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Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds

Romeo Brunetti, Klaus Fredenhagen

TL;DR

This work advances the rigorous construction of interacting quantum field theories on curved spacetimes by developing a purely local Epstein–Glaser renormalization framework grounded in microlocal analysis. It introduces a state-independent Wick polynomial construction, a local S-matrix formulation, and a comprehensive microlocal extension theory via Steinmann scaling degrees, including extensions to submanifolds and surfaces. The authors prove that polynomial interactions in four dimensions on globally hyperbolic backgrounds share the same short-distance perturbative classification as in Minkowski space, and they build a consistent net of local algebras of observables without relying on translation symmetry. The results lay a robust algebraic-analytic foundation for perturbative QFT on curved backgrounds and open avenues for addressing finite renormalizations and energy-momentum tensor renormalization in curved spacetimes.

Abstract

We present a perturbative construction of interacting quantum field theories on any smooth globally hyperbolic manifold. We develop a purely local version of the Stueckelberg-Bogoliubov-Epstein-Glaser method of renormalization using techniques from microlocal analysis. As byproducts, we describe a perturbative construction of local algebras of observables, present a new definition of Wick polynomials as operator-valued distributions on a natural domain, and we find a general method for the extension of distributions which were defined on the complement of some surfaces.

Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds

TL;DR

This work advances the rigorous construction of interacting quantum field theories on curved spacetimes by developing a purely local Epstein–Glaser renormalization framework grounded in microlocal analysis. It introduces a state-independent Wick polynomial construction, a local S-matrix formulation, and a comprehensive microlocal extension theory via Steinmann scaling degrees, including extensions to submanifolds and surfaces. The authors prove that polynomial interactions in four dimensions on globally hyperbolic backgrounds share the same short-distance perturbative classification as in Minkowski space, and they build a consistent net of local algebras of observables without relying on translation symmetry. The results lay a robust algebraic-analytic foundation for perturbative QFT on curved backgrounds and open avenues for addressing finite renormalizations and energy-momentum tensor renormalization in curved spacetimes.

Abstract

We present a perturbative construction of interacting quantum field theories on any smooth globally hyperbolic manifold. We develop a purely local version of the Stueckelberg-Bogoliubov-Epstein-Glaser method of renormalization using techniques from microlocal analysis. As byproducts, we describe a perturbative construction of local algebras of observables, present a new definition of Wick polynomials as operator-valued distributions on a natural domain, and we find a general method for the extension of distributions which were defined on the complement of some surfaces.

Paper Structure

This paper contains 19 sections, 21 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. General Theory of Quantized Fields and Microlocal Analysis
  3. Wave front sets and Hadamard states for free fields
  4. A new construction of Wick polynomials
  5. On a Local Formulation of Perturbation Theory
  6. Formulation of the local $\boldmath S$-matrix
  7. Defining properties
  8. Inductive Construction up to the Small Diagonal
  9. Steinmann Scaling Degree and the Extension of Distributions
  10. The scaling degree
  11. Extensions of distributions to a point
  12. Surfaces of Uniform Singularity and the Microlocal Scaling Degree
  13. Scaling degrees at submanifolds
  14. Invariance and properties for the scaling degrees
  15. Transversal scaling degree
  16. ...and 4 more sections

Key Result

Lemma 2.2

Let $\Phi\in{\mathcal{H}}_\omega$ induce some quasi-free Hadamard state $\omega^\prime$. Then $\Phi\in{\mathcal{D}}$.

Theorems & Definitions (23)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4: Generalized Wick expansion Theorem
  • Proposition 2.5
  • Theorem 3.1: Microlocal Theorem 0
  • Lemma 4.1
  • Proposition 4.2
  • Theorem 4.3
  • Lemma 5.1
  • ...and 13 more