Geometric Structures Induced by Deformations of the Legendre Transform

TL;DR

This paper explores how deforming the Legendre transform through symplectic geometry and complexification reveals new geometric structures, offering a unified framework for understanding physical systems beyond traditional information theory.

Contribution

It introduces a novel geometric perspective on deformed Legendre transforms, linking symplectic geometry and complexification to physical systems not captured by classic information measures.

Findings

Revealed geometric structures from Legendre transform deformations

Linked symplectic geometry to physical system modeling

Provided a unified framework for non-classical information quantities

Abstract

The recent link discovered between generalized Legendre transforms and non-dually flat statistical manifolds suggests a fundamental reason behind the ubiquity of R\'{e}nyi's divergence and entropy in a wide range of physical phenomena. However, these early findings still provide little intuition on the nature of this relationship and its implications for physical systems. Here we shed new light on the Legendre transform by revealing the consequences of its deformation via symplectic geometry and complexification. These findings reveal a novel common framework that leads to a principled and unified understanding of physical systems that are not well-described by classic information-theoretic quantities.