Renormalization and periods in perturbative Algebraic Quantum Field Theory

Kasia Rejzner
March 09, 2016
In this paper I give an overview of mathematical structures appearing in perturbative algebraic quantum field theory (pAQFT) and I show how these relate to certain periods. pAQFT is a mathematically rigorous framework that allows to build models of physically relevant quantum field theories on a large class of Lorentzian manifolds. The basic objects in this framework are functionals on the space of field configurations and renormalization method used is the Epstein-Glaser (EG) renormalization. The main idea in the EG approach is to reformulate the renormalization problem, using functional analytic tools, as a problem of extending almost homogeneously scaling distributions that are well defined outside some partial diagonals in $\mathbb{R}^n$. Such an extension is not unique, but it gives rise to a unique "residue", understood as an obstruction for the extended distribution to scale almost homogeneously. Physically, such scaling violations are interpreted as contributions to the $\beta$ function.