# Hilbert Book Model Project/Hilbert Book Model

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# Introducing the Hilbert Book Model Project

### Intention

The Hilbert Book Model is a purely mathematical model that is intended to describe the foundation and the lower levels of the structure of physical reality.

The idea behind this project is that physical reality possesses structure and that this structure must be based on one or more foundations.

Foundations must be inherently simple. Therefore, quite probably these foundations were discovered long ago by some intelligent humans.

They did not discover them as foundations of the structure of physical reality, but instead, they found an interesting structure that they probably added to their mathematical toolkit.

These foundations are particular because they must automatically extend into more complicated levels of the structure of physical reality.

For example sets are very simple structures but they lack the ability to automatically extend into a higher level structure. The trick is to rediscover a simple structure that like a seed automatically extends into a chain of increasingly complicated levels of a structure. The seed extends into a particular plant. In analogy, the sought base structure must extend in a comprehensible and clearly restricted way into higher levels of the evolving structure.

### Limitations

The foundation is defined by a set of axioms. Because the consistency can only be verified mathematically, a purely mathematical model such as the Hilbert Book Model cannot claim to provide a correct description of the foundation and of the lower levels of the structure of physical reality. Observers cannot perceive the structure and the behavior of the most primitive objects that operate at these levels. Thus, it is impossible to verify this part of the model via observations, and that includes measurements that apply the most sophisticated equipment. Only at higher levels the model introduces observers that can perceive events. Devotees of the so-called scientific method must correct their attitude before they can accept the proposed model.

### Approach

Our task is to rediscover the foundation structures in the mathematical library of structures.

The HBM project supposes that about eighty years ago the duo Garrett Birkhoff and John von Neumann discovered a suitable foundation in the form of a relational structure that they called "quantum logic."

The duo introduced their discovery in a paper in which they proved that the set of closed subspaces of a separable Hilbert space has exactly the relational structure of their discovery[1].

The duo called the relational structure "quantum logic" because its lattice structure resembles closely the lattice structure of "classical logic."

Mathematicians gave the discovered lattice a more neutral name and called it "orthomodular lattice." That is a better choice because nothing indicates that the discovered lattice is a logic system of logical propositions like the lattice that describes classical logic. Instead, the elements of the orthomodular lattice are represented by closed subspaces of the separable Hilbert space. The complete set of atoms of the lattice are represented by mutually orthogonal rays that together span the complete Hilbert space.

#### Extension of the foundation

The orthomodular lattice is an atomic lattice. A complete set of atoms generates the full lattice. Each atom in the set corresponds with a ray of the separable Hilbert space. A ray is a one-dimensional subspace. The vectors that span the rays in the representing set form an orthogonal base of the Hilbert space.

The extension to the separable Hilbert space introduces number systems into the model. The separable Hilbert space applies number systems that are division rings for specifying the values of the inner products of pairs of Hilbert space vectors[2][3][4]. The Hilbert space acts as a structured repository for the members of the number system via the notion of operators that map the Hilbert space onto itself. The eigenspaces of these operators act as storage locations for eigenvalues that are members of the number system.

All non-zero members of a division ring own a unique inverse. Only three suitable division rings exist. These are the real numbers, the complex numbers, and the quaternions. The eigenspaces of the operators in the separable Hilbert space must be countable. Thus, the separable Hilbert space can only store the rational members of these number systems. The Hilbert Book Model selects the quaternionic number system because it represents the most versatile division ring and quaternions ideally suit for the storage of dynamic geometric data in the form of the combination of a time-stamp and a three-dimensional spatial location.

A little-known fact is that quaternionic number systems exist in many versions that differ in their sequencing symmetry. A Cartesian coordinate system succeeded by a polar coordinate system can define the sequence and thus the symmetry of a version of the quaternionic number system. The version of the number system that serves for specifying the inner product plays a particular role. In the form of the eigenspace of a particular reference operator, it acts as the private parameter space of the separable Hilbert space. One of the separable Hilbert spaces acts as a background platform. Consequently a series of separable Hilbert spaces can float with the geometric center of their private parameter space over the background parameter space, which is the private parameter space of the background platform.The real value of the hovered location in the background parameter space acts as the progression value. The selected versions of the quaternionic number systems must align their Cartesian coordinate systems in parallel. This reduces the number of tolerated types of the floating versions of the number system to a small subset. The extra restrictions are far from straightforward. Not the Hilbert space, but instead the embedding mechanisms enforce this restriction.

In combination with a set of quaternionic functions, a reference operator can give rise to the specification of a category of defined operators that share the eigenvectors of the reference operator, but that apply the function values that correspond to the parameter values as their eigenvalues. This method merges Hilbert space operator technology with quaternionic function theory.

#### Continuums

Every infinite dimensional separable Hilbert space owns a unique companion non-separable Hilbert space that features operators, which own continuum eigenspaces. In this companion Hilbert space, a similar trick can be performed with reference operators and defined operators. This time all members of the number system can be applied. This procedure embeds the separable Hilbert space into its non-separable companion.

The procedure merges quaternionic function theory and quaternionic differential and integral calculus with the operator technology of the combined repository[5][6]. This procedure results in a very powerful modeling platform.

For the combination of two Hilbert spaces, multiple interpretations are possible.The non-separable Hilbert space can embed its separable companion, or it is possible to consider that the non-separable Hilbert space contains the separable Hilbert space. By describing the embedding as an ongoing process, the Hilbert Book Model exploits the first view.

#### Dynamic repository

The operator that defines the background parameter space can split into a Hermitian operator and an anti-Hermitian operator. The eigenvectors of the operator that belong to the same eigenvalue of the Hermitian operator span a subspace of the Hilbert space. Varying the selected eigenvalue let the subspace scan over the combined repository. It is possible to interpret this dynamic model such that the embedding process occurs within the scanning subspace. In this way, the embedding becomes an ongoing process. The selected eigenvalue takes the role of progression.

Apart from the floating parameter spaces and the ongoing embedding process, this base model does not show any interesting dynamics.

External mechanisms must extend the model into a fully functional model . This requires the introduction of elementary modules.

### Modular model

Looking around learns that all observable massive objects in the universe are modules or modular systems. Sets of elementary modules exist that configure all other modules. Elementary modules do not configure from lower level modules. In the base model , these elementary modules are represented by rays. This makes them point-like objects. Without additional features, this does not explain the diversity of types of elementary particles that physical reality shows. What might help, is the attachment of each elementary module to a private floating parameter space. At every instant, the point-like object may take a different location in this parameter space. However, in no mechanism exists that creates these locations. The modular model postulates that a private mechanism, which applies a stochastic process, generates the locations. Consequently, the elementary module hops around in a stochastic hopping path. The hop landing locations constitute a hop landing location swarm. Something must ensure that the mechanism generates a coherent swarm. If the stochastic process owns a characteristic function, then the Fourier transform of this characteristic function equals the location density distribution of the hop landing location swarm. The characteristic function consists of two factors. The first factor ensures the coherence of the swarm. The second factor is a gauge that implements a displacement generator. This second factor controls the displacement of the floating platform. Consequently, at first approximation, the swarm moves as one unit. This fact makes the swarm a coherently moving object.

#### Embedding the floating Hilbert spaces

The locations of the elementary modules are taken from their private separate Hilbert space. Not the geographical center is taken, but instead, the stochastic process takes a location from the surround of that geometric center such that the center of the hop landing location swarm stays near the geometric center of the parameter space that characterizes the floating Hilbert space. Thus, the hop landing location swarm floats over the background parameter space of the base model. Its also means that within the floating parameter space the hopping path of the elementary module is closed. The displacement generator displaces the geometric center of the floating platform of the swarm. With other words, without the displacement generator the characteristic function controls the stochastic process such that it produces a closed hopping path and a coherent hop location swarm.

#### Version alignment

The versions of the quaternionic number system that the model supports must align their Cartesian coordinate axes. In this way these versions only differ in the way that the coordinates are ordered along the main axes. This restriction significantly reduces the number of types of platforms that can float on top of the background parameter space. Nothing in the Hilbert space exists that requests the alignment restriction. The alignment restriction is exploited by the embedding mechanisms. This suggests that instead of the Hilbert space, the embedding mechanism imposes the alignment restriction. The embedding depends on the symmetry differences between the floating platforms and the background platform. For that reason the embedding process must be able to determine that difference. The Hilbert Book Model supposes that the mechanisms apply the extended Stokes theorem to determine the difference. The extended Stokes theorem requests alignment of the coordinate axes of the applied versions of the number system. The extended Stokes theorem can then use a cube-like encapsulation of the floating platform, such that all discrepant embedding locations fall inside the cube.

Everything that resides on the platform at that progression instant must fit in the cube. The encapsulation isolates a subspace of the vector space that acts as a separate (encapsulated) Hilbert space, which has its own inner product. The reference operator that supports the floating parameter space resides in this encapsulated Hilbert space. The operator that stores the hop landing location that is generated by the private stochastic process also resides on this encapsulated Hilbert space. The hop landing location marks the eigenvector and the ray where the encapsulated Hilbert space and the encapsulating Hilbert space join. A second location marks the geometric center of the floating parameter space. This corresponds to a another ray. Thus, the subtraction concerns a single potentially discrepant embedding location.

One of the integrals uses the internal side of the surface of the cube and the other integral uses the outside of that surface. The results of the two integrals are subtracted. Thus, the opposite surfaces of the cube deliver the same contribution. The contribution per surface side can be zero or one unit and it can be positive or negative.This offers a short list of -6, -4, -2, 0, +2, +4, and +6 as solutions of the subtraction. Dividing by 6 offers the short list of symmetry related charges. that can be found in the Standard Modelː ${\displaystyle -1,-{\frac {2}{3}},-{\frac {1}{3}},0,{\frac {1}{3}},{\frac {2}{3}},1}$. The non-integer members of this list correspond to three color charges.

The hopping location corresponds to the location where the embedding field gets triggered. The geometric center corresponds to the center of mass of the hop landing location swarm.

If monitored over the full regeneration cycle of the hop landing location swarm, then the complete swarm is contained in the enveloping cube. If monitored over many regeneration cycles the hop landing locations form a graffiti-like trace and the encapsulating cube draws a tube that may zigzag in the progression direction.

#### Symmetry-related charges and fields

This section applies the names electric charge and color charge because these concepts conform with the physical notions of electric charge and color charge. In the base model, every square root of a negative real number has a spatial direction. The selection of a quaternionic base model excludes the possibility to apply gauges in the same sense as is done in a complex number based theory. The HBM is not a gauge theory in the conventional meaning of that name. However, the stochastic mechanisms apply characteristic functions that act as displacement generators.

The difference in sequencing symmetry between a floating parameter space and the sequencing symmetry of the background parameter space defines the symmetry flavor of the platform on which the floating parameter space resides. The symmetry flavor determines its symmetry-related charge. The charge locates at the geometric center of the platform and it interacts with a symmetry-related field.

The possible Cartesian symmetry related properties are listed in the table. The first column lists the sequence order, which can be upwards or downwards. The second column lists the binary representation of the sequence orderings. The left position represents the real parts of the quaternions. The next positions indicate the three Blue-Green-Red dimensions. Thus, the three colors Red, Green and Blue relate to the three mutually perpendicular directions in which the sequencing of Cartesian coordinates can differ. This is indicated in the binary number 0000 of the first row. If one of the spatial bits is set then the corresponding character R, G, or B is set in the color column. If two bits are set, then the corresponding anti-color character R, G, or B appears in the color column. The number of different spatial bits determines the size of the electric charges. Setting the left bit, switches the sign of electric charges and switches colors into anti-colors and vice versa.

The table shows that for the up quarks, the standard model differs from this simple attribution scheme. According to the table below, SM up quarks are anti-particles. The table shows that the three down quarks do not share the same handedness and the same holds for the three up quarks. The list of electric charges -3, -2, -1, 0, +1, +2, +3 in the charge column corresponds well with the list of electric charges ${\displaystyle -1,-{\frac {2}{3}},-{\frac {1}{3}},0,{\frac {1}{3}},{\frac {2}{3}},1}$ in the Standard Model.

Charge size 1/3 belongs to the down quarks. Charge size 2/3 belongs to the up quarks. Charge size 1 belongs to the electron and the positron. The lower half of the table represents the antiparticles.

The symmetry related charge combines electric charge, color charge and spin. Electric charge and color charge relate to Cartesian ordering and spin relates to polar ordering. Color charge relates to the dimension in which anisotropy occurs.

Quarks are anisotropic. The other particles are isotropic.

The electric charge follows from the number 0000 of the dimensions in which the sequencing symmetries differ from the symmetry of the sequencing symmetry of the background parameter space. Antiparticles show opposite charges.

The electric charges can attract or repel other electric charges. For that reason, they also take part in the binding of modules.

Symmetry Flavor

sequence binary R/L color charge SM type

${\displaystyle {\color {hotpink}\blacktriangle \blacktriangle \blacktriangle \blacktriangle }}$

0000 R N 0 neutrino

${\displaystyle {\color {hotpink}\blacktriangle \blacktriangle \blacktriangle }{\color {cyan}\blacktriangledown }}$

0001 L R -1 down quark

${\displaystyle {\color {hotpink}\blacktriangle \blacktriangle {\color {cyan}\blacktriangledown }\blacktriangle }}$

0010 R G -1 down quark

${\displaystyle {\color {hotpink}\blacktriangle \blacktriangle }{\color {cyan}\blacktriangledown \blacktriangledown }}$

0011 L B -2 up quark

${\displaystyle {\color {hotpink}\blacktriangle {\color {cyan}\blacktriangledown }\blacktriangle \blacktriangle }}$

0100 R B -1 down quark

${\displaystyle {\color {hotpink}{\blacktriangle \color {cyan}\blacktriangledown }\blacktriangle {\color {cyan}\blacktriangledown }}}$

0101 L G -2 up quark

${\displaystyle {\color {hotpink}\blacktriangle {\color {cyan}\blacktriangledown }{\color {cyan}\blacktriangledown }\blacktriangle }}$

0110 R R -2 up quark

${\displaystyle {\color {hotpink}{\blacktriangle \color {cyan}\blacktriangledown }{\color {cyan}\blacktriangledown }{\color {cyan}\blacktriangledown }}}$

0111 L N -3 electron

${\displaystyle {\color {cyan}\blacktriangledown }{\color {hotpink}\blacktriangle \blacktriangle \blacktriangle }}$

1000 R N 0 A_neutrino

${\displaystyle {\color {hotpink}{\color {cyan}\blacktriangledown }\blacktriangle \blacktriangle {\color {cyan}\blacktriangledown }}}$

1001 L R 1 A_down_quark

${\displaystyle {\color {cyan}\blacktriangledown }{\color {hotpink}\blacktriangle {\color {cyan}\blacktriangledown }{\color {hotpink}\blacktriangle }}}$

1010 R G 1 A_down_ark

${\displaystyle {\color {cyan}\blacktriangledown }{\color {hotpink}\blacktriangle }{\color {cyan}\blacktriangledown }{\color {cyan}\blacktriangledown }}$

1011 L B 2 A_ up quark

${\displaystyle {\color {cyan}\blacktriangledown }{\color {cyan}\blacktriangledown }{\color {hotpink}\blacktriangle \blacktriangle }}$

1100 R B 1 A_down_quark

${\displaystyle {\color {cyan}\blacktriangledown }{\color {hotpink}{\color {cyan}\blacktriangledown }\blacktriangle {\color {cyan}\blacktriangledown }}}$

1101 L G 2 A_ up quark

${\displaystyle {\color {cyan}\blacktriangledown }{\color {cyan}\blacktriangledown }{\color {cyan}\blacktriangledown }{\color {hotpink}\blacktriangle }}$

1110 R R 2 A_ up quark

${\displaystyle {\color {hotpink}{\color {cyan}\blacktriangledown }{\color {cyan}\blacktriangledown }{\color {cyan}\blacktriangledown }{\color {cyan}\blacktriangledown }}}$

1111 L N 3 positron

The Hilbert Book Model suggests that the named particles reside on the platforms of which the ordering symmetry corresponds to the first two columns. These particles inherit the symmetry-related charges and the color charges from the platforms on which they reside. The charges locate on the geometric centers of the platforms.

These charges form sources for corresponding symmetry related fields.

Polar ordering brings extra properties. Polar ordering starts with a Cartesian coordinate system. It may start with rotating over the 2π radians of the azimuth, or polar ordering may start with rotating over the π radians of the polar angle. The rotation may occur up or down. This may explain the existence of integer and half-integer spin of elementary particles.

##### Mixed domain functions

The existence of platforms that float on top of the background parameter space and feature a private parameter space that owns a private sequencing symmetry leads to the notion of functions that define on a mixture of domains that hover on top of a background domain. Closed boundaries enclose the floating domains. Integration of the mixed domain functions must apply the extended Stokes theorem. A mixed domain function defines the embedding continuum. Convolution of Green's function of the embedding continuum with the location density distribution of a module applies the extended Stokes theorem.[7] The convolution with the Green's function of the embedding continuum explains the gravitation of the modules.

#### Binding

The elementary modules generate higher level modules. A stochastic process that owns a characteristic function controls the hop landing locations that corresponds to the module. This characteristic function equals a dynamic superposition of the characteristic functions of the constituting elementary particles. Consequently the combined swarm also owns a displacement generator, and at first approximation, it moves as a single unit. This restriction is not the only effect that binds elementary modules. Also the symmetry properties of the platforms on which the elementary modules reside, appear to play a role in the total binding process.

### Interpreting the modular model

All modules act as observers. Their behavior is subject of the observation by other modules. Vibrations and deformations of the fields that embed the modules transfer observable information. The information carriers are solutions of the homogeneous second order partial differential equations that describe the behavior of the embedding fields.

Observers cannot perceive information that comes from storage locations whose time stamp lays in the future. The transport of the information through the information carrier affects the format of the information. As long as the instant of storage precedes the value of the stored time stamp, the progression value of that instant does not matter. Thus, it is safe to assume that at the instant of the creation of the model, all information was stored in a read-only repository. The modular model can impersonate a creator that stored all relevant dynamic geometric data of his creatures at the instant of the creation. He applied stochastic processes that did all their work at that same instant. After the instant of creation of the model this creator left his creatures alone.

It is also possible to assume that the mechanisms produce their output at the at the instant, which conforms to the stored time-stamp. The HBM takes the first possibility.

#### Discrepant regions

The model assumes that embedding of discrepant point-like artifacts trigger a clamp. However, the table shows that neutrinos have no discrepant properties. Probably another property that is not shown in the table exists that allows the neutron to interact as a point-like artifact with the embedding field. For the neutrino, the number of interactions during a full regeneration cycle is very low. This results in a very low mass. Similarly the difference in mass between quarks and between quarks and leptons does not become clear from the table. Also the existence of three generations of fermions appears not to have a relation with the table.

## Model views

The module enables two different views.

##### Storage view

The first view is the creator's view. The storage view represents an equivalent name for this view. The Hilbert Book Model impersonates a creator. The creator is indicated by the initials HBM, which stand for Hilbert Book Model. At the instant of the creation of the model the HBM stores all essential information of his creatures in a read-only repository. The repository is represented by base model . It includes the results of the activity of the activity of the stochastic mechanisms that operate in the full model . In this view modules implement observers, and these observers perceive information about modules that the HBM stored with a lower valued time-stamp. The information is transferred from the storage location to the observer via a continuum that embeds both the observed event and the observer. The information is stored in a Euclidean format as combinations of time-stamps and spatial locations. Quaternions act as storage containers. The information transfer affects the format of the information. The observers perceive in spacetime format. If the continuum that transfers the information does not deform by the embedding of massive objects, then the Lorentz transform describes this format conversion. Otherwise, the path along which the information travels gets bent and influences the perceived information.

##### Observer's view

The second view limits the model to what observers can perceive. This is the view in which physicists can do their experiments and where, in principle, statements about physical reality can be verified by experiments.

Devotees of the scientific method restrict their scope to the observer's view.

##### Mixed view

Mixing both views makes sense. For example, quaternionic differential calculus makes sense in the storage view where all dynamic geometric data are available in quaternionic format and continuums can be represented by quaternionic functions. If an observer's view is wanted, then the Lorentz transform can convert the data to the perceivable spacetime format. A deformed living space can still further affect the perceived information. The mechanisms that generate the locations of the elementary modules ensure the coherent dynamic behavior of these objects. The characteristic function of the applied stochastic processes cause that the dynamics of the stored data can be described by quaternionic differential calculus and more in particular with first and second order partial differential equations.

In the storage view, elementary modules can zigzag in the direction of progression. At the reflection instants, the observers will perceive annihilation of particles and creation of corresponding antiparticles. This fact puts the creation and annihilation processes in a different light!

#### Interpreting the creator

The Hilbert Book Model impersonates a creator. At the instant of the creation, the creator stores all dynamic data of all creatures in the read-only repository. After the instant of creation this creator leaves its creatures alone. A God cares about its creatures. The HBM does not impersonate a caring individual. The creatures view their existence via the observer's view. These creatures get the impression that they have a free will. The creator's view classifies their existence as predetermined.

## Restricted platform

The Hilbert Book Model represents a very restricted scenery. Everything that is observable must be stored in the read-only repository that is represented by the base model . A window that represents the current static status quo, scans this model. The scan has a beginning and an end and the progression steps are countable. Number systems must be division rings and versions of each number system must be aligned along Cartesian coordinate axes. The applied stochastic processes must be a combination of a Poisson process and one or more binomial processes. Spatial point spread distributions must implement the binomial processes and the stochastic processes must own a characteristic function. The characteristic function equals the Fourier transform of the spatial point function.

In the model , the non-separable Hilbert space embeds its separable companion and all floating platforms. The embedding occurs inside the scanning window. All observers travel with this window and can only retrieve information that is stored with a historic time stamp. The information is carried from the stored event to the observer by a continuum that embeds both the observed event and the observer.

## References

1. G. Birkhoff and J. von Neumann, The Logic of Quantum Mechanics, Annals of Mathematics, Vol. 37, pp. 823–843
2. Quaternionic Hilbert Spaces
3. “Division algebras and quantum theory” by John Baez. http://arxiv.org/abs/1101.5690
4. Warren D. Smith, Quaternions, octonions, and now, 16-ons and 2n-ons; http://scorevoting.net/WarrenSmithPages/homepage/nce2.pdf
5. Quaternionic Field Equations
6. Quaternionic Hilber spaces
7. Extended Stokes Theorem