Gaussian distributions, Jacobi group and Siegel-Jacobi space

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 Kähler 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 Kähler functions, relationship to the Jacobi group), and are related to the quantum formalism in its geometrical form, i.e., based on the Kähler 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.

References in corpus (20)

• Relational Quantum Mechanics
• Informational derivation of Quantum Theory
• Geometric Quantum Mechanics
• Characterizing quantum theory in terms of information-theoretic constraints
• A derivation of quantum theory from physical requirements
• An Information-Geometric Reconstruction of Quantum Theory, I: The Abstract Quantum Formalism
• Invariant metrics and Laplacians on Siegel-Jacobi space
• From Information Geometry to Quantum Theory
• Generalised Hunter-Saxton equations, optimal information transport, and factorisation of diffeomorphisms
• Invariant metrics and Laplacians on Siegel-Jacobi Disk
• Coherent state embeddings, polar divisors and Cauchy formulas
• Applications of the Jacobi group to Quantum Mechanics
• Exponential families, Kahler geometry and quantum mechanics
• Generalization of Hasimoto's transformation
• Remarks on the statistical origin of the geometrical formulation of quantum mechanics
• On the geometry of Siegel-Jacobi domains
• Coherent states and geometry on the Siegel-Jacobi disk
• Generalized squeezed states for the Jacobi group
• Information geometry and the hydrodynamical formulation of quantum mechanics
• The Schroedinger-Weil Representation and Jacobi Forms of Half-Integral Weight

Cited by in corpus (2)

• Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball
• On the Kähler Geometry of Certain Optimal Transport Problems