This work presents a new theoretical and numerical framework for understanding the sharp quantum–classical crossover by introducing two internal structural degrees of freedom—the local spread S(x) and contraction C(x)—derived from the gradient and curvature of a probability distribution. These quantities form a two‑dimensional Spread–Contraction (SC) phase diagram that characterizes the transition between quantum‑dominated spreading and classical localization. A minimal covariant action motivated by information geometry is constructed, from which the SC Ordinary Differential Equation (SC‑ODE) is obtained via variational principles. The SC‑ODE integrates spreading and contracting contributions in a minimal form and provides a structural mechanism for the steep drop in interference visibility observed in experiments. To investigate the role of spatial structure, the study introduces a conservative SC Partial Differential Equation (SC‑PDE) incorporating degenerate mobility and backward diffusion. Numerical experiments under extreme initial conditions—where micro‑noise is superimposed on a uniform field—reveal a three‑phase structure consisting of stable, critical, and collapse regimes. This behavior contrasts with the always‑stable SC‑ODE and demonstrates that spatial degrees of freedom are essential for generating critical phenomena. A comparison with the Cahn–Hilliard equation shows that, within the scope of the numerical experiments, the SC‑PDE preserves non‑negativity and boundedness even under conditions where the Cahn–Hilliard model fails. This robustness arises from the automatic suppression of backward diffusion near the vacuum due to the degenerate mobility structure. Overall, this work provides a unified structural framework for describing the sharpness of the quantum–classical crossover and offers a numerically stable PDE formulation that may be relevant for surface evolution, thin‑film processes, and nonlinear field dynamics.
This work presents a new theoretical and numerical framework for understanding the sharp quantum–classical crossover by introducing two internal structural degrees of freedom—the local spread S(x) and contraction C(x)—derived from the gradient and curvature of a probability distribution. These quantities form a two‑dimensional Spread–Contraction (SC) phase diagram that characterizes the transition between quantum‑dominated spreading and classical localization. A minimal covariant action motivated by information geometry is constructed, from which the SC Ordinary Differential Equation (SC‑ODE) is obtained via variational principles. The SC‑ODE integrates spreading and contracting contributions in a minimal form and provides a structural mechanism for the steep drop in interference visibility observed in experiments. To investigate the role of spatial structure, the study introduces a conservative SC Partial Differential Equation (SC‑PDE) incorporating degenerate mobility and backward diffusion. Numerical experiments under extreme initial conditions—where micro‑noise is superimposed on a uniform field—reveal a three‑phase structure consisting of stable, critical, and collapse regimes. This behavior contrasts with the always‑stable SC‑ODE and demonstrates that spatial degrees of freedom are essential for generating critical phenomena. A comparison with the Cahn–Hilliard equation shows that, within the scope of the numerical experiments, the SC‑PDE preserves non‑negativity and boundedness even under conditions where the Cahn–Hilliard model fails. This robustness arises from the automatic suppression of backward diffusion near the vacuum due to the degenerate mobility structure. Overall, this work provides a unified structural framework for describing the sharpness of the quantum–classical crossover and offers a numerically stable PDE formulation that may be relevant for surface evolution, thin‑film processes, and nonlinear field dynamics.
Description This work introduces a structural framework for understanding the quantum–classical crossover using two internal quantities of the Thickness Structure Hypothesis (TSH): the spread deviation Δf and the shrink tension γT. These quantities characterize the competing tendencies of quantum spreading and classical or gravitational contraction. By organizing physical behaviors on the Δf–γT phase diagram, the framework provides a unified structural classification of quantum, classical, and gravitational regimes. A key result is that the sharp disappearance of quantum interference can be interpreted as a critical phenomenon. Near the boundary between the Stable (quantum) and Composite (classical) phases, the interference visibility follows a Landau‑type scaling law with critical exponent β = 1/2. This suggests that the abrupt switching observed in the quantum–classical transition arises from structural changes in Δf and γT rather than from external decoherence alone. Within this framework, wave‑packet collapse and the loss of interference visibility correspond to motion toward larger γT in the structural parameter space. The Δf–γT phase diagram thus provides a general, theory‑independent viewpoint for describing interference, localization, and the onset of classical behavior. The approach offers a structural mechanism for the sharpness of the quantum–classical crossover and connects quantum, classical, and gravitational behaviors within a single unified diagram.
This work investigates the structural origin of non‑Gaussian quantum noise by introducing two universal degrees of freedom: spread and contraction. These two components appear broadly in quantum optics, measurement theory, and statistical physics, and they characterize how a distribution acquires smoothness, nonlocality, sharpness, or irreversibility. Using a minimal 1+1 dimensional dynamical model, we show that the competition between spread and contraction determines the shape of the distribution intensity p(x). When the ratio between the two crosses a critical value, the system undergoes a continuous yet abrupt transition through three regimes: Gaussian → non‑Gaussian → two‑component. The model reproduces key signatures of non‑Gaussianity—such as changes in kurtosis, skewness, and the emergence of two‑component spectra—without introducing additional parameters. This framework provides a unified structural perspective on phenomena including the disappearance of interference fringes, sudden contraction in weak measurements, and non‑Gaussian noise observed in optical state tomography and decoherence experiments. The spread–contraction plane serves as a general classification map for understanding how non‑Gaussian noise arises from the balance between smoothing and localization mechanisms, independent of any specific physical theory.
This work develops a Lorentz-invariant variational framework in which Fisher-information geometry appears as an intrinsic structural contribution to quantum dynamics. Motivated by longstanding attempts to connect quantum mechanics with information-theoretic principles, we introduce an action functional depending on the density and phase fields in the Madelung representation. Variation of this action yields a modified Klein–Gordon equation containing a single nonlinear term proportional to the four-dimensional Fisher-information curvature of the probability density. The standard Klein–Gordon equation is recovered when the structural parameter vanishes, ensuring full compatibility with established relativistic dynamics. Taking the nonrelativistic limit, we obtain a uniquely determined nonlinear Schrödinger equation in which the correction term is the functional derivative of the Fisher information. The resulting dynamics preserve probability, maintain the Hamilton–Jacobi correspondence, and contain the linear Schrödinger equation as a special case. Analytical expressions for Gaussian and superposed states demonstrate how the structural modification scales with spatial localization and interference structure, providing clear qualitative signatures that distinguish the model from previous nonlinear extensions and offer a theoretical basis for future experimental verification. The results establish a mathematically transparent link between information geometry and quantum dynamics and provide a foundation for future extensions to fermionic, gauge, and many-body systems.
This mini-monograph proposes a new variational and information-geometric framework for understanding the quantum measurement problem, with particular emphasis on the unresolved problem of outcome selection. While decoherence theory explains the suppression of interference and the emergence of stable pointer bases, it does not by itself explain why one definite outcome is realized rather than another. This work argues that the missing element is not simply an additional dynamical mechanism, but a structural principle of selection. The central thesis is that quantum measurement may be understood as a process of entropic channel selection within a structured space of admissible outcomes. Drawing on Schrödinger bridge theory, entropy-regularized optimal transport, stochastic path-space variational principles, concentration of measure, and large deviation theory, the monograph develops a framework in which collapse-like definiteness emerges through constrained probabilistic evolution. Measurement is thereby reinterpreted not as an arbitrary interruption of unitary dynamics, but as a threshold phenomenon in the geometry of information. The work introduces an informational configuration space of possible resolution channels, defines an entropic action functional over stochastic paths, and proposes a channel-selection principle based on entropy-minimizing interpolation. It further examines the relation between Schrödinger bridges and optimal transport, explores the zero-noise limit, connects the framework to path-integral intuition, and develops explicit models of Gaussian bridge dynamics and two-channel selection. The emergence of classicality is treated through exponential concentration, threshold amplification, and phase-transition-like stabilization in high-dimensional systems. The monograph does not claim to replace standard quantum mechanics. Rather, it seeks to clarify how definite outcomes might arise from variational organization within probability itself, while remaining compatible with reduced quantum dynamics, decoherence, and open-system evolution. It is intended as a conceptual and mathematical foundation for further work at the intersection of quantum foundations, stochastic processes, information geometry, and the theory of emergent classicality.
We propose an exploratory framework for a Bohmian model of quantum matter propagating on a classical curved spacetime background. The gravitational sector is governed by classical Einstein field equations throughout; no quantisation of spacetime is attempted. The wave function emerges as the scalar contraction Ψ=ψνψν∈C of a complex-valued tensorial field ψμ, encoding quantum dynamics in a geometric object. The wave tensor interacts with spacetime via the stress–energy tensor Tμν, mediated by a real scalar field a of dimension volume, so that aTμνψμψν yields the correct potential energy. We derive a covariant Adapted Schrödinger Equation as the unique minimal covariant lift of the standard equation, justify it from four guiding principles, and verify three internal consistency checks. Under seven explicit approximations the framework reproduces the Schrödinger equation with Coulomb potential for the hydrogen atom. We also derive a dynamical equation for ψμ that entails the Adapted Schrödinger Equation by contraction. Two open problems are then resolved. First, a complete Lagrangian formulation is provided: a real-valued action for Ψ yields the Adapted Schrödinger Equation via the Euler–Lagrange equations; a separate action for ψμ, extended by a non-polynomial term, yields the full dynamical equation variationally. Second, two experimental predictions are derived. Expanding to first post-Newtonian order, the perturbation Hamiltonian has coefficients (3, 1) on the kinetic and potential operators; via the virial theorem these produce a coordinate-time blueshift, which after photon propagation yields the universal Einstein gravitational redshift δν/ν=Φ/c2, confirming consistency with the equivalence principle. The same kinetic coefficient independently predicts that free quantum wave packets spread more slowly by the fractional amount 3|Φ|/c2, a correction absent in standard non-relativistic quantum mechanics.
Entropic Dynamics (ED) provides a statistical-inferential foundation for physical laws, deriving motion and field equations from principles of entropy maximization rather than quantization postulates. ED reconstructs quantum mechanics by treating the evolution of probability distributions on configuration space as driven by information constraints, yielding the Schrodinger equation as a non-dissipative diffusion process. Building on this foundation, the present work extends the ED framework into a Unified Entropic Dynamics (UED) formulation that encompasses classical, quantum, relativistic, thermodynamic, and gravitational phenomena within a single information geometric principle. By maximizing entropy subject to constraints on diffusion, drift, and gauge covariance over a manifold endowed with a supermetric H_ab, we derive a universal field equation that merges the Fokker-Planck and Hamilton-Jacobi structures into one covariant form. When specialized to different dynamical variables, this equation reproduces the harmonic oscillator, Schrodinger, Maxwell, Klein-Gordon, and gravitational wave equations, thereby revealing a deep equivalence between probabilistic inference and dynamical law. The UED framework demonstrates that spacetime geometry, quantum coherence and thermodynamic diffusion emerge as complementary expressions of the same entropic process, establishing a unified inferential foundation for both microscopic and macroscopic physics. In this formulation, energy, probability, and entropy are intertwined aspects of information geometry, providing a consistent inferential foundation for understanding classical, quantum, and gravitational dynamics as complementary expressions of a single entropic law.