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

Michael Frank

TL;DR

The work addresses how Connes' noncommutative differential calculus specializes to classical differential geometry on a smooth compact manifold. It builds the bridge by employing Gel'fand duality and the Serre-Swan correspondence, constructing Clifford and spinor bundles via Morita theory, and formulating both universal and Connes' differential algebras in the commutative setting with the Dirac operator as core data. The central result is an isomorphism $\Omega_{D/}^p C^∞(M) \cong \Gamma^∞(\Lambda^p(M))$ for all $p$, proven through Upmeier's approach, tying Connes' calculus to the de Rham complex and showing compatibility of differential structures. This establishes a concrete, operator-algebraic realization of classical forms within the noncommutative framework and underpins potential noncommutative generalizations via spectral triples and Morita equivalence.

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.

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

TL;DR

The work addresses how Connes' noncommutative differential calculus specializes to classical differential geometry on a smooth compact manifold. It builds the bridge by employing Gel'fand duality and the Serre-Swan correspondence, constructing Clifford and spinor bundles via Morita theory, and formulating both universal and Connes' differential algebras in the commutative setting with the Dirac operator as core data. The central result is an isomorphism for all , proven through Upmeier's approach, tying Connes' calculus to the de Rham complex and showing compatibility of differential structures. This establishes a concrete, operator-algebraic realization of classical forms within the noncommutative framework and underpins potential noncommutative generalizations via spectral triples and Morita equivalence.

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.

Paper Structure

This paper contains 7 sections, 10 theorems, 50 equations.

Table of Contents

  1. The theorems by Gel'fand and Serre-Swan
  2. Hermitean structures and frames for sets of sections
  3. Clifford and spinor bundles, spin manifolds
  4. Spin connection and Dirac operator
  5. The universal differential algebra $\Omega C^\infty(M)$ and Connes' differential algebra $\Omega_{D \! \! \! \! / \;} C^\infty(M)$
  6. The exterior algebra bundle $\Lambda(M)$ and the de Rham complex
  7. $\Omega_{D \! \! \! \! / \;} C^\infty(M)$ versus $\Lambda(M)$

Key Result

Theorem 1.1

(I. M. Gel'fand) Let $A$ be a commutative C*-algebra and $X$ the set of its characters. The topology on $X$ should be that one induced by the weak* topology on the dual space $A^*$. Then $X$ is a locally compact Hausdorff space, and $X$ is compact iff $A$ is unital. The C*-algebra $A$ is $*$-isomorp

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Proposition 1.6
  • proof
  • Theorem 1.7
  • proof
  • Lemma 1.8
  • ...and 16 more