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Complex powers of analytic functions and meromorphic renormalization in QFT

Nguyen Viet Dang

TL;DR

The paper develops a rigorous, geometry-informed framework for analytic regularization and renormalization in quantum field theory. It builds meromorphic families of distributional powers via Hironaka’s resolution of singularities, then exploits a Laurent-decomposition approach to define renormalization maps that satisfy locality and factorization on analytic Lorentzian spacetimes. By proving WF-holonomicity and strong convexity controls, it shows Feynman amplitudes admit meromorphic regularization with linear poles and can be extended across diagonals while preserving microlocal structure. The resulting renormalization maps are functorial with respect to causal-category morphisms, providing a universal, covariant mechanism for renormalization in analytic settings and a route toward renormalizability proofs in convex analytic spacetimes. Overall, the work unifies analytic continuation, microlocal analysis, and algebraic renormalization into a coherent, geometrically grounded quitting-point for QFT on curved backgrounds.

Abstract

In this article, we study functional analytic properties of the meromorphic families of distributions $(\prod_{i=1}^p (f_j+i0)^{λ_j})_{(λ_1,\dots,λ_p) \in \mathbb{C}^p}$ 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 $(\prod_{i=1}^p (f_j+i0)^{λ_j})_{(λ_1,\dots,λ_p)\in\mathbb{C}^p}$ 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.

Complex powers of analytic functions and meromorphic renormalization in QFT

TL;DR

The paper develops a rigorous, geometry-informed framework for analytic regularization and renormalization in quantum field theory. It builds meromorphic families of distributional powers via Hironaka’s resolution of singularities, then exploits a Laurent-decomposition approach to define renormalization maps that satisfy locality and factorization on analytic Lorentzian spacetimes. By proving WF-holonomicity and strong convexity controls, it shows Feynman amplitudes admit meromorphic regularization with linear poles and can be extended across diagonals while preserving microlocal structure. The resulting renormalization maps are functorial with respect to causal-category morphisms, providing a universal, covariant mechanism for renormalization in analytic settings and a route toward renormalizability proofs in convex analytic spacetimes. Overall, the work unifies analytic continuation, microlocal analysis, and algebraic renormalization into a coherent, geometrically grounded quitting-point for QFT on curved backgrounds.

Abstract

In this article, we study functional analytic properties of the meromorphic families of distributions 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 in the space of distributions. We also study microlocal properties of and . 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.

Paper Structure

This paper contains 65 sections, 57 theorems, 163 equations.

Table of Contents

  1. Contents of the paper.
  2. Future projects.
  3. Acknowledgements.
  4. Meromorphic functions.
  5. Meromorphic functions in several variables.
  6. Meromorphic functions with linear poles.
  7. Distributions depending meromorphically on extra parameters.
  8. A gain of regularity: when weak holomorphicity becomes strong holomorphicity.
  9. Meromorphic functions with linear poles with value distribution.
  10. The fundamental example of hypergeometric distributions.
  11. The meromorphic family $\left(\prod_{j=1}^p (f_j+i0)^{\lambda_j}\right)_{\lambda \in\mathbb{C}^p}$.
  12. More general examples of hypergeometric distributions.
  13. Appendix to section $1$: analytic sets have measure zero.
  14. The main construction.
  15. Algebras of cylindrical functions.
  16. ...and 50 more sections

Key Result

Theorem 1

Let $f$ be a real valued analytic function s.t. $\{df=0\}\subset\{f=0\}$, $Z\subset \mathbb{C}$ a discrete subset containing the poles of the meromorphic family $((f+i0)^\lambda)_\lambda$. Set For all $z\in Z$, let $a_k$ to be the coefficients of the Laurent series expansion of $\lambda\mapsto (f+i0)^\lambda$ around $z$ Then for all $k\in\mathbb{Z}$, $WF(a_k)\subset\Lambda_f$ and if $k<0$ then $a

Theorems & Definitions (124)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Example 1
  • Definition 3
  • Definition 4
  • Proposition 1
  • proof
  • Definition 5
  • ...and 114 more