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C3-Phase Formalism and the Generalized Schrödinger Geometry

AKTAŞ, BORA

Published Nov 23, 2025• 0 citations
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Abstract

Abstract The present work develops a self-consistent quantum framework based on the algebraC3={1,j,j2∣j3=−1}C_{3} = \{1, j, j^{2} \mid j^{3} = -1\}C3={1,j,j2∣j3=−1}, in which the conventional Schrödinger dynamics, the probabilistic interpretation, and the space–time metric emerge from a single cyclic phase geometry. In this construction the wavefunctionψ=a+jb+j2c\psi = a + jb + j^{2}cψ=a+jb+j2cis not a mere three-component field but the cyclic sum of quantities that are mutually related through a derivative chain: aaa corresponds to configuration, bbb to temporal rate (phase velocity), and ccc to spatial curvature. The differential operator DDD acting as Dψ=jψD\psi = j\psiDψ=jψ closes as D3=−1D^{3} = -1D3=−1, producing the canonical cubic wave equation (D3+1)ψ=0.(D^{3} + 1)\psi = 0.(D3+1)ψ=0. The physical norm is defined by the real trace of the C3C_{3}C3 inner product, ∥ψ∥phys2=a2+b2+c2,\|\psi\|^{2}_{\mathrm{phys}} = a^{2} + b^{2} + c^{2},∥ψ∥phys2=a2+b2+c2, which remains positive definite and conserved under the first-order evolution j ∂tψ+j2κ∇2ψ+(ω0+V)ψ=0.j\,\partial_{t}\psi + j^{2}\kappa\nabla^{2}\psi + (\omega_{0} + V)\psi = 0.j∂tψ+j2κ∇2ψ+(ω0+V)ψ=0. This equation generalizes Schrödinger dynamics by embedding both time and spatial propagation in the derivative cycle generated by jjj and j2j^{2}j2. Its plane-wave solutions satisfy a cubic dispersion relation Ω3=κ3k6+ω03+3κω0k2,\Omega^{3} = \kappa^{3}k^{6} + \omega_{0}^{3} + 3\kappa\omega_{0}k^{2},Ω3=κ3k6+ω03+3κω0k2, which reduces to the standard quadratic form in the weak-phase limit but reveals a new E∝k2/3E \propto k^{2/3}E∝k2/3 regime at strong curvature. Hence, the classical and quantum regimes appear as two asymptotic sectors of the same geometric equation. Hermiticity, measurement, and probability are reformulated within the C3C_{3}C3-Hilbert space: an operator A^=A0+(j−j2)Avis+(j+j2)Ahid\hat{A} = A_{0} + (j - j^{2})A_{\mathrm{vis}} + (j + j^{2})A_{\mathrm{hid}}A^=A0+(j−j2)Avis+(j+j2)Ahid is C3C_{3}C3-Hermitian if ⟨ψ∣A^ϕ⟩3=⟨A^ψ∣ϕ⟩3.\langle \psi | \hat{A}\phi \rangle_{3} = \langle \hat{A}\psi | \phi \rangle_{3}.⟨ψ∣A^ϕ⟩3=⟨A^ψ∣ϕ⟩3. The visible channel j−j2j - j^{2}j−j2 encodes measurable phase differences (time–space curvature contrast), whereas the hidden channel j+j2j + j^{2}j+j2 contains the unobservable but dynamically effective phase sum. The probability measure is then derived from the real projection of the C3C_{3}C3 inner product, ensuring consistency with classical Born statistics in the b,c→0b, c \to 0b,c→0 limit. From the composite fields (a,b,c)(a, b, c)(a,b,c) one can further define emergent metric components N,Ni,γijN, N^{i}, \gamma_{ij}N,Ni,γij, yielding a 3+13+13+1 line element ds2=−N2dt2+γij(dxi+Nidt)(dxj+Njdt),ds^{2} = -N^{2}dt^{2} + \gamma_{ij}(dx^{i} + N^{i}dt)(dx^{j} + N^{j}dt),ds2=−N2dt2+γij(dxi+Nidt)(dxj+Njdt), where temporal lapse and spatial conformal factors follow directly from the visible and hidden curvature densities. Consequently, both the wave dynamics and the geometry of space–time arise from the same algebraic phase structure. The model closes the long-standing gap between the probabilistic and geometric foundations of quantum mechanics by replacing the external metric with an intrinsic C3C_{3}C3 phase metric. Potential experimental probes—such as triple-path interferometry or Ramsey-3 schemes—could isolate the predicted E∼k2/3E \sim k^{2/3}E∼k2/3 dispersion and the visible/hidden phase channels. This formulation provides a unified description in which quantum behavior, the time operator, and curvature are facets of a single cyclic geometry. It offers a mathematically closed, physically interpretable, and experimentally testable framework that extends the Schrödinger equation into the C3C_{3}C3 algebraic domain, opening a possible path toward higher-order phase systems C4,C5,C6C_{4}, C_{5}, C_{6}C4,C5,C6 and the full geometric hierarchy of multicarrier quantum dynamics.

DOI
10.5281/zenodo.17692479

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Abstract This work introduces the C3–Quantum Geometry as a foundational extension of quantum mechanics, built upon the cubic algebra j3=−1j^{3} = -1j3=−1. By promoting the complex plane to a three-phase geometry, the theory unifies time, space, and energy within a single algebraic and differential cycle. The state function ψ=a+jb+j2c\psi = a + j b + j^{2} cψ=a+jb+j2c is interpreted not as a mere multicomponent wave, but as a cyclic triplet of position, velocity, and curvature — the differential closure of physical motion. The C3–Hilbert space introduces a positive-definite measure ∥ψ∥phys2=a2+b2+c2,\|\psi\|^{2}_{\mathrm{phys}} = a^{2} + b^{2} + c^{2},∥ψ∥phys2=a2+b2+c2, where the visible and hidden curvature axes, j−j2j - j^{2}j−j2 and j+j2j + j^{2}j+j2, separate measurable and latent quantum contributions. The corresponding C3–Schrödinger equation (j ℏ ∂t+j2γ∇2+μ)ψ=0(j\,\hbar\,\partial_{t} + j^{2}\gamma\nabla^{2} + \mu)\psi = 0(jℏ∂t+j2γ∇2+μ)ψ=0 restores the standard Schrödinger law as its visible branch while maintaining a closed cubic dynamic (L3+1)ψ=0(L^{3} + 1)\psi = 0(L3+1)ψ=0. This structure yields a new geometrical origin for uncertainty: Δx Δp=ℏ2 eα(R(3)−K),\Delta x\,\Delta p = \frac{\hbar}{2}\,e^{\alpha(R^{(3)} - K)},ΔxΔp=2ℏeα(R(3)−K), where the difference between spatial curvature R(3)R^{(3)}R(3) and temporal curvature KKK determines the observable quantum indeterminacy. The C3 framework therefore transforms the wavefunction into a full geometric object — a phase–curvature tensor uniting time, space, and energy through differential closure, Hermitian balance, and measurable norm preservation.
This monograph develops a consistent formulation of quantum mechanics on the cyclic algebra C3C_3C3, replacing the traditional complex imaginary unit iii with the unique anti-Hermitian axis j+j2j + j^2j+j2. This replacement yields a two-channel (visible–hidden) generalization of Schrödinger evolution and extends the standard complex formalism into a richer phase geometry. The resulting framework exhibits a natural decomposition into Hermitian and anti-Hermitian directions, enabling a multi-carrier interpretation of wave dynamics. The aim of this volume is not to present a finalized or definitive theory, but to consolidate a fast-expanding set of technical results—operator identities, geometric arguments, phase-projection methods, and algebraic constructions—into a unified reference document. Much of this material originated from scattered manuscripts, research notes, and exploratory derivations. The present edition gathers these components systematically, offering a coherent foundation for more optimized and fully structured future versions. The monograph should therefore be regarded as a developmental edition. Several chapters contain preliminary proofs or interim formulations; the C₃ trigonometric system, the multi-carrier interpretation of the wavefunction, and the visible–hidden projection formalism are presented in their earliest integrated form. Future editions will refine the conceptual hierarchy, provide additional examples, and extend the framework to relativistic domains, CnC_nCn operator algebras, and geometric interpretations of phase curvature. Keywords:C₃ algebra, cyclic complex numbers, multi-carrier quantum mechanics, two-channel Schrödinger equation, anti-Hermitian axis, visible–hidden phase geometry, phase curvature, generalized Born rule, operator algebra on CnC_nCn, extended trigonometric systems. Notes: This is a working research monograph; subsequent versions will reorganize and expand several sections and incorporate additional applications.
This work presents a C3-extended formulation of Schrödinger dynamics based on a phase algebra generated by an element j satisfying j^3 = -1. In contrast to the standard complex formulation, which is organized by a two-channel phase structure, the present framework introduces a three-channel decomposition of the wavefunction, Phi = a + bj + cj^2. Starting from the generalized evolution law jhbar d/dt Phi = Hhat Phi, the paper derives the associated first-order cyclic component system, the induced second-order hierarchy, and a closed third-order evolution equation for each component. In the free-particle sector, this structure leads to cubic time closure and sixth-order spatial closure at the component level. The purpose of the work is to establish the minimal mathematical core of the C3-extended Schrödinger framework, clarify its structural relation to the standard complex case, and identify the first consequences of three-channel phase dynamics. Questions of norm, probability, measurement, and observable projection are treated as open problems within a broader research program rather than as completed assumptions in the present paper.
Abstract We present a minimal and self-contained derivation showing how observable quantum dynamics emerge from a two-channel reduction of the C6 phase algebra. The phase-valued wavefunction Φ=ψ+χj3\Phi = \psi + \chi j^3Φ=ψ+χj3 satisfies a generalized phase–Schrödinger equation whose visible and hidden components mix through the internal phase generator j3j^3j3. Using well-defined projection operators PvisP_{\mathrm{vis}}Pvis and PhidP_{\mathrm{hid}}Phid, we extract the measurable sector and show that eliminating the hidden channel produces an effective nonlocal potential in the visible Schrödinger equation. This mechanism demonstrates, in the simplest possible setting, how phase geometry can induce corrections to particle dynamics and generate effective curvature terms in physical space. The construction provides a foundational bridge between C6 phase topology, emergent space geometry, and observable motion.
Abstract We propose a geometric extension of quantum mechanics based on the C3C_{3}C3-Hilbert space, where scalar elements satisfy j3=−1j^{3} = -1j3=−1. This algebra introduces two additional phase channels, jjj and j2j^{2}j2, naturally separating quantum states into visible and hidden components. The visible axis (j−j2)(j - j^{2})(j−j2) encodes measurable space–time curvature differences responsible for observable quantum behavior, while the hidden axis (j+j2)(j + j^{2})(j+j2) encapsulates non-observable phase accumulations linked to internal curvature balance. A new probability measure is introduced as the real trace of the C3C_{3}C3-inner product, yielding a positive-definite and physically meaningful norm ∥ψ∥phys2=∥ψ1∥2+∥ψj∥2+∥ψj2∥2.\|\psi\|^{2}_{\mathrm{phys}} = \|\psi_{1}\|^{2} + \|\psi_{j}\|^{2} + \|\psi_{j^{2}}\|^{2}.∥ψ∥phys2=∥ψ1∥2+∥ψj∥2+∥ψj2∥2. This generalization preserves Born’s rule while extending Hermiticity and unitarity to multi-channel operators via the C3C_{3}C3-adjoint A^†3\hat{A}^{\dagger_{3}}A^†3. The resulting framework establishes a covariant link between geometric curvature and quantum probability, offering a route toward curvature-based interpretations of uncertainty and measurement. The model recovers standard complex Hilbert mechanics as the classical limit (ψj,ψj2→0)(\psi_{j}, \psi_{j^{2}} \to 0)(ψj,ψj2→0), while allowing curvature-induced phase entanglement when these channels are active. This approach provides a natural embedding for time operators, curved metrics, and multi-carrier extensions of quantum dynamics.
Abstract This paper presents a geometric interpretation of quantum uncertainty within the C3C_{3}C3 phase formalism, where the temporal and spatial curvatures of the wavefunction are explicitly linked through the phase operators jjj and j2j^{2}j2 satisfying j3=−1j^{3} = -1j3=−1. We show that quantum behavior naturally arises when the temporal and spatial curvatures are unequal (Rt≠Rx)(R_{t} \neq R_{x})(Rt=Rx), producing a differential phase tension along the j−j2j - j^{2}j−j2 axis. Conversely, the equality Rt=RxR_{t} = R_{x}Rt=Rx corresponds to the classical limit, where the phase curvature manifold flattens and deterministic evolution is restored. This establishes a continuous geometric bridge between quantum and classical dynamics, in which uncertainty emerges not from statistical randomness but from intrinsic spacetime curvature asymmetry.