. This is in contrast to canonical quantization, which leads to the Heisenberg uncertainty principle and the natural numbers as spectrum of the harmonic quantum oscillator. The Hamiltonian needs to be self-adjoint so that the quantization can be a realization of the Hilbert-Polya conjecture.HThe central concept is about a proposed alternative harmonic quantum energy model which enables a finite "quantum fluctuation = total energy". The model is based on a fractional (distributional) Hilbert space framework, enabling a self-adjoin Hamiltonian operator. It provides a truly infinitesimal geometry.overcoming current handicaps of the manifolds framework of Einstein`s field equations (
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The proposed mathematical framework above is supposed to provide a truly infinitesimal geometry (H. Weyl). A physical interpretation could be about "rotating differentials" ("quantum fluctuations"), which corresponds mathematically to Leibniz's concept of monads. Its mathematical counterpart are the ideal points (or hyper-real numbers). This leads to non-standard analysis, whereby the number field has same cardinality than the real numbers. It is "just" the Archimedian principle which is no longer valid. This looks like a cheap prize to be paid, especially as hyper-real numbers might provide at least a proper mathematical language for the "Big Bang" initial value "function" and its related Einstein-Hilbert action functional.Looking on hyper-real numbers from the "real" number perspective one must admit to classify the term "real" as a contraction in itself, if it is understood as
We propose a fractional (energy) Hilbert space H(1/2), which already plays an elegant role in universal Teichmüller theory. It is also related to the bounded variation functions. Ist dual space with respect to the L(2) Lebesgue Hilbert space is the H(-1/2) Hilbert space. The latter one is the proposed quantum state Hilbert space. While the Hermite polynomials (or it Hilbert transforms) build an orthogonal system of the Hilbert space L(2)=H(0) (and related discrete energy spectrum) the basis of the Hilbert space H(-1/2) requires an additional eigenfunction with continuous spectrum. This "eigen-pair" is proposed to be a model for the dark energy model, given by the common (additional) root operator of the ladder "symmetric" operators ("Erzeugungs- und Vernichtungs-operatoren"; "Bosonische und Fermionische Kletteroperatoren"). By this the "symmetry" theory is also anticipated, as the current particles zoo ("materialized" in H(0) Hilbert space) has all the time the same symmetry partner (field), i.e. the "eigen-function" with continuous spectrum, which spans the closed sub-space H(-1/2)-H(0).
Identifiying "fluids" or "sub-atomic particles" not with real numbers (scalar field, I. Newton), but with hyper-real numbers (G. W. Leibniz) enables a truly infinitesimal (geometric) distributional Hilbert space framework (H. Weyl) which corresponds to the Teichmüller theory, the Bounded Mean Oscillation (BMO) and the Harmonic Analysis theory. The distributional Hilbert scale framework enables the full power of spectral theory, while still keeping the standard L(2)=H(0)-Hilbert space as test space to "measure" particles' locations. At the same time, the Ritz-Galerkin (energy or operator norm minimization) method and its counterpart, the methods of Trefftz/Noble to solve PDE by complementary variational principles (A. M. Arthurs, K. Friedrichs, L. B. Rall, P. D. Robinson, W. Velte) w/o anticipating boundary values) enables an alternative "quantization" method of PDE models (P. Ehrenfest), e.g. being applied to the Wheeler-de-Witt operator. Some related papers
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