# Hilbert Book Model Project/Stochastic Location Generators

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## Contents

# Stochastic mechanisms[edit]

### Stochastic processes[edit]

In the Hilbert Book Model all modules own a private stochastic mechanism that ensures its coherent behavior. These modules and their components apply a stochastic process that owns a characteristic function. These process play a very crucial role. Where particle physics reasons in terms of force carriers will the Hilbert book model reason in terms of the characteristic functions of the stochastic processes.

The stochastic mechanisms do not produce a stochastic spacetime. Instead, their activity concerns the ongoing stochastic generation of locations of point-like artifacts that embed in a smooth continuum, which gets deformed by this embedding. The mechanisms also do not implement the principles of Loop Quantum Gravity.

### Elementary modules[edit]

The mechanisms that at every next instant supply a new location to elementary modules, apply stochastic processes.

The processes belong to the category of the inhomogeneous spatial Poisson point processes.

It is possible to consider these processes as a combination of a genuine Poisson process and a binomial process.

A spatial point spread function implements the attenuation effect of the binomial process. This procedure creates a new generalized Poisson process for which the efficiency varies with spatial location.

A characteristic function determines the coherence of the generated location swarm. The characteristic function and the location density distribution of the location swarm are each other's Fourier transform. Thus, the characteristic function equals the Fourier transform of the spatial point spread function, which implements the binomial process. The spatial point spread function equals the squared modulus of the wavefunction. According to current physical theories the wavefunction characterizes the behavior and the spatial properties of the client object .

The spatial point spread function does not describe the deformation of the carrier field. That deformation follows from the convolution of the spatial point spread function with the Green's function of the carrier field. This deformation characterizes the influence on gravitation. This relation can be comprehended by the effects of clamps. Clamps describe the consequences of the embedding of point-like artifacts into a continuum. Clamps are spherical shock fronts, which are solutions of a homogeneous second order partial differential equation.

The characteristic function acts as a displacement generator. Consequently, at first approximation, the swarm moves as one unit.

On the other hand, the mechanism generates a stochastic hopping path. Both the swarm and the hopping path characterize the client of the mechanism.

#### Version selection[edit]

The stochastic mechanisms appear to define the version selection and the alignment of these versions of the applied quaternionic number system. This version then resides as parameter space on the platform that the mechanism applies.

#### Hilbert space representation[edit]

The base model acts as a framework in which dynamic process can take place. The base model supports the existence of floating platforms that each carry a private parameter space. In the base model nothing exists that supports stochastic processes. In the full Hilbert Book Model the output of the stochastic mechanisms only couple to the separate Hilbert space. However, the processes are controlled by a characteristic function that locates in the non-separable Hilbert space part of the HBM. Also the location density distribution of the generated swarm resides in the non-separable Hilbert space. The stochastic processes closely relate to the embedding process. At every progression instant each active stochastic process produces a single hop landing location. The HBM assumes that the generation of the hop landings occurs at the instant of creation of the model. At that instant the generated hop locations are stored in the separable Hilbert space. The time travel and a possible time zigzag only become clear after sequencing the time-stamps of the generated hopping locations. Observers travel with the scanning window and can only perceive a corresponding time travel. They interpret the zigzag reflection instants as creation and annihilation events. For observers the embedding process travels with the scanning subspace.

The private stochastic mechanism takes locations from the platform on which the client resides. The mechanisms store the generated location together with the corresponding time-stamp in an eigenvalue of a * germ operator*, which is private to the client.

#### Gravitation[edit]

The coherent hop landing location swarm owns a location density distribution. That distribution corresponds to a defined operator. It is a private * density operator*.

The convolution of the location density distribution with the Green's function of the embedding continuum results in a function that describes the deformation of the contribution to the deformation of the embedding continuum. A private deformation function defines a corresponding * deformation operator*. The density operator reorders the eigenspace of the germ operator. The Green's function blurs the location swarm. This procedure describes the gravitation of the client.

At each embedding instant the embedding continuum is locally deformed and the whole embedding continuum is extended with the volume of the green's function.

### Modules[edit]

Also for modules, a stochastic process exist that at every instant produces the locations of the elementary modules that configure the module.

This covering stochastic process owns a characteristic function that at every instant equals a superposition of the characteristic functions of the constituting elementary modules.

With other words, in first approximation, the module also moves as a single object Thus, the overall characteristic function binds the constituents of the module. The superposition coefficients are dynamic. This allows an internal movement of the constituents.

Due to their mass, the constituents cannot hop around in a stochastic fashion. Instead, the constituting modules may oscillate.

### Coherence characterizing function.[edit]

The characteristic function characterizes the coherence efficiency of the stochastic process.

The characteristic function plays an important role in the coherence of movement and in the binding of modules. This function acts as a displacement generator and its Fourier transform defines the detection probability density distribution of its target object. If the operating mode of the characteristic function changes, then its Fourier transform "collapses" and may turn in a different probability density distribution. This often goes together with an exchange of warps and clamps.

The characteristic function has much in common with the Optical Transfer Function (OTF), which is the Fourier transform of a two-dimensional Point Spread Function (PSF). This PSF can be interpreted as the location density distribution of the projection of a three-dimensional location swarm onto a flat surface that situates perpendicular to the axis in which the swarm moves.

In optical imaging the modulus of the OTF, which is called Modulation Transfer Function (MTF) serves as imaging quality criterion of linearly operating imaging equipment.

The PSF is the convolution of the location density distribution of the swarm of generated locations with the axis along which the swarm moves. Thus the OTF corresponds to a cut through the center of the characteristic function. This cut extends over the dimensions that are perpendicular to the direction of movement.

Every cut through the center of the MTF is a symmetric function.

Usually, two perpendicular cuts suffice to describe an anisotropic MTF. It suffices to specify the right half of the curve. These curves specify the coherence efficiency of the process.

#### Displacement generator[edit]

In the Fourier space a gauge factor that may be a function of progression, represents the displacement in configuration space. This means that the characteristic function of an elementary particle is the product of the gauge factor and a static function which is the Fourier transform of the generated location density distribution of the hop landing location swarm.

The characteristic function of a higher level module equals the superposition of the characteristic functions of the components. Here the gauge factors play the role of the superposition coefficients and may be implemented by oscillating fuunctions.

### Binding[edit]

The stochastic mechanisms and especially the characteristic function of the processes that these mechanisms apply play an essential role in the dynamical coherence and in the binding of modules. This is due to the fact that the characteristic function takes the role of the displacement generator of the swarm that represents the footprint of the module. Therefore, at first approximation, the module moves as a single unit. This footprint is formed by the hop landing locations of the elementary modules that constitute the module. Both the complete module as well as it's constituting elementary modules own a characteristic function and the characteristic function of the module equals the superposition of the characteristic functions of its components. The linear superposition coefficients relate to the locations of these components inside the module. Harmonic equations describe the internal oscillations. For example, the locations of the electrons inside atomic modules are described by Helmholtz equations.

#### Color confinement[edit]

For isotropic elementary modules, the characteristic function of their stochastic process acts as a displacement generator. Consequently, these modules move as a single unit. However, the color charge of quarks prohibits that the characteristic function acts as a displacement generator and only colorless components can take part in a superposition. Colored particles cannot be observed as objects that move as a single unit. If the components of a module involve quarks, then an extra binding mechanism implements color confinement. The color charges of the quarks are shifted by color shifting pairs of quaternions such that the combination reaches a neutral color charge. The color shifting pairs of quaternions implement the activity of gluons. This color confinement mechanism installs a strong binding effect. The Hilbert Book Model sees color confinement as a property of the stochastic mechanisms. The resulting hadrons are treated as color neutral components of higher order modules.

Color shifting pairs of quaternions do not affect isotropic elementary modules.

### Probability distributions[edit]

The characteristic function of the stochastic process and the location density distribution of the produced location swarm is an example of probability distribution pairs. These distributions interact with the embedding continuum. The interaction connection point is the geometric center of the platform on which the location density distribution resides. The convolution of the location density distribution with the Green's function of the embedding continuum describes the deformation of the continuum. For interaction with other actuators this deformation acts as blur that instead of the Green's function takes part in the convolution. Convolutions in configuration space convert into multiplications in Fourier space.

The characteristic function acts as a displacement generator. It smooths the stochastic hops into a coherent displacement of the swarm. A narrower modulus of the characteristic function allows more spread in configuration space. A wider modulus decreases the resulting spread in configuration space. This is treated by Heisenberg's uncertainty principle.

The embedding continuum also owns a Fourier transform. If the modulus of that Fourier transform is not needle sharp, then it affects the result of the convolutions as an extra multiplying factor in Fourier space. Thus if a plane in the embedding space contains a mask, then swarms that pass this mask are confronted with the Fourier transform of that mask as an Optical Transfer Function. A region that is encapsulated by a box that has an entrance window and an output window acts as an imaging device, which owns an Optical Transfer Function.

#### Path correction[edit]

If the swarm moves, then its platform moves. The hop landing location generator controls this. On the platform's parameter space the hopping path must close. If the map of the hopping path on the background parameter space does not close, then the platform must move such that in the platform parameter space the hopping path closes. The displacement generator closes the gap. The mechanism delivers the required kinetic energy by emitting or absorbing warps or by decreasing or increasing the number of clamps. Increasing kinetic energy increases the mass of the particle. Emission or absorption of warps affects the kinetic energy or the mass of the particle.

Emission and absorption of warps take place at the geometric center of the platform.

## Optical Transfer Function of imaging device[edit]

The Optical Transfer Function (OTF) is the Fourier transfer of the Point Spread Function (PSF) of an imaging device.

The OTF is the frequency characteristic of the device in the residing conditions.

The PSF is the response of the imaging device on a spatial pulse.

This interpretation supposes that the device acts linearly on the intensity of the impinging object image.

The concept does not require that the PSF is invariant with respect to the location on the input surface of the device. That input surface may be curved. The same holds for the output surface.

Both the OTF and the PSF depend on the chromatic and the angular distribution of the information carriers that constitute the object.

The same holds for the homogeneity of the phase of the incoming stream of information carriers. The generator of the stream of information carriers is supposed to produce as a Poisson process. Thus the stream consists of pointlike objects. Together the distributions and the generator produce a distribution of the density of the probability to detect an object that contributes to the stream of information carriers. The OTF represents part of the characteristic function of this stochastic process. During their travel, the information carrying objects constitute a coherent swarm. This swarm behaves as one unit.

The advantage of the OTF is that if a set of linearly operating imaging devices are chained in a sequence, then the OTF of the combined system equals the product of the OTF's of the constituting devices.

This rule also holds when in intermediate screens the nature of the imaging objects changes. This situation occurs in electron optical imaging systems.

The OTF is a complex number valued function. The modulus of the OTF is the Modulation Transfer Function (MTF). The phase of the OTF is the Phase Transfer function (PTF).

Usually, only the MTF is specified. For holographic systems the PTF is important. The specification usually assumes the common application conditions as the environmental conditions.

#### Relation to characteristic function[edit]

The characteristic function of a stochastic process equals the Fourier transform of the location density function of the hop landing locations that the process produces. The location density distribution of the hop landing location swarm plays the role of a three-dimensional Point Spread Function. The characteristic function of the stochastic process plays the role of the three-dimensional Optical Transfer Function.

The two-dimensional PSF equals the convolution of the three-dimensional PSF with the axis along which the geometric center of the swarm moves

The two-dimensional OTF corresponds to a cut through the three-dimensional characteristic function.

The characteristic function specifies the performance of the stochastic process in determining the geometric center of the swarm.

Increasing width of the modulus of the characteristic function corresponds to increasing coherence of the swarm. The coherence ensures that the swarm moves as one unit.

The characteristic function acts a displacement generator. The displacement generator is a translation operator or in other words, it is a momentum operator.

#### Multiplication rule[edit]

If all elements operate linearly, then the OTF of a chain of imaging elements equals the product of the OTF's of the elements.

The movement of an elementary particle can be seen as an imaging process.

If a path of an elementary particle is divided into parts where the imaging conditions are constant, then the coherence of the particle at the end of the path follows from the product of the OTF's of the path elements and the characteristic function of the particle generation process.

The Fourier transform of the transmission pattern determines the OTF of the imaging component.

For a glass lens the refraction pattern determines its OTF. A dedicated hologram can implement a lens.

More in general the equations that describe the penetration of a field into material describe the actual situation.

A Fresnel lens acts as a normal glass lens, but produces a strong halo.

In electron optics, the OTF of a path segment is determined by the Fourier transform of the potential distribution of that segment.

The movement of a set of particles can be seen as an imaging process.

Electron optical imaging systems are designed according to these rules. The movement of a set of elementary particles can be predicted in a similar way.

#### Line spread function[edit]

In imaging theory the location density distribution of a hop landing swarm is convoluted with a flat or a curved projection screen. The resulting function is taken as the Point Spread Function (PSF) at that stage of the imaging chain. The PSF is a hypothetical point image. A subsequent or preceding convolution with a perpendicular flat screen results in a Line Spread Function (LSF). The Fourier transform of the PSF is the Optical Transfer Function (OTF) that characterizes the imaging process. The modulus of the OTF is the Modulation Transfer Function (MTF). The Fourier transform of the LSF conforms to a cut through the center of the OTF. Each cut through the center of the MTF is a symmetrical function.

#### Photons[edit]

The concept of the OTF also works when the PSF represents a detection probability distribution of photons. The imaging devices are described by a spatially varying refraction index. This defines a corresponding transmission field. The camera obscura shows that in fact every small hole or slit represents a corresponding imaging device. The double slit experiment applies these concepts.

#### Particles[edit]

During their travel elementary modules are recurrently regenerated. At avery instant the elementary module gets a new location where it can be detected. Thus, the location generation mechanism produces a detection probability distribution. The swarm moves as one unit. The location density distribution acts as the corresponding PSF and the projected PSF corresponds to a two-dimensional OTF.

#### Masks[edit]

The Fourier transform of a mask acts as the Optical Transfer Function of the mask as an imaging device. This can be used to explain the imaging for a camera obscura and the imaging in a double slit experiment. The field behaves as if it passes the mask while including the distribution of the warps and the clamps that it embeds. or carries.

### Veiling glare[edit]

The MTF curve may show a sharp peak at its center. This peak indicates that part of the transferred information is much less related to the location of the geometric center of its source. The fraction that the height of the peak takes of the total height of the curve characterizes the fraction of the elements of the swarm that do not support the coherence of the characterized module.

In fact the mass or energy that is contained in the peak is lost for information transfer. It corresponds to spurious * clamps* or

*that form a*

**warps***around the coherent information transfer.*

**halo**In physics these phenomena are named * dark matter* and

*.*

**dark energy**

- a) coherent spot with halo
- b) peak in point spread function
- c) halo bottom in point spread function
- d) modulation transfer function