On the relation between geometrical quantum mechanics and information geometry

Mathieu  Molitor

doi:10.3934/jgm.2015.7.169

Article Contents

2015, Volume 7, Issue 2: 169-202. Doi: 10.3934/jgm.2015.7.169

This issue Previous Article On the extended Euler system and the Jacobi and Weierstrass elliptic functions Next Article A new multisymplectic unified formalism for second order classical field theories

On the relation between geometrical quantum mechanics and information geometry

Mathieu Molitor^1,

1.
Instituto de Matemática, Universidade Federal da Bahia, Av. Adhemar de Barros, S/N, Ondina, 40170-110 Salvador, BA

Received: March 31, 2012

Revised: March 31, 2014

Published: May 2015

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Abstract

Let $(M,g)$ be a compact, connected and oriented Riemannian manifold with volume form $d$ ${vol}_g$. We denote by $\mathcal{D}$ the space of smooth probability density functions on $M\,,$ i.e. $\mathcal{D}:= \{\rho\in C^{\infty}(M,\mathbb{R})| \rho>0\,\,$and$\,\,\int_{M}\rho\cdot $d${vol}_{g}=1\}\,.$ We regard $\mathcal{D}$ as an infinite dimensional manifold.
In this paper, we consider the almost Hermitian structure on $T\mathcal{D}$ associated, via Dombrowski's construction, to the Wasserstein metric $g^{\mathcal{D}}$ and a natural connection $\nabla^{\mathcal{D}}$ on $\mathcal{D}$. Using geometric mechanical methods, we show that the corresponding fundamental $2$-form on $T\mathcal{D}$ leads to the Schrödinger equation for a quantum particle living in $M$. Geometrically, we exhibit a map which pulls back the Fubini-Study symplectic form to the $2$-form on $T\mathcal{D}$. The integrability of the almost complex structure on $T\mathcal{D}$ is also discussed.
These results echo other papers of the author where it is stressed that the Fisher metric and exponential connection are related (via Dombrowski's construction) to Kähler geometry and the quantum formalism in finite dimension.

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Mathematics Subject Classification: Primary: 53Z05, 58B20, 81P99, 62B10; Secondary: 76Y99, 49Q99.

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