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, forms 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 detail. 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.