Quantum Mechanics as Hamilton-Killing Flows on a Statistical Manifold

TL;DR

This paper reconstructs quantum mechanics from probability theory and information geometry, showing that key quantum features emerge naturally as Hamilton-Killing flows on a statistical manifold.

Contribution

It introduces a novel geometric framework where quantum formalism arises from flows preserving symplectic and metric structures on a statistical manifold.

Findings

Quantum features derived from geometric flows

Emergence of Hilbert space structure

Reconstruction of Schrödinger equation and Born rule

Abstract

The mathematical formalism of Quantum Mechanics is derived or "reconstructed" from more basic considerations of probability theory and information geometry. The starting point is the recognition that probabilities are central to QM: the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold -- a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini-Study metric, the linearity of the Schr\"odinger equation, the emergence of a complex numbers, Hilbert spaces, and the Born rule, are derived rather than postulated.