Positivity and Integrability

Based on past contributions by Robert Schrader and Michael Karowski I review the problem of existence of interacting quantum field theory and present recent ideas and results on rigorous constructions. 1 Historical remarks The title of this essay is identical to that of a small conference at the FU-Berlin in honor of Michael Karowski and Robert Schrader at the occasion of their sixtyfifth birthday. The history of mathematical physics and quantum field theory at the FU-Berlin, a university which was founded at the beginning of the cold war, is to a good part characterized by ”positivity and integrability” [1]. Both of my colleagues joined the FU theory group in the first half of the 70s shortly after I moved there. Robert Schrader arrived after his important contribution [2] to the birth of Euclidean field theory whose proper mathematical formulation he initiated together with Konrad Osterwalder while working at Harvard university under the guidance of Arthur Jaffe; Michael Karowski came from Hamburg where he finished his thesis under Harry Lehmann. Whereas Robert, after his arrival in Berlin, was still in the midst of finishing up the work with he begun with Konrad Osterwalder at Harvard [3], Michael was looking for new challenging post-doc problems outside his thesis work. At that time Dashen, Hasslacher and Neveu [4] (DHN) had just published their observations on the conjectured exactness of the quasiclassical particle spectrum of certain 2dimensional models. There were some theoretical indications [5] and numerical checks [6] pointing to a purely elastic S-matrix in those apparently integrable models which were strongly suggestive of an explanation in the (at that time already discredited) S-matrix bootstrap setting, but now within a more special