In order to cure the old problems of local quantum field theories it was suggested to add quantum gravity effects leading to quantized space-time. The mathematical framework relevant for the formulation of quantum fields defined over quantized space-time is noncommutative geometry.
Quantum field theory as defined by the Wightman, Haag-Kastler or Osterwalder-Schrader axioms relies on the manifold structure of space-time. In giving up the manifold one has to adapt the axioms. A reasonable replacement is not yet available. First attempts in this direction focussed on Euclidean quantum field theories on noncommutative manifolds. Because of their functional integral realization, such models are easy to define: It suffices to specify a parametrization of the fields and an action functional for them which involves the product in a noncommutative algebra. This topic started around 1996 with Filk’s Feynman rules for deformed Euclidean space, and in 1999 many authors showed perturbative one-loop renormalisability of a number of models. Shortly later Minwalla, van Raamsdonk and Seiberg discovered a severe problem at higher loop order, the so-called ultraviolet/infrared mixing.