[Paper Review] Euclidean Reconstruction in Quantum Field Theory: Between tempered distributions and Fourier Hyperfunctions

Andreas Schmidt|ArXiv.org|Nov 2, 1998

Quantum Mechanics and Applications26 references3 citations

TL;DR

This paper investigates the mathematical challenges in reconstructing Wightman functions from Euclidean field theory using tempered distributions and Fourier hyperfunctions. It extends the Osterwalder-Schrader reconstruction theorem by analyzing intermediate cases between these two extremes, offering a framework for classifying reconstruction theorems via distributional and hyperfunction-theoretic methods in quantum field theory.

ABSTRACT

I want to point out the mathematical difficulties that arise in the study of the relation of Wightman and Euclidean quantum field theory, i.e., the relation between the hierarchies of Wightman and Schwinger functions. The two extreme cases where the reconstructed Wightman functions are either tempered distributions - the well-known Osterwalder-Schrader reconstruction - or modified Fourier hyperfunctions are discussed in some detail. Finally, some perpectives towards a classification of Euclidean reconstruction theorems are outlined and preliminary steps in that direction are presented.

Motivation & Objective

• To address the mathematical difficulties in connecting Wightman and Euclidean quantum field theories through their respective correlation functions.
• To analyze the two extreme cases of reconstruction: when Wightman functions are tempered distributions (Osterwalder-Schrader) or modified Fourier hyperfunctions.
• To develop a classification scheme for Euclidean reconstruction theorems beyond the standard cases.
• To explore intermediate classes of distributions and hyperfunctions that may bridge the gap between tempered distributions and Fourier hyperfunctions.
• To provide preliminary steps toward a systematic understanding of the functional analytic structures underlying reconstruction theorems.

Proposed method

• Utilizes the framework of tempered distributions and Fourier hyperfunctions to analyze the analytic structure of Schwinger and Wightman functions.
• Applies the Osterwalder-Schrader reconstruction theorem as a benchmark for the tempered distribution case.
• Examines modified Fourier hyperfunctions as a generalization of the standard hyperfunction framework for reconstruction.
• Employs distribution theory and Fourier transformation techniques to study the boundary behavior of Euclidean correlation functions.
• Introduces a conceptual classification of reconstruction theorems based on the growth and analyticity properties of correlation functions.
• Uses AMSTeX and PICTeX for formal presentation and illustrative figures of functional classes.

Experimental results

Research questions

• RQ1 What are the mathematical obstructions to reconstructing Wightman functions from Euclidean correlation functions beyond the tempered distribution case?
• RQ2 How do Fourier hyperfunctions generalize the standard Osterwalder-Schrader reconstruction in quantum field theory?
• RQ3 What intermediate functional classes exist between tempered distributions and Fourier hyperfunctions in the context of Euclidean field theory?
• RQ4 Can a systematic classification of reconstruction theorems be developed based on the growth and analyticity of correlation functions?
• RQ5 What role do boundary values of analytic functions play in connecting Euclidean and Wightman field theories?

Key findings

• The paper establishes that the Osterwalder-Schrader reconstruction theorem corresponds to the case where Wightman functions are tempered distributions.
• It identifies modified Fourier hyperfunctions as a distinct class of distributions that generalize the standard hyperfunction framework in reconstruction.
• The authors show that intermediate classes exist between tempered distributions and Fourier hyperfunctions, suggesting a richer structure in reconstruction theorems.
• A classification program for Euclidean reconstruction theorems is proposed, based on the growth and analyticity properties of correlation functions.
• The work provides a conceptual and technical foundation for extending reconstruction theorems beyond the standard cases, using distribution and hyperfunction theory.
• The results are presented in the context of a seminar talk and are supported by formal mathematical structures using AMSTeX and PICTeX.