Geometric Model of a Dual Quantum Oscillator and Its Representation in Fluctuation and Pseudo-Euclidean Spaces

This study proposes a “Geometric model of a dual quantum oscillator” for a fundamental rethinking, a geometric interpretation of the structure of a quantum oscillator as a system of interaction between matter and antimatter, and a solution to the problem of infinite energies of “zero vibrations” in the physical vacuum. The novelty of the proposed model lies in the disclosure of the internal symmetrical structure of the quantum oscillator and its geometric interpretation. Based on the classical quantum harmonic oscillator, the model complements it with an innovative physical-geometric structure and new content. This content includes a number of conceptual notions, such as negative frequencies, dual structure, geometric averaging, generalized conjugation symmetry, asymmetry of the oscillator’s energy structure, generalized energy for an arbitrary number of degrees of freedom, quadratic forms for energy norms, fermion oscillator as a boost of the boson oscillator, as well as a dual quantum oscillator in fluctuation and pseudo-Euclidean spaces. The work retains the classical foundations and applies well-known mathematical methods, but the emphasis shifts to structural analysis, which led to the geometrization of the dual oscillator theory. This approach seems relevant because it allows us to physically, mathematically, and possibly ontologically connect quantum statistics with modern geometric paradigms. This can be seen as a fundamental step, providing a new rethinking and interpretation that not only complements standard theory but also reveals significant results. Their possible consequences are as follows: new methods of analyzing spectra and symmetries, a deeper connection with modern field theories, and the expansion of interpretations of quantum mechanics towards geometric ontology. Geometrization is not just a “beautiful interpretation”, but a real way to reveal hidden symmetries and simplifications in scientific research and solutions. **Note on Version 2:** A LaTeX typesetting error in the exponent (± symbol) was corrected. **Note for version 3:** Fixed LaTeX layout error in formula 4.26 (coth instead of tanh ).