Quantum Mechanics as Hamilton-Killing Flows on a Statistical Manifold
Abstract
The mathematical formalism of Quantum Mechanics is derived or "reconstructed" from more basic considerations of probability theory and information geometry. The starting point is the recognition that probabilities are central to QM: the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold -- a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini-Study metric, the linearity of the Schr\"odinger equation, the emergence of a complex numbers, Hilbert spaces, and the Born rule, are derived rather than postulated.
Cite
@article{arxiv.2107.08502,
title = {Quantum Mechanics as Hamilton-Killing Flows on a Statistical Manifold},
author = {Ariel Caticha},
journal= {arXiv preprint arXiv:2107.08502},
year = {2021}
}
Comments
17 pages. Presented at MaxEnt 2021, The 40th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, (July 5--9, 2021, TU Graz, Austria) In V2 some arguments are slightly simplified, some references were added and typos corrected