# Hilbert Book Model Project

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## Introducing the Hilbert Book Model Project

### Initiator

Hans van Leunen is the initiator of this project.

The Hilbert Book Model Project is an ongoing project.

All its pages and sections may be revised.

Everybody is kindly invited to help to improve and to extend this project.

If you want to criticize or have other remarks, then you are kindly requested to use the Discuss tabs that appear at the top of the pages.

You can also react on my talk page.

The initiator maintains a ResearchGate project that considers the Hilbert Book Model Project.

In the opinion of the initiator a Wikiversity project is a perfect way for introducing new science.

#### Collaborators

This list is still empty.

### Hilbert Book Model

The Hilbert Book Model Project governs the development of the Hilbert Book Model and its application.

The Hilbert Book Model is a purely mathematical model of the foundations and the lower levels of the structure of physical reality.

The model emerges from its foundations. This principle leads to the following implementation of the model.

#### Implementation

The model bases on a simple foundation. In 1936 scientists discovered the structure of this foundation. The structure of the model emerges from this foundation. Mechanisms that exist external to this structure provide the geometric data that define the dynamic behavior of the model.

The model impersonates a creator that at the instant of the creation stores all dynamic geometric data of its creatures in a read-only repository.

All observable objects in the model are modules or modular systems. A set of pointlike elementary modules exists who's members configure all other modules.

This makes the creator a modular designer and a modular constructor. At every instant, the elementary object obtains a new spatial location that the repository stores together with the corresponding timestamp. A private stochastic mechanism generates the new location. All modules act as observers and can figure as actors in an observed event. Observers can only perceive information that comes from storage locations that possess a historic timestamp. That information is transferred from the storage location to the observer by vibrations and deformations of a continuum that embeds both the storage locations of the observed event and the current storage locations of the observer. The information transfer affects the format of the information that the observer perceives. The observer perceives in spacetime format. The Lorentz transform describes the format conversion from the Euclidean storage format to the spacetime format of the perceived information. The Lorentz transform gives the correct conversion when the continuum that transfers the information is flat. However, if massive objects deform this continuum, then the path along which the information transfers gets bent. This will affect the perceived information.

#### Relation to conventional physics

The Hilbert Book Model differs in many aspects from conventional physical theories. The reason bases on the fact that the Hilbert Book Model starts at its foundations and develops by extending these foundations, while most physical theories confine to concepts that can be verified by direct observations or via experiments.

Only a tiny part of the Hilbert Book Model is accessible to observers and that includes observations that apply the most sophisticated instruments.

This situation makes the Hilbert Book Model an unconventional and unorthodox approach that offers an alternative to conventional physical theories where verification cannot apply direct or equipment aided observation.

#### Restrictions

The HBM restricts to the lowest levels of the structure of its target, which is physical reality.

The HBM does not explain the origin of the stochastic mechanisms. The HBM only applies these mechanisms.

The HBM does not explain the existence of bosons, other than warps, photons and non-elementary modules.

The HBM does not explain color confinement.

The HBM does not explain generations of elementary modules.

The HBM does not explain the diversity of masses of elementary module types.

#### Crucial differences

The HBM introduces a category of super-tiny objects that cannot be observed separately. This category contains shock fronts. The HBM calls them warps and clamps.

The HBM sees clamps as the objects that provide elementary modules with their mass.

The HBM sees strings of equidistant warps as the information messengers.

The HBM considers all observable massive objects as modules or as modular systems.

The HBM introduces the zigzag of elementary modules.

The HBM introduces the creator's view as alternative to the observer's view.

The HBM interprets the Lorentz transform in a special way

The HBM interprets its base model as a read-only repository.

The HBM introduces the scanning Hilbert space subspace.

The HBM introduces the embedding of the separable Hilbert space into its non-separable companion as an ongoing process.

The HBM introduces two quaternionic second order partial differential equations that describe the embedding process and the information transfer.

## Chapters

### Introducing the Hilbert Book Model

This entry describes the discovery of the foundation of the model and explains how the purely mathematical model can derive from this foundation.

The model extends into a powerful platform that acts as a read-only repository. This base model merges function theory and differential and integral calculus with Hilbert space operator technology. In this way, the model introduces some new mathematics.

The entry introduces modular design and construction of modules whose footprint is generated by stochastic processes.

The model accepts a storage view and an observer's view. These views can mix.

Hilbert Book Model

### Relational structures

The most important foundation of the Hilbert Book Model is a relational structure that mathematicians call an orthomodular lattice. This lattice extends into a separable Hilbert space

The orthomodular lattice incorporates a modular configuration lattice. A subspace of the separable Hilbert space represents this sublattice.

Relational Structures

### Modules and modular systems

The creator appears a modular designer and constructor. Modular system generation can occur in stochastic way and as soon as intelligent species arrive, then locally, intelligent modular design may replace part of the stochastic modular design. The creator teaches these designers some important lessons.

Modules and Modular Systems

### Quaternions

Due to the fact that the base model of the Hilbert Book Model applies quaternionic Hilbert spaces, will quaternions play a major role in the project.

Quaternions

### Quaternionic Hilbert Space

Quaternionic Hilbert spaces constitute the base model of the Hilbert Book Model.

Hilbert spaces can only cope with number spaces that are division rings. The HBM selects the most versatile division ring.

Hilbert spaces exist as separable Hilbert spaces and non-separable Hilbert spaces.

Every infinite dimensional separable Hilbert space owns a unique companion non-separable Hilbert space that embeds its separable companion.

The two selected companions constitute the base model of the Hilbert Book Model.

Quaternionic Hilbert Spaces

### The behavior of continuums

This section describes the behavior of continuums by applying the first and second order partial differential equations of the quaternionic functions that define these fields.

The document interprets the solutions of homogeneous second order partial differential equations.

The relations between volume integrals, surface integrals, loop integrals and corresponding differential equations relate balance equations to continuity equations.

Finally, the document explains the Lorentz transform.

Quaternionic Field Equations

#### Nabla operators

The quaternionic nabla and the spatial nabla play an essential role in the behavior of fields.

Nabla Operators

#### Solutions

Waves, warps, clamps and plops are solutions of the quaternionic second order partial differential equations. These solutions play an essential rule in the Hilbert Book Model.

Solutions

#### Quaternionic Fourier Transform

Fourier transforms play a significant role in the assurance of dynamical coherence and in the binding of modules.

Quaternionic Fourier Transform

### Stochastic Location Generators

Each module owns a private mechanism that at every instant generates the locations that constitute their footprint. The mechanisms apply statistic processes that own a characteristic function.

In this way the mechanisms ensure dynamical coherence.

Stochastic Location Generators

### Perceptibility and Recognition at Low Dose Rate

Measuring the perceptibility of images that generated at a low dose rate show the nature of the mechanisms that produce the objects, which constitute the image.

Perceptibility and Recognition at Low Dose Rate

### The Extended Stokes Theorem

The extended Stokes theorem extends the generalized Stokes theorem, which combines the relations between volume integrals and surface integrals.

The applied integration appears to be sensitive to the ordering symmetries of the applied parameter spaces. This effect is the source of the symmetry-related charges of the platforms on which elementary modules reside. The symmetry-related charges generate the symmetry-related field. The interaction between the symmetry-related charges and the symmetry-related field controls part of the dynamics of the model.

Extended Stokes Theorem

### Compartments

The universe can be divided into compartments.

Compartments

### Zigzag

In the creator's view, the elementary modules can zigzag in the direction of progression.

Observers perceive the reflection instants as annihilation events of a particle in combination with a creation event of the corresponding anti-particle. Both events go together with the emission or absorption of two information messengers that operate in opposite directions.

Zigzag

### Information Messengers

The Hilbert Book Model supports several types of strings of warps that act as information messengers. Each type features its own emission duration and corresponds to an elementary module type.

Information Messengers

### Multi-mix Path Algorithm

This algorithm is HBM's alternative to the well-known Path Integral.

Multi-mix Path Algorithm

### Dirac equation

Dirac investigated a way to interpret the Klein-Gordon equation in a special way. That action resulted in the Dirac equation.

This equation introduced antiparticles.

Dirac Equation

### Discoveries

The Hilbert Book Model Project uncovered and discovered several important facts.

#### Emergence & restriction

The HBMP uncovered that higher level structures automatically emerge from the foundation, which is formed by an orthomodular lattice.

The number systems that the model supports restrict to division rings.

#### Base model

The base model of the Hilbert Book Model consists of a quaternionic infinite dimensional separable Hilbert space and its unique non-separable Hilbert space that embeds its separable partner. The embedding occurs in a subspace that scans over the whole base model as a function of the progression value. The scanning subspace divides the base model between a historical part, a static status quo (the scanning subspace), and a future part.

#### Impersonation

The Hilbert Book Model impersonates a creator that at the instant of the creation of the model stores all dynamic geometric data of his creatures in eigenspaces of operators that reside in the separable Hilbert space.

#### Components

All massive discrete objects in the model are modules. A set of point-like elementary modules exist that together configure all other modules. All modules can act as observers and can figure in observed events.

#### Two views

The Hilbert Book Model offers two views. One view is the observer's view and offers access to all stored data. It is also called the storage view.

The second view is the observer's view. Observers travel with the scanning subspace and can only retrieve information that is stored with a time stamp that for them lays in history. They receive the retrieved information via vibrations and deformations of the continuum that embeds both the observed event and the observer. This information transfer affects the format of the perceived information. First of all a coordinate transform implements the required time dilation and length contraction. This conversion of the Euclidean storage format to the perceived data format is described by the Lorentz transformation, which is a hyperbolic coordinate transformation.

Further, the information transfer is affected by the deformation of the embedding continuum. The path along which the information travels is a curved geodesic, rather than a straight line.

#### Reverse bra-ket method

The reverse bra-ket method merges quaternionic Hilbert space operator technology with quaternionic function theory and indirectly with quaternionic differential and integral calculus.

#### Embedding

The base model of the Hilbert Book Model shows that an ongoing process embeds a discrete universe into a continuum universe.

The discrete universe consists of point-like artifacts and enclosed discrepant regions.

The embedding process deforms the embedding continuum.

#### Plaforms

The discrete universe supports a number of platforms that float over the background platform. Each of these platforms is covered by a private parameter space.

These parameter spaces correspond to versions of the quaternionic number system. The combination of a floating platform and the background platform defines a symmetry flavor that corresponds to symmetry-related charges of the floating platform. Everything that resides on the floating platform inherits its symmetry-related properties.

A quaternionic function defines the embedding continuum and applies the background platform as its parameter space.

#### Symmetry-related fields

The symmetry-related charges of the floating platforms act as sources for corresponding symmetry related fields. The charges locate at the geometric center of the platform.

#### Stochastic mechanisms

Each platform owns a private mechanism that applies a stochastic process, which generates hop landing locations that the process takes from the platform and, which the process embeds into the embedding continuum.

The embedding continuum responses with spherical shock fronts that integrate into its Green's function. Each hop landing results in a temporary deformation that quickly fades away. The stochastic process owns a characteristic function that acts as a displacement generator for the produced coherent hop landing location swarm. Consequently the swarm moves coherently as a single unit. The displacement generator is the Fourier transform of the location density distribution of the swarm. The swarm represents an elementary module. The squared modulus of the wave function of the elementary module equals the location density distribution.

#### Gravity

The convolution of the Green's function of the embedding field and the location density distribution of the swarm equals the deformation of the embedding field that is due to the swarm. This deformation defines the raw gravitation potential of the corresponding elementary module.

#### Quantum gravity

Since more than two and a half century the solutions of a homogeneous second order partial differential equation are known. Well-known solutions are waves. That is why the equation is known as the wave equation. Shock fronts are less well-known solutions. They did not even get a special name. The HBM calls the one-dimensional shock fronts warps and calls the three-dimensional shock fronts clamps. During travel warps keep their amplitude. Warps carry a standard bit of energy. Clamps diminish their amplitude as 1/r with distance r to the trigger location. Clamps quickly fade away, but in the mean time they integrate into the Green's function of the embedding field. They temporarily deform the embedding field. Consequently they carry a standard bit of mass. Both objects quantize the embedding continuum They form the most basic gravitational quanta. Superpositions of waves can also be quantized. This is shown by the Helmholtz equation.

#### Spectral binding

In modules the characteristic function of the module installs the spectral binding of the components of the module. The fact that this characteristic function equals the superposition of the characteristic functions of its components causes that also the module moves as a single coherent object. This way of looking at the binding is revolutionary. In the HBM, it replaces hard and weak forces. Gravitation and attractive symmetry related charges may add to the effect of spectral binding.

#### Quaternionic differential calculus

The HBM applies quaternionic partial differential equations. They describe the behavior of quaternions and quaternionic continuums that are stored in the base model.

The first order partial differential equation splits in five terms that can get different names and symbols.

The homogeneous second order partial differential equation exist in two different forms. One is the quaternionic equivalent of the wave equation. It applies the quaternionic d'Alembert's operator.

As inhomogeneous equation splits the second partial differential equation into two first order partial differential equations. It does not offer waves as part of the solutions of the homogeneous equation. However, it offers warps that show polarization.

Both homogeneous equations offer warps and clamps as solutions.

#### Quaternionic integral calculus

Quaternionic integrals reveal the influence of symmetry flavors. Especially the extended Stokes theorem puts this dependence to the front. It is THE reason that symmetry related charges exist.

#### Zigzag

A mixture of the creator's view and the observer's view reveals that what observers perceive as pair production and pair annihilation in the observer's view will be interpreted as the zigzag reflection od a single particle in the creator's view.

#### Multi-mix path algorithm

Based on the fact that during the swarm regeneration cycle the displacement generator can be considered constant, the multi-mix path algorithm couples the stochastic hopping path to the Lagrangian and the Hamilton equations. In this way the algorithm walks the reverse route of Feynman's famous path integral.