Hilbert Book Model Project

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HansVanLeunen respectfully asks that people use the discussion page, their talk page or email them, rather than contribute to this page at this time. This page might be under construction, controversial, used currently as part of a brick and mortar class by a teacher, or document an ongoing research project. Please RESPECT their wishes by not editing this page.

This project is still in preparation phase. Translation is partly finished.

Quick overview

My mottoː Think simple. If you think, then think twice

Slide show This slideshow highlights some aspects of the model

Introducing the Hilbert Book Model Project

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Hans van Leunen is the initiator of this project. ORCID

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 that concerns this Wikiversity project, 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. The ResearchGate site supports a flexible way for discussing scientific subjects.

The initiator has generated some documents that contain highlights as excerpts of the project and he stored these papers on his personal e-print archive http://vixra.org/author/j_a_j_van_leunen .

The website contains most documents both in pdf as well as in docx format. None of these documents claims copyright. Everybody is free to use the content of these papers.

The initiator was born in the Netherlands in 1941. He will not live forever. This project will contain his scientific inheritance.


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

It especially serves the needs of independent or retired scientific authors.

Introducing new science always introduces controversial and unorthodox text. The Hilbert Book Model Project is an ongoing enterprise. Its content is dynamic and is revised regularly.

The content of this project is not peer reviewed. The reader is responsible for checking the validity of what he/she reads. The peer review process cannot cope with the dynamics of revisions and extensions.

Reviewers are always biased and they are never omniscient. The peer review process is expensive and often poses barriers to renewal of science.

In comparison to openly accessible publication on internet the peer review process is a rather slow process. In addition it inhibits the usage of revision services, such as offered by vixra.org and by arxiv.org/

It is the task of the author to ensure the correctness of what he writes.

One way to check the validity of the text is to bring parts of the text to open scientific discussion sites such as ResearchGate..

The initiator challenges everybody to disprove the statements made in this report. He promises a fine bottle of XO cognac to anyone that finds a significant flaw in the presented theories.

This challenge stands already for several years. Up to so far nobody claimed the bottle.


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If you want to become a collaborator, then please contact me.

This list is still empty.

Hilbert Book Model

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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 distinguishes from other models by its emphasis to the control by stochastic mechanisms in contrast to the investigation of what forces bind the particles.

The model emerges from its foundations. In principle a model of all aspects of physical reality can evolve from these foundations. In that case the model would be a theory of everything. This is not the purpose of the Hilbert Book Model. Only the first steps of the evolution will be investigated. Higher levels of the model will become too complicated and therefore incomprehensible for human beings. These principles and restrictions lead to the following implementation of the model.


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This is a sketchy description. The full description is supplied in the chapters

The model bases on a simple foundation. In 1936 two scientists discovered the structure of this foundation. Mathematicians call the discovered base structure an orthomodular lattice. The structure of the base model of the Hilbert Book Model emerges from this foundation. The base model is a combination of a quaternionic infinite dimensional separable Hilbert space and its unique non-separable companion Hilbert space that embeds its separable partner. A background parameter space is formed by the eigenspace of a normal operator that is spanned by the members of a selected version of the quaternionic number system that specifies the values of the inner products. A subspace scans this base model as a function of a progression value that equals the real part of these eigenvalues and is spanned by the corresponding eigenvectors. The subspace represents the current static status quo and divides the historic part from the future part of the model. Mechanisms that exist external to this base model provide the geometric data that define the dynamic behavior of the model. This base mode gets its first extension by a series of separable Hilbert spaces that float on top of the background separable Hilbert space and apply the same vector space as the background platform does. Each of these separable Hilbert spaces manages their own version of the quaternionic number system in the eigenspace of their private reference operator. This eigenspace acts as a private parameter space which floats on top of the background parameter space. On each floating platform resides an elementary particle that acts as an elementary module.

The Hilbert Book Model impersonates a creator that at the instant of the creation stores all dynamic geometric data of its creatures in a read-only repository. The repository divides in a part that stores all discrete data and a part that stores continuums. This second part embeds the first part. That occurs such that it is possible to interpret the embedding as an ongoing process.

The model reveals that two categories of super-tiny objects exist that together constitute all other discrete objects that exist in the model. These super-tiny objects are shock fronts and in separation they stay inobservable. These tiny shock fronts require a point-like trigger.

All observable objects in the model are modules or modular systems. A set of point-like elementary modules exists who's members configure all other modules. Each elementary module resides on a private separable Hilbert space that applies a private version of the quaternionic number system. A reference operator manages in its eigenspace a private parameter space that is formed by the selected version of the quaternionic number system. All separable Hilbert spaces share the underlying vector space with the background separable Hilbert space. The private separable Hilbert spaces of the elementary particles act as platforms, whose private parameter spaces float over the background parameter space. The elementary particles inherit their symmetry properties from the platform on which they reside.

This structure of the base model 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. More details in Hilbert Book Model .

Relation to conventional physics

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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.This results in the fact that conventional theories offer descriptions that suit applied physics. The Hilbert Book Model offers explanations of subjects that conventional physics cannot offer.

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.

Project highlights

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Several highlights of the project are published on the open archive http://vixra.org/author/j_a_j_van_leunen

Discussions are initiated byː

Tracing the structure of physical reality by starting from its fundamentals; http://dx.doi.org/10.13140/RG.2.2.16452.07047

TheStructureOfPhysicalReality; http://dx.doi.org/10.13140/RG.2.2.10664.26885

BehaviorOfBasicFields; http://dx.doi.org/10.13140/RG.2.2.15517.20960

64 Shades of Space;: http://dx.doi.org/10.13140/RG.2.2.28012.46724

Mass: http://dx.doi.org/10.13140/RG.2.2.10268.59528

Coherence; http://dx.doi.org/10.13140/RG.2.2.36417.45925

Pure Energy; http://dx.doi.org/10.13140/RG.2.2.20498.91841

Generating the Universe from Scratch; http://dx.doi.org/10.13140/RG.2.2.12110.31043


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

The HBM does not fully explain the origin of the stochastic mechanisms. The HBM only applies these mechanisms and relates the mechanisms to the ongoing embedding process.

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

The HBM does not explain color confinement. Instead it suggests a reason for its existence.

The HBM does not explain generations of elementary modules.

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

Crucial differences

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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 and as carriers of pure energy.

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 subspace of the background Hilbert space.

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

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

The most crucial introduction concerns the stochastic processes that own a characteristic function.

Astonishing conclusions

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This project is based on the non-straightforward assumption that mathematics restricts the extension of levels of the structure of physical reality into more complicated levels of that structure. This assumption appears to be valid in the levels that the project traversed so far. With other words, the selected foundation behaves like a seed from which only one type of plant can grow.

A very astonishing conclusion is that physical reality possesses a read-only repository that stores all dynamic geometric data of all discrete objects that exist in the universe. This fact, enables the interpretation of the model as the personification of the creator of the model.

After the instant of the creation the creator left his creatures alone.

The creator appears to be a modular designer. Elementary modules constitute all other modules. All massive objects in the universe are modules. Some modules constitute modular systems.

A very astonishing conclusion is that a set of stochastic processes that generate the locations of super-tiny spherical shock fronts completely control the universe.

The fact that each of the spherical shock fronts extends the continuum that embeds them and therefore deforms this continuum at the landing locations explains the extension of the universe and it explains the origin of mass.

This leads to the very astonishing conclusion that all massive objects in the universe are recurrently regenerated. This conclusion could be interpreted as if reality generates mass from nothing, but the driving mechanism behind this is the ongoing embedding of all separable Hilbert spaces into a non-separable Hilbert space.

One-dimensional shock fronts transfer a standard bit of pure energy. Photons are strings of equidistant one-dimensional shock fronts that at the instant of emission obey the Einstein-Planck relation E = h v. Consequently, at the same instant, all photons share the same emission duration.

Astonishing is that the diversity of versions of the quaternionic number system correspond to the basic properties of elementary particles.


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Introducing the Hilbert Book Model

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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 (Clicking this reference brings you to the Hilbert Book Model page.)

Relational structures

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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 set of closed subspaces of the separable Hilbert space is lattice isomorphic to the orthomodular lattice. The underlying vector space contains a subspace that contains all modules, which exist in the universe. This subspace is spanned by rays that represent elementary modules. The elementary modules represent the atoms of a modular configuration lattice.

Relational Structures

Modules and modular systems

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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


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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.


Quaternionic Hilbert Space

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

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

Hilbert spaces exist as separable Hilbert spaces and as 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 background of the base model of the Hilbert Book Model.

Quaternionic Hilbert Spaces

The behavior of continuums

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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.

The chapter explains the relation between forces and fields.

Finally, the document explains the Lorentz transform.

Quaternionic Field Equations

Nabla operators

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

Nabla Operators


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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.


Quaternionic Fourier Transform

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

Quaternionic Fourier Transform

Stochastic Location Generators

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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

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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

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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


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The universe can be divided into compartments.



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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.


Information Messengers

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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

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

Multi-mix Path Algorithm

Dirac equation

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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


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Gravitation is a non-flat condition of the embedding continuum.


In the beginning

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In the beginning the embedding field was flat. It coincided with its parameter space. The stochastic mechanisms did not yet generate a single hop landing location. So, elementary particles did not yet exist. Only after a full generation cycle the first elementary particles existed. During this first phase our living space quickly inflated. After completion of the swarms the living space still expanded by the newly generated clamps, but that expansion occurred at a smaller pace.


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

Emergence & restriction

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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 automatically restrict to division rings. The versions of the selected number system restrict to versions that feature parallel Cartesian coordinates.

Base model

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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. On top of this background float a series of separable Hilbert spaces that use the same vector space. The embedding of the floating separable Hilbert spaces 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.


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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.


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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

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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

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


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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.


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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.

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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

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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.


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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

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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 pays attention to shock fronts that are triggered by point-like actuators and 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 and they permanently expand the volume of this field. Consequently they possess the capability of 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.

Super-tiny dark objects

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The warps and the clamps represent two categories of super-tiny objects from which all discrete objects in universe are constituted. They are shock fronts and cannot be perceived in separation. They are the dark objects that populate universe.

Spectral binding

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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

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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

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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.


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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

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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.