We establish a unified probabilistic–geometric formulation of quantum motion that extends infor- mation geometry into relativistic spacetime. The central result is the Nemoto–Curvature Correspon- dence, which identifies the Fisher–Rao scalar curvature RP with the quantum potential QP, thus linking probabilistic curvature to quantum energy density. By embedding the statistical manifold into a Lorentzian spacetime, we derive a covariant metric coupling term that predicts curvature backreaction from statistical fluctuations. A further Entropy–Curvature Duality connects informa- tion curvature with the second derivative of Shannon entropy, offering a new explanation of temporal asymmetry. Numerical analysis for Gaussian states demonstrates curvature-induced quantum dif- fusion, and the model yields testable corrections to spacetime curvature at Planck scales. This framework provides an original and predictive unification between probability, curvature, and en- tropy.
We establish a unified probabilistic–geometric formulation of quantum motion that extends infor- mation geometry into relativistic spacetime. The central result is the Nemoto–Curvature Correspon- dence, which identifies the Fisher–Rao scalar curvature RP with the quantum potential QP, thus linking probabilistic curvature to quantum energy density. By embedding the statistical manifold into a Lorentzian spacetime, we derive a covariant metric coupling term that predicts curvature backreaction from statistical fluctuations. A further Entropy–Curvature Duality connects informa- tion curvature with the second derivative of Shannon entropy, offering a new explanation of temporal asymmetry. Numerical analysis for Gaussian states demonstrates curvature-induced quantum dif- fusion, and the model yields testable corrections to spacetime curvature at Planck scales. This framework provides an original and predictive unification between probability, curvature, and en- tropy.
The absence of a common mathematical structure underlying quantum theory and relativistic physics remains a central obstacle to unification. While relativity is formulated in terms of spacetime geometry and covariance, quantum theory is intrinsically probabilistic, relying on amplitudes, uncertainty, and statistical interpretation. Existing unification programs typically attempt to quantize geometry or to geometrize quantum fields, but the probabilistic structure itself is rarely treated as a fundamental geometric object. In this paper, we propose a unified description of relativistic and quantum dynamics based on probabilistic geometry. Probability densities are taken as primary variables and are endowed with an intrinsic information–geometric structure defined by the Fisher–Rao metric. Within this framework, physical motion is described as the covariant evolution of probability distributions on a statistical manifold rather than as trajectories in a preassigned phase space or Hilbert space. The central result is a unified probabilistic–geometric field equation in which the relativistic d’Alembertian acting on the probability density is coupled to the scalar curvature of the associated information manifold. In an appropriate nonrelativistic limit, this equation reproduces the quantum potential as a curvature term, establishing a direct correspondence between quantum effects and informational geometry. Relativistic covariance is preserved throughout, allowing quantum and relativistic structures to appear as different regimes of a single probabilistic equation. The proposed framework does not introduce new particles, interactions, or quantization postulates. Instead, it provides a structural unification in which quantum and relativistic features emerge from the geometry of probability itself.
We present a probabilistic-geometric formulation of quantum motion that establishes a direct geometric correspondence between statistical uncertainty and relativistic curvature. Quantum states are represented as probability distributions on a statistical manifold equipped with the Fisher–Rao metric, whose geodesic flow reproduces the dynamical evolution of quantum motion in probabilistic form. Embedding this manifold into a Lorentzian spacetime yields a unified metric structure in which stochastic fluctuations generate microscopic deformations of spacetime geometry. This framework enables the visualization and prediction of quantum motion as probabilistic geodesics, and provides a foundation for connecting information geometry, stochastic mechanics, and general relativity within a single coherent formulation.
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.
We develop a geometric formulation of stochastic dynamics in which noise, diffusion, path probabilities, fluctuation theorems, and entropy production arise from the intrinsic geometry of an evolving manifold rather than from externally imposed randomness. Within the theory of moving manifolds, we establish a curvature-noise correspondence: fluctuations are governed by the inverse curvature tensor, while entropy production is controlled by curvature deformation. The invariant continuity law on a moving hypersurface yields a geometric Fokker-Planck equation, and curvature-velocity coupling generates a quadratic Onsager-Machlup functional determining path weights. The resulting entropy functional satisfies a curvature-driven monotonicity law, providing a geometric derivation of the Second Law. In two dimensions, the curvature invariant reduces to Gaussian curvature and encodes topology, so topological transitions produce discrete entropy jumps. When the ambient space carries a Minkowskian signature, the same curvature-kinetic quadratic form that generates dissipative thermal weights produces oscillatory phase weights, and the Laplace-Beltrami operator governing entropy evolution acquires a Schrödinger-type structure. This provides a geometric resolution of the apparent distinction between classical stochastic behaviour and quantum dynamics. These results show that stochastic behaviour, thermodynamic irreversibility, and quantum transition amplitudes are unified within the moving manifold framework. Geometry does not merely accommodate stochasticity; stochastic behaviour arises as a consequence of deterministic geometric evolution. The theory predicts curvature-controlled anisotropic diffusion, entropy jumps at topology-changing events, and a geometric thermal-quantum crossover in which classical stochastic weights and quantum amplitudes are generated by the same curvature-kinetic action.
This work analyzes the stability of probabilistic manifolds embedded in curved spacetime, extending earlier formulations in which gravity emerges from the localization of probability. By introducing an affine probabilistic connection that preserves normalization, the framework shows that curvature induces a natural correction to coherence transport on the underlying state manifold. The resulting curvature‑dependent coherence equation yields a stability criterion in which positive curvature enhances localization and suppresses divergence between probability configurations, while negative curvature drives decoherence growth. A generalized divergence metric and curvature‑modulated coherence gate unify gravitational localization, quantum coherence limits, and vacuum stability within a single geometric principle: the self‑consistency of probability under curvature. The analysis predicts curvature‑dependent decoherence rates, suppression of false‑vacuum tunneling in regions of positive curvature, and cosmological thresholds associated with coherence instability. This establishes a dynamic feedback loop between probability, coherence, and geometry, offering a unified interpretation of gravitational behavior and quantum stability.
This work presents an extended, mathematically complete formulation of the Information Geometry Unification Principle (IGUP), a foundational framework in which quantum mechanics, statistical thermodynamics, general relativity, and spacetime geometry emerge from a single information-theoretic ontology. We begin with rigorous axioms defining information as the primary physical substance, then systematically construct the Bures–Fisher–Rao metric on the manifold of quantum density matrices. We derive the universal information action functional and perform full variational derivations to obtain the unified field equations. We show explicitly how standard physical theories including the Schrödinger equation, Heisenberg picture, Liouville mechanics, classical thermodynamics, and Einstein field equations are recovered in appropriate limiting regimes. We further derive geometric expressions for entanglement, entropy, temperature, energy, and force. The theory resolves longstanding conceptual tensions between unitary quantum evolution and gravitational irreversibility, provides a natural resolution to the black hole information paradox, and yields eight concrete, experimentally testable predictions. This extended version includes full tensor calculus, covariant derivatives, curvature expansions, and limit theorems, amounting to a self-contained theory of everything based on Geometry=Information.