Multiplicative renormalization and Hopf algebras

TL;DR

This paper demonstrates how Hopf algebra structures underpin Green's functions in quantum field theory, enabling derivation of Dyson's formulas and revealing the algebraic role of Slavnov--Taylor identities.

Contribution

It establishes the existence of Hopf subalgebras generated by Green's functions, showing the coproduct's closure and deriving key renormalization relations.

Findings

Hopf subalgebras generated by Green's functions exist in quantum field theory.

Dyson's formulas are derived using Hopf algebra structures.

Slavnov--Taylor identities are crucial in non-abelian gauge theories.

Abstract

We derive the existence of Hopf subalgebras generated by Green's functions in the Hopf algebra of Feynman graphs of a quantum field theory. This means that the coproduct closes on these Green's functions. It allows us for example to derive Dyson's formulas in quantum electrodynamics relating the renormalized and bare proper functions via the renormalization constants and the analogous formulas for non-abelian gauge theories. In the latter case, we observe the crucial role played by Slavnov--Taylor identities.