Essentials of the Hilbert book model

The Hilbert book model

This model is a simple model of physics that is strictly based on the axioms of quantum logic. Quantum logic is very similar to classical logic, but one of the axioms of quantum logic is weaker than the corresponding axiom of classical logic. This concerns the modular law. The direct result of this weakening is that quantum logic has a lot more complicated structure than classical logic. Where classical logic can be displayed with a simple representation consisting of Venn diagrams, will quantum logic correspond to a complicated mathematical model that has the same lattice structure. This model consists of the set of the closed subspaces of an infinite dimensional separable Hilbert space. Hence that quantum physics is usually carried out within the framework of a Hilbert space. However, because also other types of Hilbert spaces exist, not always a separable Hilbert space is used for that purpose. An example is quantum field theory. This derogation can lead to contradictions that must be solved by a renormalization of the solution thus obtained. But, there are better ways to address fields that keep the direct relation with quantum logic intact. Such a different approach is applied in the Hilbert book model.
The Hilbert book model uses the broadest choices that can be made for this separable Hilbert space. The most important freedom of choice that still exists is the number system with the help of which the inner product between Hilbert vectors can be defined. This number system may consist of real numbers, complex numbers or quaternions. This last one is the widest choice and offers the most flexible opportunities. For this reason, the Hilbert book model also allows quaternions as eigenvalues of operators and as the values of fields and coordinates.
Both quantum logic and the corresponding separable Hilbert space offer no place for fields and can only offer a static representation. Neither quantum logic nor the separable Hilbert space can adapt the equivalent of local time. In addition the operators that work in separable Hilbert space do not possess eigenspaces that have the properties of a continuum. The eigenspaces of these operators have a countable number of eigenvalues. It is not possible to use these ingredients in order to form continuous equations of motion. So it is no wonder that physicists look to the capabilities of other Hilbert spaces. That happens especially in quantum field theories. Such a step breaks the direct relationship with quantum logic. However, there are other solutions to this dilemma.
Every separable Hilbert space is part of a Gelfand triple. This construct features operators that possess eigenspaces with the structure of a continuum. For that reason the Gelfand triple is also falsely called a "rigged Hilbert space". However, it is not a real Hilbert space. It's a sandwich, where a separable Hilbert space is part of.
The next step is that the eigenvalues of operators in the separable Hilbert space link to the continuum background eigenspace of corresponding operators in the Gelfand triple. This link will not be one on one. We allow the link to be inaccurate in a stochastic way. In other words, a probability distribution is added that for the eigenvector of the operator in the separable Hilbert space selects an exact value that at a test event is taken from the continuum background eigenspace.
Instead of directly applying a probability density distribution we use a quaternionic probability amplitude distribution. However, the square of the modulus of this distribution is a probability density distribution.
We can separate the amplitude distribution into a charge density distribution and a current density distribution. For a while the interpretation of these charges and currents will be left in the middle. In the described way we achieve several goals at one blow. It opens the possibility to apply continuity equations. Continuity equations are also called balance equations. The equations of motion of the charge-bearing quanta are in fact continuity equations. By using this approach, we have created the possibility to analyze the movement of these quanta, whatever those quanta may be.
This interpretation also determines the kind of operator that is involved. It is an operator that delivers an observable, which changes dynamically in the realm of a background continuum space.
We now have a powerful weapon in the hands to describe the behavior of quanta. Unfortunately, neither the separable Hilbert space that is extended with quaternionic probability amplitude distributions, nor the similarly extended quantum logic can represent anything else than a static status quo. Since the charge distributions and flow distributions are known in the form of probability distributions at least something is known about how the following static status quo will look. Thus, the somewhat extended model is still a static model and not a dynamic model.
The solution is obvious. It consists of an ordered sequence of consecutive static models. Each static model consists of a sandwich in the form of a Gelfand triple and the stochastic but static links that exist between the separable Hilbert space and the Gelfand triple. The picture that emerges is that of a book where the consecutive pages represent successive sandwiches. The page number acts as a progress counter. It is a global working counter. It adds a parameter that represents the Hilbert space wide progress to each Hilbert space in the Hilbert book model. So this parameter is not our common notion of time, but the progression counter has certainly much relation with it.
This model shows that no direct relationship exists between the progression parameter and the position of a quantum. The progression is represented by a Hilbert space wide parameter. The position is an eigenvalue of a corresponding operator. Only when the quantum moves, a relationship emerges between these quantities. A uniform movement can be described by a Galileo transformation or, when a maximum speed exists, by a Lorentz transformation.
We now have a dynamic model that can display the movement of quanta and can describe the behavior of related fields.
The beauty of this model is that it is literally based on pure logic. Only mathematics is used in order to extend that foundation. The main extra ingredient is the stochastic link between eigenvalues in the separable Hilbert space and eigenspaces in the Gelfand triple.
For more details see: Http://www.crypts-of-physics.eu/OntheoriginofdynamicsBoek2.pdf

Essentials of quantum movement

For an introduction, see: Essentials of the Hilbert book model.
The movements of quanta can be described by continuity equations. Dirac was the first to put the equation of motion of free electrons in the form of a continuity equation. However, he used a complex format and he used spinors and Dirac matrices in order to represent the quaternionic behavior of fields. Majorana followed him with a similar equation for Majorana particles. Both equations can be converted into quaternionic format and then they become a much easier interpretable form. It appears that the continuity equation contains a source/drain term that contains another quaternion flavor of the main transported field.
The general format of the equation that describes the free movement of quanta appears to be:
∇ψˣ = m ψʸ
Here the quaternionic nabla operator is the transporter. ψˣ is the transported quaternionic field flavor. ψʸ is the coupled quaternionic field flavor and m is the coupling factor.
The ordered pair {ψˣ, ψʸ} identifies a quantum type.
Two independent switch operations together determine four field flavors. The local coordinate system acts as a reference.
The first switch operation ψ⟹ψ* switches the sign of all three imaginary base vectors of the quaternion values. In imaginary quaternion space this operation works isotropic.
The second switch operation ψ⟹ψ¹ switches the sign of only one imaginary base vector of the quaternion values. In imaginary quaternion space this operation works anisotropic.
Both switch operations switch the handedness of the quaternion external vector product.
The four flavors differ in the sign of one, two or three imaginary base vectors. Two of the field flavors are right handed, the other two are left handed.
The flavor of the nabla operator equals the flavor of the local coordinate system of which the values act as parameters for the distributions that form the considered fields.
With each quantum that fulfills this scheme exists an anti-quantum.
That anti-quantum is formed by the pair {ψˣ*, ψʸ*} and obeys the equation of movement:
∇*ψˣ* = m ψʸ*
The real part of ψˣ represents a “charge” density distribution. The imaginary part of ψˣ represents a “current” density distribution. The “charge” for which the dynamics is described by the continuity equation stands for the switch event ψˣ⟹ψʸ in case of the quantum and stands for the switch event ψˣ*⟹ψʸ* in case of the anti-quantum.
If in a switch event the handedness is switched, then this involves an electrical charge. The strength of the charge and its sign is determined by the number of imaginary base vectors that switch and on the direction in which this switch takes place. This result in a variety of electrical charges (±e, ± ⅔ e, ± ⅓ e and 0.). Apart from electrical charges the quanta can be afflicted with color charge. This characteristic relates to the direction of the anisotropic switch.
Quanta exist for which coupling factor is zero. They obey different equations of motion.
If the transported field ψˣ is isotropic and the coupling factor not equal to zero, then the quantum concerns a fermion. Otherwise, it's a boson.
From the equation of motion another equation can be derived, from which via integration the coupling factor m can be calculated from the fields.
∫˯ (ψʸ*∇ψˣ) dV = m ∫˯ (ψʸ* ψʸ) dV= m ∫˯ |ψʸ|² dV = mg

g is a real constant g>0.
This equation shows that m is a property of the pair {ψˣ, ψʸ}, which defines the corresponding quantum.
In this way important properties, equations of motion and coupling factors for electrons, neutrinos, quarks, W bosons, Z bosons and their anti quanta can be derived.
Since photons and gluons are not coupled, apply to them other equations.
This schema determines important characteristics for all known particles from the standard model. It is also contains open places for not yet observed particles.
More details can be found in: http://www.crypts-of-physics.eu/Quaternionic_continuity_equation_for_charges.pdf

The cause of gravitation

The cause of the gravitational field is almost invariably seen as a single mass or a group of mass carriers which have a common center of gravity. We are faced with the gravity field because things have weight or due the fact that acceleration confronts us with inertia. From deeper knowledge, we learn that the gravitational field is associated with the curvature of the space that surrounds us. In our immediate environment the effects of curvature(other than weight) are barely noticeable, but there are places in the universe where the curvature is much stronger and causes significant effects.
From these experiences the false idea may emerge that the gravitational field acts as the cause of curvature. However, that is a reversal of affairs. The gravitational field is no more and no less than the accurate administrator of the curvature of the local space. Therefore the value of the gravitational field is not a simple number. It is a tensor that exactly represents the local curvature by using a matrix of characterizing numbers. This matrix can vary from place to place.
We can go a step further and argue that the gravitational field designates the points on which the cause of the curvature seems anchored. Actually, these point-shaped mass carriers do not need to exist. It means that there may also be other causes to be coined for the curvature. An example is formed by an anomaly in the local geometry that is surrounded by a different structure of the normal curvature.
A black hole can be considered as such a geometric abnormality. A black hole is surrounded by a very strong curvature field such that information can no longer pass the skin of the hole. Therefore the hole can as well be completely empty. What happens to the material that is sucked up by the black hole? Well, that will be ripped apart into its smallest possible parts. Part of the debris is used to widen the skin of the hole. The other part escapes from the absorption process and is reflected back. The hole gets bigger, but that only becomes visible via the enlargement of the skin. The surface of the skin gives an indication of the mass, which the hole represents. The curvature around the black hole is in correspondence with this mass. However, the hole itself is empty.
If this picture is correct, then nature might perform this trick more often. What to say of the idea that all elementary particles that have a mass, contain in their inner a small geometric abnormality that is responsible for the gravitational field of the particle.
The question results of what causes this abnormality. The physical fields that belong to a particle determine what is present in the domain of that quantum. These fields are quaternion valued probability amplitude distributions. The square of the modulus of these distributions is a probability density distribution of the presence of the quanta considered. At places where nothing is allowed to be present, in fact a geometric abnormality emerges in the form of a hole. The hole represents the mass of the particle.
The gravity field is thus a very different type of field than the fields that belong to the quanta.
What about inertia? This effect is caused by the collaboration of all geometric abnormalities. The most distant abnormalities play the largest role. Further away, the influence of a single abnormality falls off quickly with distance, but the number of participating abnormalities increases much faster with that same distance. Moreover, the influence of differences level out due to averaging over an increasing number of contributions. It looks as if a uniform background attracts the local object from every direction. Because the forces come in equal size from all directions nothing happens. In a similar way nothing will happen as long as the object moves in a uniform manner. If the object accelerates, then according to field theory, this goes hand in hand with the presence of an additional gravitational field that counteracts the acceleration.
So it is not surprising that physicists cannot reconcile the electromagnetic fields and the weak and strong binding fields with the gravitational field. Apparently the gravitational field is a field that is caused by the other fields. All other fields are quaternionic probability amplitude distributions that belong to quanta.