The Hilbert Book Model

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

2. 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.

3. 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

4. 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.

5. 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 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.

6. 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.

7. 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 • They only control static relations • Their specification does not cover dynamics

8. 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.

9. Correspondences • ≈1930 Garret Birkhoff and John von Neumann discovered the lattice isomorphy: • Infinite, butcountablenumber ofatoms / base vectors

10. Atoms & base vectors • Atom • Contents not important • Set is unordered • Many sets possible • Logic • Lattice • Only relations important

11. Atoms & base vectors • Atom • Contents not important • Set is unordered • Many sets possible • Base vector • Set is unordered • Many sets possible • Can be eigenvector • Eigenvalue • Real • Complex • Quaternionic • Logic • Lattice • Only relations important

12. Atoms & base vectors • Atom • Contents not important • Set is unordered • Many sets possible • Base vector • Set is unordered • Many sets possible • Can be eigenvector • Eigenvalue • Real • Complex • Quaternionic • Hilbert space • Inner product • Real • Complex • Quaternionic Constantin Piron: Inner product must be real, complex or quaternionic The eigenvalues are the same type of numbers as the inner products

13. 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

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

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

16. 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

17. Threefold hierarchy The sub-models form a threefold hierarchy

18. Three structures, three levels

19. No support for dynamics None of the three structures has a built-in mechanism for supporting dynamics

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

21. 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.

22. 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!

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

24. 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

25. 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

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

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

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

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

30. 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.

31. 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

32. 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

33. 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

34. Reference continuum • Select a reference Hilbert space • Has countable number of dimensions/base vectors • Criterion is densest packaging of enumerators*. • Take its Gelfand triple (rigged “Hilbert space”) • Has over-countable number of dimensions/base vectors • Has operators with continuum eigenspaces • Select equivalent of enumeration operator in Hilbert space • Use its eigenspace as reference continuum (*Cyclic: Densest with respect to reference continuum)

35. Atoms & base vectors • Atom • Set is unordered • Many sets possible • Contents not important • Base vector • Set is unordered • Many sets possible • Can be eigenvector • Eigenvalue • Real • Complex • Quaternionic

36. Atoms & base vectors • Hilbert space & Hilbert logic • Inner product • Real • Complex • Quaternionic • Enumerator operator • Eigenvalues • Rational quaternionic enumerators(RQE’s) • Enumerates atoms • Atom • Set is unordered • Many sets possible • Contents not important • Base vector • Set is unordered • Many sets possible • Can be eigenvector • Eigenvalue • Real • Complex • Quaternionic

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

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

39. Enumeration • 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

40. Enumeration • 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

41. 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

42. 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

43. 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

44. 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

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

46. 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

47. Real Hilbert space model • Progression separated • Use rational eigenvalues in Hilbert space • Use real eigenvalues in Gelfand triple • Cohesion not too stiff (otherwise no dynamics!) • Keep sufficient interspacing ⇛ 1D blur ? • Define lowest rational • May introduce scaling as function of progression • Rather fixed progression steps

48. Complex Hilbert space model • Progression at real axis • Use rational complex numbers in Hilbert space • Use real complex numbers in Gelfand triple • Cohesion not too stiff (otherwise no dynamics!) • Keep sufficient interspacing ⇛ 1D blur ? • Lowest rational at both axes (separately) • May introduce scaling as function of progression • No scaling or blur at progression axis

49. Quaternionic Hilbert space model • Progression at real axis • Use rational quaternions in Hilbert space and in Gelfand triple • Cohesion not too stiff (otherwise no dynamics!) • Keep sufficient interspacing⇛ 3D blur • Lowest rational at all axes (same for imaginary axes) • May introduce scaling as function of progression • No scaling and no blur at progression axis • Blur installed by correlation vehicle

50. Why blurred ? • Pre-enumerated objects (atoms, base vectors) are not ordered • No origin • Affine-like space • Enumeration must not introduce extra properties • No preferred directions