The Standard Model - the Commutative Case: Spinors, Dirac Operator and de Rham Algebra

TL;DR

This paper surveys the mathematical foundations of classical field theory in the commutative setting, focusing on algebraic structures like spinor bundles, Dirac operators, and Connes' differential algebra, emphasizing a module-based approach.

Contribution

It provides a comprehensive overview of key mathematical concepts in classical field theory, including a new proof of the isomorphism between exterior algebra sections and Connes' differential algebra.

Findings

Established the module-based framework for classical field theory structures.

Presented a new proof of the differential algebra isomorphism due to Harald Upmeier.

Clarified the relationship between geometric and algebraic structures in the commutative case.

Abstract

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^C-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.