GRAVITY COUPLED WITH MATTER AND THE FOUNDATION OF NONCOMMUTATIVE GEOMETRY

We first exhibit in the commutative case the simple algebraic relation s between the algebra of functions on a manifold and its infinitesimal length element ds. Its unitary representations correspond to Riemanni an metrics and Spin structure while a's is the Dirac propagator ds = x -x = D-1, where D is the Dirac operator. We extend these simple relati ons to the non-commutative case using Tomita's involution J. We then w rite a spectral action, the trace of a function of the length element, which when applied to the non-commutative geometry of the Standard Mo del will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations 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 this slightly non-commutative case. The group of local gauge transformations appears spontaneously as a norma l subgroup of the diffeomorphism group.