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Addendum 2 – The Fuzziness of a Point in Real Space and the Measurement Problem

Rasky, Thomas Zigmond

Published Dec 5, 2025• 0 citations
Publisher

Abstract

Addendum 2 shows that positional fuzziness in SCG arises from the continuous rotation of α under parallel transport, with measurement simply sampling one projection of this ongoing change. Massive particles remain fuzzy before and after sampling, while photons do not survive positional measurements. Quantum behavior is thus a consequence of observing single projections within an inherently dynamic, change-driven geometry.

DOI
10.5281/zenodo.17831188

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Addendum 2 shows that positional fuzziness in SCG arises from the continuous rotation of α under parallel transport, with measurement simply sampling one projection of this ongoing change. Massive particles remain fuzzy before and after sampling, while photons do not survive positional measurements. Quantum behavior is thus a consequence of observing single projections within an inherently dynamic, change-driven geometry.
Based on Klauder's continuous representations in Hilbert space, we discuss a fundamental fuzziness of the coordinate and momentum variables for a quantum particle. These fuzzy dynamical variables are intimately related to a very small radical length R(\ensuremath{\lesssim}${10}^{\mathrm{\ensuremath{-}}18}$ cm) and a very large cosmological length S (\ensuremath{\gtrsim}1010{10}^{10}1010 light years). The uncertainty \ensuremath{\Delta}q for the measurement of a particle's position alone is assumed to be restricted by S\ensuremath{\gtrsim}\ensuremath{\Delta}q\ensuremath{\gtrsim}R. Physical properties of fuzzy base states (p\ensuremath{\Vert} and (q\ensuremath{\Vert} and their general cosmological and very-high-energy implications are discussed. In particular, a new density of states and its possible effects on decay rates and scattering cross sections at large momenta p\ensuremath{\gtrsim}\ensuremath{\Elzxh}/R\ensuremath{\sim}104{10}^{4}104 GeV/c (c is the velocity of light) are considered. We also discuss the superposition of waves in the presence of R and S.
This addendum clarifies two core elements of Space-Change Geometry (SCG): internal change and measurement. It argues that a particle is a self-sustaining wave-pattern existing only because imaginary curvature supports intrinsic α-rotation. The measurement process is reframed as sampling one instant of a system’s real–imaginary evolution: a definite position arises from the real coordinate plus the instantaneous projection of the CPW coupling term. Absorption does not freeze the wave-pattern; it ends it, transferring its energy into a new configuration.
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Commutative fuzzy geometry considered as the possible mathematical framework for reformulation of quantum-mechanical formalism in strictly geometric terms. In that approach the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. Due to manifold weak topology, in 1-dimensional case m space coordinate x acquires principal uncertainty σ x and described by positive, normalized density ω ( x,t ), the analogous uncertainties appear for 3-dimensional manifold. It's shown that m states on such manifold are equivalent to Dirac vectors of complex Hilbert space, their evolution correspond to Shroedinger dynamics of massive quantum particle.