A Probabilistic–Geometric Unification of Relativistic and Quantum Dynamics Information Curvature as a Common Structural Equation
Ryusho Nemoto
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.
•Feb 20, 2026•0 citations