The Dirichlet Hopf algebra of arithmetics

TL;DR

This paper introduces the Dirichlet Hopf algebra of arithmetics, explores its weakened structure called a Hopf gebra, and connects these concepts to quantum field theory, renormalization, and number theory applications.

Contribution

It systematically develops the Dirichlet Hopf algebra, introduces the Hopf gebra concept, and links algebraic structures to quantum field theory and number theory applications.

Findings

Weakened Hopf algebra structures affect cohomology and multiplicativity.

Unrenormalized coproducts and pairings address structural deficiencies.

Applications include symmetric functions, quantum mechanics, and QFT renormalization.

Abstract

In this work, we develop systematically the ``Dirichlet Hopf algebra of arithmetics'' by dualizing addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, obeying only a weakened (multiplicative) homomorphism axiom. The consequences of the weakened structure, called a Hopf gebra, e.g. on cohomology are explored. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an `unrenormalized' coproduct and an `unrenormalized' pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra. This can be modelled alternatively by employing Rota-Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.