The Hilbert Book Model

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

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

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

4. 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.

5. Rules Restrict Complexity • Physical reality applies rules for relational structures that it accepts • These rules intent to reduce the complexity of these relational structures

6. 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

7. 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.

8. 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.

9. 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.

10. 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

11. 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.

12. First Model About 25 axioms Classical Logic Only static status quo & No fields Separable Hilbert Space Separable Hilbert Space Weaker modularity isomorphism Traditional Quantum Logic Particle location operator Countable Eigenspace

13. 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

14. Threefold hierarchy Possible interpretation of isomorphisms

15. 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.

16. Static Representation Quantum logic Hilbert space } • No full isomorphism • Cannot represent continuums • Solution: • Refine to Hilbert logic • Add Gelfand triple

17. Discrete sets and continuums • A Hilbert space features operators that have countable eigenspaces • A Gelfand triple features operators that have continuous eigenspaces

18. Static Status Quo of the Universe Separable Hilbert Space Classical Logic Separable Hilbert Space Gelfand Triple Subspaces Separable Hilbert Space Traditional Quantum Logic location ContinuumEigenspace Particle location isomorphisms Isomorphism’s Hilbert Logic embedding Countable Eigenspace vectors

19. Implementing dynamics The sub-models can only implement a static status quo

20. 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.

21. 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!

22. The Hilbert Book Model • Sequence ⇔book⇔ HBM • Sub-models ⇔ sequence members ⇔pages • Sequence number ⇔page number ⇔ progression parameter • Thisresults in a paginatedspace-progression model

23. 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

24. 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

25. Recreation • The whole universe is recreated at every progression step • If no other measures are taken,the model will represent dynamical chaos

26. Dynamic coherence 1 An external correlation mechanism must take care such that sufficient coherence between subsequent pages exist

27. Dynamic coherence 2 The coherence must not be too stiff, otherwise no dynamics occurs

28. Storage The eigenspaces of operators can act as storage places

29. 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.

30. 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

31. 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

32. 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

33. Enumeration • Hilbert space & Hilbert logic • Enumerator operator • Eigenvalues • Rational quaternionic enumerators(RQE’s)

34. Allocation • Hilbert space & Hilbert logic • Enumerator operator • Eigenvalues • Rational quaternionic enumerators(RQE’s) • Model • Allocation function • Parameters • RQE’s • Image • Qtargets

35. 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 • Spread function • Blur

36. 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 • Spread function • Blur

37. 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

38. 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

39. Blurred allocation function Curved space • Function • Blurred ⇒ Produces swarm ⇒ Qtarget • Sharp ⇒ Produces planned Qpatch • Spread function⇒ Produces Qpattern • QPDD • QuaternionicProbabilityDensityDistribution Only exists at current instance QPDD

40. 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

41. Blurred allocation function Curved space • Function • Blurred ⇒ Produces QPDD ⇒ Qtarget • Sharp ⇒ Produces planned Qpatch • S ⇒ Produces Qpattern • QPDD • QuaternionicProbabilityDensityDistribution Allocation function Swarm

42. 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

43. 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

44. 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

45. Swarming conditions The swarming conditions distinguish this type of swarm from normal swarms

46. 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

47. 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

48. 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

49. 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

50. 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