Integrability and positivity in quantum field theory on noncommutative geometry

Abstract

We review a sequence of papers in which we construct the λϕ4⋆4-model and the λϕ2,4,6⋆3-models on noncommutative Moyal space by a common method. Thereby we show that not only the Kontsevich model λΦ3 but also the λΦ44-model is integrable in a certain scaling limit which corresponds to infinitely large Moyal deformation parameter. Surprisingly, this limit gives rise to Schwinger functions on commutative Euclidean space. Our explicit formulae permit us to discuss reflection positivity of these Schwinger functions.