Exponential families, Kähler geometry and quantum mechanics

Abstract

Exponential families are a particular class of statistical manifolds which are particularly important in statistical inference, and which appear very frequently in statistics. For example, the set of normal distributions, with mean µ and deviation σ , form a 2-dimensional exponential family.In this paper, we show that the tangent bundle of an exponential family is naturally a Kähler manifold. This simple but crucial observation leads to the formalism of quantum mechanics in its geometrical form, i.e. based on the Kähler structure of the complex projective space, but generalizes also to more general Kähler manifolds, providing a natural geometric framework for the description of quantum systems.Many questions related to this "statistical Kähler geometry" are discussed, and a close connection with representation theory is observed.Examples of physical relevance are treated in details. For example, it is shown that the spin of a particle can be entirely understood by means of the usual binomial distribution. This paper centers on the mathematical foundations of quantum mechanics, and on the question of its potential generalization through its geometrical formulation.

Gaussian distributions, Jacobi group, and Siegel-Jacobi space

“…It is on the basis of the above facts (together with others that are collected in [Mol12a, Mol12b,Mol13]), that we arrived at the conclusion that the quantum formalism might have an information-theoretical origin. Now there are two possibilities:…”

“…Take a finite set Ω := {x 1 , ..., x n } and consider the space P × n of nowhere vanishing probabilities p : Ω → R , p > 0 , n k=1 p(x k ) = 1 . This is a (n−1)-dimensional exponential family, and it can be shown (see [Mol12b]) that T P × n is locally isomorphic to the complex projective space P(C n ) (see also [Mol13] for a refinement of this statement using the concept of "Kählerification").…”

“…Kähler functions are important in relation to the geometrical formulation of quantum mechanics, for they play the role of observables (see [AS99]). The geometrical formalism of quantum mechanics analysed in [Mol13] under the light of the above "Kähler decomposition" led naturally to the following definition: the spectrum of a Kähler function f : T E → R of the form Ω X(x)[(π • Φ)(p)](x)dx is Spec(f ) := Im(X), where Im(X) is the image of the random variable X ∈ A E . Using this definition, we described the spin of a particle passing through two consecutive Stern-Gerlach devices, without using physicists' standard approach based on the unitary representations of su(2).…”

Notions of the ergodic hierarchy for curved statistical manifolds

“…Relevant consequences from these researches such as the connection between Hessian structures and exponential families [38], nonextensive statistical models [34,35,36,39] and other extensions [32,33,37] has been reported.…”

Finslerian geometrization of quantum mechanics in the hydrodynamical representation

“…A geometrization of the non-relativistic quantum mechanics for mixed states was introduced in [29], making use of the Uhlmann's principal fibre bundle. A formulation of the formalism of quantum mechanics in a geometrical form based on the Kähler structure of the complex projective space was proposed in [30]. A geometric framework for mixed quantum states represented by density matrices, was discussed in [31].…”