2012/04/03 by Mathieu Molitor, Molitor, Mathieu · 1 citation
Mathematics · Physics and Astronomy · #37K99 #53B35 #58B10 #62B10 #81P99 #94A15 #Differential Geometry (math.DG) #FOS: Mathematics #Probability and Statistical Research #Quantum Mechanics and Applications #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1204.0663
openalex publication_date 2012/04/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let (M,g) be a compact, connected and oriented Riemannian manifold. We denote D the space of smooth probability density functions on M. In this paper, we show that the Frechet manifold D is equipped with a Riemannian metric gD and an affine connection ∇D which are infinite dimensional analogues of the Fisher metric and exponential connection in the context of information geometry. More precisely, we use Dombrowski's construction together with the couple (gD,∇D) to get a (non-integrable) almost Hermitian structure on D, and we show that the corresponding fundamental 2-form is a symplectic form from which it is possible to recover the usual Schrodinger equation for a quantum particle living in M. These results echo a recent paper of the author where it is stressed that the Fisher metric and exponential connection are related (via Dombrowski's construction) to Kahler geometry and quantum mechanics in finite dimension.