Geometrical structure of Faddeev–Popov fields and invariance properties of gauge theories

Let 𝒫n be the trivial principal bundle with structural group G and base space 𝒫n−1, 𝒫1 being the usual fiber bundle of gauge theories. In order to give a geometrical interpretation to the Faddeev–Popov fields, as well as to the Becchi, Rouet, and Stora transformations, we need to use the fiber bundle 𝒫3. The gauge fields and the Faddeev–Popov ghost and antighost fields appear as part of certain 1-forms defined on the base space 𝒫2. The anticommuting character of the ghost and antighost fields is essentially due to their identification with 1-forms. The Becchi, Rouet, and Stora transformations are identified with generalized infinitesimal gauge transformations on 𝒫3 of parameters related to the ghost fields. We obtain a further invariance of the action given by a similar generalized infinitesimal gauge transformation on 𝒫3 related to the antighost fields.