Minimalist translation-invariant non-commutative scalar field theory

Four-dimensional quantum field theory suffers from infrared and ultraviolet divergences as well as from the divergence of the renormalised perturbation expansion. Despite early suggestions as to modify the space-time structure [1], quantum field theory on non-commutative spaces was studied in details only after Alain Connes developed noncommutative geometry [2]. The first steps were connected to symmetry preserving regularisations of quantum field theoretical models. After it was realized that such models can be obtained from string theory in the field theory limit [3, 4], a new boom set in. The formulation of models on non-commutative spaces is quite simple and many properties have been analysed, see for example [5]. The study of the question of renormalisability is much more involved [6, 7]. The best-known non-commutative space-time is the Moyal plane. Let us review sketchily the main advances about renormalisation on such a space.