A Treatise on Information Geometry

TL;DR

This paper explores ideas from Roy Frieden's 'Physics from Fisher Information', develops these concepts further, and surveys related topics in differential geometry and geometric measure theory to connect physics with exotic manifold structures.

Contribution

It advances the understanding of Fisher information in physics and investigates exotic differentiable structures and their relevance to physical theories.

Findings

Survey of non-standard manifold structures

Introduction to semi-Riemannian geometry

Overview of geometric measure theory and PDE existence

Abstract

This is the first version of a submission made to vixra back in August 2009. It is concerned with the exploration of various ideas due to Roy Frieden of the University of Arizona in his manuscript "Physics from Fisher Information", which was originally published in December 1998 by Cambridge University Press. In addition to exploration and investigation of same, a key aim of this article is/was to further develop the ideas mentioned in the aforementioned work, as well as explore various exotic differentiable structures and their relationship to physics. Apart from the original component of this manuscript, a series of survey chapters are provided, in the interest of keeping the treatise self-contained. The first summarises the main preliminary results on the existence of non-standard structures on manifolds from the Milnor-Steenrod school. The second is a standard introduction to semi-riemannian geometry. The third introduces the language of geometric measure theory, which is important in justifying the existence of smooth solutions to variational problems with smooth structures and smooth integrands. The fourth is a short remark on PDE existence theory, which is needed for the fifth, which is essentially a typeset version of a series of lectures given by Ben Andrews and Gerhard Huisken on the Hamilton-Perelman program for proving the Geometrisation Conjecture of Bill Thurston.