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 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.
- Publication:
-
Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering
- Pub Date:
- May 2012
- DOI:
- 10.1063/1.3703625
- arXiv:
- arXiv:1108.5601
- Bibcode:
- 2012AIPC.1443...96R
- Keywords:
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- Quantum Physics;
- Mathematical Physics
- E-Print:
- 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