REMARKS ON THE STATISTICAL ORIGIN OF THE GEOMETRICAL FORMULATION OF QUANTUM MECHANICS

Abstract

A quantum system can be entirely described by the Kähler structure of the projective space [Formula: see text] associated to the Hilbert space [Formula: see text] of possible states; this is the so-called geometrical formulation of quantum mechanics. In this paper, we give an explicit link between the geometrical formulation (of finite dimensional quantum systems) and statistics through the natural geometry of the space [Formula: see text] of non-vanishing probabilities [Formula: see text] defined on a finite set En: = {x1, …, xn}. More precisely, we use the Fisher metric gF and the exponential connection ∇(1) (both being natural statistical objects living on [Formula: see text]) to construct, via the Dombrowski splitting theorem, a Kähler structure on [Formula: see text] which has the property that it induces the natural Kähler structure of a suitably chosen open dense subset of ℙ(ℂn). As a direct physical consequence, a significant part of the quantum mechanical formalism (in finite dimension) is encoded in the triple [Formula: see text].