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| Format: | Recurso digital |
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Zenodo
2026
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| Accès en ligne: | https://doi.org/10.5281/zenodo.19772687 |
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- <p><br>This work addresses the structural problem of defining physical observables in regimes where the spectral decomposition of self-adjoint operators is not purely discrete. In particular, it is demonstrated that, outside the ultraviolet regime, the identification between mass and eigenvalues ceases to be consistent when the associated spectral measure has a continuous component.</p> <p>Starting exclusively from minimal spectral hypotheses -- self - adjoint operator, resolvent, spectral measure, and its density -- it is established, without resorting to ad hoc hypotheses, that the observable physical structure is determined by the spectral measure associated with the state. As a consequence, it is proven that the physical dimension of a system is not given by the dimension of the Hilbert space nor by the cardinality of the spectrum, but by the structure of the dominant spectral regions of the density, defining the notion of effective spectral dimension.</p> <p>In this framework, physical observables emerge as functionals of the spectral measure. In particular, mass is characterized as the first-order spectral moment, while the spectral width quantifies the dispersion associated with the energy distribution. It is further shown that dominant spectral regions act as effective degrees of freedom, providing a distributed description of the dynamics.</p> <p>The developed formalism recovers the discrete limit of quantum mechanics, is compatible with the spectral representation of quantum field theory, and admits a natural interpretation in terms of scale flow. Explicit falsifiability criteria are also established, and the structural limitations of the proposed framework are delineated.</p> <p>The central result is the formulation of a principle: physical dimension and observables emerge from the effective spectral structure, implying the necessary substitution of pointlike descriptions with distributed descriptions in non-discrete regimes.</p>