Study of nonlinear structures: quantum and fractal geometries
Abstract
A few years ago, we studied a geometric object called a "sector," defined as a portion of a plane in the form of a parallelogram. We demonstrated that the set of these sectors forms an algebraic structure, which we termed a "sectorial space." This space is scale-invariant and translation-invariant; it is built upon a sequence of novel algebraic structures that are rooted in second-degree equations-which is why we have characterized them as "non-linear." Although we constructed this sectorial space using well-defined geometric objects, the lack of concrete examples-other than the sectors themselves-made it difficult to fully grasp the potential of these structures. Recently, however, in a study on quantum mechanics, we proposed a mathematical definition for a "beam" of particles. In this article, we revisit that definition-refining it into a more purely mathematical form-to examine its underlying algebraic structure. It turns out that this object possesses all of these non-linear algebraic structures. Thus, by studying the "beam" object, we are able to re-examine these non-linear structures. Furthermore, we extend the application of these structures, by noting that the set of Gaussian functions and the set of distributions (in the sense of Laurent Schwartz) could be endowed with these structures through the definition of an equivalence relation. This would make it possible to perform operations such as the multiplication of Gaussian functions or the multiplication of distributions within this sectorial space. This space might even prove to be the ideal framework for the study of fractal geometries. Finally, this study provides a mathematical demonstration that, in order for quantum mechanics to be interpreted in a local and realistic manner, it requires a "sector"-based model.
Related
- Fractal Hyper-Trees: Combinatorial Enumeration, Symmetry Properties, and Ultrametric Structures
- Multiscale Nonlinear Risk Transmission in the Carbon–Fossil Energy–Clean Energy System from a Fractal-Market Perspective: Evidence from EMD–TVP-VAR and Interpretable Machine Learning
- Geometries of Quantum Field Theories
- Quantum cluster structures on bosonic extensions
- Shifted quantum affine symmetric pairs: finite presentations and graded coideal structures
BibTeX
@misc{corre2026study,
title = {{Study of nonlinear structures: quantum and fractal geometries}},
author = {Stephane Le Corre},
howpublished = {HAL},
year = {2026},
month = nov,
url = {https://hal.science/hal-05847389v2},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Study of nonlinear structures: quantum and fractal geometries
AU - Stephane Le Corre
PY - 2026
DA - 2026/11/03
PB - HAL
UR - https://hal.science/hal-05847389v2
AB - A few years ago, we studied a geometric object called a "sector," defined as a portion of a plane in the form of a parallelogram. We demonstrated that the set of these sectors forms an algebraic structure, which we termed a "sectorial space." This space is scale-invariant and translation-invariant; it is built upon a sequence of novel algebraic structures that are rooted in second-degree equations-which is why we have characterized them as "non-linear." Although we constructed this sectorial space using well-defined geometric objects, the lack of concrete examples-other than the sectors themselves-made it difficult to fully grasp the potential of these structures. Recently, however, in a study on quantum mechanics, we proposed a mathematical definition for a "beam" of particles. In this article, we revisit that definition-refining it into a more purely mathematical form-to examine its underlying algebraic structure. It turns out that this object possesses all of these non-linear algebraic structures. Thus, by studying the "beam" object, we are able to re-examine these non-linear structures. Furthermore, we extend the application of these structures, by noting that the set of Gaussian functions and the set of distributions (in the sense of Laurent Schwartz) could be endowed with these structures through the definition of an equivalence relation. This would make it possible to perform operations such as the multiplication of Gaussian functions or the multiplication of distributions within this sectorial space. This space might even prove to be the ideal framework for the study of fractal geometries. Finally, this study provides a mathematical demonstration that, in order for quantum mechanics to be interpreted in a local and realistic manner, it requires a "sector"-based model.
ER -