Arithmetic gravity and Yang-Mills theory: An approach to adelic physics via algebraic spaces

TL;DR

This paper develops an adelic physics framework using algebraic spaces to describe space-time, integrating p-adic and real aspects, and explores solutions to arithmetic Einstein and Yang-Mills equations.

Contribution

It introduces a novel algebraic space approach to adelic physics, unifying general relativity and gauge theories in an arithmetic setting.

Findings

Space-time modeled as algebraic spaces over Dedekind schemes

Identification of X as a Neron model in the arithmetic case

Presentation of solutions to arithmetic Einstein equations

Abstract

This work is a dissertation thesis written at the WWU Muenster (Germany), supervised by Prof. Dr. Raimar Wulkenhaar. We present an approach to adelic physics based on the language of algebraic spaces. Relative algebraic spaces X over a base S are considered as fundamental objects which describe space-time. This yields a formulation of general relativity which is covariant with respect to changes of the chosen domain of numbers S. With regard to adelic physics the choice of S as an excellent Dedekind scheme is of interest (because this way also the finite prime spots, i.e. the p-adic degrees of freedom are taken into account). In this arithmetic case, it turns out that X is a Neron model. This enables us to make concrete statements concerning the structure of the space-time described by X. Furthermore, some solutions of the arithmetic Einstein equations are presented. In a next step, Yang-Mills gauge fields are incorporated.