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

Klaus Fredenhagen; Romeo Brunetti

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

Romeo Brunetti, Klaus Fredenhagen

Mathematical Physics · Submitted 12 Mar 1999

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We present a perturbative construction of interacting quantum field theories on any smooth globally hyperbolic manifold.

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arXiv IDmath-ph/9903028 (version 1)

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CitationKlaus Fredenhagen, Romeo Brunetti. "Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds." arXiv:math-ph/9903028 (1999). https://pith.science/paper/AIBSQ4C7

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.24 · mentions code or data availability 0.02.

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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.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a perturbative construction of interacting quantum field theories on any smooth globally hyperbolic manifold.	contribution 0.97	0.96	0.58	asserted 1.00
s1	We develop a purely local version of the Stueckelberg-Bogoliubov-Epstein-Glaser method of renormalization using techniques from microlocal analysis.	method 0.69	0.95	0.15	asserted 1.00
s2	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.	contribution 0.67	0.96	0.16	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.75 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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Original 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.

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References

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Introduction to Modern Canonical Quantum General Relativity
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How later papers used it

67 later papers cite it

Evidence: substantial. 66 of the citing papers share no author with it, and 20 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 155 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 4 used its method or tool.

Uses its method or tool 4 of 16 papers with this label

Followed its ideas to find elements corresponding to the extension by considering projections and applying their transpose.

On-shell extension of distributions (2012) · 1.00 · paraphrased: the licence does not allow quoting
Essentially followed this paper for the subsequent derivation.

The Sine-Gordon QFT in de Sitter spacetime (2024) · 0.96 · paraphrased: the licence does not allow quoting
Used this paper's unique extension that preserves the scaling degree, after Taylor expanding and choosing p large enough to obtain a distribution with low enough scaling degree.

The renormalized locally covariant Dirac field (2012) · 0.98 · paraphrased: the licence does not allow quoting
Adapted to the case in hand this paper's approach, which is based on techniques proper of microlocal analysis.

A microlocal investigation of stochastic partial differential equations for spinors with an application to the Thirring model (2023) · 0.85 · paraphrased: the licence does not allow quoting

How the literature uses it

56 arXiv papers cite this in 130 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 89 sentences without a confident label

Builds on it 2 papers

Yet, adapting to the case in hand the results of [ref] , in [ref] it has been proven that it is possible to extend $t_k$ to $t̂_k∈𝒟^'(M^2k+2)$.

An Algebraic and Microlocal Approach to the Stochastic Non-linear Schr\"odinger Equation · Introduction · 0.88
Definition [ref] can be generalized so to encompass the notion of scaling degree with respect to an embedded submanifold $N⊆M$ – cf. [ref] .

A Microlocal Approach to Renormalization in Stochastic PDEs · Scaling degree · 0.98

Uses its method or tool 13 papers

Using the theory of scaling degrees at submanifolds, cf. [ref] and Section [ref], one can show that this extension is unique.

Dispersion relations in quantum electrodynamics on the noncommutative Minkowski space · The two–particle spectrum · 0.98
Following the ideas of [ref] , we do so by considering projections[This means that $W^t:(^n)↦_r(^n)$ is continuous and $(W^t)^2=W^t$.] $W^t:(^n)→_r(^n)$, and applying their transpose $W:'_r(^n)→'(^n)$ to $ũ∈'_r(^n)$.

On-shell extension of distributions · Extension of distributions · 1.00
For the subsequent derivation, we essentially follow [ref] .

The Sine-Gordon QFT in de Sitter spacetime · The Haag–Kastler net · 0.96

Cites it as support 3 papers

The reason why this is possible is that the theory is local [ref] .

Dispersion relations in quantum electrodynamics on the noncommutative Minkowski space · Interactions · 0.97
As shown in [ref] , time-ordered products exist and are unique up to a well-characterized, local and covariant renormalization ambiguity which is described by the main theorem of renormalization theory.

Background independence in gauge theories · Time-ordered products · 0.97
However, if the state is not of Hadamard form, there exists a composite operator (for example the normal-ordered square of some time derivative of the scalar field) which has infinite fluctuations in this state [ref] — clearly something which should be avoided on physical grounds.

Trace anomaly for chiral fermions via Hadamard subtraction · The definition of composite operators via Hadamard subtraction · 0.96

Cites it as background 17 papers

In addition, throughout this paper, we assume that the reader is familiar with the basic concepts of microlocal analysis, see e.g. [ref] , as well as with the notion of scaling degree, see in particular [ref] and [ref] .

An Algebraic and Microlocal Approach to the Stochastic Non-linear Schr\"odinger Equation · The Stochastic Nonlinear Schrödinger Equation · 1.00
In the remainder of this section we refer mainly to [ref] , in which all proofs of the following statements are present.

A Microlocal Approach to Renormalization in Stochastic PDEs · Scaling degree · 0.85
We refer to [ref] for details.

Non-equilibrium steady states for the interacting Klein-Gordon field in 1+3 dimensions · Perturbative algebraic quantum field theory and KMS states for interacting models · 0.90

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