Quantum theory from the geometry of evolving probabilities

TL;DR

This paper derives quantum mechanics from the geometry of evolving probability distributions, introducing a Kähler structure that naturally leads to wave functions and Hilbert space, providing a geometric foundation for quantum theory.

Contribution

It presents a novel geometric framework where quantum mechanics emerges from the geometry of probability spaces with a Kähler structure, without relying on traditional postulates.

Findings

Wave functions arise as natural variables in the geometry of probabilities.

The quantum free particle Hamiltonian is derived from geometric symmetries.

A Hilbert space structure is constructed from the probability geometry.

Abstract

We consider the space of probabilities {P(x)}, where the x are coordinates of a configuration space. Under the action of the translation group there is a natural metric over the space of parameters of the group given by the Fisher-Rao metric. This metric induces a metric over the space of probabilities. Our next step is to set the probabilities in motion. To do this, we introduce a canonically conjugate field S and a symplectic structure; this gives us Hamiltonian equations of motion. We show that it is possible to extend the metric structure to the full space of the {P,S} and this leads in a natural way to a Kaehler structure; i.e., a geometry that includes compatible symplectic, metric and complex structures. The simplest geometry that describes these spaces of evolving probabilities has remarkable properties: the natural, canonical variables are precisely the wave functions of quantum mechanics; the Hamiltonian for the quantum free particle can be derived from a representation of the Galilean group using purely geometrical arguments; and it is straightforward to associate with this geometry a Hilbert space which turns out to be the Hilbert space of quantum mechanics. We are led in this way to a reconstruction of quantum theory based solely on the geometry of probabilities in motion.