The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization

TL;DR

This paper explores the mathematical structures underlying renormalization in quantum field theory, connecting Hopf algebras, normal coordinates, and Kontsevich's deformation quantization to deepen theoretical understanding.

Contribution

It establishes a novel link between the Hopf algebra of renormalization, Birkhoff decomposition, and Kontsevich's universal deformation formula using normal coordinates.

Findings

Clarifies the relation between twisted antipode axiom and Birkhoff decomposition.

Provides a new perspective on deformation quantization in the context of quantum field theory.

Enhances the mathematical framework for renormalization processes.

Abstract

Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation.