Fractal Reinterpretation of Quantum Mechanics Positioning: Implications for Hamiltonian Space and Future Quantum Realms (Intro: Vol-II) A Teaching Note
This paper reinterprets the triaxial positioning of fundamental physics as described in Mamun (2026), shifting from a linear coordinate framework to fractal geometric concepts. The current positioning of quantum mechanics thoughts, approximated at (0.7, 0.6, 0.4) across quantum, relativistic, and classical axes, is reconceptualised through fractal dimensions, self-similarity, and scale invariance, drawing on existing literature. We evaluate whether fractals can assess verticals—interpreted as specialised domains—in quantum mechanical enquiries, with a focus on Hamiltonian space where chaotic dynamics yield fractal structures. Speculations on future possibilities suggest fractals may bridge quantum realms with gravity and complexity, fostering unified theories. References are drawn from key works in fractal quantum mechanics. Mamun, SM (2026)……………………………………….1 Fractal Reinterpretation of Quantum Mechanics Positioning: Implications for Hamiltonian Space and Future Quantum Realms (Intro: Vol-II) Also available on Academia.edu: https://www.academia.edu/attachments/131513458/download_file?s=portfolio
This paper reinterprets the triaxial positioning of fundamental physics as described in Mamun (2026), shifting from a linear coordinate framework to fractal geometric concepts. The current positioning of quantum mechanics thoughts, approximated at (0.7, 0.6, 0.4) across quantum, relativistic, and classical axes, is reconceptualised through fractal dimensions, self-similarity, and scale invariance, drawing on existing literature. We evaluate whether fractals can assess verticals—interpreted as specialised domains—in quantum mechanical enquiries, with a focus on Hamiltonian space where chaotic dynamics yield fractal structures. Speculations on future possibilities suggest fractals may bridge quantum realms with gravity and complexity, fostering unified theories. References are drawn from key works in fractal quantum mechanics. Mamun, SM (2026)……………………………………….1 Fractal Reinterpretation of Quantum Mechanics Positioning: Implications for Hamiltonian Space and Future Quantum Realms (Intro: Vol-II) Also available on Academia.edu: https://www.academia.edu/attachments/131513458/download_file?s=portfolio
This report constructs a rigorous theoretical bridge connecting the deterministic evolution of isolated quantum systems, governed by the Hamiltonian operator, with the irreversible, macroscopic thermalization demanded by the second law of thermodynamics. It utilizes the Eigenstate Thermalization Hypothesis (ETH) to reconcile this contradiction, asserting that individual energy eigenstates are intrinsically thermal. To transcend ETH's descriptive limits, the Statistical Jacobi Approximation (SJA) is introduced to predict far-from-equilibrium dynamics and response functions following quantum quenches without requiring complete re-diagonalization. The document demonstrates that multiscale energy transfer, specifically the inverse energy cascade mediated by recursive zig-zag patterns observed in classical turbulence, persists within microscopic quantum superfluids and optical lattices. Critically, to stress-test these frameworks, the report employs fractal geometries as advanced simulational incubators. Fractals, by Mamun, SM (2026) ………………………….1 Quantum Mechanics, the Hamiltonian, and Thermodynamics: Predicting Eigenstates via Fractal Simulations (Volume II) K S C breaking translational symmetry, lead to the emergence of non-ergodic multifractal (NEM) states that actively defy ETH, and provide empirical validation through the high-resolution observation of Hofstadter's Butterfly in moiré superlattices. Also available on Academia.edu: https://www.academia.edu/attachments/132683179/download_file?s=portfolio
Hilbert spaces, introduced in the early twentieth century as infinite-dimensional generalisations of Euclidean space, provide the rigorous mathematical framework for quantum mechanics and numerous applied disciplines. This paper offers a comprehensive examination of their historical development, axiomatic structure, and multifaceted applications, with particular emphasis on their foundational role in quantum mechanics. We then connect these structures to quantum information geometry, where the (projective) Hilbert space is endowed with natural Riemannian metrics—most notably the Fubini–Study metric for pure states and the Bures metric for mixed states—that quantify statistical distinguishability, parameter estimation precision, and geometric phases. By exploring classical information geometry as a precursor and highlighting quantum extensions, we illustrate how Hilbert-space geometry reveals deep insights into quantum phase transitions, metrology, and quantum technologies. Nuances such as separability, rigged Hilbert spaces for continuous spectra, and infinite-dimensional subtleties are addressed, alongside implications for machine learning via reproducing kernel Hilbert spaces (RKHS) and beyond. The analysis underscores the enduring interplay between pure mathematics and quantum physics. Mamun, SM (2026) …………………………………..1 Hilbert Spaces: Foundations, Applications, and Geometric Connections to Quantum Information Theory (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132277960/download_file?s=portfolio
The measurement problem remains one of the most profound foundational issues in quantum mechanics. While the theory excels at predicting probabilities of measurement outcomes via the Schrödinger equation and the Born rule, it fails to provide a consistent, unitary description of the measurement process itself. This paper offers a comprehensive teaching-oriented review: it traces the historical development from the 1920s Copenhagen formulation through Schrödinger's cat paradox, examines the core mathematical and conceptual questions, analyses major interpretations (with a detailed exposition of the Many-Worlds Interpretation and an expanded treatment of decoherence theory), discusses Bell's theorem and its experimental tests, surveys contemporary trends as of 2025–2026 (including quantum computing implications), and evaluates the quantum-to-classical transition. The analysis highlights that the problem is not merely philosophical but has direct implications for quantum technologies, macroscopic superpositions, and our understanding of reality. No single interpretation is endorsed; instead, strengths, limitations, and open questions of each are presented to equip readers with a clear conceptual map. Mamun, SM (2026)……………………………………..1 The Quantum Measurement Problem: Historical Evolution, Fundamental Questions, Interpretations, Bell Tests, and Current Trends (Vol - I) Also available on Academia.edu: https://www.academia.edu/attachments/132618367/download_file?s=portfolio
Modern physics, grounded in quantum field theory (QFT) and general relativity, describes reality through fields, symmetries, and interactions without invoking classical mechanical substrates. Yet ontological questions persist regarding the underlying "ground" of these fields. This article examines a refined proposal that reinterprets the absolute as a meta-level generative condition for physical laws and the ether as a continuous, dynamic field substrate—aligning with contemporary theory while avoiding empirical contradictions. Drawing on the historical evolution of the ether concept from Aristotelian cosmology to the luminiferous aether of the 19th century and its apparent demise, the analysis integrates quantum implications, including the vacuum structure, measurement perturbations, and non-local correlations in entanglement. Far from a regression, this framework offers a unified ontological layer beneath existing equations, with nuances explored across classical, relativistic, and quantum perspectives. Potential implications for a theory of everything, cosmology, and philosophy of science are considered, alongside edge cases such as preferred frames and hidden variables. Mamun, SM (2026)………………………………………1 Reinterpreting the Absolute and Ether: Historical Evolution, Quantum Field Substrates, and Implications for Entanglement and Foundational Physics (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132227622/download_file?s=portfolio
This volume systematically reinterprets the security foundations of lattice-based post-quantum cryptography (PQC) for the theoretical post-post-quantum era through the mathematical lens of fractal geometry and fractal algebra. The analysis asserts that cryptographic hardness boundaries are not smooth Euclidean manifolds but complex fractal objects exhibiting scale invariance. Specifically, the research demonstrates four key fractal phenomena: 1) The distribution of viable short vectors in Learning With Errors (LWE) lattices possesses a non-integer box-counting dimension (typically KSCCN Mamun, SM (2026) ……………………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of Fractal Geometry Interpretations of Post-Post-Quantum Cryptography (Vol-VIII) K S C 1.8–2.3), yielding a tighter, scale-invariant bound on average-case hardness against quantum oracles. 2) Heavy-tailed Lévy-stable noise generates scale-free random walks (Lévy flights) with infinite Hausdorff measure, geometrically overwhelming the finite Euclidean capacity of Gottesman-Kitaev-Preskill (GKP) quantum error-correcting codes at every resolution. 3) The physical limitations of quantum attacks are bounded by thermodynamic strange attractors governing Quantum Random Access Memory (QRAM) erasure costs. 4) Noisy Intermediate-Scale Quantum (NISQ) variational algorithms fail due to chaotic optimization paths across non-integer dimensional barren plateaus. Also available on Academia.edu: https://www.academia.edu/attachments/132780584/download_file?s=portfolio
This assessment challenges the prevailing classical, isolationist paradigm of defense quantum engineering, which regards quantum states as inherently fragile and susceptible to thermal decoherence. It introduces an active, topological, and biological framework by synthesizing quantum-biological mechanisms—such as the radical-pair mechanism for coherence stabilization—with advanced mathematical constructs. The methodology integrates Entanglement-Weighted Operator Geometry (EWOG), multi-parameter Topological Data Analysis (TDA) via G-LoG bi-filtration, Information Geometry on Fisher-Rao manifolds, and the spectral realization of Riemann zeta zeros. This synthesis constructs a blueprint for resilient, emergent informational manifolds capable of active decoherence suppression. The resulting hybrid architectures accelerate the timeline for mission-relevant quantum sensing, computing, and communications, providing a robust, noise-resistant foundation that can bypass current limitations outlined in reports like the Congressional Research Service Defense Primer. KSCCN Mamun, SM (2026) …………………………….1 Quantum Defense Technologies Reimagined: A Geometric, Fractal, and Information-Theoretic Assessment (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132889757/download_file?s=portfolio