Gauge transformations and fiber bundle theory

Abstract

The relationship between the group Γ of pure gauge transformations of Atiyah, Hitchin, and Singer [Proc. R. Soc. London Ser. A 362, 425 (1978)] of a principal fiber bundle and the group 𝒢 of gauge transformations consisting of the direct product of the local gauge groups on the base space is studied. Γ is an invariant subgroup of 𝒢 and the quotient 𝒢/Γ is identified with the group of inequivalent gauge transformations. In the framework of the category of principal fiber bundles with connections, a natural explanation for the relevance of the group Γ in the classical and quantum theories of gauge fields is presented. The paper is made self-contained by an introductory discussion of the concepts of principal coordinate fiber bundle (gauge fixed principal fiber bundle) and principal fiber bundle, and of the equivalence between the three different versions of the group of vertical automorphisms of the bundle.