Second-order ordinary linear differential equations appear ubiquitously across physics, describing the behavior of systems from the quantum world of atoms to the classical world of gravitating bodies. We present a unified symmetry-based framework that provides a ``litmus-test criterion'' to determine when such a system admits a hierarchical ladder structure, and, whenever it does, explicitly constructs the ladder. This approach uncovers a previously underappreciated connection to supersymmetric quantum mechanics and a deep commonality among diverse physical problems. Applications to the quantum harmonic oscillator and dynamical tidal response of Kerr black holes are presented to illustrate the framework.
Second-order ordinary linear differential equations appear ubiquitously across physics, describing the behavior of systems from the quantum world of atoms to the classical world of gravitating bodies. We present a unified symmetry-based framework that provides a ``litmus-test criterion'' to determine when such a system admits a hierarchical ladder structure, and, whenever it does, explicitly constructs the ladder. This approach uncovers a previously underappreciated connection to supersymmetric quantum mechanics and a deep commonality among diverse physical problems. Applications to the quantum harmonic oscillator and dynamical tidal response of Kerr black holes are presented to illustrate the framework.
This article proposes a unified conceptual reconstruction of physical phenomena by examining the recurring structural patterns that connect classical and quantum systems across scales. Oscillatory behavior, potential landscapes, and binding principles form a common mathematical template that appears in mechanical oscillators, molecular vibrations, atomic orbitals, and quantized field modes. By reframing quantum phenomena as refinements of classical oscillatory structures rather than conceptual discontinuities, this work highlights the deep continuity that underlies modern physics. The framework also clarifies energy minimization, mass defect, statistical behavior, and vacuum fluctuations as manifestations of shared principles. Implications for pedagogy, theory building, and cross‑disciplinary coherence are discussed.
Extended version / Monograph-length manuscript This work proposes a unified conceptual reconstruction of physical phenomena by examining the recurring structural patterns that connect classical and quantum systems across scales. Oscillatory behavior, potential landscapes, and binding principles form a common mathematical template that appears in mechanical oscillators, molecular vibrations, atomic orbitals, and quantized field modes. By reframing quantum phenomena as refinements of classical oscillatory structures rather than conceptual discontinuities, this work highlights the deep continuity that underlies modern physics. The framework also clarifies energy minimization, mass defect, statistical behavior, and vacuum fluctuations as manifestations of shared principles. Implications for pedagogy, theory building, and cross‑disciplinary coherence are discussed. The innovative hypotheses, which depart from classical norms, will be explicitly marked with the heading HYPOTHESIS. As such, they are excluded by default from use as references in current academic or educational frameworks. They may instead be considered “revolutionary attempts to be monitored”, that is, the very “targets” mentioned by Weinberg. Albert Einstein introduced, on the one hand, a special relativity that he wished to be perfectly reversible, and on the other hand, general relativity which, when applied to cosmology, imposed on our Universe a spherical and static state—thus of constant radius—resisting gravitational collapse through the repulsion of the cosmological constant. Here, the associated relativities will instead give priority to irreversible systems evolving in accordance with the second law of thermodynamics, within their local space-time domain of one-way time flow. quantum physics, classical physics, grand unification, field theory, atomic stability, Albert Briffoz, physique quantique, unification, théorie des champs
Extended version / Monograph-length manuscript This work proposes a unified conceptual reconstruction of physical phenomena by examining the recurring structural patterns that connect classical and quantum systems across scales. Oscillatory behavior, potential landscapes, and binding principles form a common mathematical template that appears in mechanical oscillators, molecular vibrations, atomic orbitals, and quantized field modes. By reframing quantum phenomena as refinements of classical oscillatory structures rather than conceptual discontinuities, this work highlights the deep continuity that underlies modern physics. The framework also clarifies energy minimization, mass defect, statistical behavior, and vacuum fluctuations as manifestations of shared principles. Implications for pedagogy, theory building, and cross‑disciplinary coherence are discussed.The innovative hypotheses, which depart from classical norms, will be explicitly marked with the heading HYPOTHESIS. As such, they are excluded by default from use as references in current academic or educational frameworks. They may instead be considered “revolutionary attempts to be monitored”, that is, the very “targets” mentioned by Weinberg.Albert Einstein introduced, on the one hand, a special relativity that he wished to be perfectly reversible, and on the other hand, general relativity which, when applied to cosmology, imposed on our Universe a spherical and static state—thus of constant radius—resisting gravitational collapse through the repulsion of the cosmological constant.Here, the associated relativities will instead give priority to irreversible systems evolving in accordance with the second law of thermodynamics, within their local space-time domain of one-way time flow.
The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator (QMHO) carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general – except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalise the Heisenberg uncertainty principle. Coupled SUSY is explicitly realised by an infinite family of differential operators which admit orthonormal bases of eigenfunctions of generalised harmonic oscillators.
This paper presents a complete and verifiable theoretical framework that unifies quantum mechanics and classical physics/general relativity under a single mathematical function: H(S,θ,K)=Fclassical(S)⋅Q(θ)+∇K⋅(ΠK[δS])H(S,θ,K)=Fclassical(S)⋅Q(θ)+∇K⋅(ΠK[δS]) We propose that quantum mechanics is not a fundamental separate theory, but rather a structural regulatory mechanism activated by specific boundary conditions within a fundamental continuous space-time. The theory introduces three core components: S: A unified continuous structural field representing both space-time and matter Q(θ): A context-dependent switching function that transitions between classical (Q=1) and quantum (Q=0) regimes K: An inherent geometric framework that governs all structural deformations The theory operates through three mathematically rigorous conditions: Pure Quantum Space (Q=0): Geometric deformations dominate via K Classical Physics + GR (Q=1): Reduces exactly to Einstein/Newton equations Local Quantum Effects (0<Q<1): Hybrid regime preserving fundamental constants Key innovations include: Resolution of the measurement problem through natural θ-transitions Preservation of all classical laws (c, G, ħ) as global constants Elimination of the quantum-gravity coupling problem Falsifiable predictions through approximate self-similarity analysis We propose three mathematical candidates for K (E₈ symmetry, Hodge-Laplacian, Spin Network) with specific experimental predictions in atomic spectra, optomechanics, and black hole entropy. The theory provides a concrete research program for 2026-2030, transitioning physics from fragmented descriptions to a unified dynamical system. This work represents not just an equation, but an ecosystem for unified physics—a testable speculative paradigm that enriches theoretical discourse regardless of its ultimate verification status.
Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations (LSODE), including systems with friction linear in velocity such as the damped harmonic oscillator, can be related to the quantum free-particle dynamical system. This implies that symmetries and simple computations in the free particle can be exported to the LSODE-system. The quantum Arnold transformation is given explicitly for the damped harmonic oscillator, and an algebraic connection between the Caldirola-Kanai model for the damped harmonic oscillator and the Bateman system will be sketched out.