This project establishes Spectral Quantum Mechanics (SQM) as a new theoretical framework based on arithmetic structure. It is formulated within a prime-logarithmic Hilbert space, 𝐿 2 ( 𝑅 + , 𝑑 𝑥 / 𝑥 ) L 2 (R + ,dx/x), where quantum behavior emerges from spectral operators defined over the logarithms of prime numbers. SQM replaces traditional spacetime-based quantization with a purely arithmetic spectral foundation.
This project establishes Spectral Quantum Mechanics (SQM) as a new theoretical framework based on arithmetic structure. It is formulated within a prime-logarithmic Hilbert space, 𝐿 2 ( 𝑅 + , 𝑑 𝑥 / 𝑥 ) L 2 (R + ,dx/x), where quantum behavior emerges from spectral operators defined over the logarithms of prime numbers. SQM replaces traditional spacetime-based quantization with a purely arithmetic spectral foundation.
This preprint proposes an arithmetic-spectral vacuum framework in which quantum superposition is interpreted as a structural property of an underlying vacuum substrate. The model represents the vacuum as a discrete spectral state space whose elementary modes are indexed by prime numbers. Physical states are described as superpositions over these prime-indexed modes, while observables, effective time, geometry, entanglement, uncertainty, and measurement are interpreted as emerging from filtered and stabilized structures within this state space. The framework introduces a realizability filter, a structural capacity functional, and a coherence functional in order to distinguish formal mathematical configurations from physically persistent ones. The work is presented as a speculative mathematical framework, not as an experimentally established fundamental theory. Its purpose is to provide a structured research program for exploring whether quantum phenomena may be reformulated as emergent features of a deeper arithmetic-spectral organization of the vacuum.
Abstract Recent experimental results (Nature 2021; PRL 2026) have established the full empirical equivalence between real‑number quantum mechanics (RQM) and standard complex quantum mechanics (CQM), raising the foundational question of why quantum theory admits a real formulation and what the ontological origin of the complex structure is. In this work, we develop the χ‑Reality framework — an arithmetic‑spectral, operator‑theoretic paradigm in which the complex structure, physical constants, vacuum dynamics, renormalization flows, and macroscopic geometry emerge from the action of a single real operator χ on the unified substance S. The antisymmetric projection χJ=12(χ−χ†), acting as a normal operator on the invariant subspace of stable resonances Hstable, satisfies χJ2=−I, thereby generating the complex‑structure isometry within a purely real operator space. The deformed Dirac–Kähler operator Dχ possesses a dual arithmetic spectrum Spec(Dχ)={lnpk}∪{γk∣ζ(1/2+iγk)=0}, linked by Selberg/Weil dualities, binding coordinate‑domain prime modes to frequency‑domain spectral zeros. The arithmetic potential Aε(σ,t)=∑p∈Pp−σe−εplnp sin(tlnp), with scale‑invariant phase t(E)=ln(E/Λχ), governs vacuum expectation values and induces renormalization flows via the operator identity β(g)=−ε∂∂ε⟨Aε(σ,t)⟩χ, linking the running of coupling constants directly to the prime distribution π(x)∼x/lnx. Fundamental constants emerge as spectral invariants: the fine‑structure constant α as a χ‑fixed point on the critical line σ=1/2; the electron mass ratio μ as a stabilized χ‑mode under ε‑deformation; the vacuum permittivity ε0 as low‑frequency χ‑density; and Planck’s constant h as the global conformal scale of χ‑dynamics, relating the phase‑space volume of the real operator manifold to the invariant Liouville measure of χ‑flow. The theory yields three independent, falsifiable predictions across radically different energy scales: (i) logarithmic‑periodic modulations in ultra‑high‑energy cosmic rays arising from prime‑mode oscillations; (ii) sub‑Schwinger vacuum breakdown at strong‑field facilities such as ELI‑NP due to χ‑vacuum hotspots; and (iii) ultra‑low χ‑noise at Eχ≈10−33 eV with spectral density Sχ(ν)∝[νln(ν/νχ)]−1, detectable through long‑term drift in optical frequency standards and gravitational‑wave interferometers. Consequently, χ‑Reality offers a self‑consistent, operator‑anchored, arithmetic‑spectral paradigm that unifies quantum mechanics, geometry, empirical constants, and vacuum structure within a single, testable framework. Its predictions are quantitative, multi‑scale, and directly susceptible to empirical verification, rendering the theory a rigorous candidate for a foundational description of physical reality.
Modern theoretical physics is built upon several fundamental mathematical structures, including Hilbert spaces, Green functions, operator algebras, Gibbs ensembles, and spacetime geometry. These structures are traditionally introduced independently and serve as the foundations of quantum mechanics, quantum field theory, statistical mechanics, and general relativity. In this work, we propose an alternative mathematical framework based on a single Reconstruction Framework, in which these structures are generated sequentially from one primitive mathematical architecture. Beginning with an elementary transport architecture, transport operators, Green functions, Diagram Hilbert Spaces, complex quantum operators, Gibbs states, entropic projection, and finally macroscopic spacetime and gauge fields are reconstructed as successive mathematical layers. Each layer is determined by the preceding one and introduces no additional primitive mathematical assumptions. The proposed framework unifies several existing approaches to emergent physics within a common mathematical hierarchy and suggests that the diverse mathematical languages of modern theoretical physics are different representations of one underlying microscopic architecture. Rather than postulating Hilbert spaces or spacetime as fundamental, the present framework treats them as reconstructed structures arising naturally from progressively richer mathematical descriptions.
Modern theoretical physics employs several fundamental mathematical structures, including Green functions, Hilbert spaces, operator algebras, Gibbs ensembles, gauge fields, and spacetime geometry. These structures are traditionally introduced independently and serve as separate foundations for quantum mechanics, quantum field theory, statistical mechanics, and general relativity. In the present work, we propose an alternative mathematical organization based on a single Reconstruction Principle. Beginning with one primitive mathematical architecture, successive reconstruction layers generate transport operators, Green functions, Diagram Hilbert Spaces, complex quantum operators, Gibbs ensembles, entropic projections, and finally macroscopic spacetime together with gauge fields. Each layer is reconstructed uniquely from its predecessor without introducing additional primitive mathematical structures. Consequently, Hilbert space, Green functions, and spacetime are interpreted as successive mathematical representations of one underlying architecture rather than independent foundations. The resulting reconstruction hierarchy provides a unified conceptual framework connecting quantum mechanics, statistical mechanics, gauge theory, and emergent gravity while substantially reducing the number of independent assumptions.