@inproceedings{fe829fca285846d09b40c8151a744b14,
title = "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, P(x) →P(x + θ), there is a natural metric over the parameters θ 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 introducing a K{\"a}hler 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.",
keywords = "Fisher-Rao metric, Kaehler geometry, quantum particle, symplectic geometry",
author = "Marcel Reginatto and Hall, {Michael J.W.}",
year = "2012",
doi = "10.1063/1.3703625",
language = "English",
isbn = "9780735410398",
series = "AIP Conference Proceedings",
pages = "96--103",
booktitle = "Bayesian Inference and Maximum Entropy Methods in Science and Engineering - 31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, MaxEnt 2011",
note = "31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, MaxEnt 2011 ; Conference date: 09-07-2011 Through 16-07-2011",
}