Hopf Algebra Primitives in Perturbation Quantum Field Theory

TL;DR

This paper explores the algebraic structures underlying perturbative quantum field theory, revealing that certain functionals are Hopf primitives linked to Hall trees, which facilitate systematic derivation of Green functions.

Contribution

It demonstrates that power sum symmetric functionals are Hopf primitives represented by Hall trees, establishing a recursive method for computing Green functions in quantum field theory.

Findings

Hopf primitives correspond to Hall trees in perturbation series

Each Hall tree relates to sums of Feynman diagrams with fixed vertices, legs, and loops

Primitives generate a recursion relation for Green functions

Abstract

The analysis of the combinatorics resulting from the perturbative expansion of the transition amplitude in quantum field theories, and the relation of this expansion to the Hausdorff series leads naturally to consider an infinite dimensional Lie subalgebra and the corresponding enveloping Hopf algebra, to which the elements of this series are associated. We show that in the context of these structures the power sum symmetric functionals of the perturbative expansion are Hopf primitives and that they are given by linear combinations of Hall polynomials, or diagrammatically by Hall trees. We show that each Hall tree corresponds to sums of Feynman diagrams each with the same number of vertices, external legs and loops. In addition, since the Lie subalgebra admits a derivation endomorphism, we also show that with respect to it these primitives are cyclic vectors generated by the free propagator, and thus provide a recursion relation by means of which the (n+1)-vertex connected Green functions can be derived systematically from the n-vertex ones.