Open Access



The Hilbert Book Model project


This page is renewed on 19 May 2016


Author: Hans van Leunen

Retired physicist, chemist, software expert


Send your comments to “info” at this site


The owner of this site (me) uses this medium as a public work pad. You will find my latest papers here. When they mature then I upload the papers in pdf format to my e-print archive on Vixra.org. That site only accepts papers in pdf format. I use it because Vixra provides a very efficient revision service. With that service the reader can follow how the ideas behind the paper evolve in time. I prepare the papers in MS-Word docx format. MS-Word offers a powerful equation editor. For that reason I keep the latest docx version of the papers also on this website. The papers do not enforce copyrights. Everybody may use the published texts.

My papers are quite unorthodox and often very controversial. They will not easily pass a biased peer-review barrier. In addition the peer-review process cannot cope with the revision service that I use intensively. I do not want to disturb people with a request for endorsement as is required by the registration for ArXiv.org. The controversial and unorthodox character of my papers may pose too much burden for the endorsers. I also not use the open access journals that still ask a high fee to the authors. My retirement budget does not allow that. I have published a few papers on OALib.com. Their publication fee is acceptable low and they are not afraid to publish controversial papers.


Een wiskundig model van de realiteit (in Dutch)

Op zoek naar witte vlekken (in Dutch)



Keeping an e-book of about three hundred pages in sync with continuously deepening insights is a very laborious task. For that reason my newest insights are reflected in “My newest paper”. This is a set of more dedicated papers that each have a certain focus. When the new insights mature, then I store the result on my personal e-print archive on Vixra. Vixra has an excellent revision service that allows you to follow the development of my ideas.


My career as a developer of image intensifying devices offered me unique insights into the world of photons and elementary particles. See what image intensifiers reveal. Later in my career I switched to the development of software and especially to the improvement of the software generation process. This last activity guided in the direction of modular software system construction.


My newest papers


Review of the Hilbert book model


The Hilbert Book Test Model




Merging Methods


Differential and integral calculus


Quaternionic versus Maxwell based differential calculus


The generalized Stokes theorem


Dirac equation


Dirac equation in quaternionic format


Somewhat older papers


These papers are partly replaced by newer papers


The orthomodular base model:


Foundation of a mathematical model of physical reality


On the origin of physical fields


Coherent stochastic mechanisms


On the origin of electric charges


A consistent set of structured mathematical storage media


A mathematical model of physical reality



A math compendium:


Quaternions and Hilbert spaces


Merging Methods


The reverse bra-ket method


Fermion symmetry flavors



Lattice theory:


Skeleton relational structures


What drives reality?


Relativity and space-progression models:


The Lorentz transformation


Model building:


Reality contains a network of mathematical structures


About the Hilbert Book Model


The HBM is a model that is based on a
recipe for modular construction


The most fundamental law of physics cannot be stated in the form of a formula. Instead it is stated in the form of a commandment:


“Thou shalt construct in a modular way”


The Hilbert Book Model

is a paginated space-progression model


The dynamical coherence of the HBM is controlled by
a real time operating system


This Hilbert Book Model does not offer another physical reality.


The Hilbert Book Model offers an alternative view on physical reality


That view differs from the view that is offered by
contemporary physics


The new view creates new insights !



Three forms of models of physical reality can be discerned:


·         Classical physics = set based


·         Quantum physics = continuum based modularization


·         Discrete physics = discrete object based modularization


The HBM focuses on discrete object based modularization


In a simplified description:

Quantum physics treats objects and constructs that perform above the wave function

Discrete physics treats objects and constructs that perform below the wave function


“Above the wave function” = what uses the wave function

“Below the wave function” = what is described by the wave function



What exists underneath the wave function?




I do this HBM project purely for the fun of it and out of curiosity to the lower levels of physics



The most basic aspect of this model is:


Each of the discrete objects in this model can be represented by a closed subspace of an infinite dimensional separable Hilbert space.



The foundation of the model represents a recipe for modular construction.



The last e-book (nearly finished) 



The previous manuscript (completed)



Introduction to the Hilbert Book Model


The role of the observer



The traffic to this website became so intense that the costs surpassed my budget.

I am now constructing alternatives via Google drive

YouTube intro

Quick PowerPoint

Corresponding mp4 video

Quick YouTube


The full video mp4 flash(current version)

The Hilbert Book Model Game pdf (under development)


The full slide show (in preparation)

Full slideshow with voice.(Not yet ready)

The corresponding spoken text .(Not yet ready)

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The concepts of the Hilbert Book Model was explained at the DPG congress in Berlin at Tuesday march 18



What Image Intensifiers reveal

I offer a fine bottle of X.O. French cognac to the one that puts a significant argument, which disproves the Hilbert Book Model.

I also offer a fine bottle of X.O. French cognac to the one that adds a significant extension to the Hilbert Book Model

Send your responses to “info” at this site or post them on suitable LinkedIn groups


Nearly all tools that quantum physicists use are in some way based on the concept of the wave function. This means that such tools deliver a blurred view of the fine grain structures and fine grain behavior that these tools describe.

See: What is underneath the wave function.




A model that starts with an existing concept of space and an existing concept of progression is fooling us. SPACE IS NOT STATIC.


For those that deny that non-observable features and phenomena exist and keep non-observable in the future: I challenge you to design a model that contains subjects such as progression and space without just stating them as already present concepts. Thus these concepts must emerge in your model.


I bed that you cannot design such model without accepting non-observable features and phenomena that present a foundation of the model!


So, if I am right, then either all existing "physical models" are cheating us or they must also contain a foundation from which progression and space emerge.


Thus, if I am right, all models that do not support the emergence of progression and space must be classified as non-physical.

Insights change

The Hilbert Book Model started during the study of the author in the sixties on the TUE university in Eindhoven. The project paused during the author’s career in industry. After his retirement the author restarted the project in 2009. The project got its current name in 2011.

Since he restarted the Hilbert Book Model project, the insights of the author changed significantly.

You can follow this development via publications and his discussions on ResearchGate, on Science2.0 and on LinkedIn. 

Since November 2011 the author publishes on his e-print archive at http://vixra.org/author/j_a_j_van_leunen



Time ticks

Time proceeds with rather fixed progression steps. It is a parameter that is more fundamental than any other physical feature. It underlies all dynamics.


Let us define for convenience the concept of “observed time”. The observed time clock ticks at the location of the observed item and travels with that item. We suppose that all observable items own such clock.


In a paginated space-progression model all observed time clocks are synchronized. With other words in that model resides a universe wide time clock. The reason that the HBM is a paginated model is the fact that its foundation; the skeleton relation structure that is called quantum logic, offers no means to implement dynamics. A dynamic model uses an ordered sequence of these static models.


In a paginated model the setting of the observed time clock equals the current value of the progression step counter. The clock tick corresponds to a progression step.


As a consequence in that model universe can be considered to be proceeding with universe wide progression steps from each static status quo to a subsequent status quo. It means that universe can be considered to be recreated at each progression step. This recreation occurs with a super-high frequency. It is the frequency with which the universe wide time clock ticks. Phenomena that occur with that frequency cannot be observed. Only their averaged effects can be observed. It also means that every lower frequency wave must be in synchrony with the universe wide clock or is chopped and can only live on as a modulation of a super-high frequency wave. Due to the unobservable super-high frequency progression seems to flow. In fact it steps. Time ticks.


Such a view on space-progression is called a paginated space progression model. In this model each static status quo of the universe is described in a single page.


The observer and the observed item are linked via an information path. Via this path the information is transported from the observed item to the observer. Despite the fact that this path possesses characteristic attributes, these attributes are usually not known by the observer.


Let us define for convenience the concept of “observer’s time”. The observer’s time clock ticks at the location of the observer and travels with the observer. We suppose that all observers own such clock. The observer uses this clock in order to estimate the time at the location of the observed item.


The observer also owns an observed time clock. The readings of the observer’s time clock and the observed time of the observed event will usually differ. The difference depends on the characteristics of the path that information must travel from observed item to observer.


Contemporary physics uses the spacetime model. It uses the observer’s time instead of universe wide time. The observer’s time clock ticks at the location of the observer. In the spacetime model, space and observer’s time are coupled via the local speed of information transfer. In the spacetime model, the observer’s time clock can be selected freely.


In general, the observed time setting at the location of an observed item cannot be measured directly. If the path and the local speed that information takes in order to arrive from the location of the observed item to the location of the observer is known, then for a given observation it is possible to derive an equivalent observer’s time. The path depends on space curvature.


In a paginated model, for all observed items the universe wide time clock has the same value. Thus, in that model the observer’s time clock cannot be selected freely, but must be derived from the universe wide time value at the observed event. This classifies the paginated space-progression model as a fully deduced model. That does not say that the paginated model is not a valid space progression model.

The main criterion for the validity of the paginated model is the fact whether all observed time clocks can be synchronized.

If the paginated space-progression model exists, then this model and the spacetime model can be considered to be two different views of the same physical reality.


Static status quo descriptors

Another significant argument for the existence of a paginated space-progression model can be found in the foundations that were suggested at the advance of quantum physics. In 1936 John von Neumann and Garret Birkhoff wrote their famous paper about quantum logic and its lattice isomorphic companion; the set of closed subspaces of a separable Hilbert space. Inspection of these structures shows that they do not have a built-in means for implementing dynamics. These proposed models can represent a static status quo of a quantum physical system, but in order to represent dynamics these models must be extended. In contemporary physics this is done by making either wave functions or operators time dependent. However, it is also possible to attach a progression parameter to the whole model. This last choice means that the dynamic model is represented by an ordered sequence of sub-models that each represent a static status quo. With other words, this dynamic model is a paginated model.

Why quantum logic can be used as foundation

In no way a model can give a precise description of physical reality. At the utmost it presents a correct view on physical reality. But, such a view is always an abstraction.

Physical reality is very complicated. It seems to belie Occam’s razor. However, views on reality that apply sufficient abstraction can be rather simple and it is astonishing that such simple abstractions exist. Complexity is caused by the number and the diversity of the relations that exist between objects that play a role. A simple model has a small diversity of its relations.

Physical reality appears to have selected a skeleton relational structure as a means to keep the complexity of its constructs within bounds.

Mathematical structures might fit onto observed physical reality because its relational structure is isomorphic to the relational structure of these observations.

The part of mathematics that treats relational structures is lattice theory. Logic systems are particular versions of lattice theory. Classical logic has a simple relational structure. However since 1936 we know that physical reality cheats classical logic. Since then we think that nature obeys quantum logic, which has a much more complicated relational structure.

Thus quantum logic seems to represent the skeleton relational structure that physical reality has selected in order to reduce complexity.

Mathematics offers structures that are lattice isomorphic to quantum logic. One of them is the set of closed subspaces of a separable Hilbert space. However, this set cannot be interpreted as a logic system. Interpreting the elements as construction elements would fit better. The Hilbert space adds the superposition principle to the skeleton relational structure of quantum logic. Via the eigenspaces of its operators it adds a storage place for geometrical data.


The conclusion of this deliberation is that physical reality is not based on mathematics, but that it happens to feature relational structures that are similar to the relational structure that some mathematical constructs have. That is why mathematics fits so well in the formulation of physical laws. Physical laws formulate repetitive relational structure and behavior of observed aspects of nature.

The correlation mechanism

Without extra measures a paginated model will lead to dynamical chaos. An external correlation mechanism must take care that sufficient coherence exists between the subsequent sub-models. However, this coherence must not be too stiff, otherwise again no dynamics will take place.

The correlation mechanism must perform quite a lot of complicated tasks and it is strange that contemporary physics assigns these tasks to quantum state functions or operators. These actors are better suited as storage places than as controlling bodies.

The tasks of the correlation mechanism are:


·       Embedding particles in the field that acts as the curved operating space.

·       Establishing the swarming conditions for elementary particles

·       Controlling the propagation of the wave fronts that implement the potentials of the particles

·       Storing data in eigenspaces of operators and in quantum state functions.

·       Supporting entangled systems and subsystems.

Stepwise development

See: Stochastic nature of quantum physics


An original Poisson process can be coupled to an attenuating binomial process that is implemented by an isotropic 3D spread function. This combined process can be considered as a generalized Poisson process that locally has a lower production rate. In this way a 3D object distribution can be generated that at large production rates will resemble a 3D Gaussian distribution. That distribution can be described by two different descriptors. The first is a continuous object density distribution that can be interpreted as a probability density distribution. This descriptor has all aspects of the squared modulus of a wave function. The second description uses the sequence of generated objects. This sequence forms a stochastic path in 3D space. It can be interpreted as a path that is walked by a single object. Together, these descriptors describe an elementary building block that is characterized by a wave function and that during each production cycle walks along the mentioned stochastic micro-path.

You might agree that this comes close to the description of an elementary particle.

Extra restrictions set by the correlation mechanism

An extra restriction that is installed by the correlation mechanism is that the coherent discrete distribution of step stones that belong to an embedded particle can be characterized by a continuous step stone density distribution that exists in the embedding continuum. Further the mechanism ensures that this continuous object density distribution can be characterized as a probability density distribution. If this is the case, then the object density distribution can be considered as the squared modulus of the wave function of the considered object. This describes the fundamental stochastic nature of the universe wide time clock model. These extra restrictions are far from obvious. The consequence is that the stochastic micro-path is generated in a recurrent fashion such that important statistical attributes are reinstalled in a cyclic fashion.


If after walking along the full micro-path the next walk keeps the average location of the step stones at the same location, then the object is considered to stay at rest or to take part in an oscillatory movement such that the micro-path is stretched along the path of the oscillation. If that is not the case, then the object is considered to move and the micro-path is considered to be stretched along the path of that movement.

Here the correlation mechanism will put another restriction that concerns the stretching of the micro-path along the movement or oscillation paths. This must occur such that that the Fourier transform of the density distribution of the step stones will reflect the probability distribution of the momenta that characterize the motion. This restriction reflects the impact of Heisenberg’s uncertainty principle.


Together these non-obvious additional restrictions present the model as a quantum physical system and support the particle-wave nature of the objects that are controlled by the correlation mechanism.



The three extra conditions for the coherence between subsequent static status quos that are enforced by the correlation mechanism set the conditions for swarming. Swarming means that the swarm appears to move as one body. These extra conditions are:
 The coherent distribution of step stones can be described by a continuous density distribution.
 And by a corresponding continuous current density distribution
 The density distribution can be interpreted as a probability density distribution
 The (infinitesimal) movement of the whole coherent distribution can be described by a single displacement generator
This last condition can be interpreted as the fact that the probability density distribution of the infinitesimal displacements of the step stones equals the Fourier transform of the probability density distribution of the step stones. Or in first order the movement of the step stones is not hampered by the space curvature that is raised by the step stones. In second order this is no longer true for massive particles. This second order dependence is the origin of inertia.
The first order dependence is reflected by the coupling equation, which uses normalized quaternionic functions ψ and φ in order to represent density distributions.
∇ψ=m φ
After Fourier transformation this runs as
Pψ ̃=m φ ̃
Swarming conditions apply to massive elementary particles, photons, gluons and entangled composites. Photons and gluons have no step stones but they possess locations where they can be detected.
The coupling equation classifies quantum physics as a special kind of fluid dynamics. Apart from the differential continuity equation also the corresponding integral balance equation holds. 
∫_V▒〖∇ψ dV〗=m∫_V▒〖 φ dV〗
The swarming conditions result in the capability of the swarm to behave as interference patterns.
Entangled systems
Composites that are equipped with a quantum state function that at any progression step equals the superposition of the quantum state functions of its components form an entangled system. 
It means that the focus of the corresponding probability density distribution lays on the complete system and not on one of its components. As soon as this focus shifts to one of its components, then the properties of that component become observable. Getting focus means that the quantum state function is normalized to unity. 

The definition of entanglement also means that the superposition coefficients can be functions of progression. These functions may describe motions of the components that are internal to the system. These motions are restricted to quantum oscillations.

Entangled systems obey the swarming conditions. This means that they move as a single unit. The third swarming condition requires that the quantum state function has a Fourier transform. The fact that internal motions are restricted may be interpreted as the condition that the functions that describe the behavior of the superposition coefficients must be invariant under Fourier transformation.
Examples of Fourier invariant function are the Gauss function, complex even functions, complex odd functions, functions that describe spherical harmonics and the functions that describe linear quantum harmonics.

Entanglement is governed by the correlation mechanism. For entanglement it is not necessary that the corresponding system or subsystem forms a connected composite. The quantum state function of an entangled (sub)system obeys the coupling equation. Further, entangled systems are governed by the Pauli principle.

Entanglement represents a binding mechanism.
Pauli principle
If two components of an entangled (sub)system that have the same quantum state function φ, are exchanged, then we can take the system location at the center of the location of the two components. Now the exchange means for bosons that the (sub)system quantum state function is not affected: 
for all α and β{αφ(-x)+βφ(x)=αφ(x)+βφ(-x)}⇒φ(-x)=φ(x) 
and for fermions that the corresponding part of the (sub)system quantum state function changes sign. 
for all α and β{αφ(-x)+βφ(x)=-αφ(x)-βφ(-x)}⇒φ(-x)=-φ(x) 

This conforms to the Pauli principle. It also indicates that the correlation mechanism, which controls the entanglement, takes care of the fact that if one of these two twin components exposes any of its properties (e.g. its spin) that it has IMMEDIATE effect on the properties of the other component. As long as none of the components is inspected, the focus is on the complete system. Inspecting a component puts focus on the component, rather than on the system as a whole.
Universe wide time

Universe wide time ticks at a super-high frequency. Phenomena, such as waves, that run at this super-high frequency cannot be observed. Only their averaged effects can become noticeable. Potentials are typical examples of such averaged phenomena.


Other processes may run in sync with the universe wide time clock. These processes concern the recreation of parts of the universe. Most of these processes run at a lower cycle time. For example the recreation of all aspects of a particle takes a large number of progression steps.


In contemporary physics, red-shift is measured and interpreted as space expansion. Further the speed of information transport appears to be constant. The HBM takes this speed as a model constant. As a consequence space expansion goes together with a similar expansion of the progression step. With other words the universe wide time clock slows down as a function of progression.

Super-high frequency waves

Super-high frequency waves are special. Since all lower frequency waves are chopped, the super-high frequency waves are carrier waves for all other waves. These other waves are modulations or temporal averages of the super-high frequency waves. The background field that acts as our curved space is modulated by the super-high frequency waves.



The stuff from which we are made

Quantum fluid dynamics

The medium in which light propagates is space. This space can curve. The curvature is not static. So, this space moves. It can be treated as a field. Particles are embedded in this continuum.


The behavior of this combination can be analyzed by a kind of fluid dynamics. Let us call this method quantum fluid dynamics. It differs from conventional fluid dynamics in the medium that is treated. In conventional fluid dynamics this is a gas or a fluid. Fluid dynamics concerns density distributions and currents. In quantum fluid dynamics these are space location density distributions and space location current density distributions. They can be combined in quaternionic distributions, where the real part is the space density distribution and the imaginary part is the space current density distribution.


Quantum state functions are probability amplitude distributions. They can be specified as complex functions or as quaternionic functions. In the last case they fit the purpose of quantum fluid dynamics. In fact they are a special type of quaternionic distributions that we call quaternionic probability amplitude distributions.


In quantum fluid dynamics the quaternionic probability amplitude distributions act on the continuum in which they are embedded. The shared parameter space of all quaternionic probability amplitude distributions comprises the whole universe. It is the arena where everything occurs. In the HBM this arena is called Palestra.

The Hilbert Book Model

The Hilbert Book Model (HBM) is a simple model of the lowest levels of fundamental physics. The HBM is strictly based on quantum logic. The concepts in the following text are directly or indirectly derived from this foundation.

In the Hilbert Book Model (HBM) nature steps with universe wide progression steps from one static status quo to the next static status quo. Progression conforms to universe wide time. In the HBM all observed time clocks are synchronized.

In the HBM nature's building blocks (elementary particles) are represented by coherent sets of what I call step stones. The step stones are placeholders of locations where the building block can be. The set is generated by a stochastic process.

At every progression instant only one step stone is used. In this way, even at rest, the building block walks along a micro-path. At every arrival at a step stone the building block emits a wave front that carries information about the presence and the properties of the building block. This wave front propagates with light speed away from its source. The wave front slightly folds and thus curves the embedding continuum. This explains the origin of space curvature. The wave fronts that were emitted by ALL building blocks that existed in universe, together form a huge background field. This field acts as the embedding continuum that we observe as our curved space. It is not a potential. It has no unique source. The background field implements inertia. (It counteracts acceleration of embedded particles).


The wave fronts that are emitted by a single particle are thus generated at slightly different locations. Already at a small distance they seem to be generated at an super-high frequency by a source that has a rather stationary location. Together, these wave fronts form an SHF wave.

At small scales the wave fronts that are emitted by a building block interfere. Together they form a set of rather static potentials that represent the averaged effect of the wave fronts. The contribution to a potential by a wave front is characterized by a dedicated Green's function.


A sudden change of the energy of the building block goes together with a temporary modulation of the wave fronts. We know such modulations as photons. The duration of the modulation equals the duration of a complete micro-walk.

Such occasions occur with electrons inside atoms. There the electrons walk along a micro-path that is stretched along the path of a spherical harmonic oscillation. Due to this stochastic motion the electrons potentials act as if the electron is free. With other words, the oscillation is completely hidden by the stochastic stepping. Only the static potentials are shown. The extra movement is accounted in the mass of the electron. However, if the electron switches its energy level, then this goes together with the emission or absorption of a photon that corresponds to the energy jump.

The fact that the energy quantum is reflected in the frequency of the photon leads to the conclusion that the photon is created/annihilated in a fixed number of progression steps. That number conforms to the duration of a complete micro-walk.

At the start of quantum physics this phenomenon looked strange to physicists that expected EM waves that correspond to the spherical harmonic oscillation.

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The overview of the involved objects is treated in detail in the paper:


Work in progress:



Physics of the Hilbert Book Model






Sketch of the design of the Hilbert Book Model


The stochastic nature of quantum physics




PowerPoint file of Hilbert Book Model

 PowerPoint file of HBM_Intro _part I

Spoken text

PowerPoint file of HBM_part 2





Spacetime model versus paginated model


Entanglement in paginated space progression models



Oscillation of a HBM particle


Discoveries of the Hilbert Book Model



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Abstract of the manuscript

The Hilbert Book Model is the name of a personal project of the author. The model is deduced from a foundation that is based on quantum logic and that is subsequently extended with trustworthy mathematical methods. What is known from conventional physics is used as a guideline, but the model is not based on the methodology of contemporary physics. In this way the model can reach deeper into the basement of physics. The ambition of the model is rather modest. It limits its scope to the lowest levels of the physical hierarchy. Thus fields and elementary particles are treated in fair detail, but composites are treated marginally and only some aspects of cosmology are touched. Still the model dives into the origins of gravitation and inertia and explains the diversity of the elementary particles. It explains what photons are and introduces a lower level of physical objects and a new kind of ultra-high frequency waves that carry information about their emitters. It explains entanglement and the Pauli principle. Above all the HBM introduces a new way of looking at space and time. Where contemporary physics applies the spacetime model, the HBM treats space and progression as a paginated model.



The author’s e-print archive is at


See also HAL (Hyper Articles en Ligne)


Schets van het Hilbert Boek Model  (Dutch)


Natuurkundige dilemma’s (Dutch)


Sketch of the Hilbert Book Model



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Older versions:

On the hierarchy of objects



HBM_Presentation (26Mb)


Hilbert logic

Hilbert logic slides

Hilbert logic slide comments


Deep Field Theory




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Features of the Hilbert Book Model


You can download this manuscript free of charge


Beschrijving: Support independent publishing: Buy this book on Lulu.

You may wish to buy a printed copy of the manuscript.



This manuscript is of an earlier date.





Essentials of the Hilbert Book Model


Table of elementary particles




Hilbertlogica (Dutch!)


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What image intensifiers reveal

At low dose rates it becomes apparent that radiation is not built from waves, but from separate quanta that move in un-sharp corridors (clouds), which may be shaped as waves.


A short film of the output of an X-ray image intensifier made at a very low dose.

Provided by Philips Healthcare

  • The pixel size is about 200μm
  • The number of pixels is about 500 * 600
  • The average number of X-ray quanta per pixel per frame in the mid gray area is circa 1
  • The range inside these picture is about 20
  • The direct radiance is about 5 quanta per pixel per frame
  • The dark regions get <<1 quanta per frame
  • The number of pictures is 33


Other pictures:

http://en.wikipedia.org/wiki/File:Moon_in_x-rays.gif . Low dose X-ray image of the moon.

http://www.youtube.com/watch?v=U7qZd2dG8uI ; Hail storm. Warning, this is NOT a video of an external object. It is constructed by coupling two night vision goggles. There is no input image. What is shown is cold emission of the photo cathode of the first goggle at different temperatures.

http://en.wikipedia.org/wiki/Shot_noise Very low dose photos.


At low dose rates, I never saw any sign of a wave. At the utmost I saw detection patterns that looked like interference patterns.


Why is that so? Remember that the squared modulus of the wave function is a probability density distribution that indicates the probability of finding the particle at the location defined by the parameter of the wave function!

The wave function has a Fourier transform, which makes it a wave package! This establishes the wave behavior of the particle.


Photons also have wave functions that represent detection probability density distributions and have Fourier transforms. Thus the detection probability density distribution is a wave package. However these waves differ from the photon wave!

The detection probability density distribution establishes the particle behavior of a set of  photons!!


The Hilbert Book Model states that photons are formed by strings of shape keeping fronts that each carry a fixed amount of energy. In free space, the string moves with light speed and the shape keeping fronts also keep their amplitude. At a given instant of progression, the passage of the complete string takes a fixed number of progression steps.


The HBM states that everything in nature is constituted from quaternionic probability amplitude distributions (QPAD's). Photons and gluons are themselves free (oscillating) QPAD's. Elementary particles are constituted by the coupling of two QPAD's. QPAD's are fields. All other physical fields are derived from the couplings of QPAD's.

QPAD's describe the density distributions of carriers and the density distributions of the currents of these carriers. The density of the carriers is moving and the movement can be an oscillation. The density distribution can have the form of a wave but except for photons and gluons it does not need to oscillate.


The carriers can be interpreted to be tiny patches of the parameter space of the distributions. They represent the lowest level of objects that exist in nature. They represent locations where the owner of the QPAD can be detected. They are the objects that produce the spots in the image that is produced by an image intensifier in low radiation dose conditions. So what the observer of the intensified image sees is a hail storm of impinging carriers and no radiation wave.


It is one of the experiments that directly reveal the existence and some of the characteristics of (quaternionic) probability amplitude distributions.


Elementary particles

Elementary particles are constituted by the coupling of two Qpatterns that belong to the same generation. One of the Qpatterns is the quantum state function of the particle. The other Qpattern implements inertia. Apart from their sign flavors these constituting Qpatterns form the same quaternionic distribution. However, the sign flavor must differ and their progression must have the same direction. The coupling has a small set of observable properties: coupling strength, electric charge, color charge and spin. The coupling affects the local curvature of the involved Palestras.

Qpatterns that belong to the same generation have the same shape. The difference between the coupling partners resides in the discrete symmetry sets. Thus the properties of the coupled pair are completely determined by the sign flavors of the partners.


In the figure below  acts as the reference sign flavor.


Eight sign flavors 
(discrete symmetry sets)
Colors N,R,G,B,R ̅,G ̅,B ̅, W
Right or Left handedness R,L

Figure 1: Sign flavors



















Continuous quaternionic functions exist in 16 sign flavor versions. They can be bundled in sign flavor bundles. The component that has the same sign flavor as the parameter space has is the reference sign flavor of the bundle.

Our living space is represented by such a sign flavor bundle.


HYPOTHESIS: If the quaternionic quantum state function of an elementary particle couples to the reference member of the sign flavor bundle to which the particle belongs, then the particle is a fermion.

If the quaternionic quantum state function of an elementary particle couples to member which is not the reference member of the sign flavor bundle to which the particle belongs, then the particle is a boson.

For anti-particles the quaternionic conjugate of the reference sign flavor must be used.


The application of the sign flavor bundle ensures gravitational coupling between fermions and bosons. This implements the functionality that is thought to be implemented by the Higgs mechanism.


The coupling of two Qpatterns is controlled by a coupling equation

ψ=m φ

This equation is equivalent to a quaternionic differential continuity equation.


And it is equivalent to a quaternionic differential equation.



Here is the quaternionic nabla. ψ, φ and ϕ are Qpatterns that belong to the same generation.


Photons and gluons are not particles they are energy quanta that are implemented as modulations of ultra-high frequency waves that form the potentials of particles.


In the standard model the eight gluons are constructed from superpositions of the six base gluons.

Coupling Qpatterns

Qpatterns are not static. Instead they are distorted by the local curvature, they move or they oscillate. At rest, on average the Qpattern keeps its location and it keeps its size. Thus an outbound move must be followed by an inbound move. The sharp allocation function takes care of the slower part of the dynamics.


The coupling uses pairs  of two sign flavors. Thus the coupling equation runs:

Corresponding anti-particles obey

The anti-phase couplings must use different sign flavors.

Elementary particle properties


Spin relates to the fact whether the coupled Qpattern is the reference Qpattern. Each generation has its own reference Qpattern.

Electric charge

Electric charge depends on the difference and direction of the base vectors for the Qpattern pair. Each sign difference stands for one third of a full electric charge. Further it depends on the fact whether the handedness changes. If the handedness changes then the sign of the count is changed as well.

Color charge

The color charge of the reference Qpattern is white. The corresponding anti-color is black. The color charge of the coupled pair is determined by the color of its members.


Mass is related to the number of involved Qpatches.


With these ingredients we can look for agreements with the standard model.


The tables merely demonstrate that the discrete properties of elementary particles are directly related with the discrete symmetries of continuous quaternionic functions. Each time two of these quaternionic functions are involved and the properties of the particles depend on the differences of the discrete symmetry sets. The diversity in the HBM table is much greater than in the SM. This is due to the fact that the SM is based on measured properties, while color charge is not measurable. Still it is used in the HBM table in order to differentiate between particle types.

Leptons and quarks

According to the Standard Model both leptons and quarks comprise three generations. Neutrinos will be treated separately.







SM Name































































The generations contain the muon and tau generations of the electrons, the charm and top versions of the up-quark and the strange and bottom versions of the down-quark.


W-particles have indiscernible color mix. and  are each other’s anti-particle.






SM Name



























Z-particles have indiscernible color mix.






SM Name















Neutrinos are fermions and have zero electric charge. They are leptons, but they seem to belong to a separate low-weight family of (three) generations. They couple to a Qpattern that has the same sign-flavor. They have a small rest mass.


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[1] The HBM uses the name state function for the quantum state of an object or system rather than the usual term wave function because the state function may characterize flow behavior as well as wave behavior.

[2] The notion of “sign flavor” is used because for elementary particles “flavor” already has a different meaning.

[1] The notion of “sign flavor” is used because for elementary particles “flavor” already has a different meaning.