A Lie theoretic approach to renormalization

TL;DR

This paper introduces a Lie algebraic perspective to renormalization, building on Connes and Marcolli's work, using combinatorial and algebraic tools to deepen understanding and provide new proofs and approaches.

Contribution

It develops a Lie algebraic framework for renormalization, connecting Hopf algebras, descent algebras, and Galois theory, independent of geometric regularization methods.

Findings

Provides algebraic proofs of renormalization combinatorics

Introduces a Galois-theoretic approach to renormalization

Applies the framework to BPHZ scheme

Abstract

Motivated by recent work of Connes and Marcolli, based on the Connes-Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on properties of Hopf algebras encapsulated in the notion of associated descent algebras. Besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures, the new techniques give rise to an algebraic approach to the Galois theory of renormalization. In particular, they do not depend on the geometry underlying the case of dimensional regularization and the Riemann-Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme.