# Gravity coupled with matter and the foundation of non-commutative geometry

@article{Connes1996GravityCW, title={Gravity coupled with matter and the foundation of non-commutative geometry}, author={Alain Connes}, journal={Communications in Mathematical Physics}, year={1996}, volume={182}, pages={155-176} }

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond to Riemannian metrics and Spin structure whileds is the Dirac propagatords=x−x=D−1, whereD is the Dirac operator. We extend these simple relations to the non-commutative case using Tomita's involutionJ. We then write a spectral action, the trace of a function of the length element, which when applied…

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