Perturbative Quantum Field Theory and Configuration Space Integrals

TL;DR

This paper explores the algebraic and cohomological structures underlying perturbative quantum field theory, focusing on L-infinity morphisms, Feynman integrals, and their relation to renormalization and topological quantum field theories.

Contribution

It introduces a cohomological framework for formality morphism coefficients, connecting L-infinity morphisms with Hopf algebra structures and configuration space integrals.

Findings

Weights of expansions are cycles in DG-coalgebra of Feynman graphs

Develops a cohomological approach as an alternative to analytical integrals

Links Feynman integrals with topological quantum field theory concepts

Abstract

L-infinity morphisms are studied from the point of view of perturbative quantum field theory, as generalizations of Feynman expansions. The connection with the Hopf algebra approach to renormalization is exploited. Using the coalgebra structure (Forest Formula), the weights of the corresponding expansions are proved to be cycles of the DG-coalgebra of Feynman graphs. The properties of integrals over configuration spaces (Feynman integrals) are investigated. The aim is to develop a cohomological approach in order to construct the coefficients of formality morphisms using an algebraic machinery, as an alternative to the analytical approach using integrals over configuration spaces. The connection with a related TQFT is mentioned, supplementing the Feynman path integral interpretation of Kontsevich formula.