Quantum theory from the geometry of evolving probabilities

Marcel Reginatto*, Michael J.W. Hall

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

15 Citations (Scopus)

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 Kähler 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.

Original languageEnglish
Title of host publicationBayesian 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
Pages96-103
Number of pages8
DOIs
Publication statusPublished - 2012
Event31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, MaxEnt 2011 - Waterloo, ON, Canada
Duration: 9 Jul 201116 Jul 2011

Publication series

NameAIP Conference Proceedings
Volume1443
ISSN (Print)0094-243X
ISSN (Electronic)1551-7616

Conference

Conference31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, MaxEnt 2011
Country/TerritoryCanada
CityWaterloo, ON
Period9/07/1116/07/11

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