Notes on Feynman Integrals and Renormalization

TL;DR

This paper reviews Feynman integrals, regularization, and renormalization, unifying various methods through linear algebra and resolutions of singularities, and explores their motivic aspects.

Contribution

It introduces a unifying linear algebraic approach to Feynman rules and connects different renormalization techniques using resolutions of singularities.

Findings

Unifies renormalization methods via resolutions of singularities.

Highlights the motivic nature of Feynman integrals.

Provides a linear algebraic perspective on Feynman rules.

Abstract

I review various aspects of Feynman integrals, regularization and renormalization. Following Bloch, I focus on a linear algebraic approach to the Feynman rules, and I try to bring together several renormalization methods found in the literature from a unifying point of view, using resolutions of singularities. In the second part of the paper, I briefly sketch the work of Belkale, Brosnan resp. Bloch, Esnault and Kreimer on the motivic nature of Feynman integrals.