Euclidean Reconstruction in Quantum Field Theory: between Tempered Distributions and Fourier Hyperfunctions

In this short note on my talk 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 wellknown 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. 0. Introduction: Why Euclidean Reconstruction Euclidean methods are widely used in quantum field theory and other areas of mathematical physics. So let me outline some of the reasons for the attractivity of those techniques. I will use the language of axiomatic “Wightman” quantum field theory, for the fundamental notions of which I may refer the reader to my last years talk [1] or the famous books [SW] of Streater and Wightman or [GJ] of Glimm and Jaffe respectively. To use “Euclidean” methods in quantum field theory amounts formally in changing the coordinates x = (x, x, x, x) = (t, ~x) of Minkowski space to ξ = ıx = (it, ~x), where ı = (i, 1, 1, 1). By that the original Minkowski metric η = diag(−1, 1, 1, 1) prescribed by relativistic covariance of the theory becomes the trivial metric of Euclidean space, but the new time coordinate is purely imaginary. The benefits of this transformation is somewhat näıvely but