We prove that the stationary NUVO scalar transport equation λ 2 ∇ η 2 ψ+( 2 ∇ η lnλ⋅ ∇ η + ∇ η 2 lnλ )ψ=0 is gauge-equivalent, under ψ=ϕ/λ , to a Schrödinger eigenproblem ( − ℏ 2 2m Δ+ V phys + V eff [ λ ] )ϕ=Eϕ , with geometric potential V eff [ λ ]= ℏ 2 2m ( | ∇lnλ | 2 −Δlnλ ) . Under mild decay bounds on ∇lnλ and Δlnλ , H λ is semibounded and self-adjoint via the Friedrichs extension; for Kato-class V phys the operator domain equals H 2 ( ℝ 3 ) . If additionally λ, λ −1 ∈ L ∞ , the map ϕ↦ϕ/λ is a bounded similarity on L 2 . We give solvable profiles, a Birman-Schwinger/Lippmann-Schwinger representation, and a coupled partial-wave system for anisotropic λ . Consistency checks (constant λ , hydrogenic expectations) and spin terms from a conformal Dirac reduction confirm the construction. In combination with prior results of Quantization III, this paper establishes the Schrödinger framework as a scalar-geometric limit of NUVO, with V eff [ λ ] playing the role of a geometric quantum potential.
We prove that the stationary NUVO scalar transport equation λ 2 ∇ η 2 ψ+( 2 ∇ η lnλ⋅ ∇ η + ∇ η 2 lnλ )ψ=0 is gauge-equivalent, under ψ=ϕ/λ , to a Schrödinger eigenproblem ( − ℏ 2 2m Δ+ V phys + V eff [ λ ] )ϕ=Eϕ , with geometric potential V eff [ λ ]= ℏ 2 2m ( | ∇lnλ | 2 −Δlnλ ) . Under mild decay bounds on ∇lnλ and Δlnλ , H λ is semibounded and self-adjoint via the Friedrichs extension; for Kato-class V phys the operator domain equals H 2 ( ℝ 3 ) . If additionally λ, λ −1 ∈ L ∞ , the map ϕ↦ϕ/λ is a bounded similarity on L 2 . We give solvable profiles, a Birman-Schwinger/Lippmann-Schwinger representation, and a coupled partial-wave system for anisotropic λ . Consistency checks (constant λ , hydrogenic expectations) and spin terms from a conformal Dirac reduction confirm the construction. In combination with prior results of Quantization III, this paper establishes the Schrödinger framework as a scalar-geometric limit of NUVO, with V eff [ λ ] playing the role of a geometric quantum potential.
This paper derives the Schrödinger equation directly from the geometric structure of NUVO scalar space, without probabilistic postulates. Starting from the conformal metric g=λ2ηg = \lambda^2 \etag=λ2η and sinertia conservation, the analysis shows that the scalar transport law reduces in the stationary, closed-frame limit to the Schrödinger equation. The approach introduces a purely geometric interpretation of momentum, energy, and quantization, linking them to curvature and holonomy of the scalar field λ(x)\lambda(x)λ(x). Planck’s constant arises as a calibrated scale of scalar coherence.
We derive nonrelativistic quantum mechanics directly from NUVO scalar geometry without probabilistic postulates. Starting from the conformal metric g= λ 2 η and the conservation of sinertia flow, we obtain a deterministic scalar transport law whose stationary, closed-frame limit is the Schrödinger equation. Momentum and energy operators arise from the scalar-covariant derivative D μ = ∂ μ +i ∂ μ lnλ , canonical commutators follow from curvature [ D μ , D ν ]=i F μν , and the uncertainty relation is a local geometric bound. Planck’s constant appears as the calibrated scale of holonomy. Applications (hydrogenic levels, tunneling, spin holonomy, squeezing) illustrate how standard quantum phenomena emerge from curvature and holonomy of λ( x ) .
In conventional quantum mechanics, measurement and uncertainty are treated as intrinsic aspects of quantum systems, with little attention paid to the geometric structure of the observer’s frame. NUVO theory offers a new interpretation: quantum systems evolve within modulated frames defined by a conformal scalar field λ(t, r, v), while observers typically operate from a distinct reference frame. This paper introduces the concept of frame interface operators—operators that act across the boundary between a system’s internal λ-modulated frame and the observer’s external frame. We identify key operators in standard quantum mechanics that implicitly function at this interface, including the time evolution operator, boost generators, gauge transformations, geometric phase connections, basis transformations, and projective measurements. We show that many quantum phenomena, including the canonical commutator and Heisenberg uncertainty, already reflect inter-frame effects and need no further modification in NUVO. By rigorously distinguishing between frame-internal and frame-interface behavior, NUVO resolves longstanding interpretive ambiguities and establishes a foundation for a covariant operator formalism rooted in geometric modulation. This reinterpretation enables a consistent unification of quantum theory and conformal dynamics, providing new insight into measurement, decoherence, and observer-relative quantum structure.
A mathematically explicit extension of quantum hydrodynamics is developed in which a scalar coherence field ΦG modifies phase transport and is driven by the information-geometric structure of the probability density together with probability transport itself. The construction starts from the Madelung representation Ψ = √ρ eiS/ℏ, augments the phase gradient by the covariant shift ∇S ↦ ∇S − αℏ∇ΦG, and defines the dynamics through a variational (ρ, S) sector together with a dissipative Onsager-type relaxation law for ΦG. This yields a closed PDE system consisting of a continuity equation, a modified Hamilton–Jacobi equation, and a coherence-field equation that reduces in the overdamped limit to an elliptic closure. The source of ΦG is neither bare mass density nor bare probability density: it is a local Fisher-information density together with probability transport, so that ΦG measures coherence demand. The field equations are written explicitly; the coupling sign is fixed by an energy-stability argument, the transport-forcing sign by a dissipation argument, and the induced coherence-pressure term is computed exactly. A structural no-signalling condition is obtained by restricting the nonlinearity to Fisher-marginal source functionals that are blind to the entanglement Fisher excess under marginalisation. A coarse-grained Yukawa sector is shown to admit a conditional Newtonian limit under explicitly stated screening and additivity assumptions; this remains a conjectural macroscopic regime rather than a completed derivation of gravity. The paper closes with a falsifiable numerical programme—ablation studies, dispersion diagnostics, Fisher-information monitoring, and an explicit pilot solver scaffold—together with a claim-status ledger, a testability map, and a development roadmap that separate what is defined, what is derived, what is heuristic, and what remains unproven. Keywords: quantum hydrodynamics; Fisher–Rao geometry; Madelung dynamics; coherence field; no-signalling; information geometry.
We derive the canonical three-dimensional Schrödinger equation as the unique minimal local lift of Newtonian (Hamilton–Jacobi) particle dynamics on the ω-adapted spatial slice supplied by Charlton Geometry (CG). The geometric input is the CG substrate @ = (E4, δ, ω) with δ(ω, ω) = 1 and the Householder-reflected Lorentzian metric η := δ−2 ω⊗ω, which canonically defines a 3+1 split and spatial projector γ = δ−ω⊗ω. The observational input is the CG “noninvertibility route”: physically accessible macrostates form an observational quotient q : e S ↠ S by the lost-state group G = Z2 °ø U(1) with no preferred lift, so probabilities are forced as Ginvariant pushforwards and the Born density is fixed as |ψ|2. Given these inputs, and imposing a compact choice package on the lift representative ψ (U(1) covariance, norm preservation, correct ℏ → 0 Hamilton–Jacobi limit, and minimal local linear differential order: first in time, second in space), the only compatible evolution is iℏ ∂tψ =[− (ℏ^2 * ΔΣω) / 2m + V ] ψ. We also record (i) how the scalar sector generalizes canonically to a vector-potential sector by replacing ∇ with the covariant derivative of the CG-induced U(1) connection, where the electromagnetic potential arises from ω via A := η♭(ω) in the CG “reflected exterior calculus”, and (ii) falsifiers and open obligations, in particular the geometric origin of ℏ and the scale at which non-minimal corrections appear. Original DOI: 10.5281/zenodo.18225100 Version: 0.9.0
We derive the Schr¨odinger, Dirac, and Klein-Gordon equations from the scalar-time continuity law of Time-Scalar Field Theory (TSFT). By promoting evolution to the scalar-time parameter Θ and introducing a local time-rate field α(x), classical Hamiltonian and quantum operator dynamics unify under the relation dOdΘ = α{O,H} (classical) and dO dΘ = i¯hα[ ˆH ,O] (quantum). Schr¨odinger’s equation follows from ∂Θ = α∂t; the Dirac equation emerges from a Θ-weighted tetrad with a spin-connection term ∝ ∂μ ln α; and the Klein-Gordon equation arises as the relativistic scalar field equation in the effective metric induced by Θ. Standardquantum mechanics is recovered when ∇α = 0, while TSFT predicts falsifiable gradient corrections accessible to precision matter-wave and spinor interferometry. We supply quantitative estimates (e.g. δϕ ∼ 10−3 rad for Q = 109 cavities) and cross-link TSFT’s microphysics to its cosmological continuity, emphasizing multi-scale coherence and experiment-first falsifiability.