By Richard Bellman

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26) One can show that M is an associative -algebra. It contains, among others, the identity, the polynomials, the ı-distribution and its derivatives. 27) often given as a definition of the Moyal space, holds in M (but not in A‚ ). 2 The 4 -theory on R4Â . The simplest non-commutative model one may consider is the 4 -theory on the four-dimensional Moyal space. @ C m ? C ? ? 3), this action can be explicitly computed. 29) dx iD1 'D 4 X . 1/iCj C1 xi ^ xj : i

From the point of view of quantum field theory, it is certainly the most studied space. Let us start with its precise definition. ˚ « . 1 The Moyal space RD Â free algebra generated by E. Let ‚ a D D non-degenerate skew-symmetric matrix (which requires D even) and I the ideal of ChEi generated by the elements x x {‚ . The Moyal algebra A‚ is the quotient ChEi=I . Each element in A‚ x x is a formal power series in the x ’s for which the relation Œx ; x D {‚ holds. 0/ ÂD=2 0 D ÂD=2 . The preceding algebraic definition Sometimes one even set Â D Â1 D whereas short and precise may be too abstract to perform real computations.

40 42 43 44 44 49 51 53 53 55 59 62 64 64 65 69 73 78 80 This review follows lectures given by V. R. at the workshop “Renormalisation et théories de Galois”, Luminy, March 2006. 40 Vincent Rivasseau and Fabien Vignes-Tourneret 1 Introduction General relativity and ordinary differential geometry should be replaced by non-commutative geometry at some point between the currently accessible energies of about 1 – 10 Tev (after starting the Large Hadron Collider (LHC) at CERN) and the Planck scale, which is 1015 times higher, where space-time and gravity should be quantized.