Gaussian distributions, Jacobi group and Siegel-Jacobi space

TL;DR

This paper reveals that the tangent bundle of the Gaussian distribution space, with its natural Kähler structure, is equivalent to the Siegel-Jacobi space, connecting geometry, number theory, and quantum formalism.

Contribution

It establishes a geometric correspondence between Gaussian distributions and the Siegel-Jacobi space, enriching the understanding of quantum formalism through differential geometry.

Findings

The tangent bundle of Gaussian distributions forms the Siegel-Jacobi space.

The paper details geometric properties like completeness and curvature of this space.

Connections to the Jacobi group and quantum formalism are elucidated.

Abstract

Let $\mathcal{N}$ be the space of Gaussian distribution functions over $\mathbb{R}$, regarded as a 2-dimensional statistical manifold parameterized by the mean $\mu$ and the deviation $\sigma$. In this paper we show that the tangent bundle of $\mathcal{N}$, endowed with its natural K\"ahler structure, is the Siegel-Jacobi space appearing in the context of Number Theory and Jacobi forms. Geometrical aspects of the Siegel-Jacobi space are discussed in detail (completeness, curvature, group of holomorphic isometries, space of K\"ahler functions, relationship to the Jacobi group), and are related to the quantum formalism in its geometrical form, i.e., based on the K\"ahler structure of the complex projective space. This paper is a continuation of our previous work, where we studied the quantum formalism from a geometric and information-theoretical point of view.