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.

arXiv commentFeedback welcome ! arXiv admin note: text overlap with arXiv:1305.3535 by other authors

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