Abstract We develop a Finslerian extension of the Schrödinger equation in which the underlying geometry depends on both position and phase direction. This approach generalizes the curved-phase formalism introduced earlier, replacing the Riemannian metric gμνg_{\mu\nu}gμν with a direction-dependent Finsler metric gμνF(x,y)g^{F}_{\mu\nu}(x,y)gμνF(x,y). The resulting dynamics unify quantum probability, geometric curvature, and analytic (ζ-layer) structure under a single covariant operator framework. An explicit construction of the Finsler–Schrödinger Hamiltonian, its Madelung decomposition, and a small-anisotropy expansion based on the Randers metric are presented. We further analyze the self-adjointness, unitarity, and potential experimental signatures of the theory, highlighting the geometric origin of anisotropic quantum propagation and phase–curvature coupling.
Abstract We develop a Finslerian extension of the Schrödinger equation in which the underlying geometry depends on both position and phase direction. This approach generalizes the curved-phase formalism introduced earlier, replacing the Riemannian metric gμνg_{\mu\nu}gμν with a direction-dependent Finsler metric gμνF(x,y)g^{F}_{\mu\nu}(x,y)gμνF(x,y). The resulting dynamics unify quantum probability, geometric curvature, and analytic (ζ-layer) structure under a single covariant operator framework. An explicit construction of the Finsler–Schrödinger Hamiltonian, its Madelung decomposition, and a small-anisotropy expansion based on the Randers metric are presented. We further analyze the self-adjointness, unitarity, and potential experimental signatures of the theory, highlighting the geometric origin of anisotropic quantum propagation and phase–curvature coupling.
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 We present a covariant generalization of the Phase–Cone Inequality to curved geometries, formulating it within both Riemannian and Finsler frameworks. In the flat limit, the inequality reduces to vϕ2≤κn(ΔΦ)2,v_{\phi}^{2} \leq \kappa_{n} (\Delta \Phi)^{2},vϕ2≤κn(ΔΦ)2, where κn\kappa_{n}κn is the multicarrier curvature coefficient characterizing interference order. Here, curvature and analytic continuation introduce two new covariant terms: the geometric curvature penalty K[u,ω]K[u,\omega]K[u,ω], governed by the Ricci or Finsler–Ricci tensor, and the analytic continuation term An[g;Φ]A_{n}[g;\Phi]An[g;Φ], carrying the transcendental constants (π,ζ(3),ζ(5),…)(\pi, \zeta(3), \zeta(5), \ldots)(π,ζ(3),ζ(5),…) that encode the parity–driven analytic hierarchy previously identified in flat CnC_{n}Cn manifolds. The resulting covariant form, vϕ2≤κn(ΔΦ)2−K[u,ω]+An[g;Φ],v_{\phi}^{2} \leq \kappa_{n} (\Delta \Phi)^{2} - K[u,\omega] + A_{n}[g;\Phi],vϕ2≤κn(ΔΦ)2−K[u,ω]+An[g;Φ], links quantum phase transport to local spacetime curvature and analytic number–theoretic structure. In the Finsler extension, anisotropic phase propagation introduces direction–dependent curvature coupling through the Finsler–Ricci scalar. These results imply that phase dynamics in curved or anisotropic backgrounds are constrained by both geometric focusing (Ricci contraction) and analytic opening (zeta hierarchy), offering a unified geometric–analytic limit for the speed of quantum evolution.
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 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 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.