The Commutative Case: Spinors, Dirac Operator and de Rham Algebra

The present chapter is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan’theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spinℂ-structures, Dirac operators, exterior algebra bundles and Connes’ differential algebras in the commutative case, among other elements. We avoid the use of principal bundles and put the emphasis on a module-based approach using Serre-Swan’s theorem, Hermitian structures and module frames. A detailed proof of the differential algebra isomorphism between the set of smooth sections of the exterior algebra bundle and Connes’ differential algebra is presented.