Dirac Type Gauge Theories – Motivations and Perspectives

Abstract

We summarize the geometrical description of a specific class of gauge theories, called “of Dirac type”, in terms of Dirac type first order differential operators on twisted Clifford bundles. We show how these differential operators may be geometrically considered as being the images of sections of a specific principal fibering naturally associated with twisted Clifford bundles. Based on the notion of real Hermitian vector bundles, we discuss the most general real Dirac type operator on “particle-anti-particle” modules over an arbitrary (orientable) semi-Riemannian manifold of even dimension. This setting may be appropriate for a common geometrical description of both the Dirac and the Majorana equation.