The Hopf algebra of Feynman graphs in QED

TL;DR

This paper develops a Hopf algebra framework for QED renormalization, ensuring Ward-Takahashi identities are inherently satisfied, thus providing a mathematically consistent approach to quantum electrodynamics calculations.

Contribution

It introduces a Hopf algebraic structure for Feynman graphs in QED that naturally incorporates Ward-Takahashi identities, advancing the mathematical understanding of renormalization.

Findings

Ward-Takahashi identities are linear relations in the Hopf algebra

Compatibility ensures WT-identities hold for counterterms and amplitudes

Derivation of the identity Z1=Z2 from algebraic principles

Abstract

We report on the Hopf algebraic description of renormalization theory of quantum electrodynamics. The Ward-Takahashi identities are implemented as linear relations on the (commutative) Hopf algebra of Feynman graphs of QED. Compatibility of these relations with the Hopf algebra structure is the mathematical formulation of the physical fact that WT-identities are compatible with renormalization. As a result, the counterterms and the renormalized Feynman amplitudes automatically satisfy the WT-identities, which leads in particular to the well-known identity $Z_1=Z_2$.