Fr\"olicher-smooth geometries, quantum jet bundles and BRST symmetry

We attempt a clarification of geometric aspects of quantum field theory by using the notion of smoothness introduced by Fr\"olicher and exploited by several authors in the study of functional bundles. A discussion of momentum and position representations in curved spacetime, in terms of generalized semi-densities, leads to a definition of quantum configuration bundle which is suitable for a treatment of that kind. A consistent approach to Lagrangian field theories, vertical infinitesimal symmetries and related currents is then developed, and applied to a formulation of BRST symmetry in a gauge theory of the Yang-Mills type.