A simple model of fundamental physics By J.A.J. van Leunen 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. 2 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 3 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. 4 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. 5 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. 6 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 7 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. 8 Correspondences ≈1930 Garret Birkhoff and John von Neumann discovered the lattice isomorphy: Infinite, but countable number of atoms / base vectors Quantum logic Propositions: 𝑎, 𝑏 atoms 𝑐, 𝑑 Relational complexity: 𝐶𝑜𝑚𝑝𝑙𝑒𝑥𝑖𝑡𝑦 𝑎 ∩ 𝑏 Inclusion: 𝑎 ∪ 𝑏 For atoms 𝑐𝑖 : Hilbert space Vectors: |𝑎⟩, |𝑏⟩ Base vectors: |𝑐⟩, |𝑑⟩ Inner product: 𝑎|𝑏 Sum: |𝑎⟩ + |𝑏⟩ Subspace 𝑐𝑖 𝒊 𝛼𝑖 |𝑐𝑖 ⟩ 𝑖 ∀ 𝛼𝑖 9 Atoms & base vectors Atom Contents not important Set is unordered Many sets possible Logic Lattice Only relations important 10 Atoms & base vectors Atom Contents not important Set is unordered Many sets possible Base vector Set is unordered Many sets possible Can be eigenvector Logic Lattice Only relations important Eigenvalue Real Complex Quaternionic 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 Hilbert space Inner product Real Complex Quaternionic Constantin Piron: Inner product 𝑥|𝑦 must be real, complex or quaternionic 𝑎|𝑃𝑎 = 𝑎|𝑝𝑎 = 𝑎|𝑎 𝑝 The eigenvalues are the same type of numbers as the inner products 12 First Model About 25 axioms Classical Logic Separable Hilbert Space Weaker modularity isomorphism Traditional Quantum Logic Particle location operator Countable Eigenspace Only static status quo & No fields Representation Quantum logic Hilbert space } No full isomorphism Cannot represent continuums Solution: • Refine to Hilbert logic • Add Gelfand triple 14 Discrete sets and continuums A Hilbert space features operators that have countable eigenspaces A Gelfand triple features operators that have continuous eigenspaces 15 Static Status Quo of the Universe Classical Logic Separable Hilbert Separable Hilbert Space SpaceTriple Gelfand Subspaces Separable Hilbert Space Traditional Quantum Logic isomorphisms Isomorphism’s Particle location location Continuum Eigenspace embedding Hilbert Logic vectors Countable Eigenspace The sub-models form a threefold hierarchy Three structures, three levels Relational structure Isomorphisms Quantum Hilbert Logic space Quantum Logic Atomic Subspace quantum logic proposition Hilbert logic Atomic Hilbert Logic proposition Base vector Set of particles Swarm of step stones Step stone 18 No support for dynamics None of the three structures has a built-in mechanism for supporting dynamics 19 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. 21 Sequence · · · |-|-|-|-|-|-|-|-|-|-|-|-| · · · · · · · · · · |-|-|-|-|-|-|-|-|-| · · · Prehistory Reference sub-model has densest packaging 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 This results in a paginated space-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 sufficient cohesion between 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 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) 34 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 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 Hilbert space & Hilbert logic Inner product Real Complex Quaternionic Enumerator operator Eigenvalues Rational quaternionic enumerators (RQE’s) Enumerates atoms 36 Enumeration Hilbert space & Hilbert logic Enumerator operator Eigenvalues Rational quaternionic enumerators (RQE’s) 37 Enumeration Hilbert space & Hilbert logic Enumerator operator Eigenvalues Rational quaternionic enumerators (RQE’s) Model Allocation function 𝒫 Parameters RQE’s Image Qtargets 38 Enumeration Hilbert space & Hilbert logic Enumerator operator Eigenvalues Rational quaternionic enumerators (RQE’s) Model Enumeration function Parameters RQE’s Image Qtargets Function 𝒫 = ℘ ∘ 𝒮 Blurred 𝒫 Sharp ℘ Spread function 𝒮 Blur 𝜓 39 Enumeration Hilbert space & Hilbert logic Enumerator operator Eigenvalues Rational quaternionic enumerators (RQE’s) Model Enumeration function Parameters RQE’s Image Qtargets Swarm Function 𝒫 = ℘ ∘ 𝒮 Blurred 𝒫 Sharp ℘ Spread function 𝒮 Blur 𝜓 40 Blurred allocation function 𝒫 Convolution Function 𝒫 = ℘ ∘ 𝒮 Blurred 𝒫 Sharp ℘ Spread function 𝒮 QPDD Quaternionic Probability Density Distribution ⇒ Produces swarm ⇒ Qtarget ⇒ Produces planned Qpatch ⇒ Produces Qpattern ⇒ Swarm ⇓ QPDD Described by the QPDD Swarm 41 Blurred allocation function 𝒫 Convolution Function 𝒫 = ℘ ∘ 𝒮 Blurred 𝒫 Sharp ℘ Spread function 𝒮 QPDD 𝜓 Quaternionic Probability Density Distribution ⇒ Produces swarm ⇒ Qtarget ⇒ Produces planned Qpatch ⇒ Produces Qpattern Only exists at current instance QPDD 𝜓 42 Blurred allocation function 𝒫 Function 𝒫 = ℘ ∘ 𝒮 Blurred 𝒫 Sharp ℘ Spread function 𝒮 QPDD 𝜓 Quaternionic Probability Density Distribution Curved space ⇒ Produces swarm ⇒ Qtarget ⇒ Produces planned Qpatch ⇒ Produces Qpattern Only exists at current instance QPDD 𝜓 43 Blurred allocation function 𝒫 Function 𝒫 = ℘ ∘ 𝒮 Blurred 𝒫 Sharp ℘ Spread function 𝒮 QPDD 𝜓 Quaternionic Probability Density Distribution Curved space ⇒ Produces swarm ⇒ Qtarget ⇒ Produces planned Qpatch ⇒ Produces Qpattern Only exists at current instance QPDD 𝜓 44 Blurred allocation function 𝒫 Function 𝒫 = ℘ ∘ 𝒮 Blurred 𝒫 Sharp ℘ Spread function 𝒮 QPDD 𝜓 Quaternionic Probability Density Distribution Curved space ⇒ Produces QPDD ⇒ Qtarget ⇒ Produces planned Qpatch ⇒ Produces Qpattern Allocation function Swarm 𝜓 45 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 46 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 47 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 48 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 49 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 50 Swarming conditions 1 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 51 Swarming conditions 2 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 52 Swarming conditions The swarming conditions distinguish this type of swarm from normal swarms 53 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 54 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 55 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 56 Coupling For swarms the coupling equation holds Φ = 𝛻𝜓 = 𝑚 𝜑 𝜓 and 𝜑 are normalized quaternionic functions They describe quaternionic probability density distributions 𝛻 is the quaternionic nabla Factor 𝑚 is the coupling strength P𝜓 = 𝑚 𝜑 P is the displacement generator 57 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 58 Planned and actual swarm Reference continuum Swarm of step stones Placeholder generator 𝒮 Embedding continuum 𝒫 =℘∘𝒮 Qtarget Set of placeholders Qpattern Continuous allocation function ℘ Random selection 59 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 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 60 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 𝒮 At each progression step a different step stone becomes the Qtarget location of the particle 61 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 62 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 𝐸𝑟𝑓(𝑟) 𝑟 This quickly reduces to 1/𝑟 (form of gravitational potential) The result is a Qpattern 63 Blurred allocation function 𝒫 Convolution Blurred function 𝒫 = ℘ ∘ 𝒮 Sharp ℘ Spread 𝒮 maps RQE maps Qpatch ⇒ Qpatch ⇒ Qtarget Function 𝒫 Produces QPDD 𝜓 Stochastic spatial spread function 𝒮 Produces Qpattern Produces gravitation (1/𝑟) Sharp ℘ Describes space curvature Delivers local metric d ℘ 64 Micro-path The Qpatterns contain a fixed number of step stones The step stones that belong to a Qpattern form a micro-path Even at rest, the Qpattern walks along its micro-path This walk takes a fixed number of progression steps When the Qpattern moves or oscillates, then the micro-path is stretched along the path of the Qpattern This stretching is controlled by the third swarming condition 65 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 66 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 67 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 68 Photon & gluon emission A sudden decrease in the energy of the emitting particle causes a modulation of the amplitude of the emitted wave fronts The creation of this modulation lasts a full micro-walk The modulation of the SHF carrier wave becomes observable as a photon or a gluon The modulation represents an energy quantum E=ℏ∙ν The energy is shown in the modulation frequency ν 69 Embedding continuum A curved continuum embeds the elementary particles The continuum is constituted by a background field 70 Photon & gluon absorption A modulation of the embedding continuum can be absorbed by an elementary particle The modulation frequency determines the absorbable energy quantum The modulation must last during a full micro-walk 71 Photons and gluons Photons and gluons are energy quanta Photons and gluons are NOT electro- magnetic waves! Photons and gluons are NOT particles 72 Palestra Curved embedding continuum Represents universe Embedded in continuum 𝑄𝑝𝑎𝑡𝑐ℎ Collection of Qpatches The Palestra is the place where everything happens 73 Mapping 𝒫 =℘∘𝒮 Space curvature GR Quantum physics Quaternionic metric 𝑑𝒫 16 partial derivatives No tensor needed Quantum fluid dynamics • Continuity equation 𝛻𝜓 = 𝜙 • Dirac equation 𝛻0 𝜓 + 𝛁𝛂 𝜓 • In quaternion format 𝛻𝜓 = 𝑚𝜓 ∗ 74 How to use Quaternionic Distributions and Quaternionic Probability Density Distributions The HBM is a quaternionic model The HBM concerns quaternionic physics rather than complex physics. The peculiarities of the quaternionic Hilbert model are supposed to bubble down to the complex Hilbert space model and to the real Hilbert space model The complex Hilbert space model is considered as an abstraction of the quaternionic Hilbert space model This can only be done properly in the right circumstances 76 Continuous Quaternionic Distributions Quaternions 𝑎 = 𝑎0 + 𝒂 c = 𝑎𝑏 = 𝑎0 𝑏0 − 𝒂, 𝒃 + 𝑎0 𝒃 + 𝑏0 𝒂 + 𝒂 × 𝒃 Quaternionic distributions Two equations Differential equation g = 𝛻𝑓 = 𝛻0 𝑓0 − 𝛁, 𝒇 + 𝛻0 𝒇 + 𝛁𝑏0 + 𝛁 × 𝒃 Three kinds 𝜙 = 𝛻𝜓 = 𝑚 𝜑 { 𝑔0 = 𝛻0 𝑓0 − 𝛁, 𝒇 𝐠 = 𝛻0 𝒇 + 𝛁𝑏0 + 𝛁 × 𝒃 Differential Coupling Continuity } equation 77 Field equations 𝜙 = 𝛻𝜓 𝜙0 = 𝛻0 𝜓0 − 𝛁, 𝜓 𝝓 = 𝛻0 𝜓 + 𝛁𝜓0 + 𝛁 × 𝜓 Spin of a field: 𝜮𝑓𝑖𝑒𝑙𝑑 = 𝕰 × 𝝍 𝑑𝑉 𝑉 𝕰 ≡ 𝛻0 𝝍 + 𝜵𝜓0 𝕭≡𝜵×𝝍 𝝓=𝕰+𝕭 𝐸≡ 𝜙 = 𝜙0 𝜙0 + 𝝓, 𝝓 = 𝜙0 𝜙0 + 𝕰, 𝕰 + 𝕭, 𝕭 + 𝟐 𝕰, 𝕭 Is zero ? 78 QPDD’s Quaternionic distribution 𝑓 = 𝑓0 + 𝒇 Scalar field Vector field Quaternionic Probability Density Distribution 𝜓 = 𝜓0 + 𝝍 = 𝜌0 + 𝜌0 𝒗 Density distribution Current density distribution 79 Coupling equation Differential 𝜙 = 𝛻𝜓 = 𝑚𝜑 𝜓 = 𝜑 𝜓 and φ are normalized Integral 𝑚 = total energy = rest mass + kinetic energy 𝜓 2 𝑉 𝜑 𝑉 𝜙 𝑉 𝑑𝑉 = 2 2 𝑑𝑉 = 1 𝑑𝑉 = 𝑚2 Flat space 80 Coupling in Fourier space 𝛻𝜓 = 𝜙 = 𝑚 𝜑 ℳ𝜓 = 𝜙 = 𝑚 𝜑 𝜓|ℳ 𝜓 = 𝑚 𝜓|𝜑 ℳ = ℳ0 + 𝞛 ℳ0 𝜓0 − 𝞛, 𝝍 = 𝑚 𝜑0 ℳ0 𝝍 + 𝞛𝜓0 + 𝞛 × 𝝍 = 𝑚 𝝋 𝜙2 𝑉 𝑑𝑉 = ℳ𝜓 𝑉 2 𝑑 𝑉 = 𝑚2 In general 𝜓 is not an eigenfunction of operator ℳ. That is only true when 𝜓 and 𝜑 are equal. For elementary particles they are equal apart from their difference in discrete symmetry. 81 Dirac equation Approximately flat space 𝛻0 𝜓 + 𝛁𝛂 𝜓 = 𝑚𝛽 𝜓 Spinor 𝜓 Dirac matrices 𝛂, 𝛽 • 𝛻0 𝜓𝑅 + 𝛁𝜓𝑅 = 𝑚𝜓𝐿 • 𝛻0 𝜓𝐿 − 𝛁𝜓𝐿 = 𝑚𝜓𝑅 In quaternion format • 𝛻𝜓 = 𝑚𝜓 ∗ • 𝛻 ∗ 𝜓 ∗ = 𝑚𝜓 𝜓𝑅 = 𝜓𝐿∗ = 𝜓0 + 𝝍 Qpattern QPDD 82 Entanglement 83 Entanglement The correlation mechanism manages entanglement At every progression instant the quantum state function of an entangled system equals the superposition of the quantum state functions of its components Entangled systems obey the swarming conditions For entangled systems the coupling equation holds Φ = 𝛻𝜓 = 𝑚 𝜑 𝜓 and 𝜑 are normalized Entanglement acts as a binding mechanism 84 Binding The fact that superposition coefficients define internal movements can best be explained by reformulating the definition of entangled systems. Composites that are equipped with a quantum state function whose Fourier transform at any progression step equals the superposition of the Fourier transforms of the quantum state functions of its components form an entangled system. Now the superposition coefficients can define internal displacements. As a function of progression they define internal oscillations. 85 Geoditches In an entangled system the micro-paths of the constituting elementary particles are folded along the internal oscillation paths. Each of the corresponding step stones causes a local pitch that describes the temporary (singular) curvature of the embedding continuum. These pitches quickly combine in a ditch that like the micro-path folds along the oscillation path. These ditches form special kinds of geodesics that we call “Geoditches”. The geoditches explain the binding effect of entanglement. 86 Pauli principle If two components of an entangled (sub)system that have the same quantum state function are exchanged, then we can take the system location at the center of the location of the two components. Now the exchange means for bosons that the (sub)system quantum state function is not affected: For all α and β{αφ(-x)+βφ(x)=αφ(x)+βφ(-x)}⇒φ(-x)=φ(x) and for fermions that the corresponding part of the (sub)system quantum state function changes sign. For all α and β{αφ(-x)+βφ(x)=-αφ(x)-βφ(-x)}⇒φ(-x)=-φ(x) This conforms to the Pauli principle. 87 Non-locality Action at a distance cannot be caused via information transfer Non-locality already plays a role inside the realm of separate elementary particles. Hopping along the step stones occurs much faster than the information carrying waves can follow. Similar features occur inside entangled systems. Due to the exclusion principle, observing the state of a sub-module has direct (instantaneous) consequences for the state of other sub-modules. 88 Focus If in an entangled system the focus is on the system, then the whole system acts as a swarm and the correlation mechanism causes hopping along ALL step stones that are involved in the system When the focus shifts to one or more of the constituents, then the entanglement get at least partly broken The separated particles and the resulting entangled system act as separate swarms 89 Binding 90 Binding mechanism When it is used, each step stone that is involved in an entangled system produces a singularity. The influence of that singularity spreads over the embedding continuum in the form of a wave front that folds and thus curves this continuum The traces of these Qtargets mark paths where the wave fronts dig pitches into the continuum that combine into channels that act as geodesics. 91 The effect of modularization 92 Modularization Modularization is a very powerful influencer. Together with the corresponding encapsulation it reduces the relational complexity of the ensemble of objects on which modularization works. The encapsulation keeps most relations internal to the module. When relations between modules are reduced to a few types , then the module becomes reusable. If modules can be configured from lower order modules, then efficiency grows exponentially. 93 Modularization Elementary particles can be considered as the lowest level of modules. All composites are higher level modules. Modularization uses resources efficiently. When sufficient resources in the form of reusable modules are present, then modularization can reach enormous heights. On earth it was capable to generate intelligent species. 94 Complexity Potential complexity of a set of objects is a measure that is defined by the number of potential relations that exist between the members of that set. If there are n elements in the set, then there exist n·(n-1) potential relations. Actual complexity of a set of objects is a measure that is defined by the number of relevant relations that exist between the members of the set. Relational complexity is the ratio of the number of actual relations divided by the number of potential relations. 95 Relations and interfaces Modules connect via interfaces. Relations that act within modules are lost to the outside world of the module. Interfaces are collections of relations that are used by interactions. Physics is based on relations. Quantum logic is a set of axioms that restrict the relations that exist between quantum logical propositions. 96 Types of physical interfaces Interactions run via (relevant) relations. Inbound interactions come from the past. Outbound interactions go to the future. Two-sided interactions are cyclic. They take multiple progression steps. They are either oscillations or rotations of the interactor. Cyclic interactions bind the corresponding modules together. 97 Modular systems Modular (sub)systems consist of connected modules. They need not be modules. They become modules when they are encapsulated and offer standard interfaces that makes the encapsulated system a reusable object. All composites are modular systems 98 Binding in sub-systems Let 𝜓 represent the renormalized superposition of the involved distributions. 𝛻𝜓 = 𝜙 = 𝑚 𝜑 𝑉 𝜓 2 𝑑𝑉 = 𝑉 𝜑 2 𝑑𝑉 = 1 𝑉 𝜙 2 𝑑𝑉 = 𝑚2 𝑚 is the total energy of the sub-system The binding factor is the total energy of the sub- system minus the sum of the total energies of the separate constituents. 99 Random versus intelligent design At lower levels of modularization nature designs modular structures in a stochastic way. This renders the modularization process rather slow. It takes a huge amount of progression steps in order to achieve a relatively complicated structure. Still the complexity of that structure can be orders of magnitude less than the complexity of an equivalent monolith. As soon as more intelligent sub-systems arrive, then these systems can design and construct modular systems in a more intelligent way. They use resources efficiently. This speeds the modularization process in an enormous way. 100 The noise of low dose imaging Low dose X-ray imaging Film of cold cathode emission 101 Shot noise Low dose X-ray image of the moon 102 Shot noise 103 Large scale fluid dynamics 104 Physical fields-1 SHF wave modulations Photon 𝛻𝜓 = 0 Gluon 𝛻2𝜓 = 0 } harmonic 𝛻𝜓 = 𝑚𝜑 Energy quanta 𝑛𝑖 𝑒𝑖 𝜓𝑖 𝑖 𝑒𝑖 = ±𝑒 SHF wave potentials Electromagnetic field Gravitation field 𝑛𝑖 𝑚𝑖 𝜑 𝑖 𝑖 105 Physical fields-2 Fields from step stone distributions Quaternionic quantum state function QPDD Quaternionic distributions Charges are preserved 𝛻𝜓 = 𝑚𝜑 106 Inertia-1 Inertia is implemented via the embedding continuum The embedding continuum is formed by a curved background field that forms our living space 107 Inertia-2 Potential fields of distant particles Φ0 = 𝑉 𝜓 dV In a uniform background: 𝜓 = 𝜌0 𝑟 ; 𝜌0 is constant Everywhere present background field 𝜌0 Φ0 = 𝐺= 𝑉 −𝑐 2 Φ 𝜌0 𝒗 𝑟 dV = 𝜌0 𝑉 1 𝑟 dV = 2π𝑅2 𝜌0 (Dennis Sciama) ; 𝚽 = Φ0 𝒗 𝑐 𝕰 = 𝛻𝟎 𝚽 + 𝛁Φ0 = 𝚽 + 𝛁Φ0 = Φ0 𝒗 𝑐 𝚽= 𝑉 𝑐𝑟 dV = Φ 𝒗 𝑐 + 𝛁Φ0 108 Inertia-3 Φ0 is a scalar background field 𝜱 is a vector background field 𝐺 is gravitational constant 𝕰 = Φ0 𝒗 𝑐 + 𝛁Φ0 𝕰 ≈ Φ0 𝒗 = 𝐺𝒗 Acceleration goes together with an extra field 𝕰 This field counteracts the acceleration 𝑐 109 Inertia-4 Starting from coupling equation 𝛻𝜓 = 𝑚𝜑 𝜓 = χ + χ0 𝒗 χ represents particle at rest 𝜓0 = χ0 Small 𝝍 = χ + χ0 𝒗 𝛻0 𝝍 = χ0 𝒗 = 𝑚𝝋 − 𝜵𝜓0 − 𝜵× 𝝍 𝕰 ≡ 𝛻0 𝝍 + 𝜵𝜓0 Represents influence of distant particles 110 Continuity equation Balance equation Total change within V = flow into V + production inside V 𝑑 𝑑𝜏 𝑉 𝜌0 𝑑𝑉 = 𝛻𝜌 𝑉 0 0 𝑑𝑉 = 𝒗 𝒏𝜌0 𝑆 𝑐 𝑉 𝑑𝑆 + 𝛁, 𝝆 𝑑𝑉 + 𝑠 𝑉 0 𝑠 𝑉 0 𝑑𝑉 𝑑𝑉 Gauss 𝝆 = 𝜌0 𝒗/𝑐 𝜌 = 𝜌0 + 𝝆 𝑠 = 𝛻𝜌 𝑠0 = 2𝛻0 𝜌0 − 𝒗 𝑞 , 𝛁𝜌0 − 𝛁, 𝒗 𝜌0 𝒔 = 𝛻0 𝒗 + 𝛁𝜌0 +𝜌0 𝛁 × 𝒗 − 𝒗 × 𝛁𝜌0 111 Inversion surfaces 𝑑 𝑑𝜏 𝑉 𝑉 𝜌 𝑑𝑉 + 𝛻 𝜌 𝑑𝑉 = The criterion 𝑆 𝑉 𝒏𝜌 𝑑𝑆 = 𝑉 𝑠 𝑑𝑉 𝑠 𝑑𝑉 𝑆 𝒏𝜌 𝑑𝑆=0 divides universe in compartments Inversion surface 112 Compartments universe Huge BH Black holes Huge BH ⇔ s tart of new episode BH ⇔ densest packaging Merge Compartments Never ending story 113 History of Cosmology Black hole represents natal state of compartment Black holes suck all mass from their compartment A passivized huge black hole represents start of new episode of its compartment Driving force is enormous mass present outside compartment ⇒ expansion Whole universe is affine space Result is never ending story 114 Gravitation The Palestra is a curved space 𝒫𝑏𝑙𝑢𝑟𝑟𝑒𝑑 = ℘𝑠ℎ𝑎𝑟𝑝 ∘ 𝒮𝑠𝑝𝑟𝑒𝑎𝑑 𝜈 𝑑𝑠 𝑥 = 𝑑𝑠 𝑥 𝑒𝜈 = 𝑑℘ = 𝑞 𝜇 is quaternion c dτ dr 𝜕℘ 𝑑𝑥𝜇 = 𝑞 𝜇 𝑥 𝑑𝑥𝜇 𝜇=0…3 𝜕𝑥𝜇 16 partial derivatives 𝑐 2 𝑑𝑡 2 = 𝑑𝑠 𝑑𝑠 ∗ = 𝑑𝑥02 + 𝑑𝑥12 +𝑑𝑥22 +𝑑𝑥32 𝑑𝑥02 = 𝑑𝜏 2 = 𝑐 2 𝑑𝑡 2 − 𝑑𝑥12 −𝑑𝑥22 −𝑑𝑥32 ∆𝑠𝑓𝑙𝑎𝑡 = ∆𝑥0 + 𝒊 ∆𝑥1 + 𝒋 ∆𝑥2 + 𝒌 ∆𝑥3 ∆𝑠℘ = 𝑞 0 ∆𝑥0 + 𝑞1 ∆𝑥1 + 𝑞 2 ∆𝑥2 + 𝑞 3 ∆𝑥3 Pythagoras Minkowski Flat space Curved space 115 Metric 𝑑℘ is a quaternionic metric It is a linear combination of 16 partial derivatives 𝑑℘ = 𝜕℘ 𝑑𝑥𝜇 𝜕𝑥 𝜇=0…3 𝜇 = 𝜈=0,…3 = 𝑞 𝜇 𝑥 𝑑𝑥𝜇 𝜕℘𝜈 𝑒𝜈 𝑑𝑥𝜇 = 𝜕𝑥𝜇 𝜇=0…3 𝜇 𝑒𝜈 𝑞𝜈 𝑑𝑥𝜇 𝜈=0,…3 𝜇=0…3 Avoids the need for tensors 116 The primary building blocks 117 Elementary particles Coupling equation 𝛻𝜓 𝑥 = 𝑚 𝜓 𝑦 𝛻𝜓 𝑥 ∗ = 𝑚 𝜓 𝑦 ∗ Coupling occurs between pairs {𝜓 𝑥 , 𝜓 𝑦 } Colors x, y N, R, G, B, R, G, B, W Right and left handedness R,L Sign flavors 𝝍⓪ 𝑁 𝐑 𝝍① 𝑅 𝐋 Imaginary part 𝝍② 𝐺 𝐋 𝝍③ 𝐵 𝐋 𝝍④ 𝐵 𝐑 𝝍⑤ 𝐺 𝐑 𝝍⑥ 𝑅 𝐑 𝝍⑦ 𝑁 L 𝝍⓪ is the Reference QPDD Discrete symmetries 118 Spin HYPOTHESIS : Spin relates to the fact whether the coupled QPDD is the reference Qpattern 𝝍⓪ . Each generation has its own reference QPDD. Fermions couple to the reference QPDD 𝝍⓪ . Fermions have half integer spin. Bosons have integer spin. The spin of a composite equals the sum of the spins of its components. 119 Sign of spin The micro-path can be walked in two directions This determines the sign of spin 120 Electric charge HYPOTHESIS : Electric charge depends on the number of dimensions in which the discrete symmetry of Qpattern elements differ from the discrete symmetry of the embedding field. Each sign difference stands for one third of a full electric charge. Further it depends on the fact whether the handedness differs. If the handedness differs then the sign of the count is changed as well. 121 Color charge HYPOTHESIS : Color charge is related to the direction of the anisotropy of the considered QPDD with respect to the reference QPDD. The anisotropy lays in the discrete symmetry of the imaginary parts. The color charge of the reference QPDD is white. The corresponding anti-color is black. The color charge of the coupled pair is determined by the colors of its members. All composite particles are black or white. The neutral colors black and white correspond to isotropic QPPDs. Currently, color charge cannot be measured. In the Standard Model the existence of color charge is derived via the Pauli principle. 122 Total energy Mass is related to the coupling factor of the involved QPPDs. It is directly related to the square root of the volume integral of the square of the local field energy 𝐸. Any internal kinetic energy is included in 𝐸. The same mass rule holds for composite particles. The fields of the composite particles are dynamic superpositions of the fields of their components. 123 Leptons Pair s-type e-charge c-charge {𝜓 ⑦ , 𝜓 ⓪ } fermion -1 {𝜓 ⓪ , 𝜓 ⑦ } Antifermion +1 Handed ness SM Name N LR electron W RL positron 124 Quarks Pair s-type e-charge c-charge Handedness SM Name {𝜓 ① , 𝜓 ⓪ } fermion -1/3 R LR down-quark {𝜓 ⑥ , 𝜓 ⑦ } Anti-fermion +1/3 R RL Anti-down-quark {𝜓 ② , 𝜓 ⓪ } fermion -1/3 G LR down-quark {𝜓 ⑤ , 𝜓 ⑦ } Anti-fermion +1/3 G RL Anti-down-quark {𝜓 ③ , 𝜓 ⓪ } fermion -1/3 B LR down-quark {𝜓 ④ , 𝜓 ⑦ } Anti-fermion +1/3 B RL Anti-down-quark {𝜓 ④ , 𝜓 ⓪ } fermion +2/3 B RR up-quark {𝜓 ③ , 𝜓 ⑦ } Anti-fermion -2/3 B LL Anti-up-quark {𝜓 ⑤ , 𝜓 ⓪ } fermion +2/3 G RR up-quark {𝜓 ② , 𝜓 ⑦ } Anti-fermion -2/3 G LL Anti-up-quark {𝜓 ⑥ , 𝜓 ⓪ } fermion +2/3 R RR up-quark {𝜓 ① , 𝜓 ⑦ } Anti-fermion -2/3 R LL Anti-up-quark 125 Reverse quarks Pair s-type e-charge c-charge Handedness SM Name {𝜓 ⓪ , 𝜓 ① } fermion +1/3 R RL down-r-quark {𝜓 ⑦ , 𝜓 ⑥ } Anti-fermion -1/3 R LR Anti-down-r-quark {𝜓 ⓪ , 𝜓 ② } fermion +1/3 G RL down-r-quark {𝜓 ⑦ , 𝜓 ⑤ } Anti-fermion -1/3 G LR Anti-down-r-quark {𝜓 ⓪ , 𝜓 ③ } fermion +1/3 B RL down-r-quark {𝜓 ⑦ , 𝜓 ④ } Anti-fermion -1/3 B LR Anti-down-r_quark {𝜓 ⓪ , 𝜓 ④ } fermion -2/3 B RR up-r-quark {𝜓 ⑦ , 𝜓 ③ } Anti-fermion +2/3 B LL Anti-up-r-quark {𝜓 ⓪ , 𝜓 ⑤ } fermion -2/3 G RR up-r-quark {𝜓 ⑦ , 𝜓 ② } Anti-fermion +2/3 G LL Anti-up-r-quark {𝜓 ⓪ , 𝜓 ⑥ } fermion -2/3 R RR up-r-quark {𝜓 ⑦ , 𝜓 ① } Anti-fermion +2/3 R LL Anti-up-r-quark 126 W-particles {𝜓 ⑥ , 𝜓 ① } boson -1 RR RL 𝑊− {𝜓 ① , 𝜓 ⑥ } Anti-boson +1 RR LR 𝑊+ {𝜓 ⑥ , 𝜓 ② } boson -1 RG RL 𝑊− {𝜓 ② , 𝜓 ⑥ } Anti-boson +1 GR LR 𝑊+ {𝜓 ⑥ , 𝜓 ③ } boson -1 RB RL 𝑊− {𝜓 ③ , 𝜓 ⑥ } Anti-boson +1 BR LR 𝑊+ {𝜓 ⑤ , 𝜓 ① } boson -1 GG RL 𝑊− {𝜓 ① , 𝜓 ⑤ } Anti-boson +1 GG LR 𝑊+ {𝜓⑤ , 𝜓② } boson -1 GG RL 𝑊− {𝜓 ② , 𝜓 ⑤ } Anti-boson +1 GG LR 𝑊+ {𝜓 ⑤ , 𝜓 ③ } boson -1 GB RL 𝑊− {𝜓 ③ , 𝜓 ⑤ } Anti-boson +1 BG LR 𝑊+ {𝜓 ④ , 𝜓 ① } boson -1 BR RL 𝑊− {𝜓 ① , 𝜓 ④ } Anti-boson +1 RB LR 𝑊+ {𝜓 ④ , 𝜓 ② } boson -1 BG RL 𝑊− {𝜓 ② , 𝜓 ④ } Anti-boson +1 GB LR 𝑊+ {𝜓 ④ , 𝜓 ③ } boson -1 BB RL 𝑊− {𝜓 ③ , 𝜓 ④ } Anti-boson +1 BB LR 𝑊+ 127 Z-particles Pair s-type e-charge c-charge Handedness SM Name {𝜓 ② , 𝜓 ① } boson 0 GR LL Z {𝜓 ⑤ , 𝜓 ⑥ } Anti-boson 0 GR RR Z {𝜓 ③ , 𝜓 ① } boson 0 BR LL Z {𝜓 ④ , 𝜓 ⑥ } Anti-boson 0 RB RR Z {𝜓 ③ , 𝜓 ② } boson 0 BR LL Z {𝜓 ④ , 𝜓 ⑤ } Anti-boson 0 RB RR Z {𝜓 ① , 𝜓 ② } boson 0 RG LL Z {𝜓 ⑥ , 𝜓 ⑤ } Anti-boson 0 RG RR Z {𝜓 ① , 𝜓 ③ } boson 0 RB LL Z {𝜓 ⑥ , 𝜓 ④ } Anti-boson 0 RB RR Z {𝜓 ② , 𝜓 ③ } boson 0 RB LL Z {𝜓 ⑤ , 𝜓 ④ } Anti-boson 0 RB RR Z 128 Neutrinos type s-type e-charge c-charge Handedness SM Name {𝜓 ⑦ , 𝜓 ⑦ } fermion 0 NN RR neutrino {𝜓 ⓪ , 𝜓 ⓪ } Anti-fermion 0 WW LL neutrino {𝜓 ⑥ , 𝜓 ⑥ } boson? 0 RR RR neutrino {𝜓 ① , 𝜓 ① } Anti- boson? 0 RR LL neutrino {𝜓 ⑤ , 𝜓 ⑤ } boson? 0 GG RR neutrino {𝜓 ② , 𝜓 ② } Anti- boson? 0 GG LL neutrino {𝜓 ④ , 𝜓 ④ } boson? 0 BB RR neutrino {𝜓 ③ , 𝜓 ③ } Anti- boson? 0 BB LL neutrino 129 Color confinement The color confinement rule forbids the generation of individual particles that have non-neutral color charge 130 Color confinement Color confinement forbids the generation of individual quarks Quarks can appear in hadrons Color confinement blocks observation of gluons 131 Photons & gluons type s-type e-charge c-charge Handedness SM Name {𝜓 ⑦ } boson 0 N R photon {𝜓 ⓪ } boson 0 W L photon {𝜓 ⑥ } boson 0 R R gluon {𝜓 ① } boson 0 R L gluon {𝜓 ⑤ } boson 0 G R gluon {𝜓 ② } boson 0 G L gluon {𝜓 ④ } boson 0 B R gluon {𝜓 ③ } boson 0 B L gluon 132 Photons & gluons Photons and gluons are NOT particles Ultra-high frequency waves are constituted by wave fronts that at every progression step are emitted by elementary particles Photons and gluons are modulations of ultra-high frequency carrier waves. 133 Fundamental particles Due to color confinement some elementary particles cannot be created as individuals Quarks can only be created combined in hadrons Fundamental particles form a category of particles that are created in one integral action The color charge of fundamental particles is neutral 134 135 Dual space distributions A subset of the (quaternionic) distributions have the same shape in configuration space and in the linear canonical conjugated space. We call them dual space distributions These are functions that are invariant under Fourier transformation. The Qpatterns and the harmonic and spherical oscillations belong to this class. Fourier-invariant functions show iso-resolution, that is, ∆p = ∆q in the Heisenberg’s uncertainty relation. 136 Why has nature a preference? Nature seems to have a preference for this class of quaternionic distributions. A possible explanation is the two-step generation process, where the first step is realized in configuration space and the second step is realized in canonical conjugated space. The whole pattern is generated two-step by two-step. The only way to keep coherence between a distribution and its Fourier transform that are both generated step by step is to generate them in pairs. 137 Conclusion Fundament Quantum logic Book model Correlation vehicle Main features Fundamentally countable ⇛ Quanta Embedded in continuum ⇛ Fields Fundamentally stochastic ⇛ Quantum Physics Palestra is curved ⇛ Quaternionic “GR” Quaternionic metric } 138 Conclusion Contemporary physics works (QED, QCD) But cannot explain fundamental features Origin of dynamics Space curvature Inertia Existence of Quantum Physics What photons are 139 End Physics made its greatest misstep in the thirties when it turned away from the fundamental work of Garret Birkhoff and John von Neumann. This deviation did not prohibit pragmatic use of the new methodology. However, it did prevent deep understanding of that technology because the methodology is ill founded. 140 Lattices, classical logic and quantum logic 141 Logic – Lattice structure A lattice is a set of elements 𝑎, 𝑏, 𝑐, …that is closed for the connections ∩ and ∪. These connections obey: The set is partially ordered. With each pair of elements 𝑎, 𝑏 belongs an element 𝑐, such that 𝑎 ⊂ 𝑐 and 𝑏 ⊂ 𝑐. The set is a ∩ half lattice if with each pair of elements 𝑎, 𝑏 an element 𝑐 exists, such that 𝑐 = 𝑎 ∩ 𝑏. The set is a ∪ half lattice if with each pair of elements 𝑎, 𝑏 an element 𝑐 exists, such that 𝑐 = 𝑎 ∪ 𝑏. The set is a lattice if it is both a ∩ half lattice and a ∪ half lattice. 142 Partially ordered set The following relations hold in a lattice: 𝑎 ∩ 𝑏 = 𝑏 ∩ 𝑎 (𝑎 ∩ 𝑏) ∩ 𝑐 = 𝑎 ∩ (𝑏 ∩ 𝑐) 𝑎 ∩ (𝑎 ∪ 𝑏) = 𝑎 𝑎 ∪ 𝑏 = 𝑏 ∪ 𝑎 (𝑎 ∪ 𝑏) ∪ 𝑐 = 𝑎 ∪ (𝑏 ∪ 𝑐) 𝑎 ∪ (𝑎 ∩ 𝑏) = 𝑎 • has a partial order inclusion ⊂: a⊂b⇔a⊂b=a • A complementary lattice contains two elements 𝑛 and 𝑒 with each element a an complementary element a’ 𝑎 ∩ 𝑎’ = 𝑛 𝑎 ∩ 𝑛 = 𝑛 𝑎 ∩ 𝑒 = 𝑎 𝑎 ∪ 𝑎’ = 𝑒 𝑎 ∪ 𝑒 = 𝑒 𝑎 ∪ 𝑛 = 𝑎 143 Orthocomplemented lattice Contains with each element 𝑎 an element 𝑎” such that: 𝑎 ∪ 𝑎” = 𝑒 𝑎 ∩ 𝑎” = 𝑛 (𝑎”)” = 𝑎 𝑎 ⊂ 𝑏 ⟺ 𝑏” ⊂ 𝑎” Distributive lattice 𝑎 ∩ (𝑏 ∪ 𝑐) = (𝑎 ∩ 𝑏) ∪ ( 𝑎 ∩ 𝑐) 𝑎 ∪ (𝑏 ∩ 𝑐) = (𝑎 ∪ 𝑏) ∩ (𝑎 ∪ 𝑐) Modular lattice (𝑎 ∩ 𝑏) ∪ (𝑎 ∩ 𝑐) = 𝑎 ∩ (𝑏 ∪ (𝑎 ∩ 𝑐)) Classical logic is an orthocomplemented modular lattice 144 Weak modular lattice There exists an element 𝑑 such that 𝑎 ⊂ 𝑐 ⇔ 𝑎 ∪ 𝑏 ∩ 𝑐 = 𝑎 ∪ (𝑏 ∩ 𝑐) ∪ (𝑑 ∩ 𝑐) where 𝑑 obeys: (𝑎 ∪ 𝑏) ∩ 𝑑 = 𝑑 𝑎 ∩ 𝑑 = 𝑛 𝑏 ∩ 𝑑 = 𝑛 [(𝑎 ⊂ 𝑔) and (𝑏 ⊂ 𝑔) ⇔ 𝑑 ⊂ 𝑔 145 Atoms In an atomic lattice ∃𝑝 𝜖 𝐿 ∀𝑥 𝜖 𝐿 {𝑥 ⊂ 𝑝 ⇒ 𝑥 = 𝑛} ∀𝑎 𝜖 𝐿 ∀𝑥 𝜖 𝐿 {(𝑎 < 𝑥 < 𝑎 ∩ 𝑝) ⇒ (𝑥 = 𝑎 𝑜𝑟 𝑥 = 𝑎 ∩ 𝑝)} 𝑝 is an atom 146 Logics Classical 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. 147 Hilbert space The set of closed subspaces of an infinite dimensional separable Hilbert space forms an orthomodular lattice Is lattice isomorphic to quantum logic 148 Hilbert logic Back Add linear propositions Linear combinations of atomic propositions Add relational coupling measure Equivalent to inner product of Hilbert space Close subsets with respect to realational coupling measure Propositions ⇔ subspaces Linear propositions ⇔ Hilbert vectors 149 Superposition principle Linear combinations of linear propositions are again linear propositions that belong to the same Hilbert logic system 150 Isomorphism Lattice isomorhic Propositions ⇔ closed subspaces Topological isomorphic Linear atoms ⇔ Hilbert base vectors 151