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- W2524749809 abstract "Let $(M,g)$ be a compact, connected and oriented Riemannian manifold with volume form $d$ ${vol}_g$. We denoteby $mathcal{D}$ the space of smooth probability density functions on $M,,$ i.e.$mathcal{D}:= {rhoin C^{infty}(M,mathbb{R})| rho>0,,$and$,,int_{M}rhocdot $d${vol}_{g}=1},.$ We regard $mathcal{D}$ as an infinite dimensionalmanifold. In this paper, we consider the almost Hermitian structure on $Tmathcal{D}$ associated, via Dombrowski's construction, tothe Wasserstein metric $g^{mathcal{D}}$ and a natural connection $nabla^{mathcal{D}}$ on $mathcal{D}$. Using geometricmechanical methods, we show that the corresponding fundamental $2$-form on $Tmathcal{D}$ leads tothe Schrödinger equation for a quantum particle living in $M$. Geometrically, we exhibit a map which pulls back the Fubini-Studysymplectic form to the $2$-form on $Tmathcal{D}$. The integrability of the almost complex structure on $Tmathcal{D}$is also discussed. These results echo other papers of the author where it is stressed that the Fisher metric and exponentialconnection are related (via Dombrowski's construction) to Kähler geometry and the quantum formalism in finite dimension." @default.
- W2524749809 created "2016-10-07" @default.
- W2524749809 creator A5047823290 @default.
- W2524749809 date "2015-01-01" @default.
- W2524749809 modified "2023-09-26" @default.
- W2524749809 title "On the relation between geometrical quantum mechanics and information geometry" @default.
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- W2524749809 doi "https://doi.org/10.3934/jgm.2015.7.169" @default.
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