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 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 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 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 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 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.
Abstract This paper derives a quantitative relation between the probability amplitude of a wavefield and the local curvature produced by its phase congruence. Starting from the covariant continuity equation for the probability current, we show that the amplitude A(x) of a complex field Ψ = A e^{iΦ} evolves according to the expansion scalar of its phase flow, which obeys the Raychaudhuri equation. The resulting expression, R_{\mu\nu} u^\mu u^\nu ≈ 2 · (1/A) · D²A / Dτ², demonstrates that curvature acts as the differential measure of amplitude concavity along the phase trajectories. Therefore, probability is not an independent quantity but the visible manifestation of phase-induced geometric curvature. The framework provides a natural geometric interpretation of quantum probability densities as curvature projections within analytic phase geometry.
Abstract This paper develops a formal correspondence between quantum probability and geometric curvature, showing that the amplitude of the wavefunction represents the observable manifestation of microscopic curvature generated by the phase field. Within the generalized phase–metric framework, the gradient of the phase Φ(x) deforms the background geometry, producing a curvature tensor that governs the local modulation of probability density. The analysis demonstrates that the conservation of the phase current acquires curvature corrections, yielding a Raychaudhuri-like equation linking the second covariant derivative of the amplitude to the Ricci curvature. Consequently, quantum indeterminacy can be reinterpreted as an epistemic projection of phase-induced curvature rather than intrinsic randomness. This approach unites the probabilistic and geometric aspects of quantum theory and integrates naturally into the Temporal Potential (ZPAT) framework, where temporal curvature governs amplitude evolution. The resulting picture provides an analytic–geometric synthesis bridging phase dynamics, probability flow, and spacetime structure.