On the relation between geometrical quantum mechanics and information geometry

Abstract

Let (M, g) be a compact, connected and oriented Riemannian manifold with volume form dvolg. We denote by D the space of smooth probability density functions on M, i.e. D := {ρ ∈ C∞ (M, ℝ) | ρ > 0 and ∫Mρ·dvolg = 1}. We regard D as an infinite dimensional manifold. In this paper, we consider the almost Hermitian structure on TD associated, via Dombrowski's construction, to the Wasserstein metric gD and a natural connection ∇D on D. Using geometric mechanical methods, we show that the corresponding fundamental 2-form on T 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 TD. The integrability of the almost complex structure on TD 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.