Gravity coupled with matter and the foundation of non commutative geometry

We rst exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its in nitesimal length element ds. Its unitary representations correspond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds = | = D 1 where D is the Dirac operator. We extend these simple relations to the non commutative case using Tomita's involution J . We then write a spectral action, the trace of a function of the length element in Planck units, which when applied to the non commutative geometry of the Standard Model will be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian coupled to gravity. The internal uctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in the slightly non commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the di eomorphism group. Riemann's concept of a geometric space is based on the notion of a manifold M whose points x 2M are locally labelled by a nite number of real coordinates x 2 . The metric data is given by the in nitesimal length element,