Emergent Quantum Dynamics and Finite-Cutoff Area Quantization from Araki–BKM Information Geometry Information-Geometric Foundations of Quantum Dynamics and Emergent Geometry

We unify two emergent aspects of quantum theory within a single information-geometricframework based on the Araki relative entropy and the associated Bogoliubov–Kubo–Mori(BKM) metric on a von Neumann algebra state manifold.First, we show that in the commutative reduction to a spatial density subalgebra, theBKM metric becomes the classical Fisher information metric. Its trace gives, after a singlenormalization choice, the expectation value of the Bohmian quantum potential, and thecorresponding Bohmian trajectories arise as Jacobi geodesics of a conformally related metric.The guidance equation is then recovered from the polar decomposition of states satisfyingthe emergent Schr¨odinger equation.Second, we show that the modular flow of the reference state yields the Heisenbergand von Neumann equations, and that the physical Schr¨odinger equation follows under astandard unitary state implementation. In the finite regulated K¨ahler sector, geometricprequantization of the BKM state manifold leads to a discrete horizon area spectrum and touncertainty relations, with an induced gravitational coupling derived from the Seeley–DeWittcoefficient [1].No continuum limit is taken, and no identification with the observed Newton constantis assumed. The results exhibit a common information-geometric framework for Bohmianmechanics, quantum dynamics, and emergent gravitational structures.