Quantum theory from the geometry of evolving probabilities

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Title Quantum theory from the geometry of evolving probabilities
Author Reginatto, Marcel; Hall, Michael
Journal Name AIP Conference Proceedings
Year Published 2012
Place of publication United States
Publisher American Institute of Physics
Abstract We consider the space of probabilities {P(x)}, where the x are coordinates of a config- uration 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 canon- ically 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.
Peer Reviewed Yes
Published Yes
Alternative URI http://dx.doi.org/10.1063/1.3703625
Copyright Statement Copyright 2012 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in AIP Conference Proceedings, Vol. 1443, pp. 96-103 and may be found at dx.doi.org/10.1063/1.3703625.
Volume 1443
Page from 96
Page to 103
ISSN 0094-243X
Date Accessioned 2012-06-04
Language en_US
Research Centre Centre for Quantum Dynamics
Faculty Faculty of Science, Environment, Engineering and Technology
Subject Quantum Physics
URI http://hdl.handle.net/10072/47063
Publication Type Journal Articles (Refereed Article)
Publication Type Code c1

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