Abstract This work introduces a covariant reformulation of quantum dynamics in which the laboratory time ttt is replaced by a geometric parameter τ\tauτ, interpreted as a potential-time variable along a phase flow. The standard Schrödinger equation,iℏ ∂tψ=H^ψi\hbar\,\partial_t \psi = \hat{H}\psiiℏ∂tψ=H^ψ,is generalized toiℏ DψDτ=H^τψi\hbar\,\frac{D\psi}{D\tau} = \hat{H}_\tau \psiiℏDτDψ=H^τψ,withDDτ:=uμ(∇μ+iΩμ)\frac{D}{D\tau} := u^\mu(\nabla_\mu + i\Omega_\mu)DτD:=uμ(∇μ+iΩμ),where uμu^\muuμ is the unit phase flow and Ωμ\Omega_\muΩμ the time connection, whose curvatureKμν=∇μΩν−∇νΩμK_{\mu\nu} = \nabla_\mu \Omega_\nu - \nabla_\nu \Omega_\muKμν=∇μΩν−∇νΩμdefines a measurable time curvature. The resulting framework embeds Schrödinger evolution within a geometric covariant structure that remains unitary, allows a natural definition of a time operator or covariant time POVM, and recovers the standard form in the flat limit (Ωμ→0\Omega_\mu \to 0Ωμ→0, R(3)→0R^{(3)} \to 0R(3)→0). Probability conservation, phase transport, and curvature coupling are derived explicitly, showing that amplitude focusing obeysRμνuμuν≈2 D2lnA/Dτ2+(2/ℏ) DΩτ/DτR_{\mu\nu}u^\mu u^\nu \approx 2\,D^2\ln A/D\tau^2 + (2/\hbar)\,D\Omega_\tau/D\tauRμνuμuν≈2D2lnA/Dτ2+(2/ℏ)DΩτ/Dτ.This approach unifies time, geometry, and quantum evolution in a single covariant dynamical law.
Abstract This work introduces a covariant reformulation of quantum dynamics in which the laboratory time ttt is replaced by a geometric parameter τ\tauτ, interpreted as a potential-time variable along a phase flow. The standard Schrödinger equation,iℏ ∂tψ=H^ψi\hbar,\partial_t \psi = \hat{H}\psiiℏ∂tψ=H^ψ,is generalized toiℏ DψDτ=H^τψi\hbar,\frac{D\psi}{D\tau} = \hat{H}\tau \psiiℏDτDψ=H^τψ,withDDτ:=uμ(∇μ+iΩμ)\frac{D}{D\tau} := u^\mu(\nabla\mu + i\Omega_\mu)DτD:=uμ(∇μ+iΩμ),where uμu^\muuμ is the unit phase flow and Ωμ\Omega_\muΩμ the time connection, whose curvatureKμν=∇μΩν−∇νΩμK_{\mu\nu} = \nabla_\mu \Omega_\nu - \nabla_\nu \Omega_\muKμν=∇μΩν−∇νΩμdefines a measurable time curvature. The resulting framework embeds Schrödinger evolution within a geometric covariant structure that remains unitary, allows a natural definition of a time operator or covariant time POVM, and recovers the standard form in the flat limit (Ωμ→0\Omega_\mu \to 0Ωμ→0, R(3)→0R^{(3)} \to 0R(3)→0). Probability conservation, phase transport, and curvature coupling are derived explicitly, showing that amplitude focusing obeysRμνuμuν≈2 D2lnA/Dτ2+(2/ℏ) DΩτ/DτR_{\mu\nu}u^\mu u^\nu \approx 2,D^2\ln A/D\tau^2 + (2/\hbar),D\Omega_\tau/D\tauRμνuμuν≈2D2lnA/Dτ2+(2/ℏ)DΩτ/Dτ.This approach unifies time, geometry, and quantum evolution in a single covariant dynamical law.
Abstract We present a covariant extension of the Schrödinger framework in which the quantum phase is promoted to a geometric connection AμA_\muAμ on spacetime. The associated curvature Fμν=∂μAν−∂νAμF_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\muFμν=∂μAν−∂νAμ encodes local deviations from flat phase geometry and manifests as measurable spectral shifts and uncertainty modulation. This formulation unifies geometric and probabilistic interpretations of quantum mechanics by identifying phase curvature with the internal concave curvature of the quantum domain. Two minimal, testable models—a one-dimensional ring (Aharonov–Bohm type) and a two-dimensional constant-curvature plane (Landau-type quantization)—demonstrate how phase curvature translates into observable level spacing. The framework preserves probability conservation in admissible gauges and recovers standard quantum mechanics in the flat-phase limit.
Abstract The present work develops a covariant Schrödinger formalism in which phase, curvature, and probability flow emerge as interdependent aspects of a single field. Starting from a phase-induced metric deformation gμν(Φ)g_{\mu\nu}(\Phi)gμν(Φ), we construct a geometric Schrödinger equation that naturally couples to an Einstein-like tensor Gμν[g]G_{\mu\nu}[g]Gμν[g]. This yields a unified system linking the quantum amplitude AAA, the local curvature R[g]R[g]R[g], and the analytic hierarchy of transcendental constants (π, ζ(3), ζ(5), …). Within this framework, the classical probability interpretation is reinterpreted as the observable projection of phase–curvature dynamics. The resulting equations provide a self-consistent description of how quantum interference, analytic number structure, and spacetime geometry co-evolve within a single differential system.
Abstract This paper reformulates the Schrödinger dynamics in a fully geometric and gauge-theoretic language, where the phase of the quantum wavefunction acts as a U(1) connection and its curvature manifests as measurable interference and geometric deformation. By coupling the connection Aμ=ℏ−1∂μSA_\mu = \hbar^{-1}\partial_\mu SAμ=ℏ−1∂μS to a curved phase metric gμν(n)=gμν+χnAμAν+ηnA2Πμνg^{(n)}_{\mu\nu} = g_{\mu\nu} + \chi_n A_\mu A_\nu + \eta_n A^2 \Pi_{\mu\nu}gμν(n)=gμν+χnAμAν+ηnA2Πμν, the Schrödinger equation acquires a covariant form that unites probability flow, curvature, and analytic hierarchy. The corresponding field strength Fμν=∂μAν−∂νAμF_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\muFμν=∂μAν−∂νAμ defines the geometric curvature of phase space, leading to a Maxwell-like equation for the phase field and an emergent Lorentz-type force on the probability current. The framework provides a direct correspondence between phase holonomy, Cn-carrier parity, and the analytic ζ-layer structure, suggesting a measurable bridge between gauge geometry and transcendental constants in multi-path interferometry.
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.
Abstract We propose a geometric reinterpretation of the Schrödinger equation in which the phase gradient functions as a U(1) connection and its curvature defines a measurable field strength. This framework transforms quantum probability flow into a covariant gauge theory, revealing a direct equivalence between phase curvature, spatial geometry, and analytic constants such as π and ζ(3). The result suggests that the geometric structure underlying quantum interference may itself be a manifestation of curvature in a higher-dimensional phase manifold.
Description This work develops a unified operator-flow framework on Hilbert spaces in which the standard dynamical formulations of quantum theory—the Schrödinger equation, the Heisenberg operator evolution, and Dirac-type spinor dynamics—arise as curvature-selected representations of a single abstract unitary flow. Starting from an axiomatic assignment of an admissible field tensor FFF to an anti-Hermitian generator G(F)G(F)G(F), the framework establishes the existence of a strongly continuous unitary flow acting on vectors, operators, and Clifford modules. From the same field tensor, an effective metric geff(F)g_{\mathrm{eff}}(F)geff(F) and its scalar curvature κ(F)\kappa(F)κ(F) are constructed, providing a geometric structure underlying the operator evolution. The central result, the Curvature Trichotomy Theorem, shows that the sign of κ(F)\kappa(F)κ(F) uniquely determines the curvature-adapted representation of the flow: positive curvature selects the Schrödinger picture, zero curvature selects the Heisenberg picture, and negative curvature selects a Dirac-type Clifford-module representation. In this way, the three classical formulations of quantum dynamics are interpreted not merely as unitarily equivalent pictures, but as distinct geometric realizations of a single operator-flow governed by curvature. Analytically, the work relies on standard results from the theory of generators of unitary groups, non-autonomous evolution equations, operator algebras, and spin geometry. Conceptually, it provides a geometric classification principle for quantum dynamics, linking representation theory, operator flows, and curvature within a unified mathematical structure. Preprint notice This manuscript is a preprint made publicly available for the purpose of early dissemination and scholarly discussion. It has not yet undergone peer review. The content, structure, or presentation may be revised in a subsequent peer-reviewed publication, and the final published version—if accepted—may differ from this preprint.