Complex powers of analytic functions and meromorphic renormalization in QFT

Abstract

In this article, we study functional analytic properties of the meromorphic families of distributions $(\prod_{i=1}^p (f_j+i0)^{\lambda_j})_{(\lambda_1,\dots,\lambda_p) \in \mathbb{C}^p}$ using Hironaka's resolution of singularities, then using recent works on the decomposition of meromorphic germs with linear poles, we renormalize products of powers of analytic functions $\prod_{i=1}^p(f_j+i0)^{k_j}, k_j \in \mathbb{Z}$ in the space of distributions. We also study microlocal properties of $(\prod_{i=1}^p (f_j+i0)^{\lambda_j})_{(\lambda_1,\dots,\lambda_p)\in\mathbb{C}^p}$ and $\prod_{i=1}^p (f_j+i0)^{k_j}, k_j \in \mathbb{Z}$. In the second part, we argue that the above families of distributions with \emph{regular holonomic singularities} provide a universal model describing singularities of Feynman amplitudes and give a new proof of renormalizability of quantum field theory on convex analytic Lorentzian spacetimes as applications of ideas from the first part.