We prove that the self-interacting scalar field on the four-dimensional degenerate Moyal plane is renormalisable to all orders when adding a suitable counterterm to the Lagrangian. Despite the apparent simplicity of the model, it raises several non-trivial questions. Our result is a first step towards the definition of renormalisable quantum field theories on a non-commutative Minkowski space.
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Harald Grosse, Fabien Vignes-Tourneret, Quantum field theory on the degenerate Moyal space. J. Noncommut. Geom. 4 (2010), no. 4, pp. 555–576DOI 10.4171/JNCG/66