Euclidean Reconstruction in Quantum Field Theory: Between tempered distributions and Fourier Hyperfunctions

TL;DR

This paper explores the mathematical challenges in relating Wightman and Euclidean quantum field theories, focusing on cases involving tempered distributions and Fourier hyperfunctions, and discusses prospects for classifying Euclidean reconstruction theorems.

Contribution

It analyzes the mathematical difficulties in Euclidean reconstruction, particularly in the contexts of tempered distributions and Fourier hyperfunctions, and outlines steps toward a classification of these theorems.

Findings

Analysis of Wightman and Schwinger functions relationship

Discussion of tempered distributions and Fourier hyperfunctions cases

Outline of perspectives for classifying Euclidean reconstruction theorems

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.