In this article we exploit the geometric structure of statistical manifolds to compare vis-a-vis quantum and classical dynamics. We show that the fixed points of geodesic dynamics in a classical setting coincide with the distributions that minimise the Shannon's entropy. Whereas in a quantum setting this happens only for particular initial conditions, which correspond to classical submanifolds. This result can be interpreted as a manifestation of the uncertainty relation. Other differences put forward concern the (in)stability of geodesic dynamics. The classical scenario results in an isotropic manifold where the geodesic flow is stable. Conversely, on the quantum setting the instability of geodesic dynamics matches the spread out of geodesic ball volumes described by negative constant scalar curvature. spread out of geodesic ball volumes described by negative constant scalar curvature.
In this article we exploit the geometric structure of statistical manifolds to compare vis-a-vis quantum and classical dynamics. We show that the fixed points of geodesic dynamics in a classical setting coincide with the distributions that minimise the Shannon's entropy. Whereas in a quantum setting this happens only for particular initial conditions, which correspond to classical submanifolds. This result can be interpreted as a manifestation of the uncertainty relation. Other differences put forward concern the (in)stability of geodesic dynamics. The classical scenario results in an isotropic manifold where the geodesic flow is stable. Conversely, on the quantum setting the instability of geodesic dynamics matches the spread out of geodesic ball volumes described by negative constant scalar curvature. spread out of geodesic ball volumes described by negative constant scalar curvature.
The purpose of this paper is to exploit the geometric structure of quantum mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannon’s entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.
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.
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.
The quantum–classical transition is one of the most frequently invoked concepts in modern physics. Yet the notion of classicality itself is far from unique. Depending on the physical context, classical behavior may be associated with decoherence, the semiclassical limit, thermodynamic averaging, suppression of correlations, emergence of collective order, or geometric simplification of statistical state space. These viewpoints are often presented as if they described a single phenomenon, although they emphasize different physical mechanisms and different operational criteria. In this article we examine the principal notions of classicality that appear across quantum theory, statistical physics, condensed matter physics, and information geometry. We compare the corresponding mechanisms of classical emergence and analyze the physical quantities commonly used to characterize them, including coherence, entanglement, fluctuations, correlation length, Fisher information, statistical complexity, and information-geometric curvature. We argue that many apparently distinct routes toward classical behavior share a common structural feature: a reduction of effective fluctuation freedom. From this perspective, classicality may be interpreted as an emergent regime in which the accessible fluctuation manifold becomes progressively constrained, stabilized, or geometrically simplified. This viewpoint naturally unifies decoherence, semiclassical localization, thermodynamic averaging, decorrelation, and collective organization within a common conceptual framework. Rather than representing a unique physical process, the quantum–classical transition appears as a family of related mechanisms through which complex quantum fluctuation structure gives rise to effective macroscopic classical behavior.
Abstract This paper presents a geometric interpretation of quantum uncertainty within the C3C_{3}C3 phase formalism, where the temporal and spatial curvatures of the wavefunction are explicitly linked through the phase operators jjj and j2j^{2}j2 satisfying j3=−1j^{3} = -1j3=−1. We show that quantum behavior naturally arises when the temporal and spatial curvatures are unequal (Rt≠Rx)(R_{t} \neq R_{x})(Rt=Rx), producing a differential phase tension along the j−j2j - j^{2}j−j2 axis. Conversely, the equality Rt=RxR_{t} = R_{x}Rt=Rx corresponds to the classical limit, where the phase curvature manifold flattens and deterministic evolution is restored. This establishes a continuous geometric bridge between quantum and classical dynamics, in which uncertainty emerges not from statistical randomness but from intrinsic spacetime curvature asymmetry.
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.