The riemannian structure on the states of quantum-like systems

Abstract

A riemannian metric is introduced in the manifold representing the states of a generic physical system, under suitable assumptions of regularity on the “generalized transition probability” defined in [1]. From the mean values of the observables it is then possible to construct gradients and brackets, and in the special case of a system admitting a quantum-mechanical description the latter are shown to be related to the familiar commutators via a skew-symmetric tensor field which is part of the intrinsic geometry of the projective Hilbert space of physical states.