Information metric from Riemannian superspaces

Abstract

The Fisher's information metric is introduced in order to find the real meaning of the probability distribution in classical and quantum systems described by Riemaniann non-degenerated superspaces. In particular, the physical r\^{o}le played by the coefficients $\mathbf{a}$ and $\mathbf{a}^\ast$ of the pure fermionic part of a genuine emergent metric solution, obtained in previous work is explored. To this end, two characteristic viable distribution functions are used as input in the Fisher definition: first, a Lagrangian generalization of the Hitchin Yang-Mills prescription and, second, the probability current associated to the emergent non-degenerate superspace geometry. Explicitly, we have found that the metric solution of the superspace allows establish a connexion between the Fisher metric and its quantum counterpart, corroborating early conjectures by Caianiello {\em et al.} This quantum mechanical extension of the Fisher metric is described by the $CP^1$ structure of the Fubini-Study metric, with coordinates $\mathbf{a}$ and $\mathbf{a}^\ast$.