineteenth to the tweneth century, a
group of mathemacians and mathemacal physicists led by
David Hilbert had an intense discussion about set theory. [4] [5]
David Hilbert intended to establish an axiomac theory of both
mathemacs and physics. Unfortunately, he rered before he
could nish that target.
The discussion focused on the various forms of innity and
countability. The discussion partners also paid signicant
aenon to the phases and phase transions of the collecon.

20
For example, they paid aenon to the connuum hypothesis.
[6] However, they never used the words phases and phase
transions. The HBM applies these names for the special_set to
disnguish this set from other sets.
The menoned discussion ignored the container of the set and
paid no aenon to the type of objects that formed the set.
These choices are signicant in physical_reality and the Hilbert
Book Model. By choosing space as a container and locaons as
elements of the set, the number systems the HBM uses to
discover the locaons obtain added properes that human
researchers and physical_reality must consider. These added
properes are the symmetries that stand for the freedom of
choice that the calculaon rules of the number systems do not
dene. As a result, in the HBM, the number systems exist in
several versions that their symmetry disnguishes. For example,
the locaon 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 direcon_lines
in one or the opposite direcon. Physical_reality must adhere to
the calculaon rules and will use as many symmetry choices as
possible. A dierent choice of symmetry yields a dierent
version of the number system. The word symmetry has various
meanings. These disnct meanings also occur in this
publicaon. In the HBM, geometric symmetries play a
prominent role. Dierences between geometric symmetries are
essenal.

21
9 Coordinates
Three associave 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 dier in their symmetry. Recording the symmetry
is possible with coordinate markers. These markers use the
locaon that shows the value of the number. In the HBM, a
Cartesian coordinate system records all the selecon freedoms
of a version of a number system. The record removes the
selecon 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 limitaons imposed by the vector space create geometric
symmetry. Therefore, if a model designs number systems
without these limitaons, then that model does not meet
geometric symmetries.
9.1 Hops and symmetries.
A hop can split in paral 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 selecons concern the
up or down direcon along the coordinate line. These selecons

22
correspond to the symmetries that we discussed before. The
paral jumps lead to the Frenet–Serret formulas. These
formulas form the base of dierenal geometry.
Willard Gibbs promoted dierenal 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 sciensts rejected quaternionic eld theory. The
mainstream physicists spent lile aenon to the symmetries of
versions of number systems. Instead, symmetry groups and Lie
groups draw their aenon. Universies wanted to coordinate
their lectures on theorecal physics and wanted to avoid
confusion. That is why most universies 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 theorecal physics. This history explains
why theorecal physics appears to have entered a dead end.
Invesgate:
hps://www.researchgate.net/publicaon/363541991_The_set
backs_of_theorecal_physics
Sll, Gibbs and Heaviside smulated the development of
muldimensional dierenaon technology and indirectly
promoted mathemacal quaternionic dierenal analysis
development. The introducon of me as a progression
indicator produced the quaternionic dierenal 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 dierenal calculus and set
theory.
10 Mainstream Science
10.1 Warning
Mainstream science sll plays a crucial role in promong a
standard reference for teaching and comparing science. This
role limits confusion for students and scienc instuons.
However, being promoted by mainstream science is not
synonym with granng the truth.
This warning especially holds for mathemacs, theorecal
physics, and mathemacal 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.
Aer the archival in an abstract structure, the stored
quaternions retrieve in an orderly manner. A dedicated
operator manages the archival and the retrieval.
Sciensts oen 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 dicult 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
converng 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, sciensts showed lile interest in
quaternionic Hilbert spaces. However, a small eort can adapt
the procedure to apply for quaternions. Hilbert spaces can thus
work with any of the associave division rings.
The HBM restricts the archival to the second phase of the
special_set. This choice limits the dened archival capability to
the separable Hilbert spaces.
Each separable Hilbert space chooses a private version of one
of these number systems. As menoned, the separable Hilbert
space can archive collecons of elements of this version and
retrieve them in an orderly manner. This capability also applies
to the enre chosen version of this number system. There is a
devoted operator who manages this collecon. 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. Secon
Dirac’s bra-ket procedure treats the formulas.
11.1 Function space
The private parameter space turns every Hilbert space into a
funcon space. Through the funcons, Dirac's bra-ket
procedure denes new operators who manage the target space
of the sampled funcon as eigenspace.
11.2 Quantum logic
To the surprise of many mathemacians, the set of the closed
subspaces of Hilbert space appears to have a lace structure
that is slightly dierent from the lace structure of classical
logic. Some sciensts suggested that this deviaon could be the
cause of the quantum structure of the energy exchange seen in
small parcles and atoms. Therefore, they assigned the name
quantum logic to this new lace. [9] A closed subspace of a
Hilbert space is again a Hilbert space. Dierenal calculus oers
a more obvious explanaon. Dierenal calculus only comes
into eect in the third phase of number systems. Funcon
theory and dierenal calculus describe the third phase of
number systems. The Arithmec of changes secon describes
the formulas that govern the third phase of number systems.
The formula chapter treats lace theory in a separate secon.

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 mulple phases of the chosen
number system. The non-separable Hilbert space uses a
modied version of Paul Dirac's bra-ket procedure that uses
integrals of funcons instead of sums of series. This changed
version supplies insight into the workings of uncertaines and
the expectaon value of a stochascally spread series of
numbers.
The extension to non-separable Hilbert spaces uses Dirac
distribuons rather than standard funcons.
Not all features of standard funcons hold for Dirac
distribuons which are generalized funcons. This disncon is
why non-separable Hilbert spaces do not behave like separable
Hilbert spaces. This disncon 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 funconally relang 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 lace isomorphic
with quantum logic.
The version of the number system that denes 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
direcon_line in the spaal 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 funcon space.
11.3.2 Subdividing into parameter space and target space
When visualizing funcons, humans intuively 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, funcons usually act in the third phase of the
number system. However, the model applies sampled funcons
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 funcons

28
the common target space. In a separable Hilbert space, an
orthonormal set of base vectors represenng a target value of
one or more funcons 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 collecon of target
spaces of stac funcons 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 noon of me into the model. This subdivision
acts as the funconality of a book in which each page stands for
an instant of the history of the usual target space. Thus, me is
an arcial 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 arcial progression steps that
the HBM introduces as instances of me.
Humans created dierenal calculus as part of mathemacs.
The creaon of the arcial me concept allows humans to
apply dierenal calculus.

29
11.3.2.2 Keeping the relation between parameter value and target
value
The original arrangement of locaons in the parameter space
can be demolished in the target space. This demolishment
would occur when oscillaons or rotaons are involved. The
demolishment endangers the relaon 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
oscillaons and rotaons. Secon A system of Hilbert spaces
treats this. Embedding oang Hilbert spaces, or clusters of
Hilbert spaces, disrupts the relaon with the background
parameter space.
Consequently, these objects own a dierent me sequence than
the elementary oang Hilbert spaces. That me sequence
depends on the local gravitaonal potenal in the embedding
eld. See the presentaons of Carlo Rovelli about the noon of
me and gravitaonal me dilataon. The following secon
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 direcon_lines on each page of the usual target space,
superposions of cosine funcons can stand for the mirror-

30
symmetric funcons. Likewise, the superposions of sine
funcons can stand for the an-mirror-symmetric funcons.
At the geometrical center of the parameter space, the cosine
funcons have a maximum. At the geometrical center of the
parameter space, the sine funcons switch from negave to
posive. 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 funcon can combine with a sine funcon with the
same frequency into a complex number-valued exponenal
funcon. This combinaon is allowed because the imaginary
factor
i
belongs to the direcon of that same direcon_line. The
resulng complex exponenal funcon has the remarkable
property that it relates to the paral dierenal change
operator that belongs to the selected direcon_line. The secon
Fourier transform in the formula chapter presents the details.
The sine and cosine funcons use spaal frequencies as their
parameters. This applicaon introduces a frequency parameter
space parallel to the spaal posion parameter space. The
frequency parameter space covers three spaal dimensions in
the quaternionic Hilbert space. The frequency parameter space
serves spectral funcons that populate the common target
space. We also call this representaon the change space.

31
The HBM does not restrict frequencies to a single direcon_line.
It enables spaal 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
spaal funcon 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 locaon
parameter space are superposions of the base vectors of the
frequency parameter space with all coecients having the same
amplitude. This statement also holds vice-versa.
12 Potentials and forces
In physics, potenal energy is energy held by an object
because of its posion relave to other objects.
The potenal at a locaon is equal to the work (energy
transferred) per unit of actuator inuence that physics requires
to move an object to that locaon from a reference locaon
where the value of the potenal equals zero.
The Hilbert Book Model considers the potenal to be zero at
innity. Suppose the model selects innity as the reference
locaon. In that case, the potenal at a regarded locaon is

32
equal to the work (energy transferred) per unit of actuator
inuence that involves moving an object from innity to that
locaon. In that case, the potenal at a locaon stands for the
reverse acon of the combined actuator inuences that act at
that locaon.
12.1 Center of Influence of Actuators
The inuence of similar actuators can superimpose. Thus, a
geometrical center of these inuences denes the locaon of
the virtual locaon of a representant of the considered group of
actuators. In physical_reality, virtual locaons do not exist. It is
a theorecal concept.
This virtual representant has a potenal that has the same
potenal that a point-like actuator of the same inuence type
would possess. In the Hilbert Book Model, stac 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 dene stac virtual point-like actuators.
12.2 Forces
The rst-order change holds ve terms, two scalar terms, and
three spaal terms. In each of these subgroups, the terms can
compensate for each other. For example, in the group of spaal
terms, the gradient of the scalar part of the quaternionic eld
can compensate for the me variaon of the spaal component
of the quaternionic eld. If we neglect the curl of the part of the
quaternionic eld, then the gradient of a local potenal can
cause a me variaon of a spaal eld that describes the

33
movement of inuenced objects. If these are uniformly moving
massive objects, then these objects will accelerate. So, the
spaal 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 noons of aracng
and repelling by introducing progression as me.

34
Electric elds and gravitaonal elds dier fundamentally in
their start and boundary condions.
Electric charges can aract or repel each other.
Masses will always aract each other.
Spherical pulse responses in the form of spherical shock fronts
are dark maer objects. However, the qualicaon “dark” not
juses 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.

48
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

49
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