Quantum theory from the geometry of evolving probabilities
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.
Cite
@article{arxiv.1108.5601,
title = {Quantum theory from the geometry of evolving probabilities},
author = {Marcel Reginatto and Michael J. W. Hall},
journal= {arXiv preprint arXiv:1108.5601},
year = {2015}
}
Comments
12 pages. Presented at MaxEnt 2011, the 31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, July 10-15, 2011, Waterloo, Canada. Updated version: the affiliation of one of the authors was updated, minor changes were made to the text