Renormalization : A number theoretical model

TL;DR

This paper explores the number theoretic structure of the Dirichlet convolution ring, revealing its failure to form a Hopf algebra and proposing a renormalization-inspired solution to handle singularities through a rescaling process.

Contribution

It introduces a novel number theoretic model inspired by quantum field theory renormalization, connecting algebraic structures with singularity management.

Findings

The Dirichlet convolution ring does not form a Hopf algebra on the diagonal.

A related Hopf algebra overcounts the diagonal, leading to singularities.

Renormalization is modeled as a rescaling map from an auxiliary Hopf algebra to a weaker structure.

Abstract

We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.