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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 1st 2014
    • (edited Mar 1st 2014)

    there had been no references at Hilbert space, I have added the following, focusing on the origin and application in quantum mechanics:

    • John von Neumann, Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.

    • George Mackey, The Mathematical Foundations of Quamtum Mechanics A Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963

    • E. Prugovecki, Quantum mechanics in Hilbert Space. Academic Press, 1971.

  1. Some hardly known facts about Hilbert spaces are the following.
    Only a subtle difference exists between a vector space and a Hilbert space. In this way it becomes possible that a huge number of separable Hilbert spaces can share the same underlying vector space. Quaternionic number systems exist in many versions that distinguish between the Cartesian and polar coordinate systems that sequence their members. This affects the symmetry of the number system. A Hilbert space selects a version of the number system for specifying its inner product. This selects the symmetry of that Hilbert space. Each separable Hilbert space can manage a private parameter space in the eigenspace of a dedicated normal operator (that I call reference operator) by letting that eigenspace represent by the rational values in the selected version of the number system that is used to specify the inner products of vector pairs. A special category of normal operators can be defined by letting them share the eigenvectors of the reference operators and replacing the corresponding eigenvalues of the reference operator by the target values of a selected function. Each infinite dimensional separable Hilbert space owns in this way a unique non-separable Hilbert space that embeds its separable companion. In this way the special category of normal operators become field operators that combine Hilbert space operator technology with function theory, differential calculus and integral calculus. The continuum eigenspaces of these operators in the non-separable Hilbert space will implement a general field theory that in case of a quaternionic number system treats dynamic fields in a well-defined way.
    In this way a system of Hilbert spaces can act as a structured repository for discrete dynamic geometric data and dynamic continuums that act like (physical) fields.
  2. With the above additions to the concept of a Hilbert space, all elementary particles can be given a private separable Hilbert space. This platform floats with the geometric center of its private parameter space over a background parameter space that belongs to a background platform that is implemented by an infinite dimensional separable Hilbert space and its non-separable companion Hilbert space. The platforms of elementary particles feature a symmetry-related charge that is implemented by a source or sink that is located at the geometric center of the private parameter space. This fact requires restriction of the versions of the quaternionic number system that can contribute to this system of Hilbert spaces that all share the same underlying infinite dimensional vector space. Only versions of which the axes of the cartesian coordinates are parallel to the axes of the cartesian coordinates of the background platform can participate. This means that the sources and sinks require this restriction. If the charges relate to the differences in symmetry between the floating platforms and the background platform, then the resulting short list of charges resembles the short list of charges in the standard model of current physics.

    The fact that the system of Hilbert spaces acts as a repository for discrete dynamic geometric data and dynamic fields opens the possibility to convert the system into a self-creating model of physical reality in which the clock of the universe starts ticking after that the repository is filled with data that tell the complete life stories of the elementary particles. This possibility is explored in "A Self-creating Model of Physical Reality"; http://vixra.org/abs/1908.0223 and is presented in http://www.e-physics.eu/Base%20model.pptx
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