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Diversity of Floating Platforms

J.A.J. van Leunen

Published Dec 1, 2017• 0 citations
OpenAlex

Abstract

Physical reality archives its dynamic geometric data in a read-only repository. This repository emerges from its foundation which is an orthomodular lattice. The repository is a combination of a series of separable Hilbert spaces that share the same infinite dimensional vector space. For the definition of the inner product of pairs of vectors the separable Hilbert spaces apply a private version of the quaternionic number system. A non-separable Hilbert space embeds the separable Hilbert spaces. The version of the quaternionic number system acts as a parameter space. These parameter spaces float over a background parameter space.

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Physical reality archives its dynamic geometric data in a read-only repository. This repository emerges from its foundation which is an orthomodular lattice. The repository is a combination of a series of separable Hilbert spaces that share the same infinite dimensional vector space. For the definition of the inner product of pairs of vectors the separable Hilbert spaces apply a private version of the quaternionic number system. A non-separable Hilbert space embeds the separable Hilbert spaces. The version of the quaternionic number system acts as a parameter space. These parameter spaces float over a background parameter space.
This document concerns the discovery of the grand structure of physical reality in terms of mathematical constructs and mechanisms. The grand structure emerges from simple foundations and leads via spaces that show an increasing complexity to a system of Hilbert spaces that will be called Hilbert repository. This structure acts as a powerful storage medium that physical reality applies to store and retrieve prepared data. The repository can easily capture the dynamic geometric data of all objects that will ever exist in our universe. It can also archive the universe as a dynamic manifold and all other physical fields that play in the lifetime of the universe. This enables physical reality to prepare data in a creation episode in which no running time exists and play the story of all archived objects during a running episode in which a flowing time indicates the progression of that life story as an ongoing embedding process. The investigation concerns a hierarchy of spaces that show increasing degrees of complexity. Finding more complicated spaces is not a big problem. This turns out to be quite simple. The move to a more complicated platform appears always accompanied by several significant restrictions and understanding where these limitations come from is a much bigger problem. Today's mathematics is only just able to explain the restrictions of the Hilbert space. This no longer applies to the system of Hilbert spaces, which in this document is called the Hilbert repository. Current math cannot explain the restrictions and the extra features of the Hilbert repository. This fact will particularly interest those who are curious about the structure and behavior of physical reality. The approach is quite different from the usual path and provides other insights. The usual way tries to deduce new insights from what we know about classical physics. That path appears to be blocked. The story makes it clear that only mathematics cannot provide a complete picture of reality, while experiments alone also cannot expose physical reality. The combination of mathematics and experimentation produces the best results. The behavior of fields plays an important role in most theories. Basic physical fields are dynamic fields like our universe and the fields that are raised by electric charges. These fields are dynamic continuums. Most physical theories treat these fields by applying gravitational theories or by Maxwell equations. Mathematically these fields can be represented by quaternionic fields. Dedicated normal operators in quaternionic non-separable Hilbert spaces can represent these quaternionic fields in their continuum eigenspaces. Quaternionic functions can describe these fields. Quaternionic differential and integral calculus can describe the behavior of these fields and the interaction of these fields with countable sets of quaternions. All quaternionic fields obey the same quaternionic differential equations. The basic fields differ in their start and boundary conditions. The paper introduces the concept of the Hilbert repository. It is part of a hierarchy of structures that mark increasingly complicated realizations of a purely mathematical model that describes and explains most features of observable physical reality. That model is the Hilbert Book Model. Many of the subjects that are treated in this document cannot be found in standard textbooks. The paper treats the mathematical and experimental underpinning of the Hilbert Book Model.
It is possible to derive a model from a set of well selected first principles. After a series of extensions of this foundation the model shows many similarities with what we know from observing reality. The first principles formulate a skeleton relational structure that is mathematically known as an orthocomplemented weakly modular lattice. It can be considered as part of a recipe for modular construction. When starting from this foundation it is mathematically inescapable evident that this model confines to a quaternionic space-progression model that proceeds with model-wide progression steps through an ordered sequence of sub-models that each represent a static status quo of the whole model. This base model is a hybrid construct that consists of a sub model that treats all discrete objects and a continuum based sub model that embeds the discrete objects. The discrete part of the model keeps its data in an infinite dimensional separable Hilbert space. The continuum part stores its data in a corresponding non-separable Hilbert space. An extra mechanism that controls the coherence and the scheduling of dynamics applies the recurrent embedding of the discrete objects into appropriate continuums. Despite the fact that the target of the model is planned to reach a level in which it shows many features and phenomena that we know from observing reality, this model is not claimed to be a model of physics. The reason for this restriction is that many of its aspects cannot be observed and will never become observable. Physicists tend to deny completely or largely deduced models. The deduced model appears to own built-in restrictions that guard against uncontrolled growth of the complexity of the model.
The Hilbert book test model is a purely mathematical test model that starts from a solid foundation from which the whole model can be derived by using trustworthy mathematical methods. What is known about physical reality is used as a guidance, but the model is not claimed to be a proper reflection of physical reality. The mathematical toolkit still contains holes. These holes will be encountered during the development of the model and suggestions are made how those gaps can be filled. Some new insights are obtained and some new mathematical methods are introduced. The selected foundation is interpreted as part of a recipe for modular construction and that recipe is applied throughout the development of the model. This development is an ongoing project. The main law of physics appears to be a commandment: “Thou shalt construct in a modular way”. The paper reveals the possible origin of several physical concepts. This paper shows that it is possible to discover a mathematical structure that is suitable as an extensible foundation. However, without adding extra mechanisms that ensure dynamic coherence, the structure does not provide the full functionality of reality. These extra mechanisms apply stochastic processes, which generate the locations of the elementary modules that populate the model. All discrete items in universe are configured from dynamic geometric locations. These items are stored in a repository that covers a history part, the current static status quo, and a future part. The elementary modules float over the static framework of the repository. Dedicated mechanisms ensure the coherent behavior of these elementary modules. Fields exist that describe these elementary modules. An encapsulating repository supports these fields. Both repositories are formed by quaternionic Hilbert spaces. The model offers two interesting views. The first view is the creator’s view and offers free access to all dynamic geometric data that are stored in the eigenspaces of operators. The second view is the observer’s view. The observers are modules that travel with the vane, which represents the static status quo. The observers only perceive information that comes from the past and that is carried by the field that embeds them. This view sees the model as a spacetime based structure that stores its dynamic geometric data with a Minkowski signature.
Abstract It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space H H{\mathcal H}H correspond to direct product decompositions H ≃ A × A ⊥ H≃A×A⊥\mathcal{H}\simeq A\times A^\perpH≃A×A⊥ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.