Quantum field theory on noncommutative spaces

  • Raimar WulkenhaarEmail author


This survey tries to give a rigorous definition of Euclidean quantum field theory on a fairly large class of noncommutative geometries, namely nuclear AF Fréchet algebras. After a review of historical developments and current trends we describe in detail the construction of the Φ3-model and explain its relation to the Kontsevich model. We review the current status of the construction of the Φ4-model and present in an outlook a possible definition of Schwinger functions for which the Osterwalder–Schrader axioms can be formulated.



I thank Alain Connes for the initiative to write such a survey and for constant support and encouragement through the years. I am most grateful to my long-term collaborator Harald Grosse with whom nearly all of the presented results have been achieved. After the two years 2000/2001 together in Vienna, our collaboration has been supported by the Erwin-Schrödinger-Institute, by the Max-Planck-Institute for Mathematics in the Sciences and by the Deutsche Forschungsgemeinschaft (DFG) via the coordinated programmes SFB 478 and SFB 878. Sections 8 and 9 are based on recent results obtained with Harald Grosse, Akifumi Sako, Erik Panzer and Alexander Hock. The survey was finally assembled within the Cluster of Excellence7 “Mathematics Münster”.


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  1. 1.Mathematisches Institut der Westfälischen Wilhelms-UniversitätMünsterGermany

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