A mathematical framework to compare classical field theories

Lukas Barth

Published 2019-10-14Version 1

This article is a summary of the Master's thesis I wrote under the supervision of Prof. Ion Stamatescu and Prof. James Weatherall as a result of more than a year of research. The original work contained a bit more than 140 pages, while in the present summary all less relevant topics were shifted to the appendix such that the main part does not exceed 46 pages to ease the reading. However, the appendix was kept in order to show which parts were omitted. In the article, a mathematical framework to relate and compare any classical field theories is constructed. A classical field theory is here understood to be a theory that can be described by a (possibly non-linear) system of partial differential equations and thus the notion includes but is not limited to classical (Newtonian) mechanics, hydrodynamics, electrodynamics, the laws of thermodynamics, special and general relativity, classical Yang-Mills theory and so on. To construct the mathematical framework, a mathematical category (in the sense of category theory) in which a versatile comparison becomes possible is sought and the geometric theory of partial differential equations is used to define what can be understood by a correspondence between theories and by an intersection of two theories under such a correspondence. This is used to define in a precise sense when it is meaningful to say that two theories share structure and a procedure (based on formal integrability) is introduced that permits to decide whether such structure does in fact exist or not if a correspondence is given. It is described why this framework is useful both for conceptual and practical purposes and how to apply it. As an example, the theory is applied to electrodynamics and, among other things, magneto-statics is shown to share structure with a subtheory of hydrodynamics.