Geometric Structures Induced by Deformations of the Legendre Transform

Area of Science:

• Information geometry
• Statistical mechanics
• Theoretical physics

Background:

• A link between generalized Legendre transforms and non-dually flat statistical manifolds explains the prevalence of Rényi's divergence and entropy in physics.
• Existing research offers limited insight into the nature and physical implications of this relationship.

Purpose of the Study:

• To explore the deformation of Legendre transforms using symplectic geometry and complexification.
• To develop a deeper, intuitive understanding of the connection between Legendre transforms and physical systems.
• To establish a unified framework for analyzing physical systems not adequately described by classical information-theoretic measures.

Main Methods:

• Deformation of Legendre transforms.
• Application of symplectic geometry.
• Complexification techniques.

Main Results:

• Revealed consequences of deforming Legendre transforms.
• Established a novel common framework for understanding physical systems.
• Provided a principled and unified approach to systems beyond classic information theory.

Conclusions:

• The deformation of Legendre transforms offers new insights into their relationship with statistical manifolds.
• This work unifies the understanding of diverse physical systems using a novel information-theoretic framework.
• The findings pave the way for analyzing complex physical phenomena not captured by traditional information-theoretic quantities.