The Hilbert Book Model

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The Hilbert Book Model. A simple model of fundamental physics By J.A.J. van Leunen. I. http://www.e-physics.eu. The Hilbert Book Model. A simple model of fundamental physics By J.A.J. van Leunen. II. http://www.e-physics.eu. The Hilbert Book Model. A simple model of fundamental physics

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### The Hilbert Book Model

A simple model of fundamental physics

By J.A.J. van Leunen

I

http://www.e-physics.eu

### The Hilbert Book Model

A simple model of fundamental physics

By J.A.J. van Leunen

II

http://www.e-physics.eu

### The Hilbert Book Model

A simple model of fundamental physics

By J.A.J. van Leunen

III

http://www.e-physics.eu

Physical Reality
• 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.
• Mathematical structures might fit onto observed physical reality because their relational structure is isomorphic to the relational structure of these observations.
Rules Restrict Complexity
• Physical reality applies rules for relational structures that it accepts
• These rules intent to reduce the complexity of these relational structures
Complexity
• Physical reality is very complicated
• It seems to belie Occam’s razor.
• However, views on reality that apply sufficient abstraction can be rather simple
• It is astonishing that such simple abstractions exist
What is complexity?
• 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.
Rules and relational Structures

Logic

• The part of mathematics that treats relational structures is lattice theory.
• Logic systems are particular applications of lattice theory.
• Classical logic has a simple relational structure.
• However since the paper of Birkhoff and von Neumann in 1936, we know that physical reality cheats classical logic.
• Since then we think that nature obeys quantum logic.
• Quantum logic has a much more complicated relational structure.
Physical Reality & Mathematics
• Physical reality is not based on mathematics.
• Instead 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.
Logic systems
• Classical logic and quantum logic only describe the relational structure of sets of propositions
• The content of these proposition is not part of the specification of their axioms
• The logic systems only control static relations
• Their specification does not cover dynamics
Fundament
• The Hilbert Book Model (HBM) is strictly based on traditional quantum logic.
• This foundation is lattice isomorphic with the set of closed subspaces of an infinite dimensional separable Hilbert space.
First Model

Classical Logic

Only static status quo

&

No fields

Separable Hilbert Space

Separable Hilbert Space

Weaker modularity

isomorphism

Particle location operator

Countable Eigenspace

Three alarming facts
• The first level model does not support continuums
• HS operators have countable eigenspaces
• The first level model does not support dynamics
• Can only represent static status quo
• The Hilbert space contains deeper details than quantum logic does
• QL ⟹ propositions ↭ HS ⟹ sub-spaces
• HL ⟹ refined propositions ↭ HS ⟹ vectors
Threefold hierarchy

Possible interpretation of isomorphisms

Physical model
• The isomorphism introduces a set of particles, where each particle is represented by a swarm of step stones.
• Particles are represented by atomic quantum logical propositions.
• Step stones are represented by Hilbert space vectors that are eigenvectors of operators of the Hilbert space.
Static Representation

Quantum logic

Hilbert space

}

• No full isomorphism
• Cannot represent continuums
• Solution:
• Refine to Hilbert logic
Discrete sets and continuums
• A Hilbert space features operators that have countable eigenspaces
• A Gelfand triple features operators that have continuous eigenspaces
Static Status Quo of the Universe

Separable Hilbert Space

Classical Logic

Separable Hilbert Space

Gelfand Triple

Subspaces

Separable Hilbert Space

location

ContinuumEigenspace

Particle location

isomorphisms

Isomorphism’s

Hilbert Logic

embedding

Countable Eigenspace

vectors

### Implementing dynamics

The sub-models can only implement

a static status quo

Representation

Quantum logic

Hilbert logic

Hilbert space

}

• Cannot represent dynamics
• Can only implement a static status quo

Solution:

An ordered sequence of sub-models

The model looks like a book where the sub-models are the pages.

Sequence

· · · |-|-|-|-|-|-|-|-|-|-|-|-| · · · · · · · · · · |-|-|-|-|-|-|-|-|-| · · ·

Reference sub-model has

densest packaging

Prehistory

current

future

Reference Hilbert space delivers via its enumeration operator the

“flat” Rational Quaternionic Enumerators

Gelfand triple of reference Hilbert space delivers via its enumeration operator the reference continuum

HBM has no Big Bang!

The Hilbert Book Model
• Sequence ⇔book⇔ HBM
• Sub-models ⇔ sequence members ⇔pages
• Sequence number ⇔page number ⇔ progression parameter
• Thisresults in a paginatedspace-progression model
Paginated space-progression model
• Steps through sequence of static sub-models
• Uses a model-wide clock
• In the HBM the speed of information transfer is a model-wide constant
• The step size is a smooth function of progression
• Space expands/contracts in a smooth way
Progression step
• The dynamic model proceeds with universe wide progression steps
• The progression steps have a rather fixed size
• The progression step size corresponds to an super-high frequency (SHF)
• The SHF is the highest frequency that can occur in the HBM
Recreation
• The whole universe is recreated at every progression step
• If no other measures are taken,the model will represent dynamical chaos
Dynamic coherence 1

An external correlation mechanism must take care such that sufficient coherence between subsequent pages exist

Dynamic coherence 2

The coherence must not be too stiff, otherwise no dynamics occurs

Storage

The eigenspaces of operators can act as storage places

Storage details
• Storage places of information that changes with progression
• The countable eigenspaces of Hilbert space operators
• The continuum eigenspaces of the Gelfand triple
• The information concerns the contents of logic propositions
• The eigenvectors store the corresponding relations.
Correlation Vehicle
• Supports recreation of the universe at every progression step
• Must install sufficientcohesionbetween the subsequent sub-models
• Otherwise the model will result in dynamic chaos.
• Coherence must not be too stiff, otherwise no dynamics occurs
Correlation Vehicle Details
• Establishes
• Embedding of particles in continuum
• Causes
• Singularities at the location of the embedding
• Supported by:
• Hilbert space (supports operators)
• Gelfand triple (supports operators)
• Huygens principle (controls information transport)
• Implemented by:
• Enumeration operators
• Blurred allocation function
• Requires identification of atoms / base vectors
Correlation vehicle requirements
• Requires ID’s for atomic propositions
• ID generator
• Dedicated enumeration operator
• Eigenvalues ⇒ rational quaternions ⇒ enumerators
• Blurred allocation function
• Maps parameter enumerators onto embedding continuum
• Requires a reference continuum

RQE = Rational

Quaternionic

Enumerator

Enumeration
• Hilbert space & Hilbert logic
• Enumerator operator
• Eigenvalues
• Rational quaternionic enumerators(RQE’s)
Allocation
• Hilbert space & Hilbert logic
• Enumerator operator
• Eigenvalues
• Rational quaternionic enumerators(RQE’s)
• Model
• Allocation function
• Parameters
• RQE’s
• Image
• Qtargets
Enumeration & Allocation
• Hilbert space & Hilbert logic
• Enumerator operator
• Eigenvalues
• Rational quaternionic enumerators(RQE’s)
• Model
• Enumeration function
• Parameters
• RQE’s
• Image
• Qtargets
• Function
• Blurred
• Sharp
• Blur
Enumeration & Allocation & Blur
• Hilbert space & Hilbert logic
• Enumerator operator
• Eigenvalues
• Rational quaternionic enumerators(RQE’s)
• Model
• Enumeration function
• Parameters
• RQE’s
• Image
• Qtargets

Swarm

• Function
• Blurred
• Sharp
• Blur
Blurred allocation function

Convolution

• Function
• Blurred ⇒ Produces swarm ⇒ Qtarget
• Sharp ⇒ Produces planned Qpatch
• Spread function ⇒ Produces Qpattern ⇒ Swarm
• QPDD
• QuaternionicProbabilityDensityDistribution

QPDD

Described by the QPDD

Swarm

Blurred allocation function

Convolution

• Function
• Blurred ⇒ Produces swarm ⇒ Qtarget
• Sharp ⇒ Produces planned Qpatch
• Spread function ⇒ Produces Qpattern
• QPDD
• QuaternionicProbabilityDensityDistribution

Only exists at current instance

QPDD

Blurred allocation function

Curved space

• Function
• Blurred ⇒ Produces swarm ⇒ Qtarget
• Sharp ⇒ Produces planned Qpatch
• QPDD
• QuaternionicProbabilityDensityDistribution

Only exists at current instance

QPDD

Blurred allocation function

Curved space

• Function
• Blurred ⇒ Produces swarm ⇒ Qtarget
• Sharp ⇒ Produces planned Qpatch
• S⇒ Produces Qpattern
• QPDD
• QuaternionicProbabilityDensityDistribution

Only exists at current instance

QPDD

Blurred allocation function

Curved space

• Function
• Blurred ⇒ Produces QPDD ⇒ Qtarget
• Sharp ⇒ Produces planned Qpatch
• S ⇒ Produces Qpattern
• QPDD
• QuaternionicProbabilityDensityDistribution

Allocation function

Swarm

Hilbert space choices
• The Hilbert space and its Gelfand triple can be defined using
• Real numbers
• Complex numbers
• Quaternions
• The choice of the number system determines whether blurring is straight forward
Swarming conditions 1, 2 and 3
• In order to ensure sufficient coherence the correlation mechanism implements swarming conditions
• A swarm is a coherent set of step stones
• A swarm can be described by a continuous object density distribution
• That density distribution can be interpreted as a probability density distribution
Swarming condition 4
• A swarm moves as one unit
• In first approximation this movement can be described by a linear displacement generator
• This corresponds to the fact that the probability density distribution has a Fourier transform
• The swarming conditions result in the capability of the swarm to behave as part of interference patterns
Swarming conditions

The swarming conditions distinguish this type of swarm from normal swarms

Mapping Quality Characteristic
• The Fourier transform of the density distribution that describes the planned swarm can be considered as a mapping quality characteristic of the correlation mechanism
• This corresponds to the Optical Transfer Function that acts as quality characteristic of linear imaging equipment
• It also corresponds to the frequency characteristic of linear operating communication equipment
Quality characteristic
• Optics versus quantum physics
• In the same way that the Optical Transfer Function is the Fourier transform of the Point Spread Function
• Is the Mapping Quality Characteristic the Fourier transform of the QPDD, which describes the planned swarm. (The Qpattern)
• This view integrates over the set of progression steps that the embedding process takes to consume the full Qpattern, such that it must be regenerated
Target space
• The quality of the picture that is formed by an optical imaging system is not only determined by the Optical Transfer Function, it also depends on the local curvature of the imaging plane
• The quality of the map produced by quantum physics not only depends on the Mapping Quality Characteristic, it also depends on the local curvature of the embedding continuum
Coupling
• For swarms the coupling equation holds
• By requiring that the two sides of the quaternionic differential equation contain normalized functions, this equation turns into a coupling equation.
• and are normalized quaternionic functions
• They describe quaternionic probability density distributions
• is the quaternionic nabla
• Factor is the coupling strength
• P is the displacement generator
Swarms 1
• The correlation mechanism generates swarms of step stones in a cyclic fashion
• The swarm is prepared in advance of its usage
• This planned swarm is a set of placeholders that is called Qpattern
• A Qpattern is a coherent set of placeholders
• The step stones are used one by one
• In each static sub-model only one step stone is used per swarm
• This step stone is called Qtarget
• When all step stones are used, then a new Qpattern is prepared
Planned and actual swarm

Reference continuum

Placeholder generator

Embedding continuum

Swarm of step stones

Qtarget

Continuous allocation function

Set of placeholders

Qpattern

Random selection

Swarms 2
• At each progression step, an image of the planned swarm (Qpattern) is mapped by a continuous allocation function onto the embedding continuum
• At each progression step, via random selection a single step stone is selected, whose image becomes the Qtarget
• In fermions that step stone is not used again
• A swarm has a “center position”, called Qpatch that can be interpreted as the expectation value of location of the swarm
• The Qtargets form a stochastic micro-path
Placeholders and Step stones
• Together with the allocation function a placeholder defines where a selected particle can be
• That location is a step stone
• A coherent collection of these placeholders represent the Qpattern
• The placeholders are generated by the stochastic spatial spread function
• At each progression step a different step stone becomes the Qtarget location of the particle
Generation of placeholders and step stones
• Per progression step only ONE Qtarget is generated per Qpattern
• Generation of the whole Qpattern takes a large and fixed amount of progression steps
• When the Qpatch moves, then the pattern spreads out along the movement path
• When an event (creation, annihilation, sudden energy change) occurs, then the enumeration generation changes its mode
Qpattern generation example(no preferred directions)
• Random enumerator generation at lowest scales
• Let Poisson process produce smallest scale enumerator
• Combine this Poisson process with a binomial process
• This is installed by a 3D spread function
• Generates a 3d “Gaussian” distribution (is example)

The distribution represents an isotropic potential of the form

This quickly reduces to 1(form of gravitational potential)

• The result is a Qpattern
Blurred allocation function

Convolution

• Blurred function
• SharpmapsRQE⇒ Qpatch
• Function
• Produces QPDD
• Produces Qpattern
• Produces gravitation ()
• Sharp
• Describes space curvature
• Delivers local metric d
Micro-path
• The Qpatterns contain a fixed number of step stones
• The step stones that belong to a swarm form a micro-path
• Even at rest, the Qtarget walks along its micro-path
• This walk takes a fixed number of progression steps
• When the swarm moves or oscillates, then the micro-path is stretched along the path of the swarm
• This stretching is controlled by the third swarming condition
Wave fronts
• At every arrival of the particle at a new step stone the embedding continuum emits a wave front
• The subsequent wave fronts are emitted from slightly different locations
• Together, these wave fronts form super-high frequency waves
• The propagation of the wave fronts is controlled by Huygens principle
• Their amplitude decreases with the inverse of the distance to their source
Wave front
• Depending on dedicated Green’s functions, the integral over the wave fronts constitutes a series of potentials.
• The Green’s function describes the contribution of a wave front to a corresponding potential
• Gravitation potentials and electrostatic potentials have different Green’s functions
Potentials & wave fronts
• The wave fronts and the potentials are traces of the particle and its used step stones.
• The superposition of the singularities smoothens the effect of these singularities.
• Neither the emitted wave fronts, nor the potentials affect the particle that emitted the wave front
• Wave fronts interfere
• The wave fronts modulate a field
Palestra
• Curved embedding continuum
• Represents universe

The Palestra is the place where everything happens

Embedded in continuum

Collection of Qpatches

Mapping

Quantum fluid dynamics

Space curvature

Quantum physics

• Continuity equation
• Dirac equation
• In quaternion format

GR

Quaternionic metric

Quaternionic metric

16 partial derivatives

No tensor needed

Logic Systems

Lattices, classical logic and

quantum logic

Logic – Lattice structure
• A lattice is a set of elements that is closed for the connections ∩and ∪. These connections obey:
• The set is partially ordered. With each pair of elements belongs an element , such that and .
• The set is a ∩half lattice if with each pair of elements an element exists, such that .
• The set is a ∪half lattice if with each pair of elements an element exists, such that .
• The set is a lattice if it is both a ∩half lattice and a ∪half lattice.
Partially ordered set
• The following relations hold in a lattice:
• has a partial order inclusion ⊂:
• a ⊂ b ⇔ a ⊂ b = a
• A complementary lattice contains two elements and with each element a an complementary element a’
Orthocomplemented lattice
• Contains with each element an element such that:

Distributive lattice

Modular lattice

Classical logic is an orthocomplemented modular lattice

Weak modular lattice
• There exists an element such that
• where obeys:

Quantum logic obeys the weak modular law

Atoms
• In an atomic lattice

is an atom

Logics
• Classical logic has the structure of an orthocomplemented distributive modular and atomic lattice.
• Quantum logic has the structure of an orthocomplentedweakly modular and atomic lattice.
• Also called orthomodular lattice.
Hilbert Space
• The set of closed subspaces of an infinite dimensional separable Hilbert space forms an orthomodular lattice
• Is lattice isomorphic to quantum logic
Hilbert Logic
• Linearcombinations of atomicpropositions