The Standard Model – the Commutative Case: Spinors, Dirac Operator and De Rham Algebra

The present paper 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, Spin-structures, Dirac operators, exterior algebra bundles and Connes’ differential algebras in the commutative case, among other elements. We avoid the introduction of principal bundles and put the emphasis on a module-based approach using Serre-Swan’s theorem, Hermitian structures and module frames. A new proof (due to Harald Upmeier) of the differential algebra isomorphism between the set of smooth sections of the exterior algebra bundle and Connes’ differential algebra is presented. The content of the present paper reflects a talk given at the Workshop ’The Standard Model of Elementary Particle Physics from a mathematical-geometrical viewpoint’ held at the Ev.-Luth. Volkshochschule Hesselberg near Gerolfingen, Germany, March 14-19, 1999. In the first two sections we explain the Gel’fand and the Serre-Swan theorems to explain the background of ideas leading to noncommutative geometry. In section three Hermitean structures on vector bundles and generalized module bases called frames are introduced to have some more structural elements for proving. Furthermore, we give a short introduction to the theory of Clifford and spinor bundles over compact smooth Riemannian manifolds M . Following J. C. Várilly [26] we use the duality between vector bundles and projective finitely generated C(M)-modules as described by the Serre-Swan theorem to give a comprehensive account to the commutative theory. The spectral triple is derived and the crucial properties of the Dirac operator are listed without proof. Further, we define the differential algebra of Connes’ forms in the commutative setting and compare it to the set of all smooth sections of the exterior algebra bundle which forms also a differential algebra. The isomorphism of both these differential algebras is demonstrated by a new proof appearing here with the kind permission of its inventor Harald Upmeier. 1. The theorems by Gel’fand and Serre-Swan One of the corner stones of the beginning of noncommutative geometry was I. M. Gel’fand’s theorem published in 1940. He established an equivalence principle between some topological objects and algebraic-axiomatic structures that can be expressed in the following way (cf. [3, 20]): 1991 Mathematics Subject Classification. Primary 81R25; Secondary 53A50, 53B35, 58A12, 58B30, 46H25, 46L87. Supported in part by the Volkswagen Stiftung. 1