The Hilbert Book Model

The Hilbert Book ModelA simple model of fundamental physicsBy J.A.J. van Leunenhttp://www.e-physics.eu

II am a retired physicist. I spent my career in high-tech industry. This lecture introduces you to the Hilbert Book Model.The paper that describes the model can be found at http://www.e-physics.eu/PhysicsOfTheHilbertBookModel.pdf .The Hilbert Book Model represents my personal view on the lower layers of fundamental physics.The model deviates considerably from the models that constitute contemporary physics.The model is largely deduced. For that reason it is strictly based on a well-accepted foundation.The Hilbert Book Model selects quantum logic for that purpose. As a consequence the underlying space-progression model becomes paginated. It consists of an ordered sequence of static sub-models.1The Hilbert Book ModelA simple model of fundamental physicsBy J.A.J. van Leunenhttp://www.e-physics.eu

III started my investigations during my studies in the sixties, when I wondered about the strong differences in the way that classical physics was done and the way that quantum physics was done. I also wondered about the fact that observables were restricted to integers or real numbers, while our world is obviously a three plus one dimensional structure. In those years I was able to partly resolve these questions.

2The Hilbert Book ModelA simple model of fundamental physicsBy J.A.J. van Leunenhttp://www.e-physics.eu

IIIAfter my studies I started my career in high-tech industries in the development of image intensifiers. There I was intensively confronted with quanta. On the output screen of these devices I could only see hail storms of impinging quanta. I did not observe any sign of impinging waves. This intrigued me, but there was not much time to deliberate about these facts. It had to wait until after my retirement until I got sufficient time to (re)start my investigations. In 2009, when I was 68 old, I started a personal research project that in 2011 got its final name The Hilbert Book Model project.

3Physical RealityIn 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.

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No model of physics can fully describe physical reality. It offers an abstracted view.4Rules Restrict ComplexityPhysical reality applies rules for relational structures that it acceptsThese rules intent to reduce the complexity of these relational structures5

Models can handle a limited degree of complexity. Thus models must restrict to an abstraction of physical reality.5ComplexityPhysical reality is very complicatedIt seems to belie Occams razor. However, views on reality that apply sufficient abstraction can be rather simpleIt is astonishing that such simple abstractions exist6

Models can handle a limited degree of complexity. Thus models must restrict to an abstraction of physical reality.6What is complexity?Complexity is caused by the number and the diversity of the relations that exist between objects that play a roleA simple model has a small diversity of its relations.

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The abstraction must reduce the original complexity.7Rules and relational StructuresThe 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. 8Logic

Lattice theory treats isomorphism's between relational structures.Logic systems are special types of relational structures.

8Physical Reality & MathematicsPhysical 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.

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Some mathematical relational structures appear to be lattice isomorphic to relational structures that describe observations of physical reality.9Logic systemsClassical logic and quantum logic only describe the relational structure of sets of propositionsThe content of these proposition is not part of the specification of their axiomsThe logic systems only control static relationsTheir specification does not cover dynamics

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The specification of classical logic and the specification of quantum logic does not cover the contents of propositions. Neither classical logic nor quantum logic contains a mechanism that supports dynamics.10FundamentThe 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.11

The HBM is strictly based on quantum logic. That foundation is lattice isomorphic with the set of closed subspaces of an infinite dimensional separable Hilbert space. This second (sub)model is far more suited for formulating physical laws.11First ModelTraditional Quantum LogicSeparable Hilbert SpaceCountable EigenspaceOnly static status quo&No fieldsClassical LogicisomorphismSeparable Hilbert SpaceParticle location operatorWeaker modularityAbout 25 axioms

The HBM uses an infinite dimensional separable quaternionic Hilbert space. The Hilbert space operators have a countable number of eigenvalues.12Three alarming factsThe first level model does not support continuumsHS operators have countable eigenspacesThe first level model does not support dynamicsCan only represent static status quoThe Hilbert space contains deeper details than quantum logic doesQL propositions HS sub-spacesHL refined propositions HS vectors13

IsomorphismsRelational

structureQuantum

LogicHilbert

spaceSet

of

particlesQuantum LogicAtomic

quantum

logic

proposition

SubspaceParticle

is swarm

of step

stonesHilbert

logicAtomic

Hilbert

Logic

propositionBase vectorStep stone14Threefold hierarchyPossible interpretation of isomorphisms

Three different structures exist that each show three levels of the hierarchy.14Physical modelThe 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.

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Static RepresentationQuantum logic

Hilbert space

}No full isomorphismCannot represent continuumsSolution:

Refine to Hilbert logicAdd Gelfand triple

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The isomorphism is not complete and does not offer continuums16Discrete sets and continuumsA Hilbert space features operators that have countable eigenspaces

A Gelfand triple features operators that have continuous eigenspaces

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The model must support discrete sets as well as continuums.17Static Status Quo of the UniverseTraditional Quantum LogicSeparable Hilbert SpaceSeparable Hilbert SpaceSeparable Hilbert SpaceGelfand TripleCountable EigenspaceContinuumEigenspaceParticle locationlocationClassical LogicHilbert LogicisomorphismsIsomorphismsSubspacesvectorsembedding

Both the quantum logic and the Hilbert space are extended18Implementing dynamicsThe sub-models can only implement a static status quo

The sub-models are static. A dynamic model consists of an ordered sequence of these sub-models.19RepresentationQuantum logic

Hilbert logic

Hilbert space

}Cannot represent dynamicsCan only implement a static status quoSolution:

An ordered sequence of sub-models

The model looks like a book where the sub-models are the pages. 20

The result looks like the sequence of the pages of a book. That is why the model is called the Hilbert Book Model20Sequence |-|-|-|-|-|-|-|-|-|-|-|-| |-|-|-|-|-|-|-|-|-| PrehistoryReference sub-model hasdensest packagingcurrentfutureReference 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 continuumHBM has no Big Bang!21

We will select a state of highest packaging as the reference sub-model. The reference Hilbert space delivers flat Rational Quaternionic Enumerators (RQEs) that are eigenvalues of an enumerator operator. The corresponding Gelfand triple will deliver the reference continuum, which is eigenspace of the corresponding enumeration operator in the Gelfand triple.21The Hilbert Book ModelSequence book HBMSub-models sequence members pages

Sequence number page number progression parameter

This results in a

paginated space-progression model22

In order to avoid complete chaos, a correlation vehicle must install sufficient coherence between subsequent elements. Coherence must not be too stiff otherwise no dynamics will occur.22Paginated space-progression modelSteps through sequence of static sub-modelsUses a model-wide clockIn the HBM the speed of information transfer is a model-wide constantThe step size is a smooth function of progressionSpace expands/contracts in a smooth way23

A paginated space-progression model uses a model-wide clock. In the HBM the speed of information transfer is a constant.23Progression stepThe dynamic model proceeds with universe wide progression stepsThe progression steps have a rather fixed sizeThe progression step size corresponds to an super-high frequency (SHF)The SHF is the highest frequency that can occur in the HBM24

The model proceeds with universe wide progression steps.The progression step size is rather fixed and corresponds to an ultra-high frequency, which is the highest frequency that occurs in the model24RecreationThe whole universe is recreated at every progression step

If no other measures are taken,the model will represent dynamical chaos25

In a paginated space-progression model, the model can be considered to be recreated at every progression step.25Dynamic coherence 1An external correlation mechanism must take care such that sufficient coherence between subsequent pages exist

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In order to prevent dynamical chaos, an external mechanism must establish sufficient coherence between subsequent steps.26Dynamic coherence 2The coherence must not be too stiff, otherwise no dynamics occurs

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A too stiff coherence will prevent that dynamics occurs.27StorageThe eigenspaces of operators can act as storage places28

The countable eigenspaces of Hilbert space operators and the continuum eigenspaces of the Gelfand triple can act as storage places of information that changes with progression. The eigenvectors store the corresponding relations.28Storage detailsStorage places of information that changes with progressionThe countable eigenspaces of Hilbert space operators The continuum eigenspaces of the Gelfand triple The information concerns the contents of logic propositionsThe eigenvectors store the corresponding relations.

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Storage places contain information that changes with progression. That information concerns content of logic propositions29Correlation VehicleSupports recreation of the universe at every progression stepMust install sufficient cohesion between the subsequent sub-modelsOtherwise the model will result in dynamic chaos.Coherence must not be too stiff, otherwise no dynamics occurs30

In order to avoid dynamical chaos, a correlation vehicle must install sufficient coherence between subsequent members of the sequence. However, the coherence must not be too stiff otherwise no dynamics occurs. 30Correlation Vehicle DetailsEstablishesEmbedding of particles in continuumCausesSingularities at the location of the embeddingSupported by:Hilbert space (supports operators)Gelfand triple (supports operators)Huygens principle (controls information transport)Implemented by:Enumeration operatorsBlurred allocation functionRequires identification of atoms / base vectors

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The correlation mechanism uses the properties of the static sub-models in order to implement its task. It applies the eigenspaces of operators as storage places and a blurred allocation function in order to offer a location to the particles. It uses the Huygens principle in order to control the spread of the influence of the singularities that are caused by the embedding process.Its functionality requires the ability to identify the atoms of the quantum logic system. This will be implemented via an enumeration scheme. The enumerator mechanism must not introduce extra functionality or properties for the enumerated objects

31Correlation vehicle requirementsRequires IDs for atomic propositionsID generatorDedicated enumeration operatorEigenvalues rational quaternions enumeratorsBlurred allocation functionMaps parameter enumerators onto embedding continuumRequires a reference continuum

RQE = RationalQuaternionicEnumerator32

The correlation vehicle must be able to identify the atoms. This will be arranged by an enumerator mechanism that uses the eigenvalues of an operator in the isomorphic companion of the quantum logic system. The action of the operator can be described by an enumeration function. Both the operator and the function produce rational quaternionic enumerators.32EnumerationHilbert space & Hilbert logicEnumerator operatorEigenvaluesRational quaternionic enumerators(RQEs)

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The enumerator operator is not a common normal operator.33Allocation34

The allocation operator corresponds to an allocation function that uses RQEs as its parameters and Qtargets as its images.34Enumeration & AllocationHilbert space & Hilbert logicEnumerator operatorEigenvaluesRational quaternionic enumerators(RQEs)ModelEnumeration functionParametersRQEsImageQtargets

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The blurred allocation function is the convolution of a sharp continuous allocation function and a spread function.35Enumeration & Allocation & BlurHilbert space & Hilbert logicEnumerator operatorEigenvaluesRational quaternionic enumerators(RQEs)ModelEnumeration functionParametersRQEsImageQtargets

Swarm36

Locally the enumeration produces a Qpattern that acts as a blur and is deformed by the sharp part of the enumeration function.36

Swarm37ConvolutionDescribed by the QPDDQPDD

Like the overall (blurred) allocation function, the sharp allocation function use RQEs as parameters. The sharp allocation function produces planned Qpatches as its targets. The spread function parameters creates a Qpattern. The overall function creates Qtargets that together are described by a QPDD.A QPDD is a quaternionic probability density distribution.

37Only exists at current instance

38Convolution

Qtargets exist only at single instance. At that instance an object can be detected at that location. This corresponds with the idea that the enumeration generator generates only one enumerator of the Qpattern per progression step.38Only exists at current instance

Curved space39

The subsequent Qpatches are locations that are smoothly, but non-uniformly positioned in a multidimensional continuum. In this way they create the illusion of a curved space.39Only exists at current instance

Curved space40

Swarms define the local shape of the blur. The swarm is deformed by the local space curvature.40

Curved space41Allocation function

Qpatches indicate the location of swarm.41Hilbert space choicesThe Hilbert space and its Gelfand triple can be defined usingReal numbersComplex numbersQuaternions

The choice of the number system determines whether blurring is straight forward

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Three Hilbert space types are possible42Swarming conditions 1, 2 and 3In order to ensure sufficient coherence the correlation mechanism implements swarming conditionsA swarm is a coherent set of step stonesA swarm can be described by a continuous object density distributionThat density distribution can be interpreted as a probability density distribution

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Quantum physics uses a special kind of swarming that is determined by three swarming conditions43Swarming condition 4A swarm moves as one unitIn first approximation this movement can be described by a linear displacement generatorThis 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

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Like natural swarms the QP swarm in first approximation moves as one unit44Swarming conditionsThe swarming conditions distinguish this type of swarm from normal swarms45

No comment45Mapping Quality CharacteristicThe Fourier transform of the density distribution that describes the planned swarm can be considered as a mapping quality characteristic of the correlation mechanismThis corresponds to the Optical Transfer Function that acts as quality characteristic of linear imaging equipmentIt also corresponds to the frequency characteristic of linear operating communication equipment46

The swarming activity of the correlation mechanism can be characterized by a mapping quality characteristic.46Quality characteristicOptics versus quantum physicsIn the same way that the Optical Transfer Function is the Fourier transform of the Point Spread FunctionIs 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 regenerated47

The Fourier transform of the spread function is a quality characteristic.47Target spaceThe 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 planeThe 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 continuum48

The curvature of the target space also determines the quality of the produced image/map.48Coupling49

By requiring that the two sides of the quaternionic differential equation contain normalized functions, this equation turns into a coupling equation.

Since swarms move as one unit, a displacement generator exists that describes infinitesimal movement. This means that the quaternionic nabla forms a coupling equation.49Swarms 1The correlation mechanism generates swarms of step stones in a cyclic fashionThe swarm is prepared in advance of its usageThis planned swarm is a set of placeholders that is called QpatternA Qpattern is a coherent set of placeholdersThe step stones are used one by oneIn each static sub-model only one step stone is used per swarmThis step stone is called QtargetWhen all step stones are used, then a new Qpattern is prepared50

A large part of the correlation mechanism is the regular recreation of swarms.50Planned and actual swarm51Swarm of step stonesSet of placeholdersQpatternRandom selectionQtargetEmbedding continuumReference continuum

51Swarms 2At each progression step, an image of the planned swarm (Qpattern) is mapped by a continuous allocation function onto the embedding continuumAt each progression step, via random selection a single step stone is selected, whose image becomes the QtargetIn fermions that step stone is not used againA swarm has a center position, called Qpatch that can be interpreted as the expectation value of location of the swarmThe Qtargets form a stochastic micro-path52

At each progression step a new step stone is selected in order to become the current Qtarget. This step stone is not used again. After consuming all step stones, a new Qpattern is generated. The Qpatches form a movement path.The Qtargets form a micro path. The micro-path stretches along the particles movement path.52Placeholders and Step stones53

Building blocks are represented by a coherent set of step stones that are generated by a stochastic process. The step stones are place holders of locations where the building block can be. The step stones form a stochastic micro-path.53Generation of placeholders and step stonesPer progression step only ONE Qtarget is generated per QpatternGeneration of the whole Qpattern takes a large and fixed amount of progression stepsWhen the Qpatch moves, then the pattern spreads out along the movement pathWhen an event (creation, annihilation, sudden energy change) occurs, then the enumeration generation changes its mode54

At each progression step a new step stone becomes the Qtarget of the building block.The Qpattern covers a fixed number of step stones that together form a stochastic micro-path. The micro-path stretches along the movement path of the building block.An event means the generation, annihilation or a sudden energy change of the building block.54Qpattern generation example (no preferred directions)55

The solution is found in randomizing the low scale enumeration process. After sufficient enumerator generations, any order will be destroyed.The stochastic enumeration generator is implemented by a Poisson process. That process is attenuated by a binomial process that is installed by a 3D spread function. The result approaches a 3D Gaussian distribution. That distribution corresponds to a potential that off-center has the shape of the single charge gravitation potential. 55Convolution56

Now the enumeration function gets a double functionality. It describes space curvature via de differential of its sharp part and it describes the local blur that is caused by the random enumeration process. When Qpatterns are (relatively) static, then the pattern corresponds to a single charge gravitation potential.56Micro-pathThe Qpatterns contain a fixed number of step stonesThe step stones that belong to a swarm form a micro-pathEven at rest, the Qtarget walks along its micro-pathThis walk takes a fixed number of progression stepsWhen the swarm moves or oscillates, then the micro-path is stretched along the path of the swarmThis stretching is controlled by the third swarming condition57

Even at rest the building block walks along its micro-path. The duration of this walk is fixed. The micro-path is stretched along the movement path of the building block. Thus the building block walks its path in a stochastic way. The micro-path has a fixed cycle time. After that cycle the set of step stones is regenerated by the stochastic process. Only the statistical characteristics of the set keep the same.57Wave frontsAt every arrival of the particle at a new step stone the embedding continuum emits a wave frontThe subsequent wave fronts are emitted from slightly different locationsTogether, these wave fronts form super-high frequency wavesThe propagation of the wave fronts is controlled by Huygens principleTheir amplitude decreases with the inverse of the distance to their source

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At every arrival at a step stone, the building block emits a wave front that carries a message about it presence and its properties. These properties become visible in the potentials that are formed by the wave fronts. 58Wave frontDepending on dedicated Greens functions, the integral over the wave fronts constitutes a series of potentials.The Greens function describes the contribution of a wave front to a corresponding potentialGravitation potentials and electrostatic potentials have different Greens functions59

A dedicated Greens function describes the contribution of the wave front to the potential.59Potentials & wave frontsThe 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 interfereThe wave fronts modulate a field

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The emitted wave fronts leave the building block and do not affect that building block. Still they offer noticeable traces of the existence of the building block.

Together, the wave fronts form super-high frequency waves. The averaged effect of these SHF waves form the potentials of the particle.60PalestraCollection of QpatchesEmbedded in continuum61

The Palestra consists of the embedding continuum, which is covered by a set of Qpatches. 61MappingSpace curvatureQuantum physicsGRQuaternionic metricQuaternionic metric16 partial derivativesNo tensor neededQuantum fluid dynamics62

The blurred allocation function is the base of both a quaternionic gravitation theory and a quaternionic quantum physics that is based on quantum fluid dynamics.62Logic SystemsLattices, classical logic and quantum logic63

Classical logic and quantum logic are lattices.

63Logic Lattice structure64

Lattices are define by a set of axioms.

64Partially ordered set The following relations hold in a lattice:

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Lattices are partially ordered sets.65Orthocomplemented lattice66

An orthocomplemented lattice is a special kind of lattice.A distributive lattice obeys the distributive law.A modular lattice obeys the modular law.

66Weak modular lattice67

A weak modular lattice obeys the weak modular law.

67Atoms68

Atomic lattices contain atomic propositions.68LogicsClassical logic has the structure of an orthocomplemented distributive modular and atomic lattice.Quantum logic has the structure of an orthocomplented weakly modular and atomic lattice. Also called orthomodular lattice.

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No comment69Hilbert SpaceThe set of closed subspaces of an infinite dimensional separable Hilbert space forms an orthomodular latticeIs lattice isomorphic to quantum logic70

No comment70Hilbert LogicAdd linear propositionsLinear combinations of atomic propositionsAdd relational coupling measureEquivalent to inner product of Hilbert spaceClose subsets with respect to relational coupling measure

Propositions subspacesLinear propositions Hilbert vectors 71

No comment71Superposition principleLinear combinations of linear propositions are again linear propositions that belong to the same Hilbert logic system72

IsomorphismLattice isomorhicPropositions closed subspaces

Topological isomorphicLinear atoms Hilbert base vectors73