Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Probability amplitude wikipedia, lookup

Theoretical and experimental justification for the Schrödinger equation wikipedia, lookup

Wave function wikipedia, lookup

Copenhagen interpretation wikipedia, lookup

Wave–particle duality wikipedia, lookup

Density matrix wikipedia, lookup

Canonical quantization wikipedia, lookup

Quantum state wikipedia, lookup

Interpretations of quantum mechanics wikipedia, lookup

Path integral formulation wikipedia, lookup

Hidden variable theory wikipedia, lookup

Scalar field theory wikipedia, lookup

Renormalization group wikipedia, lookup

Symmetry in quantum mechanics wikipedia, lookup

Relativistic quantum mechanics wikipedia, lookup

Bra–ket notation wikipedia, lookup

Tight binding wikipedia, lookup

Quantum teleportation wikipedia, lookup

Lattice Boltzmann methods wikipedia, lookup

Quantum decoherence wikipedia, lookup

Ensemble interpretation wikipedia, lookup

Self-adjoint operator wikipedia, lookup

Transcript

A simple model of fundamental physics By J.A.J. van Leunen I http://www.e-physics.eu A simple model of fundamental physics By J.A.J. van Leunen II http://www.e-physics.eu A simple model of fundamental physics By J.A.J. van Leunen III http://www.e-physics.eu Physical Reality In no way a model can give a precise description of physical reality. At the utmost it presents a correct view on physical reality. But, such a view is always an abstraction. Mathematical structures might fit onto observed physical reality because their relational structure is isomorphic to the relational structure of these observations. 4 Rules Restrict Complexity Physical reality applies rules for relational structures that it accepts These rules intent to reduce the complexity of these relational structures 5 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 6 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. 7 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. 8 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. 9 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 10 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. 11 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 Three alarming facts The first level model does not support continuums 1. HS operators have countable eigenspaces 2. The first level model does not support dynamics Can only represent static status quo 3. The Hilbert space contains deeper details than quantum logic does QL ⟹ propositions ↭ HS ⟹ sub-spaces HL ⟹ refined propositions ↭ HS ⟹ vectors 13 Threefold hierarchy Relational structure Quantum Logic Hilbert logic Isomorphisms Quantum Hilbert Logic space Atomic Subspace quantum logic proposition Atomic Base vector Hilbert Logic proposition Set of particles Particle is swarm of step stones Step stone Possible interpretation of isomorphisms 14 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. 15 Static Representation Quantum logic Hilbert space } No full isomorphism Cannot represent continuums Solution: • Refine to Hilbert logic • Add Gelfand triple 16 Discrete sets and continuums A Hilbert space features operators that have countable eigenspaces A Gelfand triple features operators that have continuous eigenspaces 17 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 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. 20 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! 21 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 22 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 23 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 24 Recreation The whole universe is recreated at every progression step If no other measures are taken, the model will represent dynamical chaos 25 Dynamic coherence 1 An external correlation mechanism must take care such that sufficient coherence between subsequent pages exist 26 Dynamic coherence 2 The coherence must not be too stiff, otherwise no dynamics occurs 27 Storage The eigenspaces of operators can act as storage places 28 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. 29 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 30 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 31 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 32 Enumeration Hilbert space & Hilbert logic Enumerator operator Eigenvalues Rational quaternionic enumerators (RQE’s) 33 Allocation Hilbert space & Hilbert logic Enumerator operator Eigenvalues Rational quaternionic enumerators (RQE’s) Model Allocation function 𝒫 Parameters RQE’s Image Qtargets 34 Enumeration & Allocation 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 35 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 𝒫 = ℘ ∘ 𝒮 Blurred 𝒫 Sharp ℘ Spread function 𝒮 Blur 𝜓 36 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 37 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 38 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 39 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 40 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 41 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 42 Swarming conditions 1, 2 and 3 In order to ensure sufficient coherence the correlation mechanism implements swarming conditions 1. A swarm is a coherent set of step stones 2. A swarm can be described by a continuous object density distribution 3. That density distribution can be interpreted as a probability density distribution 43 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 44 Swarming conditions The swarming conditions distinguish this type of swarm from normal swarms 45 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 46 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 47 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 48 Coupling For swarms the coupling equation holds Φ = 𝛻𝜓 = 𝑚 𝜑 By requiring that the two sides of the quaternionic differential equation contain normalized functions, this equation turns into a coupling equation. 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 49 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 50 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 51 Swarms 2 At each progression step, an image of the planned swarm (Qpattern) is mapped by a continuous allocation function onto the embedding continuum At each progression step, via random selection a single step stone is selected, whose image becomes the Qtarget In fermions that step stone is not used again A swarm has a “center position”, called Qpatch that can be interpreted as the expectation value of location of the swarm The Qtargets form a stochastic micro-path 52 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 53 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 54 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 55 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 ℘ 56 Micro-path The Qpatterns contain a fixed number of step stones The step stones that belong to a swarm form a micro- path Even at rest, the Qtarget walks along its micro-path This walk takes a fixed number of progression steps When the swarm moves or oscillates, then the micropath is stretched along the path of the swarm This stretching is controlled by the third swarming condition 57 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 58 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 59 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 60 Palestra Curved embedding continuum Represents universe Embedded in continuum 𝑄𝑝𝑎𝑡𝑐ℎ Collection of Qpatches The Palestra is the place where everything happens 61 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 ∗ 62 Lattices, classical logic and quantum logic 63 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. 64 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 65 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 66 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 Quantum logic obeys the weak modular law 67 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 68 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. 69 Hilbert Space The set of closed subspaces of an infinite dimensional separable Hilbert space forms an orthomodular lattice Is lattice isomorphic to quantum logic 70 Hilbert Logic Add linear propositions Linear combinations of atomic propositions Add relational coupling measure Equivalent to inner product of Hilbert space Close subsets with respect to relational coupling measure Propositions ⇔ subspaces Linear propositions ⇔ Hilbert vectors 71 Superposition principle Linear combinations of linear propositions are again linear propositions that belong to the same Hilbert logic system 72 Isomorphism Lattice isomorhic Propositions ⇔ closed subspaces Topological isomorphic Linear atoms ⇔ Hilbert base vectors 73