From M\"obius inversion to renormalisation

TL;DR

This paper explores the conceptual connection between classical M"obius inversion and modern Hopf-algebraic renormalisation, offering new insights and perspectives on their mathematical relationship.

Contribution

It provides a clear abstraction pathway linking M"obius inversion to renormalisation, including new perspectives on known results and their generalisations.

Findings

Equivalence between Bogoliubov recursion and Atkinson formula

Generalisation of Weisner--Rota recursion and Hall--Leroux formula

Expository insights into the algebraic structures involved

Abstract

This paper traces a straight line from classical M\"obius inversion to Hopf-algebraic perturbative renormalisation. This line, which is logical but not entirely historical, consists of just a few main abstraction steps, and some intermediate steps dwelled upon for mathematical pleasure. The paper is largely expository, but contains many new perspectives on well-known results. For example, the equivalence between the Bogoliubov recursion and the Atkinson formula is exhibited as a direct generalisation of the equivalence between the Weisner--Rota recursion and the Hall--Leroux formula for M\"obius inversion.